Short-Maturity Asymptotics for Fast Mean-Reverting Stochastic - PowerPoint PPT Presentation

Short-Maturity Asymptotics for Fast Mean-Reverting Stochastic Volatility Models Jean-Pierre Fouque University of California Santa Barbara Joint work with Jin Feng (Kansas Univ.) and Martin Forde (Dublin City Univ.) Conference on small time asymptotics, perturbation theory and heat kernel methods in mathematical finance February 10-12, 2009 Wolfgang Pauli Institute (WPI) Vienna, Austria 1

Heston Model In this talk, we will mainly deal with the Heston model. Under the risk-neutral measure it is: rS t dt + σ t S t dW 1 dS t = t , � σ t = Y t , � Y t dW 2 dY t = κ ( θ − Y t ) dt + ν t , where W 1 , W 2 are two standard Brownian motions with covariation d � W 1 , W 2 � t = ρdt , where | ρ | < 1. We assume that 2 κθ>ν 2 , ν, κ, θ, Y 0 = y > 0, so that the square-root (or CIR) process ( Y t ) stays positive at all times. 2

Implied Volatility Black-Scholes implied volatility: Call Heston ( S, y ; K, T ) = Call BS ( S ; K, T ; σ imp ( S, y ; K, T )) In what follows we use log-moneyness : x = log( K/S 0 ) and the notation σ imp ( x, y, T ) where T is time-to-maturity . 3

Implied Volatility at Short Maturity Large deviation result. Geodesic distance: yd 2 x + 2 ρνyd x d y + ν 2 yd 2 y = 1 d ( x = 0 , y ) = 0 d ( x, y ) > 0 for x > 0 At short maturity : x σ imp ( x, y, T = 0) = d ( x, y ) where d (which does not depend on κ ) is computed numerically. Avellaneda-BoyerOlson-Busca-Friz (2003) Berestycki-Busca-Florent (2004) 4

Fast Mean-Reverting Heston Model By fast mean-reverting stochastic volatility, we mean that the rate of mean reversion κ is large . In order to ensure that volatility is not “dying” or “exploding” we also impose that the vol-vol parameter ν is large of the order of √ κ . In order to achieve this scaling, we introduce a small parameter 0 < ǫ ≪ 1, and we replace ( κ, ν ) by ( κ/ε 2 , ν/ε ) so that the model becomes: � Y t dW 1 dS t = rS t dt + S t t , ε 2 ( θ − Y t ) dt + ν κ � Y t dW 2 dY t = t . ε The small quantity ε 2 represents the intrinsic time scale of the volatility process ( Y t ), or, in other words, its de-correlation time. � 2 is equivalent to 2 κθ > ν 2 and Observe that the condition 2 � κ θ > � ν � ε 2 ε therefore independent of ε . 5

Asymptotics at Fixed Maturities Fouque-Papanicolaou-Sircar (2000) for general FMR SV models: “implied vol is affine in LMMR” � x � + O ( ε 2 ) , σ imp ( x, y, T ) = ¯ σ + ερ C T where σ is the effective volatility � σ 2 = σ ( y ) 2 Φ Y ( dy ) , Φ Y being the invariant distribution of the process Y . In the case of Heston , σ ( y ) = √ y , Φ Y = Γ( θ, θν 2 2 κ ), and σ 2 = θ . C is a constant which depends on the model parameters. 6

Prices and Pricing PDE’s E ⋆ � � P ε ( t, x, y ) = I e − r ( T − t ) h ( S ε T ) | S ε t = x, Y ε t = y ∂P ε 2 yx 2 ∂ 2 P ε ε xy ∂ 2 P ε ∂x∂y + ν 2 2 ε 2 y ∂ 2 P ε ∂t + 1 ∂x 2 + ρν ∂y 2 x∂P ε ε 2 ( m − y ) ∂P ε � � + 1 ∂x − P ε + r = 0 ∂y to be solved for t < T with the terminal condition P ε ( T, x, y ) = h ( x ) 7

Operator Notation � 1 ε 2 L 0 + 1 � P ε = 0 ε L 1 + L 2 with 2 ν 2 y ∂ 2 1 ∂y 2 + ( m − y ) ∂ L 0 = = L CIR ∂y ρνxy ∂ 2 L 1 = ∂x∂y 2 yx 2 ∂ 2 ∂t + 1 ∂ � x ∂ � = L BS ( √ y ) L 2 = ∂x 2 + r ∂x − · 8

Formal Expansion Expand: P ε = P 0 + εP 1 + ε 2 P 2 + ε 3 P 3 + · · · Compute: � 1 ε 2 L 0 + 1 � � P 0 + εP 1 + ε 2 P 2 + ε 3 P 3 + · · · � ε L 1 + L 2 = 0 Group the terms by powers of ε : 1 1 ε 2 L 0 P 0 + ε ( L 0 P 1 + L 1 P 0 ) + ( L 0 P 2 + L 1 P 1 + L 2 P 0 ) + ε ( L 0 P 3 + L 1 P 2 + L 2 P 1 ) + · · · = 0 9

Diverging terms • Order 1 /ε 2 : L 0 P 0 = 0 L 0 = L CIR , acting on y = ⇒ P 0 = P 0 ( t, x ) with P 0 ( T, x ) = h ( x ) • Order 1 /ε : L 0 P 1 + L 1 P 0 = 0 L 1 takes derivatives w.r.t. y = ⇒ L 1 P 0 = 0 = ⇒ L 0 P 1 = 0 As for P 0 : P 1 = P 1 ( t, x ) P 1 ( T, x ) = 0 with • Important observation: P 0 + εP 1 does not depend on y 10

Zero Order Term L 0 P 2 + ( L 1 P 1 = 0) + L 2 P 0 = 0 Poisson equation in P 2 with respect to L 0 and the variable y . Solution: P 2 = ( −L 0 ) − 1 ( L 2 P 0 ) Only if L 2 P 0 is centered with respect to the invariant distribution of Y . 11

Poisson Equations L 0 χ + g = 0 Expectations w.r.t. the invariant distribution of the CIR process: � � χ ( y )( L ⋆ � g � = −�L 0 χ � = − ( L 0 χ ( y ))Φ( y ) dy = 0 Φ( y )) dy = 0 t → + ∞ I lim E { g ( Y t ) | Y 0 = y } = � g � = 0 (exponentially fast) � + ∞ χ ( y ) = I E { g ( Y t ) | Y 0 = y } dt 0 checked by applying L 0 12

Leading Order Term Centering: �L 2 P 0 � = �L 2 � P 0 = 0 � ∂ 2 yx 2 ∂ 2 ∂t + 1 � x ∂ �� �L 2 � = ∂x 2 + r ∂x − · 2 � y � x 2 ∂ 2 ∂t + 1 ∂ � x ∂ � = ∂x 2 + r ∂x − · σ 2 = � y � = θ Effective volatility: ¯ The zero order term P 0 ( t, x ) is the solution of the Black-Scholes equation L BS (¯ σ ) P 0 = 0 with the terminal condition P 0 ( T, x ) = h ( x ) 13

Back to P 2 ( t, x, y ) The centering condition �L 2 P 0 � = 0 being satisfied: x 2 ∂ 2 P 0 L 2 P 0 = L 2 P 0 − �L 2 P 0 � = 1 σ 2 � � y − ¯ ∂x 2 2 2 L 0 φ ( y ) x 2 ∂ 2 P 0 = 1 ∂x 2 for φ a solution of the Poisson equation: L 0 φ = y − � y � Then 2 ( φ ( y ) + c ( t, x )) x 2 ∂ 2 P 0 ( L 2 P 0 ) = − 1 P 2 ( t, x, y ) = −L − 1 0 ∂x 2 14

Terms of order ε Poisson equation in P 3 : L 0 P 3 + L 1 P 2 + L 2 P 1 = 0 Centering condition: �L 1 P 2 + L 2 P 1 � = 0 Equation for P 1 : ( φ ( y ) + c ( t, x )) x 2 ∂ 2 P 0 �L 2 P 1 � = −�L 1 P 2 � = 1 � � �� L 1 ∂x 2 2 P 1 independent of y and L 1 takes derivatives w.r.t. y x 2 ∂ 2 P 0 � � σ ) P 1 = 1 = ⇒ L BS (¯ 2 �L 1 φ ( y ) � ∂x 2 with P 1 ( T, x ) = 0 15

The correction P ε 1 ( t, x ) = εP 1 ( t, x ) ρxy ∂ 2 x 2 ∂ 2 P 0 �� � � � � 1 − εν σ ) P ε L BS (¯ φ ( y ) = 0 ∂x 2 2 ∂x∂y x 2 ∂ 2 P BS � � 3 x ∂ σ ) P ε 1 + V ε L BS (¯ = 0 ∂x 2 ∂x BS equation with source and zero terminal condition with the small parameter V ε 3 given by: − ερν V ε � yφ ′ ( y ) � = 3 2 16

Explicit Formula for the Corrected Price x 2 ∂ 2 P BS 3 x ∂ � � P ε ( T − t ) V ε = 1 ∂x 2 ∂x where V ε 3 is a small number of order ε . The corrected price is given explicitly by x 2 ∂ 2 P BS � � 3 x ∂ P 0 + ( T − t ) V ε ∂x 2 ∂x √ where P 0 is the Black-Scholes price with constant volatility ¯ σ = θ 17

Comments • The small constant V ε 3 is a complex functions of the original model parameters ( θ, ν, ρ, ε ) σ, V ε • Only (¯ 3 ) are needed to compute the corrected price • Probabilistic representation of ( P 0 + P ε 1 )( t, x ): � T � � ¯ e − r ( T − t ) h ( ¯ e − r ( s − t ) H ( s, ¯ X s ) ds | ¯ I E X T ) + X t = x t • Put-Call Parity is preserved at the order O ( ε ) • The V ε 3 term is the skew effect ⇒ V ε ρ = 0 = 3 = 0 18

Corrected Call Option Prices h ( x ) = ( x − K ) + and P 0 ( t, x ) = C BS ( t, x ; K, T ; ¯ σ ) Compute the Delta , the Gamma and the Delta-Gamma Deduce the source x 2 ∂ 2 P BS 3 x ∂ � � H = V ε ∂x 2 ∂x and the correction 1 ( t, x ) = ( T − t ) H ( t, x ) = xe − d 2 1 / 2 � � d 1 P ε √ − V 3 σ σ 2 π 19

Expansion of Implied Volatilities Recall C BS ( t, x ; K, T ; I ) = C observed Expand I = ¯ σ + εI 1 + · · · Deduce for given ( K, T ): ∂C BS σ ) + · · · = P 0 ( t, x ) + P ε C BS ( t, x ; ¯ σ ) + εI 1 ( t, x ; ¯ 1 ( t, x ) + · · · ∂σ σ )] − 1 εI 1 = P ε = ⇒ 1 ( t, x ) [ Vega (¯ √ 1 / 2 √ Compute the Vega = ∂C BS /∂σ = xe − d 2 T − t/ 2 π and deduce 20

Calibration Formulas The implied volatility is an affine function of the LMMR : log-moneyness-to-maturity-ratio = log( K/x ) / ( T − t ) I = a [ LMMR ] + b + O ( ε 2 ) with V ε 3 a = σ 3 ¯ σ − V ε � r − 1 � 3 2 σ 2 b = σ 3 or for calibration purpose : b + a ( r − b 2 σ = 2 ) V ε ab 3 = 3 21

Fast Mean-Reverting Short-Maturity Scaling We combine fast mean reversion and short maturity by considering maturities of order ε , that is short maturities but long compared to the intrinsic volatility time scale ε 2 . We set T = ε t where t > 0. A typical situation: T ∼ 15 days and ε 2 ∼ 2 days. t �→ ǫt gives (in distribution) the rescaled process : ǫY ǫ,t dW 1 � dS ǫ,t = ǫrS ǫ,t dt + S ǫ,t t , κ ǫ ( θ − Y ǫ,t ) dt + ν Y ǫ,t dW 2 � √ ǫ dY ǫ,t = t , preserving the constant correlation ρ . The log-price X ǫ,t = log S ǫ,t is given by � t � t Y ǫ,s ds + √ ε x 0 + ǫrt − ǫ Y ǫ,s dW 1 � X ǫ,t = s . 2 0 0 22

Asymptotic Results • Large deviation principle • Option pricing • Implied volatilities Proofs based on explicit computation of moment generating functions . 23