Fourier, Wavelet and Monte Carlo Methods in Computational Finance

Transcription

1 Fourier, Wavelet and Monte Carlo Methods in Computational Finance Kees Oosterlee 1,2 1 CWI, Amsterdam 2 Delft University of Technology, the Netherlands AANMPDE-9-16, 7/7/26 Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

2 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 / 51

3 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 / 51

4 Feynman-Kac Theorem The linear partial differential equation: v(t, x) t with operator + Lv(t, x) + g(t, x) =, v(t, x) = h(x), Lv(t, x) = µ(x)dv(t, x) σ2 (x)d 2 v(t, x). Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

5 Feynman-Kac Theorem The linear partial differential equation: v(t, x) t with operator + Lv(t, x) + g(t, x) =, v(t, x) = h(x), Lv(t, x) = µ(x)dv(t, x) σ2 (x)d 2 v(t, x). Feynman-Kac theorem: [ T v(t, x) = E where X s is the solution to the FSDE t ] g(s, X s )ds + h(x T ), dx s = µ(x s )ds + σ(x s )dω s, X t = x. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

6 HJB equation, dynamic programming Suppose we consider the Hamilton-Jacobi-Bellman (HJB) equation: v(t, x) t + sup{µ (x, a)dv(t, x) + 1 a A 2 Tr[D2 v(t, x)σσ (x, a)] + g(t, x, a)} =, v(t, x) = h(x). It is associated to a stochastic control problem with value function [ T ] v(t, x) = sup E x t g(s, Xs α, α s )ds + h(xt α ), α where X s is the solution to the controlled FSDE t dx α s = µ(x α s, α s )ds + σ(x α s, α s )dω s, X α t = x. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

7 Semilinear PDE and BSDEs The semilinear partial differential equation: v(t, x) t + Lv(t, x) + g(t, x, v, σ(x)dv(t, x)) =, v(t, x) = h(x). We can solve this PDE by means of the FSDE: dx s = µ(x s )ds + σ(x s )dω s, X t = x. and the BSDE: dy s = g(s, X s, Y s, Z s )ds + Z s dω s, Y T = h(x T ). Theorem: Y t = v(t, X t ), Z t = σ(x t )Dv(t, X t ). is the solution to the BSDE. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

8 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 / 51

9 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 / 51

10 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 / 51

11 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 T t s K Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

12 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 s K t T v(t, S) = max(k S T, ) =: h(s T ) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

13 Feynman-Kac Theorem (option pricing context) Given the final condition problem v + t 1 2 σ2 S 2 2 v 2 + rs v S S v(t, S) = h(s T ) = given Then the value, v(t, S), is the unique solution of rv =, v(t, S) = e r(t t) E Q {v(t, S T ) F t } with the sum of first derivatives square integrable, and S = S t satisfies the system of SDEs: ds t = rs t dt + σs t dω Q t, Similar relations also hold for (multi-d) SDEs and PDEs! Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

14 A pricing approach; European options v(t, S ) = e r(t t ) E Q {v(t, S T ) F } Quadrature: v(t, S ) = e r(t t ) R v(t, S T )f (S T, S )ds T Trans. PDF, f (S T, S ), typically not available, but the characteristic function, f, often is. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

15 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 / 51

16 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 / 51

17 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 / 51

18 Multi-asset options Multi-asset options belong to the class of exotic options. v(s, T ) = max(max {S 1,..., S d } T K, ) (max call) v(s, t ) = e r(t t ) R v(s, T )f (S T S )ds High-dimensional integral or a high-d PDE s T t K Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

19 Increasing dimensions: Multi-asset options The problem dimension increases if the option depends on more than one asset S i (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: v t d 2 v [σ i σ j ρ i,j S i S j ] + S i S j i,j=1 d i=1 [rs i v S i ] rv =. Required information is the volatility of each asset σ i and the correlation between each pair of assets ρ i,j. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

20 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 / 51

21 Financial engineering; Work at Banks Financial engineering, pricing approach: 1. Start with some financial product 2. Model asset prices involved (SDEs) 3. Calibrate the model to market data (Numerics, Opt.) 4. Model product price correspondingly (PDE, Integral) 5. Price the product of interest (Numerics, MC) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

22 Financial engineering; Work at Banks Financial engineering, pricing approach: 1. Start with some financial product 2. Model asset prices involved (SDEs) 3. Calibrate the model to market data (Numerics, Opt.) 4. Model product price correspondingly (PDE, Integral) 5. Price the product of interest (Numerics, MC) 5a. Price the risk related to default (SDE, Opt.) 6. Understand and remove risk (Stoch., Opt., Numer.) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

23 Motivation Fourier Methods A characteristic function of a continuous random variable X, equals the Fourier transform of the density of X. Derive pricing methods that are computationally fast are not restricted to Gaussian-based models should work as long as we have a characteristic function, f (u; x) = e iux f (x)dx; (available for Lévy processes and also for SDE systems). Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

24 Class of Affine Jump Diffusion (AJD) processes Suppose we have given a following system of SDEs: dx t = µ(x t )dt + σ(x t )dω t + dz t, For processes in the AJD class drift, volatility, jump intensities and interest rate components are of the affine form, i.e. µ(x t ) = a + a 1 X t for (a, a 1 ) R n R n n, λ(x t ) = b + b T 1 X t, for (b, b 1 ) R R n, σ(x t )σ(x t ) T = (c ) ij + (c 1 ) T ij X t, (c, c 1 ) R n n R n n n, r(x t ) = r + r T 1 X t, for (r, r 1 ) R R n. Duffie, Pan, Singleton (2): For affine jump diffusion processes the discounted characteristic function can be derived! Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

25 Exponential Lévy Processes Lévy process {X t } t : process with stationary, independent increments. Brownian motion and Poisson processes belong to this class, as well jump processes with either finite or infinite activity Asset prices can be modeled by exponential Lévy processes small jumps describe the day-to-day noise ; big jumps describe large stock price movements. The characteristic function of a Lévy process is known: f (u; X t ) = E[exp (iux t )] = exp (t(iµu 1 2 σ2 u 2 + (e iuy 1 iuy1 [ y <1] ν(dy))), the celebrated Lévy-Khinchine formula. R Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

26 Fourier-Cosine Expansions, COS Method (with Fang Fang) The COS method: Exponential convergence; Greeks (derivatives) are obtained at no additional cost. All based on the availability of a characteristic function. The basic idea: Replace the density by its Fourier-cosine series expansion; Coefficients have simple relation to characteristic function. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

27 Series Coefficients of the Density and the ChF Fourier-Cosine expansion of density function on interval [a, b]: ( f (x) = F n cos nπ x a ), b a n= with x [a, b] R and the coefficients defined as F n := 2 b ( f (x) cos nπ x a ) dx. b a b a a Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

28 Series Coefficients of the Density and the ChF Fourier-Cosine expansion of density function on interval [a, b]: ( f (x) = F n cos nπ x a ), b a n= with x [a, b] R and the coefficients defined as F n := 2 b ( f (x) cos nπ x a ) dx. b a b a F n has direct relation to the ChF, f (u) := R f (x)eiux dx ( R\[a,b] f (x) ), ( 2 F n P n := f (x) cos nπ x a ) dx b a b a = a 2 b a R R { f ( nπ b a ) exp ( naπ i b a )}. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

29 Recovering Densities Replace F n by P n, and truncate the summation: f (x) 2 { N 1 b a R f ( nπ a ) exp (inπ n= b a b a ) } cos(nπ x a b a ). Example: f (x) = 1 2π e 1 2 x2, [a, b] = [ 1, 1] and x = { 5, 4,, 4, 5}. N error e e-16 cpu time (sec.) Exponential error convergence in N. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

30 Pricing European Options Start from the risk-neutral valuation formula: v(t, x) = e r t E Q [v(t, y) F ] = e r t Truncate the integration range: v(t, x) = e r t [a,b] R v(t, y)f (y, x)dy + ε. v(t, y)f (y, x)dy. Replace the density by the COS approximation, and interchange summation and integration: N 1 ˆv(t, x) = e r t n= { ( ) nπ R f b a ; x e inπ a b a } H n, where the series coefficients of the payoff, H n, are analytic. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

31 Pricing European Options Log-asset prices: x := log(s /K) and y := log(s T /K). The payoff for European call options reads v(t, y) max (K(e y 1), ). For a call option, we obtain H call k = = For a vanilla put, we find 2 b ( K(e y 1) cos kπ y a ) dy b a b a 2 b a K (χ k(, b) ψ k (, b)). H put k = 2 b a K ( χ k(a, ) + ψ k (a, )). Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

32 Results, Heston stochastic volatility PDE v t = 1 2 S 2 y 2 v S 2 + ργsy 2 v S y γ2 y v v + rs y 2 S + κ(σ y) v y rv. GPU computing: Multiple strikes for parallelism, 21 IC s. Heston model N MATLAB msec max.abs.err 6.991e e-8 < 1 14 GPU msec Table 1: Maximum absolute error when pricing a vector of 21 strikes. Exponential convergence, Error analysis in our papers. Also work with wavelets instead of cosines. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

33 SWIFT Wavelet Option Pricing (with Luis Ortiz) Approximate density f by a finite combination of Shannon scaling functions and recovering the coefficients from its Fourier transform. ϕ m,k (x) = 2 m/2 sin(π(2m x k)) π(2 m, k Z. (1) x k) For m = k =, we have the father wavelet, ϕ(x) = sinc(x) = sin(πx). πx Wavelets can be moved (by k), stretched or compressed (by m) to accurately represent local properties of a function. Shannon wavelets have a slow decay in the time domain but a very sharp compact support in the frequency (Fourier) domain. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

34 Using the classical Vieta formula, the cardinal sine can be approximated: sinc(t) J ( πt ) cos 2 j = 1 2 J 1 ( ) 2j 1 2 J 1 cos 2 J πt. (2) j=1 Following the wavelets theory, function f can be approximated at a level of resolution m, i.e., j=1 f (x) P m f (x) = k Z c m,k ϕ m,k (x) f m (x) := for certain accurately chosen values k 1 and k 2. k 2 k=k 1 c m,k ϕ m,k (x), (3) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

35 Summary v(x, t) = e r(t t) R v(y, T )f (y x) dy, f (y x) k 2 k=k 1 c m,k (x)ϕ m,k (y), With: c m,k = f, ϕ m,k = R f (x)ϕ(2m x k)dx. or: c m,k c m,k := 2m/2 2 J 1 2 J 1 j=1 c m,k cm,k = 2m/2 2 J 1 { 2 J 1 R ˆf R j=1 ( ) 2j 1 f (x) cos 2 J π(2 m x k) dx. (4) ( (2j 1)π2 m 2 J ) } e ikπ(2j 1) 2 J. (5) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

36 Example: Fat-tailed densities: We can control density values and the mass as a byproduct. ( ) ( ) h h f 2 m P m f 2 m = 2 m 2 c m,k δ k,h = 2 m 2 cm,h, (6) k Z...plus... A = 1 c m,k 1 2 m/2 2 + c m,k + c m,k2, (7) 2 k 1 <k<k 2 [a, b] f ( k 1 ) 2 m ( f k2 ) 2 m Error (area) [ 1, 1] 5.97e-2 4.2e e- [ 5, 5] 1.3e- 1.6e-3 3.4e- [ 2, 2] 1.1e e e-9 Difficult to accurately determine [a, b] by cumulants. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

37 Replace f by its approximation f m : k 2 v(x, t) e r(t t) k=k 1 c m,k (x) H m,k, with the pay-off coefficients, H m,k := v(y, T )ϕ m,k (y) dy. I k With k 1 := max(k 1, ), pay-off coefficients, H m,k, for a European call option are approximated by, { K2 m/2 [ ( ) ( )] 2 J 1 Hm,k := 2 J 1 j=1 I k1 1 2, k m 2 2 I k1 m 2 2, k m 2 2, if k m 2 >,, if k 2, where I 1 and I 2 are closed formulae. FFT can be applied to compute the pay-off coefficients. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

38 Example: Multiple strikes valuation: 21 strikes 1 strike scale m CPU time (milli-seconds) Max. absolute error 2.4e e e 6 scale m CPU time (milli-seconds) Absolute error 4.78e e e 7 Table 2: Simultaneous valuation of 21 call options with K from 5 to 15, and of only one European call option under the Heston dynamics with parameters S = 1, µ =, λ = , η =.5751, ū =.398, u =.75, ρ = Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

39 Optimal stopping, dynamic programming The controller only has control over his terminal time. dx t = µ(x t )dt + σ(x t )dω t, With t [, T ], and stopping times T t,t, the finite horizon optimal stopping problem is formulated as [ τ ] v(t, x) = sup E e r(t t) g(s, X s )ds + e r(τ t) h(x τ ). τ T t,t t The value function v is related to the HJB variational inequality: min[ v t Lv g, v h] =, The problem is called a free boundary problem. C is the continuation region, the complement set is the stopping or exercise region (receive the reward g). Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

40 Pricing Options with Early-Exercise s s m m+1 M K t T The pricing formulas for a Bermudan option with M exercise dates reads, for m = M 1,..., 1: { c(s, tm ) = e r tm E [v(s, t m+1 ) S tm ], v(s, t m ) = max (h(s, t m ), c(s, t m )) and v(s, t ) = e r t.e [v(s, t 1 ) S t ] Use Newton s method to determine early exercise point xm+1, i.e. the root of h(x, t m+1 ) c(x, t m+1 ) =. Use the COS formula for v(x, t ). Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

41 COS pricing method for early exercise options The pricing formulas, for m = M 1,..., 1: { c(x, tm ) = e r t R v(y, t m+1)f (y x)dy, followed by v(x, t m ) = max (h(x, t m ), c(x, t m )), v(x, t ) = e r t R v(y, t 1 )f (y x)dy. Here x, y are state variables of consecutive exercise dates. In order to get the option value, a backward recursion procedure is performed on the Fourier Cosine coefficient H k (t m+1 ): H k (t m+1 ) = 2 b max(c(x, t m+1 ), h(x, t m+1 )) cos (kπ y a b a a b a )dy, and then use H k (t 1 ) in the COS formula. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

42 Bermudan puts with 1 early-exercise dates Table 3: Test parameters for pricing Bermudan options Test No. Model S K T r σ Other Parameters 2 BS CGMY C = 1, G = 5, M = 5, Y = BS COS, L=8, N=32*d, d=1:5 CONV, δ=2, N=2 d, d=8: CGMY COS, L=8, N=32*d, d=1:5 CONV, δ=2, N=2 d, d=8: log 1 error log 1 error milliseconds milliseconds Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

43 Towards higher dimensions: Monte Carlo Formulation Dynamic Programming: COS approximation is fine for up to 3, 4 dimensions! SWIFT works as well for these options. Higher D: Monte Carlo simulation techniques, like SGBM. The Bermudan option at time t m and state S tm is given by v tm (S tm ) = max(h(s tm ), c tm (S tm )). (8) The continuation value c tm, is : c tm (S tm ) = e r tm E [ v tm+1 (S tm+1 ) S tm ]. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

44 Towards higher dimensions: Monte Carlo Formulation Dynamic Programming: s s m m+1 M K t T COS approximation is fine for up to 3, 4 dimensions! SWIFT works as well for these options. Higher D: Monte Carlo simulation techniques, like SGBM. The Bermudan option at time t m and state S tm is given by v tm (S tm ) = max(h(s tm ), c tm (S tm )). (8) The continuation value c tm, is : c tm (S tm ) = e r tm E [ v tm+1 (S tm+1 ) S tm ]. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

45 Stochastic Grid Bundling Method, with Shashi Jain Step 1: The grid points in SGBM are generated by simulation, {S t (n),..., S tm (n)}, n = 1,..., N, Step 2: Compute the option value at terminal time. Step 3: Bundle the grid points at t m into B tm (1),..., B tm (ν) non-overlapping bundles. Step 4: For B tm (β), β = 1,..., ν, compute Z(S tm+1, α β t m+1 ). Z : R d R K R, is a parametrized function which assigns values to states S tm+1. Step 5: The continuation values for grid points in B tm (β), β = 1,..., ν, are approximated by ĉ tm (S tm (n)) = E[Z(S tm+1, α β t m+1 ) S tm (n)] Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

46 Intuition In the limiting case, of number of bundles ν, and paths N, the Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51 The objective is to choose, corresponding to each bundle β at t m, a parameter vector αt β m+1 so that, V tm+1 (S tm+1 ) Z(S tm+1, αt β m+1 ). We use OLS, to define K Z(S tm+1, α t β m+1 ) = α t β m+1 (k)φ k (S tm+1 ). (9) k=1 Asset price Time step t m

47 Computing the continuation value The continuation value is computed as: ĉ tm (S tm (n)) = E[Z(S tm+1, α t β m+1 ) S tm = S tm (n)], (1) where S tm (n) B tm (β). This can be written as: [( K ) ] ĉ tm (S tm (n)) = E α t β m+1 (k)φ k (S tm+1 ) S tm = S tm (n) k=1 K = α t β m+1 (k)e [ φ k (S tm+1 ) S tm = S tm (n) ]. k=1 Choose basis functions φ so that E [ φ k (S tm+1 ) S tm = X ] has an analytic solution. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

48 Arithmetic Basket Option on 15 assets Convergence and computational time: Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

49 Arithmetic Basket Option on 15 assets Convergence and computational time: k means direct est k means path est. RB2 direct est. RB2 path est. Vt (S t ) Bundles ν Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

50 Arithmetic Basket Option on 15 assets Convergence and computational time: RB2 k means 21 Time (sec) Bundles ν Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

51 Expected exposure in CVA (joint work with Qian Feng) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

52 Exposure and Counterparty risk Exposure at a path at time t m with stock S tm { c tm (S tm ), not exercised E(S tm, t m ) =, exercised (11) Expected Exposure (EE) at time t m EE(t m ) = E [E(S tm, t m )] 1 N E(S tm (n), t m ) (12) N n=1 c tm (S tm ) is calculated at each time step, along the path, by SGBM. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

53 Semilinear PDE and BSDEs, with Marjon Ruijter The semilinear partial differential equation: v(t, x) t + Lv(t, x) + g(t, x, v, σ(x)dv(t, x)) =, v(t, x) = h(x), We can solve this PDE by means of the FSDE: dx s = µ(x s )ds + σ(x s )dω s, X t = x. and the BSDE: dy s = g(s, X s, Y s, Z s )ds + Z s dω s, Y T = h(x T ). Theorem: Y t = v(t, X t ), Z t = σ(x t )Dv(t, X t ). is the solution to the decoupled FBSDE. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

54 Backward SDE dy t = g(t, Y t, Z t )dt + Z t dω t, Y T = ξ. g is the driver function. ξ is F T -measurable random variable. A solution is a pair adapted processes (Y, Z) satisfying T T Y t = ξ + g(s, Y s, Z s )ds Z s dω s. t t Y is adapted if and only if, for every realization and every t, Y t is known at time t. Adapted process cannot see into the future. A BSDE is not a time-revered FSDE: at time t (Y t, Z t ) is F t -measurable and the process does not know the terminal condition yet. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

55 Backward SDE dy t = g(t, Y t, Z t )dt + Z t dω t, Y T = ξ. g is the driver function. ξ is F T -measurable random variable. A solution is a pair adapted processes (Y, Z) satisfying T T Y t = ξ + g(s, Y s, Z s )ds Z s dω s. t t Y is adapted if and only if, for every realization and every t, Y t is known at time t. Adapted process cannot see into the future. A BSDE is not a time-revered FSDE: at time t (Y t, Z t ) is F t -measurable and the process does not know the terminal condition yet. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

56 Y component X t := (X t, Y t, Z t ). tm+1 tm+1 Y m = Y m+1 + g(s, X s )ds Z s dω s. (13) t m t m Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

57 Y component X t := (X t, Y t, Z t ). tm+1 tm+1 Y m = Y m+1 + g(s, X s )ds Z s dω s. (13) t m t m Taking conditional expectation E m [.] = E[. F tm ]: tm+1 [ tm+1 ] Y m = E m [Y m+1 ] + E m [g(s, X s )]ds E m Z s dω s t m t m (14) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

58 Y component X t := (X t, Y t, Z t ). tm+1 tm+1 Y m = Y m+1 + g(s, X s )ds Z s dω s. (13) t m t m Taking conditional expectation E m [.] = E[. F tm ]: tm+1 [ tm+1 ] Y m = E m [Y m+1 ] + E m [g(s, X s )]ds E m Z s dω s t m t m (14) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

59 Y component X t := (X t, Y t, Z t ). tm+1 tm+1 Y m = Y m+1 + g(s, X s )ds Z s dω s. (13) t m t m Taking conditional expectation E m [.] = E[. F tm ]: tm+1 Y m = E m [Y m+1 ] + E m [g(s, X s )]ds t m (14) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

60 Y component X t := (X t, Y t, Z t ). tm+1 tm+1 Y m = Y m+1 + g(s, X s )ds Z s dω s. (13) t m t m Taking conditional expectation E m [.] = E[. F tm ]: tm+1 Y m = E m [Y m+1 ] + E m [g(s, X s )]ds t m E m [Y m+1 ] + tg(t m, X m ). (14) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

61 Z component Y m = Y m+1 + t m+1 t m g(s, X s )ds t m+1 t m Z s dω s. (15) Multiplying by ω m+1, taking the conditional expectation gives Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

62 Z component Y m = Y m+1 + t m+1 t m g(s, X s )ds t m+1 t m Z s dω s. (15) Multiplying by ω m+1, taking the conditional expectation gives Z m 1 t E m[y m+1 ω m+1 ]. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

63 Discretization scheme FBSDE X = x, for m =,..., M 1 : X m+1 = X m + µ(x m ) t + σ(x m ) ω m+1, Y M = h(x M ), for m = M 1,..., : Z m = 1 t E m[y m+1 ω m+1 ], Y m = E m [Y m+1] + g(t m, X m ) t. Ingredients: Characteristic function X m+1, given X m = x E x m[h(t m+1, X m+1 )] E x m[h(t m+1, X m+1 ) ω m+1] Recover Fourier cosine coefficients backward in time FFT algorithm Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

64 Discretization scheme FBSDE X = x, for m =,..., M 1 : X m+1 = X m + µ(x m ) t + σ(x m ) ω m+1, Y M = h(x M ), for m = M 1,..., : Z m = 1 t E m[y m+1 ω m+1 ], Y m = E m [Y m+1] + g(t m, X m ) t. Ingredients: Characteristic function X m+1, given X m = x E x m[h(t m+1, X m+1 )] E x m[h(t m+1, X m+1 ) ω m+1] Recover Fourier cosine coefficients backward in time FFT algorithm Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

65 COS formula I := E x m[h(t m+1, Xm+1)] = R b a h(t m+1, ζ)f (ζ x)dζ h(t m+1, ζ)f (ζ x)dζ. (16) Replace the density function and function h by their Fourier cosine series expansions. I b a N 1 2 H k(t m+1 )P k (x). (17) k= The COS formula: E x m[h(t m+1, Xm+1)] ( N 1 ( ) ) H k(t m+1 )R f kπ k= X m+1 b a X m a ikπ = x e b a. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

66 Example: European Call Option - GBM - P-Measure Asset price: ds t = µs t dt + σs t dω t. Hedge portfolio Y t with: a t assets S t and Y t a t S t bonds dy t = r(y t a t S t )dt + a t ds t Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

67 Example: European Call Option - GBM - P-Measure Asset price: ds t = µs t dt + σs t dω t. Hedge portfolio Y t with: a t assets S t and Y t a t S t bonds dy t = r(y t a t S t )dt + a t ds t Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

68 Example: European Call Option - GBM - P-Measure Asset price: ds t = µs t dt + σs t dω t. Hedge portfolio Y t with: a t assets S t and Y t a t S t bonds dy t = r(y t a t S t )dt + a t ds t ( = ry t + µ r ) σ σa ts t dt + σa t S t dω t, Y T = h(s T ) = max(s T K, ). Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

69 Example: European Call Option - GBM - P-Measure Y t corresponds to the value of the option and Z t is related to the hedging strategy. The option value is given by v(t, S t ) = Y t and σ(s Kees Oosterlee )v (t, (CWI, S ) = TU Z Delft). Comp. Finance AANMPDE / 51 Asset price: ds t = µs t dt + σs t dω t. Hedge portfolio Y t with: a t assets S t and Y t a t S t bonds dy t = r(y t a t S t )dt + a t ds t ( = ry t + µ r ) σ σa ts t dt + σa t S t dω t, Y T = h(s T ) = max(s T K, ). If we set Z t = σa t S t, then (Y, Z) solves a BSDE with driver, g(t, x, y, z) = ry µ r σ z.

70 Results European call option Exact solutions Y = v(, S ) = 3.66 and Z = σs v S (, S ) = error Y error Y theta=1, Euler theta=1, Euler theta=1, Milstein theta=.5, Milstein theta=1, 2. weak Taylor theta=.5, 2. weak Taylor M M Figure 1: Euler, Milstein and order 2. weak Taylor scheme, θ = 1 and θ =.5. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

71 Summary and conclusion COS and SWIFT methods for pricing European options BCOS method for nonlinear PDEs Second-order convergence met 2. weak Taylor scheme and θ = 1/2 discretization of integrals! Higher dimensions: Generalization to Monte Carlo method SGBM Challenges: Higher dimensions (MC, SGBM), stochastic control, portfolio selection Towards accurate, efficient and robust solution methods. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

72 Computational finance Efficient valuation of financial options COS method is used at many (financial) institutions world-wide; works efficient for a variety of options (barrier, Asian, multi-asset,... ) High-D American options, Monte Carlo simulation, SGBM Interpolation methods, Stochastic Collocation Monte Carlo methods (SCMC sampler) Under recent asset price model dynamics, for different asset classes Risk management Accurate hedge parameters; Numerical estimation of tail probabilities and expected shortfall, Value-at-Risk; (Counterparty) Credit risk and other types of risk; Inflation options for pension funds. Portfolio optimization (2-...) Energy portfolio, real options analysis Dynamic portfolios for pensions (target based vs time-consistent mean-variance strategy) Challenge: Incorporate (big) financial data Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

73 Characteristic function We can write the Euler, Milstein, and 2. weak Taylor discretization schemes in the following general form X m+1 = x + m(x) t + s(x) ω m+1 + κ(x)( ω m+1 ) 2, X m = x. For the Euler scheme: m(x) = µ(x), s(x) = σ(x), κ(x) =. For the Milstein scheme: m(x) = µ(x) 1 2 σσ x(x), s(x) = σ(x), κ(x) = 1 2 σσ x(x). For the order 2. weak Taylor scheme: m(x) = µ(x) 1 2 σσ x(x) + 1 ( 2 µµx (x) µ xxσ 2 (x) ) t, s(x) = σ(x) + 1 ( 2 µx σ(x) + µσ x (x) σ xxσ 2 (x) ) t, κ(x) = 1 2 σσ x(x). Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51

74 Characteristic function ( Xm+1 = x + m(x) t + κ(x) ) 2 1 s 2 (x) 4 κ(x) ω m s(x) 2 κ(x) d = x + m(x) t 1 s 2 (x) ( 4 κ(x) + κ(x) t U m λ(x)), s 2 (x) with λ(x) := 1 4 κ 2 (x) t, U m+1 N (, 1). (U m+1 + λ(x)) 2 χ 2 1 (λ(x)) non-central chi-squared distributed. The characteristic function of X m+1, given X m = x [ f X (u X m+1 m = x) = E = exp ( iux + ium(x) t ( ] exp iuxm+1) X m = x 1 2 u2 s 2 (x) t 1 2iuκ(x) t ) (1 2iuκ(x) t) 1/2. Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE / 51