Valuation of credit default swaptions using Finite Difference Method
Motshabi, Karabo Mirriam
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Credit default swaptions (CDS options) are credit derivatives that are widely used by ﬁnan-cial institutions such as banks and hedging companies to manage their credit risk. These options are usually priced using Black-Scholes model, but the assumptions underlying this model do not always hold especially when solving complex ﬁnancial problems. The proposed solution is to use numerical methods such as ﬁnite diﬀerence method (FDM) to approximate the solution of the Black-Scholes PDE in cases where closed form solutions cannot be obtained. The pricing of swaptions are important in ﬁnancial markets, hence we speciﬁcally discuss the pricing of interest rate swaptions, CDS options, commodity swaptions and energy swap-tions using Black-Scholes model. Simple parabolic PDE known as heat equation given at (Higham, 2004) forms a foundations to understand the application of FDM when solving a PDE. Since, Black-Scholes PDE is also a parabolic equation it is transformed to a form of a heat equation (diﬀusion equation) by applying change of variables technique. FDM, speciﬁcally Crank-Nicolson method can be applied to the heat equation but in this dissertation it is applied directly to the Black-Scholes PDE to approximate its solution. Therefore, it is preferable to use Crank-Nicolson method because it is known to be second- order accurate, unconditionally stable, very ﬂexible, suitable and can accommodate varia- tions in ﬁnancial problems, (Duﬀy, 2008). The stability of this method is investigated using a matrix approach because it accommodates the eﬀect of boundary conditions. To test the convergence of Crank-Nicolson method, it is compared with the Black-Scholes method used in (Tucker and Wei, 2005) to price CDS options. Conclusively the results obtained by Crank-Nicolson method to price CDS options are similar to those obtained using Black-Scholes method.