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