Lecture 7: Computation of Greeks

Transcription

1 Lecture 7: Computation of Greeks Ahmed Kebaier HEC, Paris

2 Outline 1 The log-likelihood approach

3 Motivation The pathwise method requires some restrictive regularity assumptions on the payoff function of the option price, at least one time differentiable. This is not the case of some particular payoffs such as the digital options with a payoff function h T = h(x x T ) where h(x) = 1 {x K}. The Greek of such option cannot be evaluated using the pathwise method. Note also that for the same arguments we cannot use the pathwise method to evaluate the Greek payoff Γ of a given vanilla Call option.

4 General idea Let us assume that the family of random variables (X (θ)) indexed by a parameter θ Θ (Θ an open set of R), admits positive density function p(θ, y).

5 General idea Let us assume that the family of random variables (X (θ)) indexed by a parameter θ Θ (Θ an open set of R), admits positive density function p(θ, y). f (θ) = Eφ(X (θ)) = φ(y)p(θ, y)dy. R d

6 General idea Let us assume that the family of random variables (X (θ)) indexed by a parameter θ Θ (Θ an open set of R), admits positive density function p(θ, y). f (θ) = Eφ(X (θ)) = φ(y)p(θ, y)dy. R d Theorem 1 Assume that the density function p(θ, y) taking values in r Θ R d satisfies i) θ p(θ, y) is differentiable on Θ almost every where

7 General idea Let us assume that the family of random variables (X (θ)) indexed by a parameter θ Θ (Θ an open set of R), admits positive density function p(θ, y). f (θ) = Eφ(X (θ)) = φ(y)p(θ, y)dy. R d Theorem 1 Assume that the density function p(θ, y) taking values in r Θ R d satisfies i) θ p(θ, y) is differentiable on Θ almost every where ii) g : R d R a measurable function such that R d φ(y)g(y)dy < and θ Θ θ p(θ, y) g(y),

8 General idea Let us assume that the family of random variables (X (θ)) indexed by a parameter θ Θ (Θ an open set of R), admits positive density function p(θ, y). f (θ) = Eφ(X (θ)) = φ(y)p(θ, y)dy. R d Theorem 1 Assume that the density function p(θ, y) taking values in r Θ R d satisfies i) θ p(θ, y) is differentiable on Θ almost every where ii) g : R d R a measurable function such that R φ(y)g(y)dy < and θ Θ d θ p(θ, y) g(y),then ( θ Θ, f (θ) = E φ(x (θ)) log p ) (θ, X (θ)) θ

9 Proof. According to assumptions i) and ii) we have f (θ) = θ R Eφ(X (θ)) = φ(y) p(θ, y)dy. d θ Rewriting the above expression we get ( f θ p(θ, y) (θ) = φ(y) p(θ, y)dy = E φ(x (θ)) R d p(θ, y) which completes the proof. ) p(θ, X (θ)), p(θ, X (θ)) θ

10 Black Scholes Delta Recall that the Black-Scholes model with parameters r, σ, T is given by the explicit solution ( S T = S 0 exp (r σ 2 /2)T + σ ) T G, where G N (0, 1). Consequently, for any given measurable function we get ( ( Ef (S T ) = Ef S 0 exp (r σ 2 /2)T + σ )) T G ( ( = f S 0 exp (r σ 2 /2)T + σ )) T y R 1 exp ( y 2 ) dy. 2π 2

11 Black Scholes Delta Using the following change of variable ( u = S 0 exp (r σ 2 /2)T + σ ) T y with du = σ ( T S 0 exp (r σ 2 /2)T + σ ) T y dy = uσ T dy and y = ζ(u) where ζ(u) = ( log(u/s 0 ) (r σ 2 /2)T ) /σ T

12 Black Scholes Delta Using the following change of variable ( u = S 0 exp (r σ 2 /2)T + σ ) T y with du = σ ( T S 0 exp (r σ 2 /2)T + σ ) T y dy = uσ T dy and y = ζ(u) where ζ(u) = ( log(u/s 0 ) (r σ 2 /2)T ) /σ T we get Ef (S T ) = R f (y) 1 uσ T 2π exp ( ζ(u)2 2 ) du.

13 Black Scholes Delta Therefore we deduce that the density of the random variable S T is given by ) 1 g(x) = ( xσ T 2π exp ζ(x)2. 2

14 Black Scholes Delta Therefore we deduce that the density of the random variable S T is given by ) 1 g(x) = ( xσ T 2π exp ζ(x)2. 2 Then log g(x) S 0 = g S 0 (x) = log(x/s 0) (r σ 2 /2)T g(x) S 0 σ 2. T

15 Black Scholes Delta Hence, according to the above theorem = e rt E(S T K) + S 0 ( = e rt log(s T /S 0 ) (r σ 2 ) /2)T E (S T K) + S 0 σ 2 T ( ) = e rt G E (S T K) + S 0 σ. T

16 Example: Path-dependent deltas Let us consider the case of an Asian option. Since the associated payoff involves the vector (S t1, S t2,, S tn )

17 Example: Path-dependent deltas Let us consider the case of an Asian option. Since the associated payoff involves the vector (S t1, S t2,, S tn ) we need to make explicit the joint density of this vector if we aim to apply the log-likelihood ratio method for the computation of sensitivities associated to the Asian option price.

18 Example: Path-dependent deltas Let us consider the case of an Asian option. Since the associated payoff involves the vector (S t1, S t2,, S tn ) we need to make explicit the joint density of this vector if we aim to apply the log-likelihood ratio method for the computation of sensitivities associated to the Asian option price. Using the Markovian property of the Brownian motion we can rewrite the density associated to the above vector as follows g(x 1,, x n ) = g 1 (x 1 S 0 )g 2 (x 2 x 1 ) g n (x n x n 1 ), with g j (x j x j 1 ) = 1 x i σ t j t j 1 ϕ(ζ j (x j x j 1 )), ϕ(u) = 1 2π exp ) ( u2 2 where

19 Example: Path-dependent deltas ζ j (x j x j 1 ) = log(x j/x j 1 ) (r σ 2 /2)(t j t j 1 ) σ t j t j 1.

20 Example: Path-dependent deltas ζ j (x j x j 1 ) = log(x j/x j 1 ) (r σ 2 /2)(t j t j 1 ) σ t j t j 1. Consequently, log g S 0 (S t1,, S tn ) = log g 1 S 0 (S t1 S 0 ) = G 1 S 0 σ t 1, where G 1 is the same Gaussian used for the simulation of S t1.

21 Example: Path-dependent deltas ζ j (x j x j 1 ) = log(x j/x j 1 ) (r σ 2 /2)(t j t j 1 ) σ t j t j 1. Consequently, log g S 0 (S t1,, S tn ) = log g 1 S 0 (S t1 S 0 ) = G 1 S 0 σ t 1, where G 1 is the same Gaussian used for the simulation of S t1. Finally, the value of the delta Asian call option using the Log-likelihood ratio method is given by ( ) = e rt E ( 1 n G 1 S ti K) + n S 0 σ. t 1 i=1