Fixed Income

Market Risk

  1. Market Risk Data

Effective market risk management requires all markets risks to be appropriately identified, measured and controlled. Market risk itself is defined as the potential (adverse) change in portfolio value from changes in the market inputs. A well-functioning risk control process reflects the foundational role of valuation and the direct linkage with risk from end-to-end with appropriate control points and feedback loops.

ResearchGate data construction pdf

ResearchGate curve construction

Zenodo repo curve

2. Profit and Loss

Risk measurement will create a strong linkage between pricing models used for profit and loss (P&L) and pricing models used for risk measurement. All market inputs are available for incorporation to risk management systems though some require alternative historical data for calibration procedures. Consistent valuation models will facilitate the identification of market factors driving valuation.

Core collateral

ResearchGate rho pdf

ResearchGate rho

3. Backtest Exception

Backtest exceptions occur when Clean P&L and/or Model P&L exceed Model VaR on an absolute basis. The use of both model P&L and clean P&L for exception reporting gives granularity in the drivers of the exception

Cases where the market move is the only cause of the exception, no further validations are required. The conclusion can be further proved by a comparison of the market data change with its historical trend, showing that the move is in a greater magnitude than the historical data, as well as what’s captured in the VaR window.

Core worst of option

ResearchGate fair pdf

ResearchGate fair

4. Independent Price Verification

At each month-end, the Valuation Product Control Group (VPC) performs an independent fair valuation of all portfolios within Trading Products. This process results in independent price verification adjustments (IPV) and valuation adjustments (VA). IPV reflect the valuation difference between the line of business (LOB) view of valuation and the independent VPC view. VA’s are adjustments used to achieve fair value and include Close-Out, Uncertainty, Liquidity, Model Risk, and Administrative.

Bitbucket price verification

ResearchGate bet pdf

ResearchGate bet

5. Gamma Shocks

In the Commodity framework, risk factors for NG pipelines are currently defined with respect to base and spread (over NYM_NG).  The delta sensitivities in RDR are given for the base and spread, but the gamma is an “all-in”, meaning it calculated with respect to base plus spread curve.  Because we do not have separate risk factors for the all-in curves, we modify shocks generated for the base and spread curves to construct those for the all-in curves. 

Bitbucket gamma shock

Core rainbow

6. Commodity Simulation Translation

Market instruments may be daily, for instance, but simulated risk factors may be monthly. Therefore, we need to convert monthly simulated risk factors into daily market instruments. Here we use linear interpolation within the range of risk factors and extrapolation beyond the range.

bitbucket commodity translation

Core abs

7. Commodity Volatility Closing Rate Calculation

To populate the closing rate for implied volatility skew risk factors, it will be necessary to implement the forward-delta based Omega skew models and use the Brent root finding algorithm to solve for the closing rate.  This function spec will outline this procedure.

bitbucket commodity vol

ResearchGate binary pdf

ResearchGate binary

8. Precious Metal Futures

Futures contracts are typically valued as a spread to the forwards curves.  This spread is the EFP (Exchange of Futures for Physical).  This function spec outlines the introduction of this spread as a basis risk factor into system and the transformation of sensitivities with respect to futures contracts to those with respect the spot contract, forward offered rates, and basis.

bitbucket pm futures

ResearchGate futures pdf

ResearchGate futures

9. Dividend Risk

Due to the difference between stock dividend risk factor model and stock price risk factor model, it is more natural to carry out the simulation of these two types of risk factors separately. Currently, stock price risk factor is simulated in Simulation 3 based on beta and idiosyncratic volatility assigned to it. The derived risk factor for price of stock basket or index is simulated in Simulation 4 based on closing rate and weights of components of the basket or index.

bitbucket dividend risk

ResearchGate dividend pdf

ResearchGate dividend

10. Dividend Exposure Measurement and Management

We model dividend risk by introducing a new risk factor – the innovation in dividend expectation. It is to describe the day-to-day relative changes to the expected dividend amount of the underlying stock, basket of stocks, or equity index. The new risk factor facilitates the calculation of the market expected dividend when pricing an equity derivative at future time.

bitbucket dividend exposure

ResearchGate forward dividend pdf

ResearchGate forward dividend

11. Dividend Risk Modeling and Methodology

Pricing of equity derivatives on single stock usually takes expected dividend amount as input. Sensitivity with respect to expected dividend amount is readily available. Directly modeling changes to dividend expectation is a very convenient way to capture risk via DGV approach.

bitbucket dividend model

Github dividend

12. Dividend Risk Model Calibration

As described in model description and calibration of the model, the daily innovation to expected dividend of an index is simulated from double exponential distribution with specified correlation with all other risk factors. The daily innovation to expected dividend of a single stock is simulated on the fly without correlation with other risk factors.

bitbucket dividend calibration

ResearchGate model pdf

ResearchGate model

Fixed Income


  1. LMM Implementation

In order to drive the model with fewer factors, the rank-reduced pseudo square roots of the states’ integrated covariance are required. The rank-reduced integrated states’ covariance is also required for calculations during calibration. However, calibration varies the states covariance and is it not practical to repeatedly perform rank reduction. Instead, the states’ instantaneous correlation (which is not varied) is rank reduced and used to generate an approximation to the rank reduced integrated covariance.

The volatility of the model’s states (the spanning Libors) can be specified through calibration or by inputting an instantaneous volatility surface and an instantaneous correlation matrix or by inputting parameters for a functional form volatility and correlation.

Gitbook LMM

Github libor

ResearchGate lmm pdf

ResearchGate lmm

2. Local Market Valuation

The current process to mark FX Forwards on a corresponding curve is largely a holdover from the days before efficient and liquid derivative markets existed in Mexico. During that time, FX Forwards were the lynchpins of liquidity in FX and interest rate markets. In recent years however, the market standard for calculating and trading FX Forwards has become to interpolate interest rates and create a synthetic forward curve from Mexico zero rates, USD zero rates, binded with a cross currency basis curve (using applicable FX spot rate).

The proposed methodology suggests that Mexico Products should be marked on a single curve to enhance transparency between products, and avoid potential arbitrage between internal systems. The proposed method would offer more accurate and stable PL calculations as we would be using the most applicable curve across products.

Gitbook LMV

Github local gaussian

Core fx

3. Cancellable Instrument

This is a rather broad definition, covering both trigger-type products and callable products. In practice even for callable products the decision to exercise will depend on the current state of the market, and so these are often modeled by introducing some kind of exercise boundary 6 , i.e. a function of market observables describing a multidimensional boundary beyond which it is optimal to exercise. This has the advantage of separating two problems – making a decision to exercise and calculating the value of the cancellation leg.

In general, there may be any number of cancellation legs in a product, and a cancellation leg will cancel a fixed number of other legs. Legs that can be cancelled as an effect of valuation of a cancellation leg will be referred to as cancellable legs. It is possible for a cancellation leg to be cancellable by another cancellation leg. We will however assume that in cases where there are more than one cancellation legs cancelling same legs the term-sheet defines clearly an order of precedence between them (i.e. which decision is made first), and that if two cancellation legs cancel a single leg in common, then their actions are mutually exclusive (i.e. cancellation can only occur on one of them).

Gitbook cancellable

Github callable

Core callable

4. Bermudan Note

A Bermudan callable structure is a structure consisting of two sets of cashflows, one paid and one received, which can be valued in the usual Monte Carlo setting, and a set of dates (notice dates) when the structure can optionally be cancelled (or called). When this happens all payments for future periods are stopped, and possibly a penalty payment is made

The essential difficulty in estimating the best values for pi lays in the backwards induction nature of the optimal decision at every notice date – it depends on the hold value, which in turn depends on the optimal exercise decision on the next notice date. In other words: if hold values were known, we would know what the optimal decision is on each path, so we could in principle find values for pi that get our decision function as close to it as possible.

Gitbook Bermudan

Github convertible

Github Bermudan

5. Market Risk Measurement

The Market Risk Measurement and Management Process establishes the linkages which are required in an effective risk management system. All data composing these linkages can theoretically trace dependencies back to market inputs and position inputs. Market inputs are defined as the inputs required in the valuation model that are dynamic in nature and are sourced from markets. As an input to the risk process, the positions require thorough review and validation processes to ensure completeness and consistency. The set of inputs required for valuation should also be employed for other calculations such as P&L attribution (PAA) and should be simulated when computing Value at Risk (VaR).

The Market Risk Measurement and Management Review needs to consider all potential risk measurement and data process gaps during the life cycle of a trade in the risk system. From inception, when a trade is first executed, the trade needs to be modelled and captured accurately in the risk system. This includes capturing the appropriate market data and market factor sensitivities for the specific trade. Additionally, all market factors driving valuation need to be modelled in the VaR system. Thus the process will comprise five review streams to connect the necessary valuation, risk measurement and risk capture processes.

Gitbook market risk measurement

Github risk

ResearchGate risk pdf

ResearchGate risk

6. Market Risk Factors

Risk factors in the VaR model define the parameters which are simulated in the Monte Carlo engine. The starting point for defining risk factors is reviewing the pricing models market parameters. These market factors and inputs are candidates to be included as risk factors to VaR. The inclusion of risk factors will be chosen to minimize the Unexplained P&L (see Unexplained P&L section below). The inclusion of risk factors also requires the generation of sensitivities (to which simulated returns are to be applied) and market data for calibration.

The selection of factors may require an evaluation of the simulation modelling under various definitions (such as simulating relative or absolute returns). To the extent that not all market inputs for valuation are represented in the VaR risk factor simulation set, this should be investigated and the justification documented. Often market inputs are rendered more coarse as they are translated to risk factors (i.e. a 27 point yield curve for valuation may be translated to a 5 point yield curve for VaR simulation). Such translations should also be reviewed and documented as to evidence that the underlying characterization of the market risk is preserved.

Gitbook market risk factors

Github market curve

ResearchGate market pdf

ResearchGate market

7. Unexplained Profit and Loss

The impact of the sensitivity approach indicates a gap between full revaluation P&L and sensitivity based P&L. Reducing this difference requires the addition of new sensitivities to the model. A move to a full revaluation would completely remove this error in the model P&L. This difference will be tracked over time at the limit letter level and can be used to evaluate the potential improvement from changing the valuation approach in the model.

The impact of risk factor selection indicates the gap between a direct market instrument representation in valuation compared to mapping market data to risk factors and back to market instruments. As risk factors represent a selective choice of market instruments, differences attributed to risk factor selection will require a review of the risk factors and risk factor modelling assumptions. This difference will be tracked at the limit letter level.

Gitbook unexplained P&L

Github var

ResearchGate sensitivity pdf

ResearchGate sensitivity

8. Market Risk Backtest

The backtest P&L calculations are based on the actual day-over-day changes in market inputs observed. The market inputs must be the same as those used for official valuation thereby establishing a direct linkage to P&L.

Exceptions may be classified as legitimate, or false. For instances where the exceptions are deemed as false, such as spurious market data input, and IT system issues, appropriate operational procedures need to be followed for issue resolution including reruns, market data reloading/recalibration, etc.

Gitbook backtest

Github exposure

Core risk

9. Market Risk Validation

Value at Risk (VaR) is computed using risk sensitivities from the official risk systems. Given that the IPV and VA are applied outside the system, consideration must be paid to what impact these adjustments may have, if any, to these risk sensitivities. For example, a large IPV may signal a material difference between the market data in the source system and the independent market data. The source system data is used to compute the risk sensitivities for VaR. Differences in these market data may result in changes in risk sensitivities, particularly for portfolios that exhibit non-linearity, where the risk sensitivity itself changes with changes in market data.

In certain cases, fair valuation of financial instruments may require capabilities that are not present in the source system valuation models or valuation environment. In these cases, VPC will use existing vetted models outside the source systems to compute fair value. For material IPV adjustments, the factor(s) will be assessed with respect to the implications on risk sensitivities.

Gitbook risk validation

Github close out

Core cva

10. Market Risk Modeling

A review of modelling assumptions has been incorporated in the Nextgen work. For example, in the work leading up to the Nextgen project, the simulation of base metal commodity futures was changed from an all-in representation to a spread against commodity forwards. The revised modelling assumption helped to improve the accuracy of the basis between futures and forwards, which previously exhibited nearly unbelievable scenarios that were far outside the realm of a 99% confidence level. As part of the commodities, equities and fixed income portions of Nextgen, joint efforts between VPC, RO, and Risk Models have reviewed the basic premises in the risk factor definitions including the use of constant maturity risk factors in commodities and zero interest rate yields for simulation.

The Risk Models quarterly benchmarking exercise between full revaluation and Greek-based VaR should be reviewed to understand the degree of approximation. A divergence greater than 5% should be investigated to understand the implications of the ‘missing’ VaR Greeks.

Gitbook market risk model

Github interpolation

Core lmm

11. Counterparty Exposure

Counterparty credit risk (CCR) relies on exposure profiles. They are the product of pricing all deals into the future under Monte Carlo simulation and aggregating using all relevant netting and collateral agreements. Another important feature that is shared with VaR calculation is the simulation of underlying market factor that is required in order to evaluate those deals; however for CCR the time horizon for simulation is in years rather than days or weeks for VaR.

In the CCR context, simulation models have the objective to forecast within a reasonable range and horizon market factors such as equity prices, interest and FX rates, CDS curves and so on. In order to capture a realistic view of our exposure going forward, and because CCR is not directly hedgeable, those models are typically calibrated using historical data (~3 years) and are not systematically implied from today’s market prices.

Srsweb credit risk

Gitbook ccr exposure

Github rate lock

Core jump

12. Counterparty Risk Stress Test

CCR stress test results can be more difficult to interpret than market risk VaR – there is no single 95th percentile loss to focus on, but instead we must consider the impact on the individual exposures to thousands of different counterparties. We can make this this more manageable by for example focusing on the top 50 counterparties, or aggregating by country or industry sector.

In the context where Stress tests are based on exceptional but possible scenarios, the origin of a stress scenario is the economics department. It then gets transmitted to the Stress Test group for translation into market factor shocks that CCR models can interpret. At this point, calibration takes place: stressed market data and historical prices are taken as input to that process; it yields stressed parameters (recall kappa, sigma and theta from before) which are then passed on the CCR engine where EE, PFE are calculated for each portfolio. Depending on the current application, whether it is used for ICAAP or regulatory stress tests, the results are compiled and sent to the relevant team.

Gitbook ccr stress test

Github portfolio

Core credit

13. Counterparty Risk Measure

Credit exposure is the amount a bank can potentially lose in the event that one of its counterparties defaults. Note that only OTC deals (and security financing transactions) are subject to counterparty risk. We define replacement risk in the context of this report as the maximum of the PFE at a set of pre-specified valuation time buckets.

Note that the valuation methodologies used to calculate exposure could be very different from the front office pricing since for credit exposure calculations, what is important in this project is the distribution of deal values under the real world measure at different times in the future. The valuation methodologies need to be optimized in order to perform sufficiently large number of calculations required to obtain such distribution. Because of the computational intensity required to calculate counterparty exposures, compromises are usually made with regard to the number of simulation times buckets and the number of scenarios.

Gitbook ccr measure

Github cash flow

ResearchGate credit pdf

ResearchGate credit

14. Add-on Exposure

Add-on factor tables (profile basis) are uploaded to the production system to monitor the replacement risk. The system could easily pick up the tables for exposure calculation. A complete term profile of add-on factors for FX Forward trades and FX Option trades (including buy/sell domestic currency and sell/buy foreign currency, with gross exposure and collateralized exposure) are stored in production system. Also the system stores the add-on factors of Repo, Reverse Repo, Security Bought & Sold, and Security Borrow and Lending with all currencies, issuer types, credit rating, underlying type and underlying terms.

A counterparty’s exposure limit could be time-dependent and set up in other currencies rather than USD. Also, a counterparty’s exposure profile is time-dependent. In this way, the exposure calculated at each time bucket should be compared with the limit set up at the corresponding time interval. If the exposure is higher than the limit, there should be a limit breach warning triggered. Also, the limit should be compared with the exposure calculated at the related time bucket. If the limit is lower than the exposure, the system should trigger a warning/violation.

Gitbook addon

Github fair value

ResearchGate fx option pdf

ResearchGate fx option

15. Intraday Replacement Risk

For new and amended deals completed intraday, the MTM values or premiums reflecting these values, will either be retrieved directly from the product systems (assuming that appropriate pricing parameters and market data were specified at the time of input) or entered manually by the trader. The referenced FPE, calculated with risk factor based on a transaction’s product type and underlying attributes, is then added to this MTM value to come up with the replacement risk for the deal. The overall replacement risk calculation is restricted to MAX [(0, MTM)] + FPE, such that in no instances will a negative MTM be considered in the calculation.

The percentage replacement risk factor is determined using the ratio of the upward diffused price over the strike price. For long puts, short equity forwards and short mutual fund forwards, the current price of the equity is diffused downwards with drift equal to 0 (i.e., no directional bias) and volatility set to the greatest of the 1, 3, and 5-year standard deviations. The percentage replacement risk factor is determined using the ratio of the downward diffused price over the strike price.

Gitbook intraday

Github asset

Core irc

16. Collateralized Exposure

This collateral method is built on a mixture of backward and forward looking style. The counterparty exposure is measured on a date when the counterparty is deemed to be in default. This is consistent with the terminology and concept of “Exposure at Default” in CCR. Standing at a reporting time bucket t, the collateral assets has been posted in the past, and the collateralized exposure depends on the “liquidation” value of the derivative portfolio and collateral assets at some future time.

To measure the counterparty exposure at a future time t , first we need to calculate the portfolio value. The portfolio valuation will be consistent no matter if there is a collateral agreement or not. Time t is at the end of the settlement period and the beginning of the liquidation period. The Bank faces higher market risk when it needs more time to liquidate (or replace) the portfolios. The length of the liquidation period depends on trade types and traits (notional, term, etc.). It also depends on market conditions, as some products may become very illiquid during financial stress. So the liquidity period should be defined at the trade level according to some prescribed rules, and should be allowed to be changed (e.g. for stress testing purpose).

Gitbook collateral exposure

Github collateral swap

ResearchGate collateral pdf

ResearchGate collateral

17. Collateral Methodology

When the Bank determines that the counterparty is in default, it will start to negotiate new trades to replace exist derivative portfolio. At the same time, it will take hold the collateral asset and try to sell these assets in the market. The value fluctuations of the portfolio and the collateral asset during their liquidation periods create risk to the Bank. In a CCR model which inherently incorporates the Wrong Way risk, both trades and collateral assets liquidation value need to be calculated conditional on the fact that the counterparty is in default.

Although our method is logically more consistent with the counterparty exposure definition, it can also be changed by “shifting” the exposure calculation time t along the timeline. If the time t is set at the end of the liquidation (or closeout) period, then we have a backward looking model. Or if t is set at the beginning of the settlement period, we will have a forward looking model.

Gitbook collateral methodology

Github principal

ResearchGate irc pdf

ResearchGate irc

18. Counterparty Credit Risk BackTest

Backtesting is a statistical test with the significance of any result depending on the amount of data used. A backtesting data set is a set of forecasts and the corresponding realisations of those forecasts, ie what actually occurred. This backtesting data set can be put together in a number of ways.

The backtesting data set can be aggregated over time, over trades/risk factors or over both time and trades/risk factors. The time period over which data is aggregated is referred to as the observation window. There are a number of methodologies for generating a backtesting data set over a given observation window. A selection of frequently used methodologies are set out below.

Gitbook ccr backtest

Github index

ResearchGate impact pdf

ResearchGate impact

19. Counterparty Credit Risk Jobs

A job is a specific instance that will be sent to the compute framework. It associates a job spec with a specific anchor timestamp and trade timestamp. These determine the precise bi-temporal version of market/reference data and trades respectively.

A market data path represents a possible evolution of market data through time. Generally, all paths start at the same place with the real world market data, but evolve differently to each other over time. Future market data points on a path may be generated either through a simulation model (Monte Carlo paths), through application of pre-specified ‘shocks’ to each market data point, or may be real world values if the path is being generated retrospectively (e.g. for back testing).

Gitbook ccr job

Github convertible factor

ResearchGate exam pdf

ResearchGate exam

20. Counterparty Credit Risk Limit Monitoring

Limits are set to limit the allowable exposure for an ‘Exposure Definition’ while Trading Restrictions are set to ensure adherence to rules/policy that is not an exposure versus limit check, for example, to ensure that maximum allowable tenors are not exceeded or business rules are not broken. For example, any Repo trade must have an enforceable legal agreement governing transactions between the organization and the Counterparty of the Agreement.

It is assumed that every trade will be either directly or indirectly mappable to all Aggregation Set Dimensions. However it should be noted the Aggregation Set trade membership rules are sometimes specific to the kind of Aggregation Set (the Aggregation Set Type) along with the Risk Metric that is linked to the Aggregation Set.

Gitbook ccr limit

Github fx chooser

ResearchGate jump pdf

ResearchGate jump

21. Pre-Deal Check of FX Forward

Foreign Exchange Forward Contract is an instrument that allows the buyer to lock in a foreign exchange rate for a specified date in the future. For instance, the 2-year forward rate for USD/CAD is 0.951067573351087 and 0.942640335579959 for 3-year. In order to get a forward rate for a deal that matures in 2.5Y (the system doesn’t provide 2.5Y forward rate), we can use linear interpolation method describe in Appendix to derive the appropriate forward FX rate.

After the system calculates the individual FX Forward’s exposure based on add-on factor, it will add the exposure profile on top of the pre-deal counterparty-level exposure to get post-deal counterparty-level exposure. However, the time buckets from Pre-Deal counterparty-level exposure might be defined differently from the time buckets of individual FX Forward deal’s exposure profile.

Gitbook pre-check

Github xccy

Core frtb

Fixed Income


  1. Brownian Bridge

The Brownian Bridge algorithm belongs to the family of Monte Carlo or Quasi-Monte Carlo methods with reduced variance. It generates sample paths which all start at the same initial point and end, at the same moment of time, at the same final point.

In the context of stress testing this algorithm is used for efficient generation of specific scenarios subject to certain extreme and generally unlikely conditions. If paths were generated by a conventional Monte-Carlo method only a very small portion of all the paths would satisfy such conditions.


Hcommons bb

ResearchGate option pdf

ResearchGate option

2. Hull White Volatility

Hull White model needs to be calibrated to the market price, i.e., one needs to map implied Black’s at the money (ATM) European swaption volatilities into corresponding Hull-White (HW) short rate volatilities.

At each grid point, we compared respective Black’s and HW trinomial tree payer swaption pricing benchmarks. Specifically, using the interest rate and implied Black’s volatility .

OSF HW vol

ResearchGate option vol pdf

ResearchGate option vol

3. Bond Curve Bootstrapping

A method is discussed for bootstrapping a set of zero rates from an input set of US government money market securities and bonds. The government bond bootstrapping procedure requires to input a set of financial instruments, of the type below, sorted by order of increasing time to maturity.

Government Bond Bootstrapping proceeds in two phases. The first phase uses short term instruments, which typically mature in one year or less. Consider, for example, a US government money market instrument is used.

OSF bond curve

ResearchGate curve pdf

ResearchGate curve

4. Martingale Preserving Tree

An important feature of the popular three factor trinomial tree is that it uses a deterministic approximation of the interest rates for constructing the stock tree. The preservation of the martingale property of the stock price is thus not guaranteed.  and may potentially represent a problem.

A new tree model that preserves the martingale property of the stock for sufficiently long terms (with accuracy better that 10-8 for terms of at least 10 years) is present.

OSF preserving tree

Core callable

Hcommons tree

5. Black-Karasinski Tree

The Black-Karasinski model is a short rate model that assumes the short-term interest rates to be log-normally distributed. The one factor  Black-Karasinski model is usually implemented by a binomial or trinomial tree.

OSF BK tree

Hcommons bk tree

Core reverse

6. LIBOR Rate Model

A Libor rate model is presented for pricing Libor-rate based derivative securities including caps, floors, and cross-currency Bermudan swaptions. Although referred to as a BGM model, the model is actually based on Jamshidian’s approach towards Libor rate modeling (i.e., where Libor rates are modeled simultaneously under the spot Libor measure).

LIBOR Rate Model is used for pricing Libor-rate based derivative securities. The model is applied, primarily, to value instruments that settle at a Libor-rate reset point.  In order to value instruments that settle at points intermediate to Libor resets, we calculate the numeraire value at the settlement time by interpolating the numeraire at bracketing Libor reset points.

OSF libor model

Core arn

7. Hedge Fund VaR

A VaR calculation method is present for options written on a basket of hedge funds, with minor changes and the methodology for calculating the VaR of the LTV (loan to value) ratio for loans to funds-of-funds.

The portfolio diversification and leverage limits were found to be consistent with increasing conservatism as the number of funds in a basket decreases. It should be noted that these limits cannot be considered as ‘stand alone’ since the characteristics of hedge funds change with strategy and management style–this table must be used in conjunction with other risk-management tools.

Zenodo HF VaR

Zenodo HF VaR home

ResearchGate cds pdf

ResearchGate cds

8. Hedge Fund Index

Hedge fund index is unusual in the sense that it is tracking an asset class with reduced liquidity (hedge funds), and the performance of the index tracks the actual processes involved in hedge fund investing–in particular the timing of fund redemptions.

This results in the index return being recalculated at various times with different estimates of the fund returns, until the finalized value of the index is calculated: 45 calendar days after the end of the month. Even then there may be some funds that have not reported finalized NAVs, and the index administrator may have estimated the return.

Zenodo HF index

Zenodo HF index home

ResearchGate convertible pdf

ResearchGate convertible

9. Cash Flow Hedge

For the already existing recognized assets or liabilities cashflow hedges can be designated only if cashflows of such item/s are linked to floating rates (as opposed to fixed rates). For example, one can hedge on a cashflow-hedging basis cashflows from floating rate mortgages / loans or on floating rate deposits.

An entity can also hedge the variability of cashflows related to a forecasted transaction. A “forecasted transaction” is a transaction that is probable of occurring but for which an entity has not entered into a firm commitment. Observable facts and circumstances should support the probability of the transaction occurring.

Zenodo CF hedge

Zenodo CF hedge home

ResearchGate soft pdf

ResearchGate soft

10. Fair Value Hedge

Hedgers may elect to hedge all or a specific identified portion of any potential hedged item. Fair value hedge accounting is not automatic. Specific criteria must be satisfied both at the inception of the hedge and on an ongoing basis. If, after initially qualifying for fair value accounting, the criteria for hedge accounting stop being satisfied, hedge accounting is no longer appropriate.

At inception and on an ongoing basis (at least quarterly), the hedge must be expected to be highly effective as a hedge of the identified item. The effectiveness in achieving offsetting changes to the risk being hedged must be assessed consistently with the originally documented risk management strategy.

Zenodo FV hedge

Zenodo FV hedge home

ResearchGate refix pdf

ResearchGate refix

11. Performance Deferred Share Program

The Performance Deferred Share Program (PDSP) has been established by an organization to compensate eligible employees for their contribution to the long term performance of the organization.

In order to value the payout of the performance deferred shares, one needs to model how the TSR will compare to the peer group at some future date. In order to do this, a correlated log-normal model was used to model the share price of each organization.

Zenodo PDSP

Zenodo PDSP home

Core FRN

12. Balance Sheet Model

The balance sheet model is used to determine the risks of various assets, liabilities and balance sheet items. Primarily, the model calculates the interest rate risk profile of these instruments.

The instruments on (and off) the balance sheet are split into various subaccounts, and these subaccounts are mapped to accounts. It is at the subaccount level that many of the instrument characteristics are defined, including cash flows, behavioral assumptions and valuation models.

Zenodo BSM

Zenodo BSM home

Core bond

13. Close-out Reserve

A model is present to calculate the monthly Close-Out Reserve of the structured interest rate derivatives. Products cover vanilla swaptions, Bermudan swaptions, callable swaps, variable notional swaptions, cap and floor and Treasury bond options.

Let us consider an option (vanilla or non-vanilla). Given a swaption term, an underlying term and a strike price, if we change the volatility from the above volatility cubic, we can get one Vega by using the definition of Vega.

Zenodo close out

Zenodo close out home

Core Himalaya

14. Local Volatility Gaussian

The local volatility Gaussian model represents a significant improvement over the existing Lognormal Gaussian Model in its ability to incorporate FX volatility skew effects and value FX-IR hybrid swaps in line with market consensus.

The local volatility Gaussian model assumes that the instantaneous volatility of the instantaneous FX rate is a deterministic function of only time and the instantaneous FX rate. The model assumes that local volatility is piecewise constant in time and piecewise quadratic in the logarithm of the instantaneous FX rate.

Zenodo LVG

Zenodo LVG home

ResearchGate basket pdf

ResearchGate basket

15. Curve Interpolation

The interpolation of curve bootstrapping, including both linear spline and cubic spline, is studied. Although there are a number of advantages to using piecewise cubic splines, there is one major drawback which leads us to go in favour of linear splines.  This drawback stems from the fact that the perturbation of one point will affect another point.

One can then use this to approximate other points on the curve.  The advantage of linear interpolation is its simplicity and, in many cases, it provides an adequate approximation.  A disadvantage is that the approximating curve is not smooth (since the derivative is in general discontinuous at given data points) even though the real curve may in fact be smooth. 

Zenodo interpolation

Zenodo interpolation home

Hcommons bond curve

Core amortizing

16. Short Term Curve

Short term curve construction may contain both regular and serial futures contracts that results in a significant amount of underlying term overlapping. The overlapping may lead to widely oscillating Partial Differential Hedge (PDH) numbers

If we have the discount factor at the first offset date of a mod group, then, using the forward wealth factor multiplicative property and normalization process, one can construct the discount factors at the rest of offset dates in the group. The cash deposits produce discount factors at their underlying term start and end dates, call them cash dates. The discount factor at the first offset date of a mod group, so-called seed, will be deduced via interpolation from the discount factors at the nearest (left and right) cash dates.

Zenodo short term curve

Zenodo short term curve home

Core puttable

17. LGM Calibration

The traditional calibration routine in the model works only in a domestic market, in other words, it is not applicable to cases with funding in a foreign currency. The new calibration routine corrects the old one in LGM European swaption price calculation when a basis spread adjusted zero curve is applied for a non-reference currency.

In an IR term structure model calibration to European swaption market, it is always prefer to have the swaption model price in an analytical close form so that the calibration routine can be effective and accurate. In the LGM calibration, European swaption pricing model prices follows the so-called Jamshidian bond option formula, which has been accepted as market convention for the LGM calibration to the swaption market.

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Core cds

18. Time-Weighted Quadratic Interpolation

A Time-Weighted Quadratic Average (TWQA) Interpolated Enhanced Swap Curve building algorithm is proposed. All major properties one expects from the curve (arbitrage-free, locality of sensitivity, and smooth forward curve), and achieved by the current model, are still guaranteed.

In order to obtain a unique function f, there is a need to impose meaningful conditions on the values of f at boundary points. This is done such that the locality property of the curve (when shocking an input instrument, the shock spreads to the neighbors only, not to the whole curve) is guaranteed.

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ResearchGate forward option pdf

ResearchGate forward option

19. Interest Rate TARN

An interest rate TARN swap is a structured swap contract with a regular funding leg and a structured leg. The coupons in the structured leg are defined as the same as in the corresponding interest rate TARN. Moreover, the swap has a mandatory termination once the accumulated structure coupon breaches a pre-determined barrier.

In fact, an interest rate TARN swap can be decomposed into a regular cap-floor swap and a so-called target redemption component. The target redemption component can be treated as a separable derivative product to cancel the remaining swap.

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ResearchGate option model pdf

ResearchGate option model

20. Black Option Model

Black’s option pricing model, which is in a closed-form formula, can be applied to vanilla European type options under the Black-Scholes framework. Black’s option pricing formula has been widely applied in fixed income derivative market for years.

Black’s vanilla option pricing model can be applied to pricing a variety of instruments including caps/floors, European swaptions, bond options, bond futures options and IR futures options. In the case of caps/floors and European swaptions1, X is the forward term rate and forward swap rate, respectively. For European bond options, the rate X represents the bond price. For European bond futures options and European IR futures options, X stands for bond futures price and Euro-Dollar futures price, respectively.

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ResearchGate option pdf

ResearchGate option

21. Digital Option

A pricing model for skewed European interest rate digital option is present. The traditional pricing model is under the Black-Scholes framework. The new skew-adjusted model replicates a digital option by a portfolio of vanilla call options, and/or zero-coupon bonds and/or floating rate notes (FRNs). The new model provides a better approach to pricing skewed European interest rate digital options.

One may see that a skew-adjusted digital option can be approximately evaluated by a portfolio of vanilla call options, and/or zero-coupon bonds and/or FRNs. There are three ways to use this model

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ResearchGate cva pdf

ResearchGate cvs

22. Double CMS

A double CMS derivatives represents a European type derivatives whose matured payoff depends on two CMS rates.1 For most important products in the fixed income market, the payoff function can be an affine-linear with respect to two CMS rates and may be possibly capped and/or floored.

Under an appropriate forward measure, the value of each structured coupon is equal to the discounted expectation of the coupon. For some trivial cases when the coupon rate is just a linear combination of two CMS rates, then the expectation can be calculated by using single CMS rate European vanilla option pricing model. Therefore, it suffices to consider the calculation of the expectation of CMS average/spread call-payoff

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ResearchGate correlation pdf

ResearchGate correlation

Fixed Income

Derivatives Modeling

  1. Forward Starting Option

Option price or implied volatility surfaces are available at points on a relatively sparse grid of strike and tenor pairs. Using analytical expressions to determine the local volatility function then likely yields inaccurate results due to the numerical instability from calculating first, and especially, second derivatives.

A forward starting option is an option whose strike price is not fully determined until an intermediate date before expiration.

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2. Three Factor Convertible Bond Model

The stock price process is then expressed under the bond’s coupon currency risk neutral probability measure by means of a quanto adjustment.   Under the bond’s coupon currency risk neutral probability measure, then, the short interest rate, stock price and foreign exchange rate processes respectively follow geometric Brownian motion with drift, but are driven by pair-wise correlated Brownian motions. 

We next define three related random variables, which are each taken to be particular linear combinations of the original short interest rate, stock price and foreign exchange rate random variables.  Here the respective linear combinations are chosen such that the processes for the new random variables are now driven by pairwise uncorrelated Brownian motions.

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Hcommon hw

3. Hull-White Convertible Bond

Based on the Hull-White single-factor tree building approach, respective trinomial trees are constructed for the short-term interest rate and stock’s price processes.  Using the Hull-White two-factor tree building procedure, a combined tree is constructed by matching the mean, variance and correlation corresponding to each combined tree node.  The convertible bond price is given from the combined tree by backward induction. 

Here the issue time refers to the coupon payment immediately prior to, or including, the valuation time; otherwise it corresponds to the bond’s issuance.  Since the valuation time is taken to be zero, the issue time must be less than or equal to zero.

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4. Exchangeable Convertible Bond

A convertible bond issuer pays periodic coupons to the convertible bond holder. The bond holder can convert the bond into the underlying stock within the period(s) of time specified by the conversion schedule. The bond issuer can call the bond and the holder can put it according to the call and put provisions. The Exchangeable feature assumes that the convertible bond and the underlying stock are issued by different parties.

Assume that the stock conversion is vulnerable. If the bond-issuer has defaulted by a time, t , then the stock price is zero. If, on the other hand, the bond-issuer has not defaulted by time t , then the stock price is given by St or 0.

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5. Extendable Swap

An extendable swap represents a forward swap agreement with an option of extending the swap for another term (swaption). The valuation model assumes the swap rates for different terms to be correlated log-normally distributed random variables and uses the Haselgrove integration method for pricing the deal.

The model estimates the swap price as a risk-neutral expectation of the difference between the bond price whose yield-to-maturity is the swap rate and the bond’s par. The swap rate is considered a log-normally distributed random variable.

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6. Callable Inverse Swap

A Callable Inverse Floating Rate Swap is a forward swap agreement with an option of canceling the swap each year starting from several years in future. The deal is priced with a two factor Black-Karasinski model.

The calibration procedure takes only an interest rate curve as input (ignoring volatility surfaces) and results in adjusting the “alpha” parameter of the model. To test the calculations over a range of parameters, we used  the “piece-wise constant parametrization” mode.

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7. Bond American Option

The model assumes the yield of an American Treasury bond to be a log-normally distributed stochastic process and uses Monte-Carlo simulation to price the deal as a European call option.

The model builds a trinomial tree for the yield process to price the deal as an American option. The time slices of the tree are evenly spaced. Node transition probabilities and the time interval between slices are determined by matching the first four moments of the underlying Brownian motion. The option is priced using the backward induction.

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8. Arrear Quanto CMS Swap

An arrear quanto constant-maturity-swap (CMS) is a swap that pays coupons in a different currency from the notional and in arrears. The underlying swap rate is computed from a forward starting CMS.

We note that the common currency unit in Europe is now taken to be the EURO.  Furthermore, the exchange rate from the EURO to an associated currency (e.g., FRF) is fixed, so there is no foreign exchange risk.  Therefore, FP London uses a common curve, EURIBOR, for discounting

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9. Variable Rate Swap

Variable rate swap is a special type of interest rate swap in which one leg of the swap corresponds to fixed rate payments while the other involves fixed rate payments for an initial period of time and a floating rate for the rest. The floating rate on that portion is defined as a minimum of two index rates.

Variable rate swap is an interest rate swap that has two legs: one fixed rate leg and a variable rate leg. The variable leg involves fixed rate payments for an initial period of time and a floating rate for the rest. The floating rate on that portion is defined as a minimum of two index rates.

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10. CMS Spread Option

A constant maturity swap (CMS) spread option makes payments based on a bounded spread between two index rates (e.g., a GBP CMS rate and a EURO CMS rate).  The GBP CMS rate is calculated from a 15 year swap with semi-annual, upfront payments, while the EURO CMS rate is based on a 15 year swap with annual, upfront payments.

We assume that both the forward GBP and EURO CMS rates follow geometric Brownian motion under their respective -forward measures.  Here respective initial forward CMS rates are calculated.  The forward rates are then convexity adjusted from respective parallel bonds specified using

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11. Early Start Swap

An early start swap is a swap that has an American style option for the counterparty of starting the swap early, within a period of three month. Otherwise, the swaps are plain vanilla fixed-for-floating swaps.

The internal rates of return of the two swaps, one starting at the beginning and the other at the end of the exercise period, are generated for the earliest exercise date, assuming that the two rate are practically perfectly correlated. Then the difference of the present values of the two swaps, if positive, is taken as the option value. This value is averaged over a number of scenarios.

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12. Digital Swap

A daily digital LIBOR swap is an interest rate swap whose reference interest rate is three-month USD Libor BBA. For each accrual period in the swap, one party receives the reference rate, and pays the reference rate plus a positive spread, but weighted by the ratio of the number of calendar days in the period that the reference rate sets below an upper level to the total number of calendar days in the period.

We assume that Libor rates follow geometric Brownian motion with no drift and constant volatility under their respective forward measures. In order to value a daily Libor-based digital payoff, the respective Libor rates at the daily setting time and at the accrual period start must then be expressed under the same forward measure.

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13. Ratchet Swap

The ratchet floating rate coupon is based on an index, e.g., 6-month EURIBOR. The rate is further subject to a minimum decrease of 0 bps and a maximum increase of a threshold, such as, 15 bps. These rates are reset two business days prior to the first day of each coupon period.

The valuation methodology is based on the Monte Carlo spot LIBOR rate model. The model generates spot rates which log-normally distributed at each reset date. These spot rates are derived from corresponding forward rates whose stochastic behavior is constructed in an arbitrage-free manner. Outcomes for the spot rate are generated for each reset date. These rates are then applied to the ratchet-type payoff structure. The ratchet instrument is then valued by discounting and averaging these payoffs.

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14. Hedge Fund Barrier Option

A hedge fund barrier call option is a note whose payoff is based on a basket of hedge funds. The deals are structured so that once the barrier (usually set at 95% of the notional) is hit, the funds in the basket are sold off, with the realized fund value depending on the redemption period of each fund..

The goal here is to estimate the market risk of the entire portfolio of such deals through analysis of a small representative sample of the portfolio and scaling up to the entire portfolio. While simulating the entire portfolio would result in a more accurate determination of the capital, the result is small enough that the dominant risk factors likely arise from sources other than market risk, and an order-of magnitude determination is likely sufficient.

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15. Canada Housing Trust Swap

The Canada Housing Trust Swap includes variable rate mortgages and involves reinvestments made by the principal payments. The variable rate mortgages that appear in the deal are a result of these reinvestment. The model was in the context of the much more complicated problem where the notional on the mortgages was not fixed, and reinvestments were made at prevailing market prices.

The valuation of the variable rate mortgage, as it contributes to the CHT swap is very simple. It is simplified by the fact that the interest rate payable is fixed at 30 day BA, and that we are discounting using the same curve that is used to evaluate the interest received. Furthermore, the principal payments on the variable rate mortgages, including prepayments, are reinvested at par so that the principal remains constant.

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16. BMA Knockout Swap

BMA Ratio Swap with BMA Knockout is a two-legged BMA ratio swap where one leg pays a contract specified fixed rate and the other leg pays Libor times a contract specified ratio (plus a contract specified constant spread).

If we consider a deal called Libor Swap with BMA Knock-Inwhere the knockout condition is defined by a maximum level for average BMA, the coupon payments at time i S are the following:

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17. BMA Swap

In a generic fixed for floating BMA swap, the floating side is estimated using averages of the BMA Municipal Swap Index, which is published on a weekly basis. The fixed side pays interest quarterly on a 30/360 yield basis and payment dates adjusted using modified following basis. The floating side pays interest quarterly in ACT/ACT yield basis and payment dates adjusted using the modified following basis.

Three types of swap legs are available, including BMA leg as well as typical LIBOR (floating) leg and fixed leg with variable notional. BMA leg pays (or receives) weighted average of weekly BMA indices over specified periods, based on ACT/ACT day count basis (DCB). LIBOR leg receives (or pay) LIBOR rate multiplied by a fixed ratio using ACT/360 DCB.

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18. Chooser Cap

A chooser cap (floor) is different from the traditional European/Bermudan option that the owner of the chooser option has multiple chances to exercise. The rigorous definition of chooser option is given in the appendix section of this report. From the definition of the chooser option, a lower bound of the value of the chooser cap (floor) is the sum of first k maximal values of (European) caplets (floorlets). To get a good upper bound is not trivial.

From a rigorous view point, the dynamics may not be completely arbitrage-free. However, it perfectly re-produces all European caplets (floorlets) market prices automatically. Therefore, the dynamics can be considered as approximately arbitrage-free without any additional calibration. It should also be noted that volatility skewness is not considered in this dynamics.

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19. CMS Cliquet

A CMS cliquet option has two legs: One leg of this deal is based on (regular) floating rates. The other leg links to CMS swap rates. Due to the “set-in-arrear” feature in the structured leg, convexity and timing adjustments have to be considered.

Pricing the second leg of the contract is a little bit more involved. Firstly, there is an optionality which is embedded in the contract, and secondly, this leg does not incorporated a natural time lag, which implies that the convexity adjustment is needed.

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20. Arrear Forward Rate Agreement

A general FRA is a European forward rate derivative with a maturity which is not earlier than the beginning of the forward period. A vanilla FRA is the same type of security except its maturity is right at the end of the forward period. While, a set-in-arrear FRA is the one whose maturity is right at the beginning of the forward period.

Generally, convexity adjustments are required for pricing these FRAs except vanilla FRAs. Under the assumption of single factor driving force, for a FRA whose maturity is before the end of the forward period, the convexity adjustment is positive while for a FRA whose maturity is after the end of the forward period, the convexity adjustment is negative.

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Fixed Income

FX And Retail Derivatives

  1. Quanto Himalayan Option

Himalayan options are a form of European-style, path-dependent, exotic option on a basket of equity underliers, in which intermediate returns on selected equities enter the payoff, while the equities are subsequently removed from the basket.

We employ the definitions of the respective hedge ratios, as stated in the section on the WM pricing method. With the exception of the theta, calculated through the finite difference technique, all hedge ratios are computed using the Malliavin weight approach. Additional considerations arise from the impact of the calibration procedure on the sensitivity ratios, as describe in the next section.

The equity price model is based on a discrete dividend treatment and results in shifted lognormal distributions for the equity price. A calibration step is required to obtain the required shifted-lognormal volatility parameters from the Black’s term implied volatility inputs. Monte Carlo technique is employed to compute the price and the hedge ratios.

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2. FX Asset Hedge

The foreign currency assets hedge model is employed to conduct the hedge effectiveness test for the foreign currency denominated floating rate LIBOR assets using CAD funding. The hedging derivative is a cross currency interest rate swap. The hedging is designated as the Cash Flow Hedging in which both interest rate risk and foreign exchange risk are hedged.

Note that the method of generating foreign exchange rate shock adopted here is same as that for the interest rate in the model.  The correlation between the interest rates and the foreign exchange rate are imbedded in the scenario, since exactly the same historical date is used in each scenario. Note that, unlike the usual cash flow hedging in which the fixed leg of swap is not considered, all the values of the four legs are taken into account so that the foreign exchange risk is fully captured.

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3. FX Choice Option

A derivative security considered here is a European type option whose holder, at the maturity, can either not exercise or exercise by choosing to enter one and only one of the two underlying securities: a cross-currency swap or a cross-currency forward contract. Let us call the option an FX choice option.

To elaborate, the following parameters are given at time-0: the bond term, the bond effective date is time T, the coupon rate RC and the day-count-basis (DCB). Similarly, we define BU as the bond price process of a pre-determined fixed coupon bond with unit face value in the U-currency. Further, let NU be the principal amount in the U-currency and ˆ S be a fixed exchange rate. Both are given at time-0.

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4. Local Volatility for Quanto

A model is presented for computing the price, in the domestic currency, of European standard call and put options on an underlying foreign equity (stock or index) with tenor of up to 7 years. The function implements a local volatility based pricing method.

We employed three calibration schemes for valuation. One scheme determines a constant exchange rate correlation parameter by matching with Balck’s forward equity price dynamics.

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5. GIC Pooling

Pooling is an integral part of the product book approach to managing the hedging and funding activities of the retail bank. Both the Canadian and US governing accounting bodies acknowledge pooling as a tool for the application of hedge effectiveness.  This section deals with the rules surrounding pooling as well as the practical application of pooling for our hedging and testing of hedge effectiveness.

Once the preliminary pool has been constructed based on the selected qualitative traits, each item in the pool must pass the quantitative test in order to remain a constituent of the pool. The quantitative criterion is that the proportionate change in fair values of each item to be included in the pool must be expected to be within 10% of the overall change in fair value of the pool attributable to the hedged risk.

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6. Mutual Fund Securitization

The purpose of the model is to determine, from a projected stream of future cashflows, whether all Commercial Paper used to fund the commissions to brokers for the sale of mutual funds will be repaid within a period.  Here a broker charges the Partnership a commission on the net asset value of the mutual funds sold.  The buyer of the mutual funds, however, pays nothing up front; instead, a deferred sales charge, which depends on when the mutual funds are redeemed, is assessed.

The model assumes that the monthly net asset value of the mutual funds follows a deterministic process.  Administration and program fees, as well as mutual fund redemptions are then based on the monthly net asset value.  Issued Commercial Paper is amortized into equal monthly payments over a period of six years.  Here the cashflows generated from the administration and redemption fees are paid monthly to the Partnership, to be used for the payment of outstanding Commercial Paper and associated interest.  Furthermore, the model includes a test to determine whether a collateral infusion is required to aid in re-paying the Commercial Paper.

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7. Asset Backed Senior Note

Asset backed senior note allows the holder to purchase co-ownership interests in a revolving pool of credit card receivables. To fund the acquisition of the interests in the revolving pool, the trust issued Asset-Backed Notes, in a number of different series. A share of future collections of credit charge receivables, to which the trust is entitled, is used to pay the interest and the principal of the notes.

The valuation makes the assumption that the future values of these parameters will be unchanged until the final payment date. Subsequently, the calculator performs a deterministic computation consisting of calculating the future cashflows in the waterfall and discounting them.


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8. GIC Pricing

The payoff at maturity from a GIC can be shown equal to the invested principal plus these principal times the sum of the minimum guaranteed interest rate and the payoff from a European call option on the arithmetic average of a basket price at the 12 points above, where the basket price is given by a weighted sum of the index levels above. 

We consider the pricing of this call option.  We assume that each of the underlying stock and bond market indices in the basket follows geometric Brownian motion with drift under their respective risk neutral probability measures.  Each index process is then expressed under the Canadian risk neutral probability measure by means of a corresponding quanto adjustment. 

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9. Flexible GIC

We price the option of a flexible GIC with a one factor Hull-White model via a trinomial tree. The Hull-White model assumes a normal distribution for the rates. Our solution constructs a Hull-White tree. The calibration procedures take an interest rate curve as input (ignoring volatility surfaces) and assume volatility and mean reversion parameters as constants.

A flexible GIC represents a financial instrument paying an annual coupon and provides an option for the holder to redeem the principal and accrued interest during the thirty days following the first and second coupons.

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10. Adjustable Rate Mortgages

Adjustable rate mortgages (ARMs) model has a significant amount of parameter/model risk In particular, there are many input parameters (many of them market variables) and functions of these parameters that can have add a significant amount of risk.

The prepayment model used for these instruments is a four-factor model. While the parameters and factors used were supplied to us, it is difficult to assess the accuracy of the parameters since it is a based on extensive statistical analysis of historical data by the vendor. However, the model seemed reasonable and well motivated even though direct verification of the parameters used was outside the scope of this vetting.

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11. MBS Deferred

The MBS Deferred Asset model is used for fair value assets that have been transferred to the Canada Housing Trust (CHT) through participation in the Canada Mortgage Bond (CMB) program. In particular, the model calculates the fair value of the retained interest of the MBS.

The introduction of interest rate dependent prepayments (and hence cash flows) requires the use of an dynamic interest rate model, and would complicate the model substantially. However, for the Canadian market it is deemed not nearly as important, and hence prepayments are assumed to be constant.

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12. Mortgage Commitment

A mortgage commitment can be approximately regarded as an American look-back option that gives the holder the right to effectively enter into a pay fixed leg of an amortizing swap.

In practice the holder of the commitment does not actually exercise the option optimally with regards to the American feature of the option. To capture the non-optimality of the exercise, they propose to model these commitments as a series of European swaptions, where the expiry dates on the options is determined using historical closing percentages.

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13. MBS

We begin with a review of mortgage mathematics and outlines variables that are used in subsequent sections.  Payment schedules of mortgages and the cashflows accruing to an MBS holder are also discussed in this section. A discounted cash flow (DCF) model constitutes the main pricing engine of the MBS, however, the main theoretical aspects of the model pertain to the prepayment assumptions corresponding to the underlying mortgage. A discussion of the two prepayment models is outlined in the next section..

To price an MBS we need to evaluate the monthly payments made to the underlying mortgage. These payments are divided into scheduled and unscheduled payments. The scheduled payments consist of principle and interest payments and the unscheduled payments consist solely of principle prepayments.

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14. Quanto Total Return LIBOR Swap

A quanto total return Libor Swap is a swap where one leg is a regular floating leg paying LIBOR less a constant spread and the other leg makes a single payment at the swap’s maturity equal to a leveraged non-negative return on USD-for-EURO exchange rate paid in CAD. The main focus of the valuation model is the quantoed total return on the FX rate.

A quanto total return Libor Swap is a swap with two legs. One leg of the swap pays LIBOR less a constant spread and the other leg makes a single payment at the swap’s maturity equal to a leveraged non-negative return on USD-for-EURO exchange rate paid in CAD.

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15. Principal Protected Note

A principal protected deposit note consists of zero plus call option and linear amortizing bond floor structure. these models are essentially accrual models, determining the current value of the option due to fee accrual and historical hedge fund performance in accordance with the documentation.

The value of the note to the investor (the ‘note value’) has an ultimate floor of the current price of a (CAD) zero coupon bond maturing at the maturity of the option, providing principal protection. The valuation of this structure essentially reduces to the valuation of the Zero plus a leveraged investment in an accreting strike option.

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16. Collateralized Swap

The Collateralized swap structure is an option where the client, rather than making an upfront cash payment, puts up collateral instead–in this case in the form of a basket of hedge fund investments. For the most part the option acts as an equity swap, with the client paying the returns on a basket of hedge funds and receiving a spread over Libor, with the notional amount resetting periodically.

The underlying equity consists of a basket of hedge funds (and cash and potentially other

securities). At any time t within any valuation period starting at T (which are year long in this case), the ‘Equity Amount’ is the sum of dollar changes in basket value since the beginning of the valuation period: Et = ΣΔB, with the sum taken over months since the beginning of the valuation period.

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17. Callable Range Accrual Digital Adjustments

A range accrual swap is composed of two interest rate streams, a structured stream and a funding stream. The funding stream is a set of standard floating cashflows paying LIBOR + a spread. The index for the floating cashflow resets in advance and pays in arrear according to standard market conventions.

As the range accrual swap is just a linear sum of each of the digital contributions it is not necessary to use a term structure model. Closed form approximations can be used to incorporate corrections arising from non-standard payment delays. The situation for callable range accrual swaps is not so simple.

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18. Constant Maturity Swap

A constant maturity swap (CMS) is an interest rate swap where floating rate equals the swap rate for a swap with a certain life (CMS tenor). For example, the floating payments on a CMS swap might be made every six months at a rate equal to the five-year swap rate (CMS tenor = 5 year). For convexity and timing value calculation for CMS rates, Hull-White formula with correlation coefficient, between CMS rate and forward rate, set at 0.7 is used.

This swap has same payment structure as in the floating leg of CMS swap and its value is derived from the forward swap rate t f as the internal rate of return.

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19. Inflation Swap

The valuation of inflation linked asset swap considers only the case where the cash flows matching the bond coupon and principal repayments are linked to inflation by a scaling factor. When the indexation lag for the inflation swap is not the same as that for the zero coupon swaps there is an additional convexity correction.

The valuation of a floating LIBOR stream is standard and this report discusses only the valuation of inflation linked leg

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20. CMS Cap

A traditional method of pricing CMS products involves so-called convexity adjustment. There are three major drawbacks of that approach, not even mentioning the lack of rigorousness of the method.

A new method proposed is based on the replication of CMS caps/floors by using a portfolio of IR swaptions with all strikes, which implies that CMS derivative hedging is clearly provided. In this approach, there is no any parameter which is market imperceptible. Further, the market smile and skewness is naturally embedded into the swaption portfolio.

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Fixed Income

Risk Management

Risk management is a process to identify and measure risk. The goal of risk management solution is to ensure that risk is under limit and there is no surprise in future. In capital markets, risk management is accountable for oversighting and monitoring the profit and loss, market risk, credit risk, liquidity risk and valuation risk activities of a firm.


1. Counterparty Credit Risk

Counterparty credit risk (CCR) refers to the risk that a counterparty to a bilateral financial derivative contract may fail to fulfill its contractual obligation causing financial loss to the non-defaulting party. It will be incurred in the event of default by a counterparty.


If one party of a contract defaults, the non-defaulting party will find a similar contract with another counterparty in the market to replace the default one. That is why counterparty credit risk sometimes is referred to as replacement risk.


Only over-the-counter (OTC) derivatives and financial security transactions (e.g., repo) are subject to counterparty risk. If one party of a contract defaults, the non-defaulting party will find a similar contract with another counterparty in the market to replace the default one. That is why counterparty credit risk sometimes is referred as replacement risk. The replacement risk is the MTM value of a counterparty portfolio at the time of the counterparty default.

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2. Counterparty Credit Risk Simulation Methodology

Counterparty credit risk (CCR) is the risk of loss that will be incurred in the event of default by a counterparty. It will be incurred in the event of default by a counterparty. Only over-the-counter (OTC) derivatives and financial security transactions (e.g., repo) are subject to counterparty risk. If one party of a contract defaults, the non-defaulting party will find a similar contract with another counterparty in the market to replace the default one. That is why counterparty credit risk sometimes is referred as replacement risk. The replacement risk is the MTM value of a counterparty portfolio at the time of the counterparty default.

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3. Collateral Management

Collateral is a property or an asset that a borrower offers as a way for a lender to secure the loan. Collateral arrangement is a risk reduction tool that mitigates risk by reducing credit exposure. Collateral doesn’t turn a bad counterparty into a good one and doesn’t eliminate credit risk. Instead, it just reduces the loss at the time of default. Collateral arrangement is an essential element in the plumbing of the financial system.

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4. Credit Valuation Adjustment

Credit valuation adjustment (CVA) is the market price of counterparty credit risk that has become a central part of counterparty credit risk management. By definition, CVA is the difference between the risk-free portfolio value and the true/risky portfolio value. In practice, CVA should be computed at portfolio level. That means calculation should take Master agreement and CSA agreement into account.

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5. Funding Valuation Adjustment

Funding Valuation Adjustment (FVA) is introduced to capture the incremental costs of funding uncollateralized derivatives. It can be referred to as the difference between the rate paid for the collateral to the bank’s treasury and rate paid by the clearinghouse. Also FVA can be thought of as a hedging cost or benefit arising from the mismatch between an uncollateralized client trade and a collateralized hedge in the interdealer market. FVA should be also calculated at portfolio level.

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6. Fundamental Review of the Trading Book

The Fundamental Review of the Trading Book (FRTB) is a new Basel committee framework for the next generation market risk regulatory capital rules. It is inspired by the undercapitalisation of trading book exposures witnessed during the financial crisis. FRTB aims to address shortcoming of the current Basel 2.5 market risk capital framework.

FRTB provides a clear definition of the boundary between the trading book and the banking book. It consists of an overhaul of the internal model approach (IMA) to focus on tail risk and an overhaul of the standardized approach (SA) to make it more risk sensitive. Each approach also explicitly captures default risk and other residual risks. Liquidity risk is explicitly included for different asset classes via liquidity horizons. This presentation provides an overview of the standardised approach.

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7. Historical VaR

Value at Risk (VaR) is the regulatory measurement for assessing market risk. It reports the maximum likely loss on a portfolio for a given probability defined as x% confidence level over N days. VaR is vital in market risk management and control. Also regulatory and economic capital computation is based on VaR results. Although VaR measure is objective and intuitive, it doesn’t capture tail risk. There are three commonly used methodologies to calculate VaR – parametric, historical simulation and Monte Carlo simulation. This section focuses on historical VaR.

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8. Standard Initial Margin Model

Initial Margin (IM) is the amount of collateral required to open a position with a broker or an exchange or a bank. The Standard Initial Margin Model (SIMM) is very likely to become the market standard. It is designed to provide a common methodology for calculating initial margin for uncleared OTC derivatives. Initial margin calculation is counterparty-portfolio-based. Given this standardized approach, counterparties can easily reconcile the results.

Initial margin calculation is counterparty-portfolio-based. It applies to non-cleared OTC derivatives only. Derivative trades belonging to a counterparty will be divided into cleared-trade portfolio and non-cleared-trade portfolio. The initial margin is computed for the non-cleared portfolio. This presentation provides many practical details of the SIMM.

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9. Incremental Risk Charge

The incremental risk charge (IRC) is a regulatory requirement from the Basel Committee in response to the financial crisis. It supplements existing Value-at-Risk (VaR) and captures the loss due to default and migration events at a 99.9% confidence level over a one-year capital horizon.

The liquidity of a position is explicitly modeled in IRC through liquidity horizon and constant level of risk. The constant level of risk is a new concept in IRC. It assumes banks hold portfolio constant over a liquidity horizon. At the beginning of the next horizon, they rebalance any default, downgraded, or upgraded positions and roll over any matured trades. This presentation describes methodology and implementation details of IRC.

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10. Financial Market

A financial market is a market where people trade financial products. Typical financial markets are the fixed income and interest rate market, the currency market, the equity market, the commodity market and the credit market.

One of the central tenets of financial economics is the necessity of some tradeoff between risk and expected return. This presentation gives an overview of financial market basics

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11. Market Risk Economic Capital

Financial business is exposed to many types of risks due to the nature of business. To guard against the risk, financial institutions must hold capital in proportion to the potential risk. Market risk economic capital is intended to capture the value change due to changes in market risk factors. It is an internal capital reserve to cover unexpected loss due to market movement.

Economic capital falls into the category of Value at Risk (VaR) measures as both try to capture value change due to market movement. Most institutions use the existing VaR system to compute economic capital. VaR captures the market risk of 1-day time period at 99% confidence level whereas Economic capital measures the market risk of 1-year time period at 99.95 confidence level. Therefore, scaling methodology is the key to compute economic capital, i.e., scaling 1-day to 1-year and 99% to 99.95%. This presentation is intended to answer several fundamental economic capital questions: what is economic capital? What is the difference between economic capital and regulatory capital? How to compute economic capital?

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12. Parametric VaR

Value at Risk (VaR) is the regulatory measurement for assessing market risk. It reports the maximum likely loss on a portfolio for a given probability defined as x% confidence level over N days. VaR is vital in market risk management and control. Also regulatory and economic capital computation is based on VaR results. Although VaR measure is objective and intuitive, it doesn’t capture tail risk. There are three commonly used methodologies to calculate VaR – parametric, historical simulation and Monte Carlo simulation. This section focuses on parametric VaR.

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13. Risk Sensitivity

Risk sensitivities, also referred to as Greeks, are the measure of a financial instrument’s value reaction to changes in underlying factors. The value of a financial instrument is impacted by many factors, such as interest rate, stock price, implied volatility, time, etc. Sensitivities are risk measures that are more important than fair values.

Risk sensitivities or Greeks are vital for risk management. They can help financial market participants isolating risk, hedging risk and explaining profit & loss. This presentation gives certain practical insights onto this topic.

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14. Monte Carlo VaR

Value at Risk (VaR) is the regulatory measurement for assessing market risk. It reports the maximum likely loss on a portfolio for a given probability defined as x% confidence level over N days. VaR is vital in market risk management and control. Also regulatory and economic capital computation is based on VaR results. Although VaR measure is objective and intuitive, it doesn’t capture tail risk. There are three commonly used methodologies to calculate VaR – parametric, historical simulation and Monte Carlo simulation. This section focuses on Monte Carlo VaR.

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15. Collateralized Derivatives

This paper presents a new model for pricing OTC derivatives subject to collateralization. It allows for collateral posting adhering to bankruptcy laws. As such, the model can back out the market price of a collateralized contract. This framework is very useful for valuing outstanding derivatives. Using a unique dataset, we find empirical evidence that credit risk alone is not overly important in determining credit-related spreads. Only accounting for both collateral arrangement and credit risk can sufficiently explain unsecured credit costs. This finding suggests that failure to properly account for collateralization may result in significant mispricing of derivatives. We also empirically gauge the impact of collateral agreements on risk measurements. Our findings indicate that there are important interactions between market and credit risk.

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16. CVA and Wrong Way Risk

This paper presents a Least Square Monte Carlo approach for accurately calculating credit value adjustment (CVA). In contrast to previous studies, the model relies on the probability distribution of a default time/jump rather than the default time itself, as the default time is usually inaccessible. As such, the model can achieve a high order of accuracy with a relatively easy implementation. We find that the valuation of a defaultable derivative is normally determined via backward induction when their payoffs could be positive or negative. Moreover, the model can naturally capture wrong or right way risk.

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17. Convertible Bond

This paper presents a new model for valuing hybrid defaultable financial instruments, such as, convertible bonds. In contrast to previous studies, the model relies on the probability distribution of a default jump rather than the default jump itself, as the default jump is usually inaccessible. As such, the model can back out the market prices of convertible bonds. A prevailing belief in the market is that convertible arbitrage is mainly due to convertible underpricing. Empirically, however, we do not find evidence supporting the underpricing hypothesis. Instead, we find that convertibles have relatively large positive gammas. As a typical convertible arbitrage strategy employs delta-neutral hedging, a large positive gamma can make the portfolio highly profitable, especially for a large movement in the underlying stock price.

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18. LIBOR Market Model

The LIBOR Market Model has become one of the most popular models for pricing interest rate products. It is commonly believed that Monte-Carlo simulation is the only viable method available for the LIBOR Market Model. In this article, however, we propose a lattice approach to price interest rate products within the LIBOR Market Model by introducing a shifted forward measure and several novel fast drift approximation methods. This model should achieve the best performance without losing much accuracy. Moreover, the calibration is almost automatic and it is simple and easy to implement. Adding this model to the valuation toolkit is actually quite useful; especially for risk management or in the case there is a need for a quick turnaround.

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19. Jump-Diffusion Model

This paper argues that the reduced-form jump diffusion model may not be appropriate for credit risk modeling. To correctly value hybrid defaultable financial instruments, e.g., convertible bonds, we present a new framework that relies on the probability distribution of a default jump rather than the default jump itself, as the default jump is usually inaccessible. As such, the model can back out the market prices of convertible bonds. A prevailing belief in the market is that convertible arbitrage is mainly due to convertible underpricing. Empirically, however, we do not find evidence supporting the underpricing hypothesis. Instead, we find that convertibles have relatively large positive gammas. As a typical convertible arbitrage strategy employs delta-neutral hedging, a large positive gamma can make the portfolio highly profitable, especially for a large movement in the underlying stock price.

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20. Collateralization

This paper attempts to assess the economic significance and implications of collateralization in different financial markets, which is essentially a matter of theoretical justification and empirical verification. We present a comprehensive theoretical framework that allows for collateralization adhering to bankruptcy laws. As such, the model can back out differences in asset prices due to collateralized counterparty risk. This framework is very useful for pricing outstanding defaultable financial contracts. By using a unique data set, we are able to achieve a clean decomposition of prices into their credit risk factors. We find empirical evidence that counterparty risk is not overly important in credit-related spreads. Only the joint effects of collateralization and credit risk can sufficiently explain unsecured credit costs. This finding suggests that failure to properly account for collateralization may result in significant mispricing of financial contracts. We also analyze the difference between cleared and OTC markets.

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21. Default Dependency

This article presents a comprehensive framework for valuing financial instruments subject to credit risk. In particular, we focus on the impact of default dependence on asset pricing, as correlated default risk is one of the most pervasive threats in financial markets. We analyze how swap rates are affected by bilateral counterparty credit risk, and how CDS spreads depend on the trilateral credit risk of the buyer, seller, and reference entity in a contract. Moreover, we study the effect of collateralization on valuation, since the majority of OTC derivatives are collateralized. The model shows that a fully collateralized swap is risk-free, whereas a fully collateralized CDS is not equivalent to a risk-free one.

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22. IRC Calculation

The incremental risk charge (IRC) is a new regulatory requirement from the Basel Committee in response to the recent financial crisis. Notably few models for IRC have been developed in the literature. This paper proposes a methodology consisting of two Monte Carlo simulations. The first Monte Carlo simulation simulates default, migration, and concentration in an integrated way. Combining with full re-valuation, the loss distribution at the first liquidity horizon for a subportfolio can be generated. The second Monte Carlo simulation is the random draws based on the constant level of risk assumption. It convolutes the copies of the single loss distribution to produce one year loss distribution. The aggregation of different subportfolios with different liquidity horizons is addressed. Moreover, the methodology for equity is also included, even though it is optional in IRC.

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23. Multilateral Credit Risk

This article presents a new model for valuing financial contracts subject to credit risk and collateralization. Examples include the valuation of a credit default swap (CDS) contract that is affected by the trilateral credit risk of the buyer, seller and reference entity. We show that default dependency has a significant impact on asset pricing. In fact, correlated default risk is one of the most pervasive threats in financial markets. We also show that a fully collateralized CDS is not equivalent to a risk-free one. In other words, full collateralization cannot eliminate counterparty risk completely in the CDS market.

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24. Bilateral Credit Risk

The one-side defaultable financial derivatives valuation problems have been studied extensively, but the valuation of bilateral derivatives with asymmetric credit qualities is still lacking convincing mechanism. This paper presents an analytical model for valuing derivatives subject to default by both counterparties. The default-free interest rates are modeled by the Market Models, while the default time is modeled by the reduced-form model as the first jump of a time-inhomogeneous Poisson process. All quantities modeled are market-observable. The closed-form solution gives us a better understanding of the impact of the credit asymmetry on swap value, credit value adjustment, swap rate and swap spread.

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25. Defaultable Swap

This paper presents an analytical model for valuing interest rate swaps, subject to bilateral counterparty credit risk. The counterparty defaults are modeled by the reduced-form model as the first jump of a time-inhomogeneous Poisson process. All quantities modeled are market-observable. The closed-form solution gives us a better understanding of the impact of the credit asymmetry on swap value, credit value adjustment, swap rate and swap spread.

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26. Defaultable CDS

This article presents a new model for valuing a credit default swap (CDS) contract that is affected by multiple credit risks of the buyer, seller and reference entity. We show that default dependency has a significant impact on asset pricing. In fact, correlated default risk is one of the most pervasive threats in financial markets. We also show that a fully collateralized CDS is not equivalent to a risk-free one. In other words, full collateralization cannot eliminate counterparty risk completely in the CDS market.

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27. Defaultable Derivatives

This article presents a generic model for pricing financial derivatives subject to counterparty credit risk. Both unilateral and bilateral types of credit risks are considered. Our study shows that credit risk should be modeled as American style options in most cases, which require a backward induction valuation. To correct a common mistake in the literature, we emphasize that the market value of a defaultable derivative is actually a risky value rather than a risk-free value. Credit value adjustment (CVA) is also elaborated. A practical framework is developed for pricing defaultable derivatives and calculating their CVAs at a portfolio level.

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Fixed Income

Fixed Income Products

Fixed income products are type of investment that typically generates predictable payments in future. They consist of fixed income securities, such as bonds, and fixed income derivatives, such as, bond futures. Fixed income instruments bear interest rate risk that is an important part of market risk.


1. Amortizing Bond

An amortizing bond is a bond whose principal (face value) decreases due to repaying part of the principal along with the coupon payments. Each payment to the amortizing bond holder consists of a portion of interest and a portion of principal. While an accreting bond is a bond whose principal increases during the life of the deal. Each payment to the accreting bond holder is just a part of interest. The other part of coupon is added to the principal of the bond.

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2. Bond

A bond is a debt instrument in which an investor loans money to the issuer for a defined period of time and receives coupons paid by the issuer at fixed interest rate. The bond principal will be returned at maturity date. Bonds are usually issued by companies, municipalities, states/provinces and countries to finance a variety of projects and activities.

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3. Bond Futures

A bond future is a future contract in which the asset for delivery is a government bond. Any government bonds that meet the maturity specification of a future contract are eligible for delivery. All eligible delivery bonds construct the delivery basket where each bond has its own conversion factor. Conversion factors are used to equalize the coupon and accrued interest differences of all the deliverable bonds.

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4. Bond Future Option

A bond future option is an option contract that gives the holder the right but not the obligation to buy or sell a bond future at a predetermined price. The writer/seller receives a premium from the buyer for undertaking this obligation. Options are leveraged instruments that allow the owner to control a large amount of the underlying asset with a smaller amount of money. Bond future options offer significant advantages for reducing costs, enhancing returns and managing risk. They could be European style or American style.

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5. Callable Bond

A callable bond is a bond in which the issuer has the right to call the bond at specified times from the investor for a specified price. At each callable date prior to the bond maturity, the issuer may recall the bond from its investor by returning the investor’s money. The underlying bonds can be fixed rate bonds or floating rate bonds. A callable bond can therefore be considered a vanilla underlying bond with an embedded Bermudan style option. Callable bonds protect issuers. Therefore, a callable bond normally pays the investor a higher coupon than a non-callable bond.

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6. Floating Rate Notes

A floating rate note has variable coupons, depending on a money market reference rate, such as LIBOR, plus a floating spread. When interest rate raises, the coupons of an FRN increases in line with the increase of the forward rates, which means its price remains relatively constant. Therefore, FRNs bear small interest rate risk. On the other hand, FRNs carry lower yields than fixed rate bonds of the same maturity. They also have unpredictable coupon payments.

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7. Inflation Indexed Bond

The main idea of inflation indexed bonds is that investing in the bond will generate a certain real return. Inflation indexed bonds pay a periodic coupon that is equal to the product of the daily inflation index and the nominal coupon rate. Therefore, even though the nominal value of the coupons and principal may change, the real return of these remains the same. Unlike regular (nominal) bonds, inflation indexed bonds assure that your purchasing power is maintained regardless of the future rate of inflation.

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8. Puttable Bond

A puttable bond is a bond in which the investor has the right to sell the bond back to the issuer at specified times for a specified price. At each puttable date prior to the bond maturity, the investor may get the investment money back by selling the bond back to the issuer. The underlying bonds can be fixed rate bonds or floating rate bonds. A puttable bond can therefore be considered a vanilla underlying bond with an embedded Bermudan style option. Puttable bonds protect investors. Therefore, a puttable bond normally pays investors a lower coupon than a non-callable bond.

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9. Zero Coupon Bond

Zero coupon bonds are issued at a deep discount and repaid the face value at maturity. The greater the length of the maturity is the cheaper price a bond has. Unlike other bonds, the investor’s return is the difference between the purchase price and the face value. An investor preferring a long-term investment may purchase zero coupon bonds such as saving money for children’s college tuition. The deep discount helps the investor grow a small amount of money into a sizable sum over several years. Normally investors buy zero coupon bonds when interest rates are high.

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10. Callable Floating Coupon Note

A floating coupon note is a very flexible and generic funding product. The issuer pays the buyer periodic floating coupons based on a spread-adjusted reference rate, such as LIBOR. The buyer pays an upfront fee to the issuer. Also, the buyer pays the issuer a notional amount at inception and the issuer returns it upon cancellation or maturity of the deal.


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Fixed Income

Equity Derivatives

An equity derivative is a financial product whose payoff is derived from an underlying equity or a basket of equities or an equity index. Investors and traders can use equity derivatives to take a long or short position in a stock without actually buying or shorting the stock. This is advantageous because taking a position with derivatives allows the investor/trader more leverage in that the amount of capital needed is much less than a similar outright long or short position on margin. Investors/traders can therefore profit more from a price movement in the underlying stock.


Equity derivatives provide investors a way to hedge risk or speculate. Also option trading can limit an investor’s risk and leverage investing potential. Derivative investors have a number of strategies they can utilize, depending on risk tolerance and expected return.


The following products are most heavily traded in the equity market.


1. Equity Futures

An Equity Futures contract traded over an organized exchange. In this contract parties commit to buy or sell a specified amount of an individual stock or a basket of stocks or a stock index at an agreed contract price on a specified date. Generally there are two types of Equity Futures: Index Future and Stock Future. Stock markets Index Futures are futures contracts used to replicate the performance of an underlying stock market index. They can be used for hedging against an existing equity position, or speculating on future movements of the index. Indices for futures include well-established indices such as S&P 500, FTSE 100, DAX, CAC 40 and other G12 country indices. Indices for OTC products are broadly similar, but offer more flexibility.

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2. Equity Swap

An equity swap is an agreement where one party makes payments based on a set rate, either fixed or variable, while the other party makes payments based on the total return of an equity, a basket of equities, or an equity index. Equity swap is a good vehicle for counterparties to transfer risk. One party makes cash payments based on a predefined fixed or floating rate, whereas the other party makes payments based on the total return of an underlying asset. The party receiving the total return gains exposure to the performance of the reference underlying asset without actually owning it. Therefore, this product can be used to obtain a leveraged exposure. On the opposite of the transaction, the counterparty receive payments of a reference interest rate payments that provide some protection against a potential loss of the underlying asset.

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3. Correlation Swap

A Correlation Swap is an instrument where the option buyer receives the difference between the observed correlation and the strike correlation on a basket of assets, observed over a specified time interval. It can be thought of as a forward contract on realized correlation. Its payoff is simply the difference between the realized correlation over the stated period and the strike times the notional of the contract

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4. European Option

An European options give an investor the right but not the obligation to buy a call or sell a put at a set strike price prior to the contract’s expiry date. European options are derivatives that means their value is derived from the value of an underlying equity.

European options provide investors a way to hedge risk or speculate. Also option trading can limit an investor’s risk and leverage investing potential. Option investors have a number of strategies they can utilize, depending on risk tolerance and expected return.

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5. American Option

American options provide investors a way to hedge risk or speculate. Also option trading can limit an investor’s risk and leverage investing potential.
Stock option investors have a number of strategies they can utilize, depending on risk tolerance and expected return. Buying call options allows you
to benefit from an upward price movement. The right to buy stock at a fixed price becomes more valuable as the price of the underlying stock increases.
Put options may provide a more attractive method than shorting stock for profiting on stock price declines. If you have an established profitable long stock position, you can buy puts to protect this position against short-term stock price declines. An option seller earns the premium if the underlying stock price would not change much.

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6. Asian Option

Asian options allow the buyer to purchase (or sell) the underlying asset at the average price instead of the spot price. Asian options are commonly seen options over the OTC markets. Average price options are less expensive than regular options and are arguably more appropriate than regular options for meeting some of the needs of corporate treasurers. Average can be calculated in a number of ways (daily, weekly, monthly, etc.).

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7. Basket Option

A basket option can be used to hedge the risk exposure to or speculate the market move on the underlying stock basket. Because it involves just one transaction, a basket option often costs less than multiple single options. The most important feature of a basket option is its ability to efficiently hedge risk on multiple assets at the same time. Rather than hedging each individual asset, the investor can manage risk for the basket, or portfolio, in one transaction. The benefits of a single transaction can be great, especially when avoiding the costs associated with hedging each and every component of the basket or portfolio.

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8. Digital Option

A digital option is an option with a predetermined payoff, triggered only if the underlying price meets the strike price. These are also commonly referred to as “all or nothing”. Digital call pays a fixed amount if the underlying price ends up above the strike price, while binary put pays off a fixed amount if the underlying price is below the strike price at option maturity.

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9. Quanto Option

Quantos are attractive because they shield the purchaser from exchange rate fluctuations. If a US investor were to invest directly in the Japanese stocks that comprise the Nikkei, he would be exposed to both fluctuations in the Nikkei index and fluctuations in the USD/JPY exchange rate. Essentially, a quanto has an embedded currency forward with a variable notional amount. It is that variable notional amount that give quantos their name—”quanto” is short for “quantity adjusting.”

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10. Warrant

Warrants frequently attached to fixed rate bonds or preferred stock as a sweetener can be used to enhance the yield of the fixed rate bond and make them more attractive to potential buyers. Most commonly issued warrants are often detachable, meaning that they can be separated from the fixed rate bond and sold on the secondary market before expiration. Wedded or wedding warrants are not detachable. The investor must surrender the fixed rate bond or preferred stock the warrant is “wedded” to in order to exercise it. Naked warrants are issued on their own, without accompanying fixed rate bonds or preferred stock.

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11. Variance Swap

A Variance Swap contract has a payoff which is dependent on realized variance. In order to calculate the fair value of future delivery variance, the variance swap can be replicated by a forward contract and a portfolio of appropriately weighted European call and put options with a continuous range of strikes. This continuous spectrum of options is not observable and hence a (finite) discrete approximate replication must be performed in practice.

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12. Convertible Bond

Convertible bonds typically have lower yields than the yields on similar fixed rate bonds without the convertible option. Reverse convertible bonds usually have shorter terms to maturity and higher yields than most other fixed rate bonds.

Most convertible bonds are subordinated debt of the issuer. In the event of bankruptcy, the claims of other bondholders take priority over convertible bondholders, who themselves have priority over owners of the preferred and common stock.

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13. Lookback Option

Lookback options are designed to provide investors with the opportunity for an enhanced return while reducing the downside risks with partial protection that “buffers” any negative performance of the Reference Index over the term of the Notes. The “look-back” feature offers holders an optimal entry point for their investment.

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14. Barrier Option

There are two types of contracts, knock-in barrier options and knock-out barrier options, each respond differently when the barrier is “triggered”. If a knock-out option’s underlying touches the barrier, the option is eliminated and the holder receives a rebate. Conversely, a knock-in option touches the barrier to activate the option. If the knock-in option never reaches the barrier, the holder will receive a rebate.

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15. Cliquet Option

Cliquet options are widely traded in many retail-structured products. They consist of financial derivatives which provide a guaranteed minimum return in exchange for a capping of the maximal return over the life of the contract. A cliquet option is equivalent to a series of forward-starting at-the-money options, which may be globally and locally floored and capped. .

Cliquet options are appealing to investors because they can protect themselves against downside risks. Possible variants include reverse cliquet which amounts to a cash flow minus a capped cliquet of puts, and digital cliquet, where the forward-starting options are digital options.

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16. Spread Option

The use of spread options is widespread for speculation, basis risk mitigation, or even asset valuation. Spread options allow investors to simultaneously take positions in two are more assets and profit from their price difference over some spread.

Because of their generic nature, spread options are used in markets as varied as equity markets, fixed income markets, currency and foreign exchange markets, commodity futures markets, and energy markets.

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17. Callable Notes

A callable note allows the issuer to exercise a call option on the note on a specified date or set of dates prior to maturity. Callable note is among the most challenging derivatives to price. These products are loosely defined by the provision that the holder or issuer has the right to call the product or exercise into various underlying instruments after a lock-out period expires.

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18. Best/Worst of Option

Best of Option or Worst of Option is a chooser option that returns the best/worst performing among several baskets of funds or indices that reflect growth, moderate and conservative investment styles. The returns could be based on average (Asian), single currency or quanto. The final payoff could be capped and floored.

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19. Rainbow Option

Rainbow options are appealing to investors due to its natural risk diversification, cost efficiency, and weighted average on the best or worst performing assets. The best version offers higher returns, whereas the worst version is normally cheap.

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20. Callable Range Accrual Note

A callable range accrual note contains an embedded option that allows the issuer to exercise a call on the note on a particular date or set of dates prior to maturity. This can be used by the issuer to limit the return that is paid to the note holder. On these exercise dates, the issuer may purchase the note back from the holder for a predetermined cash amount, which is typically equal to or greater than the face value of the note.

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21. Reverse Convertible Autocallable Swap or Bond

A reverse convertible autocallable swap allows two parties exchange floating coupons with fixed coupons on certain future dates. On some coupon dates, the swap may be cancelled. Should the swap be cancelled on coupon date t, the coupons due on coupon date t will be paid and all further cash flows are terminated.

A reverse convertible bond is a bond with an embedded put option that allows the issuer to purchase the note back for a predetermined quantity of cash, debt, or stock. The decision to exercise the option is made based on the performance of an underlying asset, index, or basket of assets.

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22. Callable Yield Note

Callable yield note usually pays a higher coupon or interest rate to investor. However, the event of the call of the note is uncertain to the investor. It is note only linked to the level of a basket of underlying assets, but also to volatility, correlation, dividends, and yield curve.

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23. Accelerated Return Note

An Accelerated Return Note (ARN) is a structured instrument that offers a potentially higher return linked to the performance of a reference entity that could be an equity, an index, or a basket of assets. The payoff depends on the performance of the underlying assets. Usually, it is capped but not floored, that means it does not offer any downside protection.

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24. Autocallable Note

Autocallable notes offer a coupon that is higher than regular fixed rate bond. It is suitable for investors who are seeking enhanced yield opportunities. Auto-call investments provide a contingent downside protection that protects the principal as long as the reference asset has not traded below the downside barrier.

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25. Constant Proportion Portfolio Insurance (CPPI)

A constant proportion portfolio insurance (CPPI) is a trading strategy where an initial investment is dynamically reallocated between a risky asset and risk-free bond such that a minimum payoff is guaranteed at maturity. The risky asset could be from equities, funds, or commodities.

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26. Accelerated Share Repurchase

An accelerated share repurchase (ASR) agreement is a contract or an investment strategy used by a publicly traded company to buy back shares of stocks expeditiously from the market. In these agreements, firms are able to repurchase a significant number of their shares upfront. The intermediary must then repurchase the shares over a given time window that is equivalent to enter into a forward contract.

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27. Equity-Linked Bonus Coupon Note

A bonus coupon note, also referred to as coupon growth note or bonus enhanced note or basket coupon note, is an equity-linked note that provides guaranteed coupons over the life of the note with potential for a bonus coupon based on the underlying asset trading above a specified barrier level.

The note pays a series of coupons based on the weighted performance of all assets in the basket on each Coupon Determination Date. The coupons are usually capped and floored.

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28. Himalaya Option

A Himalaya option is an option on the sum of the returns of the best (worst) performing assets from a basket on predefined observation dates. It has a set of observation dates are defined. The number of observation dates are equal to the number of underlying assets. The unique feature of Himalaya option is the withdrawal of the best performer asset from the basket at each pre-defined observation date, until only one stock remains.

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