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.

Zenodo LGM calibration

Zenodo LGM calibration home

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.

Zenodo quadratic

Zenodo quadratic home

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.

Zenodo IR TARN

Zenodo IR TARN home

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.

Zenodo black

Zenodo black home

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

Zenodo digital option

Zenodo digital option home

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

Zenodo double cms

Zenodo double cms home

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.

OSF forward start

Zenodo early start

Hcommon clquet

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.

OSF three factor model

Zenodo three factor

Hcommon convertible

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.

OSF hw convertible

Zenodo hw

Hcommons hw

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.

OSF exchangeable convertible

Zenodo exchangeable

Hcommons exchangeable

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.

OSF extendable swap

Zenodo extendable

Hcommons extendable

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.

OSF callable inverse

Zenodo callable inverse

Hcommons callable inverse

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.

OSF bond American

Zenodo bond American

Hcommons bond America

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

OSF arrear quanto cms

Zenodo arrear

Hcommons arrear quanto

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.

OSF variable swap

Zenodo variable

Hcommons variable

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

OSF cms spread

Zenodo cms

Hcommons cms spread

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.

OSF early start swap

Zenodo forward

Hcommons early start

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.

OSF digital swap

Zenodo digital

Hcommons digital swap

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.

OSF ratchet swap

Hcommons ratchet

Zenodo ratchet

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.

Zenodo hf barrier

Zenodo hf barrier home

Github hf

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.

Zenodo trust swap

Zenodo trust swap home

Github trust

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:

Zenodo knockout swap

Zenodo knockout swap home

Github knockout

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.

Zenodo bma swap

Zenodo bma swap home

Github swap

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.

Zenodo chooser cap

Zenodo chooser cap home

Github chooser

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.

Zenodo cms cliquet

Zenodo cms cliquet home

Github cliquet

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.

Zenodo arrear fra

Zenodo arrear fra home

Github arrear

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.

OSF Himalaya

Github spread

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.

Zenodo FX Hedge PDF

Zenodo FX Hedge

Github FX forward

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.

Zenodo FX Choice PDF

Github fx

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.

OSF Quanto Vol

Github vol

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.

OSF Pooling PDF

Github gic

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.

OSF MF Securitization

Github risk

Hcommon MF

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.


Github note

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. 

OSF GIC Pricing

Github market

Hcommon GIC

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.

OSF flexible gic

Github local

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.

Zenodo adjustable mortgage

Github mortgage

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.

Zenodo mbs deferred

Github mbs

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.

Zenodo mortgage commitment

Zendo mortgage commitment home

Github mortgage

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.

Zenodo mbs

Zenodo mbs home

Github mbs

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.

OSF quanto trs

Github return

Hcommon quanto himalaya

Hcommon quanto local vol

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.

Zenodo principal note

Zenodo principal note home

Github market risk

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.

Zenodo collateral swap

Zenodo collateral swap home

Github trigger

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.

Zenodo digital adjustment

Zenodo digital adjustment home

Github lgm

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.

Zenodo cms

Zenodo cms home

Github ratio

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

Zenodo inflation swap

Zenodo inflation swap home

Github xccy

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.

Zenodo cms cap

Zenodo cms cap home

Github cap