Numerical Evaluation of American Options Written on Two Underlying Assets using the Fourier Transform Approach

Transcription

1 1 / 26 Numerical Evaluation of American Options Written on Two Underlying Assets using the Fourier Transform Approach Jonathan Ziveyi Joint work with Prof. Carl Chiarella School of Finance and Economics, UTS CEF Conference 15 July 2009

2 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 2 Outline Problem Statement Solution Procedure The Fourier Transform Approach The American Spread Call example Numerical Results Comparison with Monte Carlo Method The American Max Call example Numerical Results

3 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 3 Brief Survey McKean (1965) first solved the American pricing problem as a free boundary value problem. The pricing PDE is transformed to the corresponding ODE using the incomplete Fourier Transforms. Chiarella & Ziogas (2004) solve this American pricing problem for different payoff functions. We attempt to extend this approach to options with more than one underlying asset. Here we use Jamshidian s (1992) approach to set up the PDE as an inhomogeneous problem and present the corresponding general soln. Fourier transforms are applied to the PDE for the density function.

4 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 4 The Problem Statement Under the risk - neutral probability measure, S 1 and S 2 are driven by the GBM processes, ds 1 = (r q 1 )S 1 dt + σ 1 S 1 d W 1 ds 2 = (r q 2 )S 2 dt + σ 2 S 2 d W 2. (1.1) By letting S i = e x i for i = 1, 2, we represent the American option price as C(τ, x 1, x 2 ), where the price satisfies the inhomogeneous PDE, C τ = LC rc ½ {cont}lc(x 1, x 2 ), (1.2) L is the Dynkin operator of (1.1) defined in equation (1.3) below.

5 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 5 The Problem Statement The Dynkin operator is defined as, L = (r q σ2 1 ) + (r q 2 1 x 1 2 σ2 2 ) + 1 x 2 2 σ ρσ 1 σ x 1 x 2 2 σ2 2 2 x 2 2 We solve (1.2) subject to ICs and BCs, 2 x 2 1. (1.3) C(0, x 1, x 2 ) = c(x 1, x 2 ), < x 1, x 2 < (1.4) C(τ, 0, 0) = 0, 0 τ T, (1.5) 1 and smooth pasting conditions imposed to ensure continuity at the early exercise boundary. 2 These will be specified for particular payoff functions.

6 The 1 st part of (1.6) is the European Option component and the 2 nd is the Early Exercise premium. U(τ, x 1, x 2 ; u 1, u 2 ) is the Green s function satisfying the Kolmogorov PDE. Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 6 General Solution of Inhomogeneous PDE By using Duhamel s principle, the general solution of the PDE (1.2) can be represented as, C(τ, x 1, x 2 ) = C E (τ, x 1, x 2 ) + C P (τ, x 1, x 2 ), (1.6) where, C E (τ, x 1, x 2 ) = e rτ c(u 1, u 2 ) U(τ, x 1, x 2 ; u 1, u 2 )du 1 du 2, C P (τ, x 1, x 2 ) = τ 0 e r(τ ξ) f(ξ, u 1, u 2 ) U(τ ξ, x 1, x 2 ; u 1, u 2 )du 1 du 2 dξ.

7 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 7 Transformed PDE of the Density Function The transformed PDE for the density function becomes, ( U τ = r q 1 1 ) ( U 2 v 1 + r q 2 1 ) U x 1 2 v 2 x U 2 U 2 σ2 1 x1 2 + ρσ 1 σ U x 1 x 2 2 σ2 2 x2 2. (1.7) This is solved subject to the initial condition, U(τ; x 1, x 2 ; x 1,0, x 2,0 ) = δ(e x 1 e x 1,0 )δ(e x 2 e x 2,0 ). In solving equation (1.7), we first transform to the corresponding ODE by using transform techniques. We give key results of the Fourier Transform approach.

8 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 8 The Fourier Transforms The Fourier Transform of a function, U(τ, x 1, x 2 ), is defined as, F{U(τ, x 1, x 2 )} = e iη 1x 1 +iη 2 x 2 U(τ, x 1, x 2 )dx 1 dx 2 = Û(τ,η 1,η 2 ). (1.8) The inverse of the Fourier Transform is represented as, F 1 {Û(τ, η 1, η 2 )} = 1 4π 2 e iη 1x 1 iη 2 x 2 Û(τ, η 1, η 2 )dη 1 dη 2 = U(τ, x 1, x 2 ). (1.9) Given these definitions, we apply to the PDE (1.7) for the density function.

9 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 9 Applying the Fourier Transform to the PDE Proposition The Fourier Transform of the PDE (1.7) satisfies the ODE, Û [ τ (τ,η 1,η 2 ) + iη 1 κ 1 + iη 2 κ σ2 1 η2 1 + η 1η 2 ρσ 1 σ σ2 2 η2 2]Û(τ,η1,η 2 ) = 0. (1.10) Equation (1.10) is solved subject to the initial condition, Û(0,η 1,η 2 ) = e iη 1x 1,0 +iη 2 x 2,0. (1.11)

10 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 10 Inverting the Fourier Transform The Inverse Fourier Transform of (1.10) is given by, { 1 U(τ, x 1, x 2 ) = 2πσ 1 σ 2 τ 1 ρ exp 1 [( x1 x 1,0 + κ 1 τ ) 2 2 2τ(1 ρ 2 ) σ 1 ( x1 x 1,0 + κ 1 τ 2ρ σ 1 )( x2 x 2,0 + κ 2 τ ) ( x2 x 2,0 + κ 2 τ ) 2 ] + σ 2 σ 2 (1.12) Equation (1.12) is the bivariate transition density function of the two stochastic processes, x 1 and x 2. Given this, we now have full representation of the American call option price, equation (1.6). We present the American Spread and Max Option examples. }

11 Price of the American Spread Option The American Spread call is an option written on the difference of two underlying assets. The American Spread Option problem is solved subject to initial and boundary conditions 1, 1 c(x 1, x 2 ) = max(0, e x 1 e x 2 K) which is the payoff at maturity, 2 C(τ,, x 2 ) = 0 which is an absorbing state. 3 C(τ, B(τ, x 2 ), x 2 ) = c(b(τ, x 2 ), x 2 ), = value matching condition for continuity at the early exercise boundary. 4 C lim = 0 x 1 B(τ,x 2 ) x 1 = the smooth pasting condition imposed to avoid arbitrage opportunities. 1 Note B(τ, x 2 ) = ln b(τ, S 2 ) Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 11

12 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 12 Price of the American Spread Option cont... There are different forms of Spread options trading on the markets which includes: Fixed income markets Derived from differences in interest rates in different countries Agricultural Futures Markets Soybean complex spread (CBOT) Underlying comprises of the futures contracts of soybean oil and soybean meal. Energy markets Used to quantify the cost of production of refined products from the complex raw materials used to produce them,

13 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 13 Price of the American Spread Option cont... Proposition The early exercise surface of the American Spread Call option solves, B(τ, x 2 ) e x 2 K = C(τ, B(τ, x 2 ), x 2 ) (1.13) Corresponding price C(τ, x 1, x 2 ) = C E (τ, x 1, x 2 ) + C P (τ, x 1, x 2 ). (1.14) A non linear Volterra integral eqn; solve numerically using extd Simpson s rule. Root finding yields free boundary at each time step.

14 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 14 Numerical Implementation of the American Spread Option Using MC Algorithm To validate our approach, we use Monte Carlo algorithm of Ibañez & Zapatero (2004). The American option is treated like a Bermudan Option. The early exercise surface is tracked by using DP. Valuation algorithm heavily dependent on the value - matching condition, that is, V(τ, b(τ, S 2 ), S 2 ) = v(b(τ, S 2 ), S 2 ), (1.15) Discretize the time domain such that τ i = ih for i = 0, 1,, M. At each time step, use knowledge of both the price and exercise surface at the previous time steps. So the value of the American option can be treated like a European option Thus we can apply the plain vanilla Monte Carlo simulation. At maturity, τ 0, the American option value is (S 1,τ0 S 2,τ0 K) +.

15 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 15 Numerical Implementation of the American Spread Option Using Monte Carlo Algorithm cont... Figure: Possible Paths of S 1

16 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 16 Numerical Results Parameters K = 5, r = 5%, q 1 =3%, q 2 = 2%, σ 1 = 40%, σ 2 = 49%, ρ = 0.5, and T = 1. Early Exercise Surface of the American Spread Option Using Fourier Transform Approach B(τ,S 2 ) τ S

17 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 17 Numerical Results cont Early Exercise and Continuation Regions 100 Early Exercise Region 80 B(T,S 2 ) B(τ,S 2 ) B(0,S 2 ) Continuation Region S 2

18 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 18 Numerical Results cont... Early Exercise Surface Differences 0.3 Relative Differences τ S

19 Numerical Results cont... Price Differences 0.8 Price Differences S 1 S 2 When S 1 = 35, S 2 = 30 and K = 5, = Relative Price Difference = 7.796%. When S 1 = 60, S 2 = 50 and K = 5, = Relative Price Difference = 6.364%. Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 19

20 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 20 Price of the American Max Call Option The American max call option is a contact whose payoff is derived from the maximum of two assets. The American max option has two exercise regions which we call B 1 (τ, x 2 ) and B 2 (τ, x 1 ). The American max option problem is solved subject to: The payoff = c(x 1, x 2 ) = max[max(e x 1, e x 2 ) K, 0], Absorbing state C(τ,, ) = 0 Value matching condition, C(τ, B 1 (τ, x 2 ), B 2 (τ, x 1 )) = c(b 1 (τ, x 2 ), B 2 (τ, x 1 )),

21 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 21 Price of the American Max Call Option cont... Proposition The early exercise surface of the American max call option solves, B 1 (τ, x 2 ) K = C E (τ, B 1 (τ, x 2 ), x 2 ) + C P (τ, B 1 (τ, x 2 ), x 2 ) B 2 (τ, x 1 ) K = C E (τ, x 1, B 2 (τ, x 1 )) + C P (τ, x 1, B 2 (τ, x 1 )), and the terminal values of the two boundary functions are defined as, [ ( B 1 (0, x 2 ) = max e x 2 r )], max K, K (1.16) q 2 [ ( B 2 (0, x 1 ) = max e x 1 r )], max K, K. (1.17) q 1 Corresponding price C(τ, x 1, x 2 ) = C E (τ, x 1, x 2 ) + C P (τ, x 1, x 2 ). (1.18)

22 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 22 Numerical Results for the Max Option Parameters K = 100, r = 5%, q 1 =18%, q 2 = 18%, σ 1 = 20%, σ 2 = 20% and T = 3. Assume 5 exercise opportunities. Early Exercise Boundary at the First Exercise Opportunity Early Exercise Boundary at the First Exercise Opportunity Exercise Region Exercise Region S Continuation Region Exercise Region 2 S Continuation Region Exercise Region S S 2 Figure: The 1 st Exercise Opportunity when ρ = 0.5 & 0.5

23 S 1 S 1 S 1 S 2 S 1 S 1 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 23 S 2 S 2 Numerical Results cont Early Exercise Boundary at the Second Exercise Opportunity 200 Early Exercise Boundary at the Second Exercise Opportunity Exercise Region Exercise Region Continuation Region Exercise Region Continuation Region Exercise Region Figure: The 2 nd Second Exercise Opportunity when ρ = 0.5 & Early Exercise Boundary at the Third Exercise Opportunity 250 Early Exercise Boundary at the Third Exercise Opportunity 200 Exercise Region Exercise Region Continuatio Region Exercise Region 2 50 Continuatio Region Exercise Region Figure: The 3 rd Exercise Opportunity when ρ = 0.5 & 0.5

24 S 1 S 1 S 1 S 2 S 1 S 1 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 24 S 2 S 2 Numerical Results cont Early Exercise Boundary ay the Fourth Exercise Opportunity 200 Early Exercise Boundary at the Fourth exercise Opportunity Exercise Region Exercise Region Continuation Region Exercise Region Continuation Region Exercise Region Figure: The 4 th Exercise Opportunity when ρ = 0.5 & Early Exercise Boundary at the Fifth Exercise Oppotunity 200 Early Exercise Boundary at the Last Exercise Opportunity Exercise Region Exercise Region Continuation Region Exercise Region Continuation Region Exercise Region Figure: The 5 th Exercise Opportunity when ρ = 0.5 & 0.5

25 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 25 Numerical Results cont... Figure: Price Profile of the Max Option

26 Chiarella & Ziveyi: 15th International Conference on Computing in Economics and Finance 26 Conclusion We have managed to derive and solve the integral representation of American options written on two underlying assets. Details of how to solve the free boundary at each time step. Presented two examples for the Spread call and Max option.