Valuing American Options by Simulation

Transcription

1 Valuing American Options by Simulation Hansjörg Furrer Market-consistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Valuing American Options

2 Course material Slides Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American Options by Simulation: A Simple Least-Squares Approach. The Review of Financial Studies, Vol. 14, No. 1, pp The above two documents can be downloaded from Valuing American Options 1

3 American Options Definition: Contract between two parties giving the buyer the right to, say, purchase one unit of a security for the amount K at any time on or before maturity T. Recall that a European option, in contrast, can only be exercised at a fixed date General facts: An American option can only be exercised once The buyer of the option has the choice when to stop American options are more valuable than their European counterparts Valuing American Options 2

4 General Facts (cont d) Valuing an American option entails 1. finding the optimal exercise rule 2. computing the expected discounted payoff under this rule Exercise decision can only be based on price information up to the present moment filtration, stopping times The price of an American call option equals the price of the European call option. Hence, it is optimal to wait until the option expires. Valuing American Options 3

5 Problem Formulation Y = {Y (t) : 0 t T } with Y (t) representing the payoff from exercise at time t. Example: Y (t) = (K S(t)) + B = {B(t) : 0 t T } with B(t) = exp{ t 0 r u du} money market account and {r t : 0 t T } instantaneous short rate process U = {U(t) : 0 t T } price process If the option seller knew in advance which stopping time τ 0 the investor will use, then U(0) = E Q [ Y (τ0 ) B(τ 0 ) ], Y (t) = ( K S(t) ) + Valuing American Options 4

6 Problem Formulation (cont d) Since τ is not known, the option seller should prepare for the worst possible case, and charge the maximum value [ ] Y (τ) U(0) = sup E Q, τ T B(τ) where T are the stopping times taking values in [0, T ] Often, restriction is made to options that can be exercised only at a fixed set of dates t 1 < t 2 < < t m Restriction can be part of the option contract ( Bermudan Options ) Or the restriction can be regarded as an approximation to a contract allowing continuous exercise Valuing American Options 5

7 Main result Proposition. Suppose there is Q P and define Z = {Z(t) : 0 t T } by [ Y (τ) Z(t) = sup E Q τ T t,t B(τ) F t ] B(t). (1) Then Z(t)/B(t) is the smallest Q-supermartingale satisfying Z(t) Y (t). Moreover, the supremum in (1) is achieved by an optimal stopping time τ(t) that has the form τ(t) = inf { s t : Z(s) = Y (s) } (2) In other words, τ(t) maximises the right hand side of (1): E Q [ Y (τ(t)) B(τ(t)) ] [ Y (τ) F t = sup E Q τ T t,t B(τ) ] F t. Valuing American Options 6

8 Dynamic Programming Formulation Sketch of the proof: idea is to work backwards in time Explicit construction of Z(t) by means of dynamic programming: V (t) := Y (t), t = T { [ V (t + 1) ] } max Y (t), E Q F t B(t), t T 1 (3) B(t + 1) }{{} expected payoff from continuation V = {V (t) : 0 t T } is called snell envelope. It is the smallest supermartingale dominating Y and it follows that Z = V. Valuing American Options 7

9 Decision Rules Note: dynamic programming rules (3) focus on option values It is also convenient to view the pricing problem through stopping rules: at any exercise time, compare payoff from immediate exercise with the value of continuation. Exercise if the immediate payoff is higher Continuation value: value of holding rather than exercising the option: C(t i ) = E Q [ V (ti+1 ) B(t i+1 ) F ti ] B(t i ) (4) Note: estimating these conditional expectations is the main difficulty in pricing American options by simulation Valuing American Options 8

10 Regression-Based Methods: The LSM Algorithm Idea: Use regression methods to estimate the continuation values from simulated sample paths Each continuation value C(t i ) is the regression of the (discounted) option value V (t i+1 ) on the current state S(t i ) In practice: 1. approximate C(t i ) by a linear combination of known functions of the current state S(t i ): ( C(t i ) = α ij L j S(ti ) ), j=0 where α ij R and L j (x) are basis functions (e.g. Laguerre, Legendre, Hermite polynomials) Valuing American Options 9

11 2. use regression to estimate the coefficients α ij in this approximation. The coefficients α ij are estimated from pairs ( S(ti, ω), V (t i+1, ω) ) consisting of the value of the underlying at time t i and the corresponding option value at time t i+1 Remarks: - the accuracy depends on the choice of basis functions - obviously, a finite sum will have to do it: C(t i ) = M ( α ij L j S(ti ) ) - the coefficients α ij are determined by means of least-squares ˆα ij - The Longstaff-Schwartz LSM-algorithm is a fast and broadly applicable algorithm (beyond classical American put options) j=0 Valuing American Options 10

12 Pricing Algorithm (i) Simulate n independent paths ( S(t1, ω k ), S(t 2, ω k ),..., S(t m, ω k ) ), k = 1, 2,..., n under the risk neutral measure Q (ii) At terminal nodes, set ˆV (t m, ω k ) = Y (t m, ω k ) Valuing American Options 11

13 Pricing Algorithm (cont d) (iii) Apply backward induction: for i = m 1,..., 1 Given estimated values ˆV (t i+1, ω k ), use regression to calculate ˆα i1,..., ˆα im Set ˆV (t i, ω k ) = Y (t i, ω k ), Y (t i, ω k ) Ĉ(t i, ω k ), ˆV (t i+1, ω k ), Y (t i, ω k ) < Ĉ(t i, ω k ), with Ĉ(t i) = M j=0 ˆα ij L j ( S(ti ) ) (iv) Set ˆV (0) = 1 n n k=1 ˆV (t 1, ω k ) Valuing American Options 12

14 Numerical Example Ω = {ω 1,..., ω 8 }, K = 1.1 and S(t i, ω k ) as follows: t 0 = 0 t 1 = 1 t 2 = 2 t 3 = 3 ω ω ω ω ω ω ω ω Valuing American Options 13

15 Stock Price Evolution Valuing American Options 14