On validity of a singular perturbation expansion of European options - PowerPoint PPT Presentation

On validity of a singular perturbation expansion of European options and implied volatility Masaaki Fukasawa CSFI, Osaka Univ. 1

[Stochastic volatility model: Notation] Suppose that S t : an asset price, Z t = log( S t ) : log price, r : risk free rate, W , W ′ : Brownian motions, � W , W ′ � t = ρ t , dZ t = ( r − ϕ ( X t ) 2 / 2) dt + ϕ ( X t ) dW t , dX t = b ( X t ) dt + c ( X t ) dW ′ t under risk neutral probability P . The option price for payoff h is given as P ( t , z , x ) = E [ e − r ( T − t ) h ( S T ) | Z t = z , X t = x ] . 2

In a stochastic volatility model, stylized facts such as • the (Black-Scholes) implied volatility skew/smile, • time varying volatility, • leverage effect, • mean-reverting volatility are explained. It is a natural extension of the Black-Scholes model, including the Heston model, SABR model. Unfortunately, no simple analytic formula for the option prices is available in general. 3

[Singular perturbation expansion] Fouque, Papanicolaou and Sircar (2000): consider dZ t = ( r − ϕ ( X t ) 2 / 2) + ϕ ( X t ) dW t , dX t = − aX t − b dt + Λ ( X t ) √ ǫ dt + c √ ǫ dW ′ t ǫ where ǫ is small. They obtained, by a formal calculation, that P ( t , z , x ) = P 0 ( t , z ) + √ ǫ P 1 ( t , z ) + O ( ǫ ) , where P 0 is the Black-Scholes price. 4

The validity of the singular perturbation expansion is proved by • Fouque, Papanicolaou, Sircar and Solna (2003) • Conlon and Sullivan (2005) - higher order expansion for call option • Khasminskii and Yin (2005) - ergodic diffusion on compact set and smooth payoff • Fukasawa (2008) - general ergodic diffusion and payoff - Edgeworth expansion for ergodic diffusions 5

[Extended fast mean-reverting model] More generally, consider dZ t = ( r − ϕ ( X t ) 2 / 2) dt + ϕ ( X t ) dW t , dX t = b ( X t ) + Λ ǫ ( X t ) dt + c ( X t ) √ ǫ dW ′ t , ǫ where Λ ǫ → 0 (in a weak sense) as ǫ → 0 . W ′ = ǫ − 1 / 2 W ′ Intuition: put ˆ X t = X ǫ t and ˆ ǫ t . Let Λ ǫ = 0 for brevity. Then X t /ǫ ) 2 / 2) dt + ϕ ( ˆ dZ t = ( r − ϕ ( ˆ X t /ǫ ) dW t , W ′ d ˆ X t = b ( ˆ X t ) dt + c ( ˆ X t ) d ˆ t . Notice that the law of ˆ X does not depend on ǫ . 6

Intuition (continued): putting W ′ = ǫ − 1 / 2 W ′ W = ǫ − 1 / 2 W ǫ t , ˆ ˆ ˆ X t = X ǫ t , ǫ t , we have ∫ t /ǫ ∫ t /ǫ X s ) 2 / 2) ds + √ ǫ ( r − ϕ ( ˆ ϕ ( ˆ X s ) d ˆ Z t = Z 0 + ǫ W s , 0 0 where W ′ d ˆ X t = b ( ˆ X t ) dt + c ( ˆ X t ) d ˆ t . W ′ ) does not depend on ǫ , so that Note that the law of ( ˆ X , ˆ W , ˆ ∫ t /ǫ X s ) 2 ds → t π [ ϕ 2 ] as ǫ → 0 ϕ ( ˆ � Z � t = ǫ 0 for the ergodic distribution π of ˆ X . Consequently, by the martingale CLT, √ Z t ⇒ Z 0 + ( r − π [ ϕ 2 ] / 2) t + π [ ϕ 2 ] W t . 7

[Edgeworth expansion for ergodic diffusions: Fukasawa(2008, PTRF)] For a diffusion X with scale function S : ∫ x { } ∫ b ( y ) dx S ′ ( x ) = s ( x ) , s ( x ) = exp − 2 c ( y ) 2 dy , M = c ( x ) 2 s ( x ) < ∞ x 0 we have ∫ T √ I T [ ϕ ] = 1 T ( I T [ ϕ ] − π [ ϕ ]) ⇒ N (0 , σ 2 ) , ϕ ( X t ) dt → π [ ϕ ] , T 0 where π ( dx ) / dx = 1 / Mc ( x ) 2 s ( x ) . As a refinement of this CLT, we can show √ E [ H ( T ( A ( I T [ ϕ ]) − A ( π [ ϕ ])))] ∫ ∫ H ( w ) N ( dw ) + T − 1 / 2 H ( w ) p ( w ) N ( dw ) + O ( T − 1 ) = under a moment condition and a smoothness condition. 8

[First result: Edgeworth approach] Recall that dZ t = ( r − ϕ ( X t ) 2 / 2) dt + ϕ ( X t ) dW t , dX t = ǫ − 1 b ǫ ( X t ) dt + ǫ − 1 / 2 c ( X t ) dW ′ t , where b ǫ = b + Λ ǫ . • the state space of X is supposed to be R . • assume that sup ǫ ≥ 0 | b ǫ | , c , 1 / c , ϕ are locally bounded. • the option price for payoff h is given as P ( t , z , x ) = E [ e − r ( T − t ) h ( S T ) | Z t = z , X t = x ] . 9

∫ x dy ϕ ( y ) / c ( y ) , [Assumptions] L ǫ ; the generator of ˆ X , ˆ X t = X ǫ t ψ ( x ) = • L ǫ ψ → L 0 ψ on an interval I and L 0 ψ is continuous on I , • L 0 ψ is not constant on I , • there exist κ ± > 0 such that 1 + ϕ ( x ) 2 1 + ϕ ( x ) 2 γ ± > 2 κ ± , sup e κ + x c ( x ) 2 < ∞ , sup e − κ − x c ( x ) 2 < ∞ , x ≥ 0 x ≤ 0 where b ǫ ( x ) b ǫ ( x ) γ + = − lim sup c ( x ) 2 , γ − = lim inf c ( x ) 2 . x →−∞ ,ǫ → 0 x →∞ ,ǫ → 0 10

[Theorem] If h is bounded, then for every ( t , z , x ) , P ( t , z , x ) = P 0 ( t , z ) + √ ǫ P 1 ( t , z ) + O ( ǫ ) , where P 0 is the Black-Scholes price with volatility σ = √ π ǫ [ ϕ 2 ] ; h (exp( z + ( r − σ 2 / 2) τ + σ √ τ w )) N ( dw ) , ∫ P 0 ( t , z ) = e − r τ τ = T − t , and h (exp( z + ( r − σ 2 / 2) τ + σ √ τ w )) ∫ P 1 ( t , z ) = e − r τ  1 − w 2 + w 3 − 3 w   × δ    N ( dw ) ,  σ √ τ  σ 2   ∫ ∞ ∫ x ( ϕ ( y ) 2 − σ 2 ) π ǫ ( dy ) ϕ ( x ) δ = − ρ c ( x ) dx . −∞ −∞ 11

[Corollary] The put option price admits P ( t , z , x ) = P 0 ( t , z ) − δ √ ǫ σ 2 e z φ ( d 1 ) d 2 + O ( ǫ ) , where P 0 is the Black-Scholes put price, and ( d 1 , d 2 ) is the usual one with σ . The Black-Scholes implied volatility IV admits δ √ ǫ d 2 IV = σ − + O ( ǫ ) . σ 2 √ T − t Note that log( K / S t ) IV = c 1 + c 2 + O ( ǫ ) T − t for strike K , where c 1 and c 2 are constants of O ( √ ǫ ) and O (1) respectively. 12

[Example: the Heston model] √ dZ t = ( r − V t / 2) dt + V t dW t , dt + c √ V t dV t = − aV t − b √ ǫ dW ′ t ǫ with � W , W ′ � t = ρ t . We have σ 2 = b / a and δ = ρ bc / 2 a 2 , so that a − ρ c √ ǫ d 2 √ b IV = √ + O ( ǫ ) . 2 a T − t 13

[Alternative conditions for a diffusion being not exponentially mixing] • L ǫ ψ → L 0 ψ on an interval I and L 0 ψ is continuous on I , • L 0 ψ is not constant on I , • there exist p ≥ 0 such that 1 + ϕ ( x ) 2 | x | p − 2 c ( x ) 2 < ∞ , 2 γ + 1 > 4 p , lim sup | x |→∞ where xb ǫ ( x ) γ = − lim sup c ( x ) 2 . x →∞ ,ǫ → 0 The same conclusion as before is obtained under these conditions. 14

[What is new?] • a probabilistic approach. • the Heston model (or, a model with irregular coefficients) is incorporated. • digital options (or, irregular payoff functions) are incorporated. • volatility process X with slow decay of the autocorrelation function is in- corporated. For example, X t dX t = − α + β dW t . 1 + X 2 t 15

[Second result: martingale approach] Let us consider a more general model for Z = log( S ) , Z = Z 0 + A + M + g · W + h · C , where M is a local martingale, g and h are adapted cag processes, C is a time-changed compound Poisson process with intensity Λ , W is a std BM in- dependent ( M , g , h , Λ ) , ∫ t ∫ t ∫ A t = rt − 1 2 � M � t − 1 ( e h s z − 1) ν ( dz ) Λ ( ds ) . | g s | 2 ds − 2 0 0 R We consider the following perturbation: M = M n , g = g n , h = h n , Λ = Λ n , ν = ν n with � Z � T / Σ n → T for a sequence Σ n as n → ∞ so that the martingale CLT is applied. Hence, the Black-Scholes model is recovered as n → ∞ . 16

In this framework, putting r n = ǫ 2 n → 0 , • slowly varying volatility: (Sircar, Lee, ...) dX n t = ( r n b 1 ( X n t ) + r 2 n b 2 ( X n t )) dt + r n c ( X n t ) dW t , • small volatility of volatility: (Hull and White, Lewis, S ø rensen et.al., ..) dX n t = b ( X n t ) dt + r n c ( X n t ) dW t , • fast mean-reverting: (Fouque et.al., Khasminskii et.al., Fukasawa ) t = r − 2 t ) dt + r − 1 t ) dt + r − 1 dX n n b ( X n n λ ( X n n c ( X n t ) dW t , are included. 17

[Theorem] Under several conditions, we have for any bounded function H , ∫ √ E [ H ( Z T )] = H ( rT − Σ n / 2 + Σ n z ) φ n ( z ) dz + o ( r n ) R with a sequence r n → 0 as n → ∞ , ( ) 1 + r n γ + 1 { { β ( z 2 − 1) + ( z 3 − 3 z ) φ n ( z ) = φ ( z ) 3 α 2 √ Σ n {( ) }}} z + γ ( z 2 − 1) √ − Σ n β + 3 α . Here α, β, γ are functions of T . [Proof] Apply Yoshida’s formula for martingale expansion. 18

[Corollary] The put option price e − rT E [( K − S T ) + | S 0 = S ] is expanded as √ √ { } P 0 ( σ n ) + 1 β − γ d 2 − 1 2 r n σ n TS φ ( d 1 ) 3 α ( d 2 − σ n T ) + o ( r n ) , where P 0 ( σ n ) is the Black-Scholes put price with volatility σ n and σ 2 n T = Σ n The Black-Scholes implied volatility IV is expanded as √ { { }} 1 + r n β − γ d 2 − 1 σ n 3 α ( d 2 − σ n T ) + o ( r n ) . 2 Note that IV = c 1 log( K / S 0 ) + c 2 + o ( r n ) for strike K , where c 1 and c 2 are functions of T , of O ( r n ) and O (1) respectively. 19

[What is new?] • a probabilistic approach. • various perturbations are treated in a unified way. • non-Markovian model is incorporated. • jump is also incorporated. • multi-dimensional volatility process is incorporated. Remark: the Edgeworth approach is more useful as far as considering fast mean reverting volatility with one-dimensional diffusion. 20