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Pricing Model
Hedge Fund Exposure A model is proposed for quantifying daily returns and corresponding NAV changes of individual hedge funds. NAV values of hedge funds are typically available on monthly basis. It estimates the daily NAV for a hedge fund is based on modeling daily returns of the hedge fund as a weighted sum of returns of a combination of several market indices and factors. These indices and factors typically include: a) US and international equity market indices; b) US and international fixed income indices. c) Currencies; d) Other market factors, such as credit spreads, volatilities, etc. 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 They are not forward-looking, that is, the do not attempt to value the present value of future fee income, nor do they attempt account for uncertainties in hedge fund redemption. 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 value of the collateral affects the current leverage ratio and can trigger re-/de-leveraging or redemption, but otherwise has no affect on the option valuation itself. Nevertheless, to monitor the leverage ratio daily the collateral needs to be MTM daily, and ultimately the dealer’s risk in a situation where the deal must be closed out depends on the closeout value of the collateral. Hedge Fund VaR A model is presented to determine the value at risk (VaR) due to hedge fund volatility during closeout of hedge fund positions, as well as pricing calculations for options written on a basket of funds. It contains the implemented VaR calculations 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. It contains the implemented VaR calculations 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. 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 difference between the redeemed value of the basket of funds and the strike price of the call (usually set at 75% of the notional) is forwarded to the investor. When the redeemed basket value falls below the strike price the investor loses money on the deal. 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. 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. So rather then being a weighted sum over some easily replicable market indices which is re-balanced periodically, this index is run more like a quantitative fund-of-funds. This will allow investors to directly hedge its exposure to index-linked products, which is important given that hedge funds are an asset class with reduced transparency, and with returns and have traditionally been difficult to replicate using liquid instruments. 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 issue of valuation is very much simplified by the fact that these variable rate mortgage principals pay interest based on one month BA, and this rate resets monthly. The prepayment is not an issue with these mortgages, since the principal is replaced (at par) in the event of prepayment and the floating rate payments remain one month BA. Therefore, valuing these assets is a simple exercise of valuing them equivalent to a floating rate bond. Foreign Currency Assets Hedge Model 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. A hypothetical cross currency interest rate swap (XCIRS), with the floating leg terms matching the critical terms of the hedged FC floating rate assets, is used as the proxy measure. The actual hedging derivative is another XCIRS. Specifically, the trade, upon which the model is intended to target, is substituted by a cross currency swap, which is hedged by another XCIRS. Cash Flow Hedge Effectiveness A cash-flow hedge is defined as “hedging the exposure to variability in expected future cash flows that is attributable to a particular risk. That exposure may be associated with an existing recognized asset or liability (such as all or certain future interest payments on variable-rate debt) or a forecasted transaction (such as a forecasted purchase or sale). An entity can designate the variability of cashflows such as interest receipts or payments on variable-rate assets or liabilities or a forecasted transaction as the hedged item in a cashflow hedge. Fair Value Hedge Model A fair value hedge is a hedge of the exposure to changes (that are attributable to a particular risk) in the fair value of a recognized asset, liability, or unrecognized firm commitment. Changes in fair value of derivatives that do not meet the criteria of one of these three categories of hedges are included in income. When hedging exposures associated with the price of an asset, liability, or firm commitment, the total gain or loss on the derivative is recorded in earnings. In addition, the underlying exposure due to the risk being hedged must also be marked-to-market to the extent of the change due to the risk being hedged; and these results flow through current income as well. This treatment is called a fair value hedge. Portfolio Acquisition Model The portfolio acquisition model is essentially a decision-making tool used to produce a bid that investors will make on an existing portfolio of loans. In this case, the loans are Home Equity Lines Of Credit (HELOC) (or ‘second mortgage’ because they are secured by the borrowers property, as is the ‘first mortgage’) typically with a maximum term of 15 years, with interest-only payments terminating in a final bullet payment. Borrowing is allowed up to the credit limit at any time at a variable rate tied to prime–can be fixed or floating–partial principal repayments are allowed, but not scheduled. Prior to modeling, the portfolio will have been carved up into reasonably homogeneous pieces–which the present relatively simple model might be expected to represent fairly well. As part of this ‘risk screening’, the charge-off rates are estimated from the credit quality (and other characteristics) of the loans in the portfolio. How well this is done will affect the results produced by the model–this process will not be examined here. MBS Deferred Asset Model 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 model is a simple static cash flows model for the underlying MBS. The simplicity is possible due to the assumption that the prepayments of the underlying pool are not dependent on changes in interest rates. Generally, prepayments do depend on interest rates, and this is particularly true for US mortgages. Mortgage Commitment Model The mortgage Commitment model is being used to measure the risk of mortgage commitments. The model is actually based on a two-step process, and in the end allows investor to hedge the inherent interest rate risk imbedded in these mortgage commitments. In the first stage, the actual mortgage commitment pipeline is “mapped” to a simplified portfolio of European swaptions. In the second stage, this simplified portfolio of swaptions is used to determine an appropriate portfolio of swaps to hedge the swaptions. Performance Deferred Share Program Model 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. The final payout to the employee depends on the organization’s performance against its peer group. The PDSP valuation model is an attempt to value this liability taking this performance aspect into account. 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 model involves defining an underlying interest rate process, valuation models for all the instruments that depends on the interest rates, and various scenarios for the interest rates in order to compute “risk profiles”. Mortgage Model Mortgage model is used to calculate the risk profile of various interest rate risk sensitive instruments, such as fixed rate mortgage and adjustable risk mortgage. The interest rate risk measures depend on the specific characteristics of the instrument (for example, the coupon of a fixed rate mortgage), and these are calculated different for the various instruments assuming an underlying interest rate process together with various valuation models for the instruments. We will begin by outlining the interest rate process, then proceed to discuss the various instruments separately. Rate Lock Model Rate lock is a derivative on fixed rate and adjustable rate mortgages. Fallout functions are specified for various types of rate locks, and depend on several factors including source of rate lock and mortgage type. Like the prepayment models, these fallout functions are based on analysis of historical data and these have a significant impact on the risk measures of these rate locks. Rate locks are specified using various categories upon origination. Additionally, there is a current status that may change daily during the course of the rate lock. Depending on the category and status of the rate lock, a fallout function is applied. This fallout function is supposed to be a measure of how many of these loans will close at expiry. Like the prepayment functions, these are based on historical observations. MBS Pass Through Model Mortgage Backed Securities (MBS) are essentially interest rate derivatives, this requires both a robust interest rate model as well as a model for the prepayment behavior. The prepayment model is dynamic, since prepayments (and ultimately MBS cash flows) depend very strongly on the dynamics of prevailing mortgage rates. These mortgage rate dynamics are driven by the dynamics of the yield curve. MBS cash flows depend very strongly on the prepayment assumptions of the underlying pools. The prepayments are both substantial and heavily dependent on the prevailing mortgage rates. Hence, it is very important to use a reliable prepayment model if one hopes to “accurately” capture the risk of these instruments. Callable Range Accrual Digital Adjustments Model A single factor linear Gaussian model (LGM) is proposed for pricing callable range accrual swaps (for both the fixed and floating coupon variants), and additional adjustments required to capture the implied volatility smile observed in the market. The single factor LGM is equivalent to a single factor Hull-White model. 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. BMA Knockout Swap Model The 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). On any contract payment date both the fixed and Libor coupon payments may cancel if the historical average BMA rate observed during the coupon period is above (or below, as specified by the contract) the knockout strike. For example, consider a contract where one party receives the fixed rate and pays 68% of 1M Libor. Both payments are cancelled if on a coupon date the 1M average of the 1W observations of the BMA rate (i.e., the average of the prior 4 observations of 1W BMA rate) is greater 5%. Arrear Cap Model An arrear cap consists of caplets whose payoffs are not paid at the end of underlying index period. It belongs to the category of unconventional caplets/floorlets. A non-vanilla IR cap/floor is composed of a portfolio of unconventional IR caplets/floorlets. As before, it suffices to price an unconventional caplet/floorlet. However, the conventional Black’s formula may not be directly applied to price the unconventional caplet/floorlet. A so-called convexity adjustment is needed under an arbitrage-free approach. The value of a vanilla cap/floor is the sum of values of corresponding caplets/floorlets. The value of a conventional caplet/floorlet can be given by the well-known conventional Black’s formula1. Let v be the value of the conventional caplet/floorlet. Then we may re-write the matured payoffs of the conventional caplet/floorlet. MBA Swap Model The BMA Municipal Swap Index, produced by Municipal Market Data (MMD), is a single rate released every Wednesday after 4:00 pm and effective on next business date (usually Thursday). The rate is a representative index made up of an average of rates supplied by traders within the industry. It is a simple compounding annual rate with the day count convention of ACT/ACT. In a generic fixed for floating BMA swap, the floating side is estimated using averages of weekly BMA Municipal Swap Index. While actual par Municipal Swap Rates are not available in the market, BMA ratios, which are actively traded, are applied to LIBOR par swap rates to derive the par BMA swap rate. BMA ratios are used in conjunction with LIBOR swap rates to calculate the various instruments traded in the BMA market. Capped FRN Swap Model A model is presented for pricing European/Bermudan type callable capped floating rate note (FRN) swaps. The capped FRN swap is a contract to swap cash-flows between a vanilla floating rate leg and a capped floating rate leg. The option gives the right to call the swap back in favor of the option owner. Pricing the capped FRN swap is relatively simple as the value can be expressed in a close-form analytical formula. To price the option with Bermudan type, the capped FRN swap has been reduced to a contract to swap cash-flows between a fixed leg and a caplet leg. To make the reduction valid, we need to assume that the date-setting in both legs of the capped FRN swap are almost consistent. Caption Model Caption is an option on caps and floors. Since we can view cap as a call option and floor as a put option, caption can be modeled by compounding option techniques. There are four basic types: call option on cap, put option on cap, call option on floor, and put option on floor. Here we only explain how to value European or American call option on cap in details. Other three cases can be done in a similar way. FX Choice Option Model 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. Let t ¸ 0 be a generic valuation time point and T > 0 be the maturity of the FX choice option. Let C, U and S be a base currency, a quote currency and exchange rate, respectively, where S is expressed in the American convention, i.e., S is the number of the base currency C per unit of the quote currency U. Let BC be the bond price process of a pre-determined fixed coupon bond with unit face value in the C-currency. Chooser Cap Model 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. Chooser option on interest forward rates may also be called chooser cap and chooser floor, respectively. A cap (floor) is a portfolio of caplets (floorlets). For a given number k, a chooser cap (floor) is an option which entitles the option owner the right to exercise at most k caplets (floorlets) out of the total n caplets (floorlets). CMS Cliquet Model 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. Due to the embedded option and convexity adjustments in the structure leg, we need swap rate volatilities and forward rate volatilities. The former can be interpolated from the implied swaption volatility surface. The latter should be interpolated from the so-called implied forward volatility surface. Arrear Forward Rate Agreement Model It has been well known that a convexity adjustment must be considered in pricing a set-in-arrear forward rate agreement (FRA). It also becomes market convention to do so. We use a very good approximation under the condition that the reset date of a floating index rate is very close to a payment date. In CAD, the reset date can be identical to the payment date. In other currencies, the reset date can be only two business days prior to the payment date. It must be noted that the reset date may not exceed the payment date and the payment date may not exceed the end date of the index forward period. Double CMS Derivative Model A double CMS derivatives represents a European type derivatives whose matured payoff depends on two CMS rates. 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. There are three basic types of payoffs: (a) an affine-linear on two CMS rates, (b) a call on a linear combination of the two and (c) a put on the combination. For the first type payoff (a), the present value can be calculated by using single CMS cap/floor model. Option Close-out Reserve Model A model is presented 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. Cross Currency Basis Swaption Model A Cross-currency basis swap, which is also called FX basis swap, is a contract between two parties that one side receives a floating rate (plus a possible spread) of currency A and the other receives a floating rate (also plus a possible spread) of currency B. Between the two currencies, suppose that currency A is more liquid than currency B. For example, USD is currency A and CAD is currency B. That is, in the currency A world, there is a market difference in supply and demand of currency B. Thus the traditional interest rate parity in the currency A world may not valid without any adjustment. Constant Maturity Swap Model 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. Between the two currencies, suppose that currency A is more liquid than currency B. For example, USD is currency A and CAD is currency B. That is, in the currency A world, there is a market difference in supply and demand of currency B. Thus the traditional interest rate parity in the currency A world may not valid without any adjustment. Inflation Swap Model An inflation linked asset swap (ILAS) is a security in which a stream of inflation linked cash flows, that match the payments of an inflation linked bond, are swapped for a stream of cash flows that pay LIBOR plus a spread, or a fixed rate. 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. Rate TARN Swap A target accumulated redemption note (TARN) is a structured coupon bond that will be compulsively terminated one the accumulated coupon breaches a pre-determined barrier. If structured coupons in a TARN are functions of some selected index interest rates, the TARN is called 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. Local Volatility Gaussian Model 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. MBS Model 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. Curve Interpolation Analytics This article focuses on interpolation subject, in particular, the interpolation of curve bootstrapping. Both linear spline and cubic spline are 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. MVF Swap Marginal Value of Fund (MVF) rate is an internal rate used for any internal lending or borrowing of funds within an organization. A MVF swap is a swap whose floating legs is based on the MVF rate. MVF rates are blended and excessive averaged LIBOR rates over specified moving historical periods. The MVF swap’s floating leg is settled monthly with daily average of MVF rates over each period. The swap can be correctly booked in Infinity system provided that the floating forward rate can be estimated with satisfactory. CMS Cap CMS stands for constant maturity swap. A CMS cap/floor consist of a number of caplet/floorlet on the index rate of a CMS rate. Pricing a CMS cap/floor is equivalent to price a portfolio of a continuum of vanilla interest rate swaptions. The SABR model is applied to evaluate the replicating interest rate swaption portfolio. Thus, in some cases, this forward rate is not even well defined in the market, i.e., it has to be implied from a zero rate curve, which implies that it is also difficult to obtain the statistic correlation through historical data. This drawback makes the application of the method in a subjective way since a user may manipulate the result by changing the correlation. Quadratic Average Enhanced Curve Quadratic Average Enhanced Curve Bootstrapping Algorithm (QAECBA) is intended to replace the existing Enhanced Curve Bootstrapping Algorithm (ECBA) and algorithm. The new approach achieves the ”smoothness” of both spot and forward zero-curves, and produces the instantaneous forward rate curve directly. Because the bootstrapping algorithms differ, we expect different sensitivities per instrument, but the portfolio’s total delta sensitivities should be close. Also, the sensitivities per instrument should have the same sign, implying hedge numbers having the same sign (positive for long positions, negative for short positions). Money Market Basis Curve A money market basis swap is an exchange of two floating rate notes referencing two different rate indices (indices can differ just by the length of underlying term, like USD 3 Month LIBOR versus USD 1 Month LIBOR, or by nature altogether, like USD 3 Month LIBOR versus USD 3 Month CD). Basis spreads can be applied to the indices (typically, to just one of them, the other being considered the reference index). The principals of the notes are denominated in the same currency (in our example, USD). Principals may be paid/received at maturity, or not (this last case is typical for money market basis swaps). (This contrasts with cross-currency basis swap situation where principals do change hands both at the effective date and maturity date, with principals related by a fixed contractual FX rate. Municipal Bond Option Municipal Bond Options are European-style vanilla options on coupon bonds. The Municipal Bond Option Pricing Model is used for daily calculations of P&L and risk numbers. These options are of vanilla European type. Black’s model (Black-76 model) is applied in the bond option pricing model. The discount curve is composed of a set of short tern rates, LIBORs up to two years and a set of medium to long term (2 years – 30 years) AAA yields with linear interpolation. Bond yield volatilities consist of one month (30 days) and one year (252 days) volatilities for a sequence of bond maturities with two dimensional linear interpolation. All those parameters are market observable. In application of Black’s model to price bond option, an option tenor should be much shorter than the term of an underlying bond. Black Model Black’s vanilla option pricing model can be applied to a wide range of vanilla European options such as caps/floors, European swaptions, bond options, bond futures options and interest rate (IR) futures options. 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. SABR Calibration Handling these market skews and smiles correctly is important to fixed income desks. Local volatility models have been used to manage these skews and smiles. Although local volatility models are self-consistent, arbitrage-free and can be calibrated to match observed market skews and smiles, it has been observed that the dynamic behaviour of smiles and skews predicted by local volatility models contradicts the one observed in the marketplace. In reality, options with different strikes require different volatilities to match their market prices. This is the market skew and smile. Usually, with respect to the strike, the word skew is for the slope of volatility and smile is for its curvature. The inherent contradiction of using different volatilities for different strikes in Black’s model makes it difficult to rigorously handle skew/smile risks. Digital Option This article presents a pricing model for skewed European interest rate digital option. 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. 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. Once we independently apply the above procedure for each of the three mod groups, we end up with three series of discount factors at three disjoint series of offset dates. We also have the discount factors at the cash dates. We form the final set of curve anchor dates from all offset dates and all cash dates. So, we obtain discount factors at all anchor dates. At any intermediate date we produce a discount factor by interpolating between the discount factors at the nearest (left and right) neighboring anchor dates. LGM Calibration The Gaussian HJM interest rate (IR) term structure model, also called the Linear Gaussian Model (LGM), is used to pricing some IR structured products. To keep it being consistent with the IR option market, the model needs to be calibrated to the IR European swaption market. 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.. |