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Fourier, Wavelet and Monte Carlo Methods in Computational Finance

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Fourier, Wavelet and Monte Carlo Methods in Computational Finance Fourier, Wavelet and Monte Carlo Methods in Computational Finance Kees Oosterlee1;2 1CWI, Amsterdam 2Delft University of Technology, the Netherlands AANMPDE-9-16, 7/7/2016 Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 1 / 51 Joint work with Fang Fang, Marjon Ruijter, Luis Ortiz, Shashi Jain, Alvaro Leitao, Fei Cong, Qian Feng Agenda Derivatives pricing, Feynman-Kac Theorem Fourier methods Basics of COS method; Basics of SWIFT method; Options with early-exercise features COS method for Bermudan options Monte Carlo method BSDEs, BCOS method (very briefly) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 1 / 51 Agenda Derivatives pricing, Feynman-Kac Theorem Fourier methods Basics of COS method; Basics of SWIFT method; Options with early-exercise features COS method for Bermudan options Monte Carlo method BSDEs, BCOS method (very briefly) Joint work with Fang Fang, Marjon Ruijter, Luis Ortiz, Shashi Jain, Alvaro Leitao, Fei Cong, Qian Feng Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 1 / 51 Feynman-Kac theorem: Z T v(t; x) = E g(s; Xs )ds + h(XT ) ; t where Xs is the solution to the FSDE dXs = µ(Xs )ds + σ(Xs )d!s ; Xt = x: Feynman-Kac Theorem The linear partial differential equation: @v(t; x) + Lv(t; x) + g(t; x) = 0; v(T ; x) = h(x); @t with operator 1 Lv(t; x) = µ(x)Dv(t; x) + σ2(x)D2v(t; x): 2 Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 2 / 51 Feynman-Kac Theorem The linear partial differential equation: @v(t; x) + Lv(t; x) + g(t; x) = 0; v(T ; x) = h(x); @t with operator 1 Lv(t; x) = µ(x)Dv(t; x) + σ2(x)D2v(t; x): 2 Feynman-Kac theorem: Z T v(t; x) = E g(s; Xs )ds + h(XT ) ; t where Xs is the solution to the FSDE dXs = µ(Xs )ds + σ(Xs )d!s ; Xt = x: Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 2 / 51 HJB equation, dynamic programming Suppose we consider the Hamilton-Jacobi-Bellman (HJB) equation: @v(t; x) 1 + supfµ0(x; a)Dv(t; x) + Tr[D2v(t; x)σσ0(x; a)] @t a2A 2 + g(t; x; a)g = 0; v(T ; x) = h(x): It is associated to a stochastic control problem with value function Z T x α α v(t; x) = sup Et g(s; Xs ; αs )ds + h(XT ) ; α t where Xs is the solution to the controlled FSDE α α α α dXs = µ(Xs ; αs )ds + σ(Xs ; αs )d!s ; Xt = x: Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 3 / 51 Semilinear PDE and BSDEs The semilinear partial differential equation: @v(t; x) + Lv(t; x) + g(t; x; v; σ(x)Dv(t; x)) = 0; v(T ; x) = h(x): @t We can solve this PDE by means of the FSDE: dXs = µ(Xs )ds + σ(Xs )d!s ; Xt = x: and the BSDE: dYs = −g(s; Xs ; Ys ; Zs )ds + Zs d!s ; YT = h(XT ): Theorem: Yt = v(t; Xt ); Zt = σ(Xt )Dv(t; Xt ): is the solution to the BSDE. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 4 / 51 You can buy insurance against falling stock prices. This is the standard put option (i.e. the right to sell stock at a future time point). The uncertainty is in the stock prices. Application Suppose you have stocks of a company, and you'd like to have cash in two years (to buy a house). You wish at least K euros for your stocks, but the stocks may drop in the coming years. How to assure K euros in two years? Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 5 / 51 The uncertainty is in the stock prices. Application Suppose you have stocks of a company, and you'd like to have cash in two years (to buy a house). You wish at least K euros for your stocks, but the stocks may drop in the coming years. How to assure K euros in two years? You can buy insurance against falling stock prices. This is the standard put option (i.e. the right to sell stock at a future time point). Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 5 / 51 Application Suppose you have stocks of a company, and you'd like to have cash in two years (to buy a house). You wish at least K euros for your stocks, but the stocks may drop in the coming years. How to assure K euros in two years? You can buy insurance against falling stock prices. This is the standard put option (i.e. the right to sell stock at a future time point). The uncertainty is in the stock prices. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 5 / 51 v(T ; S) = max(K − ST ; 0) =: h(ST ) Financial derivatives Call options A call option gives the holder the right to trade in the future at a previously agreed strike price, K. s s0 K 0 0 t T Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 6 / 51 Financial derivatives Call options A call option gives the holder the right to trade in the future at a previously agreed strike price, K. s s0 K 0 0 t T v(T ; S) = max(K − ST ; 0) =: h(ST ) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 6 / 51 Feynman-Kac Theorem (option pricing context) Given the final condition problem 8 2 < @v + 1σ2S2 @ v + rS @v − rv = 0; @t 2 @S2 @S : v(T ; S)= h(ST ) = given Then the value, v(t; S), is the unique solution of −r(T −t) Q v(t; S) = e E fv(T ; ST )jFt g with the sum of first derivatives square integrable, and S = St satisfies the system of SDEs: Q dSt = rSt dt + σSt d!t ; Similar relations also hold for (multi-D) SDEs and PDEs! Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 7 / 51 A pricing approach; European options −r(T −t0) Q v(t0; S0) = e E fv(T ; ST )jF0g Quadrature: Z −r(T −t0) v(t0; S0) = e v(T ; ST )f (ST ; S0)dST R Trans. PDF, f (ST ; S0), typically not available, but the characteristic function, fb, often is. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 8 / 51 Banks sell insurance against changing FX rates. The option pays out in the best currency each year. Uncertain processes are the exchange rates, interest rate, but also the counterparty of the contract may go bankrupt! Application A firm will have some business in America for several years. Investment may be in euros, and payment in the local currency. Firm's profit can be influenced negatively by the exchange rate. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 9 / 51 but also the counterparty of the contract may go bankrupt! Application A firm will have some business in America for several years. Investment may be in euros, and payment in the local currency. Firm's profit can be influenced negatively by the exchange rate. Banks sell insurance against changing FX rates. The option pays out in the best currency each year. Uncertain processes are the exchange rates, interest rate, Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 9 / 51 Application A firm will have some business in America for several years. Investment may be in euros, and payment in the local currency. Firm's profit can be influenced negatively by the exchange rate. Banks sell insurance against changing FX rates. The option pays out in the best currency each year. Uncertain processes are the exchange rates, interest rate, but also the counterparty of the contract may go bankrupt! Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 9 / 51 Multi-asset options Multi-asset options belong to the class of exotic options. v(S; T ) = max(max fS1;:::; Sd gT − K; 0) (max call) −r(T −t ) R v(S; t0) = e 0 v(S; T )f (S jS0)dS R T ) High-dimensional integral or a high-D PDE. s K 0 0 t T Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 10 / 51 Increasing dimensions: Multi-asset options The problem dimension increases if the option depends on more than one asset Si (multiple sources of uncertainty). If each underlying follows a geometric (lognormal) diffusion process, Each additional asset is represented by an extra dimension in the problem: d d @v 1 X @2v X @v + [σi σj ρi;j Si Sj ] + [rSi ] − rv = 0 : @t 2 @Si @Sj @Si i;j=1 i=1 Required information is the volatility of each asset σi and the correlation between each pair of assets ρi;j . Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 11 / 51 Numerical Pricing Approach One can apply several numerical techniques to calculate the option price: Numerical integration, Monte Carlo simulation, Numerical solution of the partial-(integro) differential equation Each of these methods has its merits and demerits. Numerical challenges: The problem's dimensionality Speed of solution methods Early exercise feature (! free boundary problem) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 12 / 51 5a. Price the risk related to default (SDE, Opt.) 6. Understand and remove risk (Stoch., Opt., Numer.) Financial engineering; Work at Banks Financial engineering, pricing approach: 1. Start with some financial product 2.
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