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

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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

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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

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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

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Github cap