Derivative pricing in fractional SABR model Tai-Ho Wang Conference - PowerPoint PPT Presentation

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Derivative pricing in fractional SABR model Tai-Ho Wang Conference Honoring Jim Gatheral’s 60th Birthday Courant Institute of Mathematical Sciences New York University, October 13, 2017

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Outline Quick review on the SABR model and the SABR formula Lognormal fractional SABR (fSABR) model A bridge representation for probability density of lognormal fSABR Small time approximations of option premium and implied volatility in lognormal fSABR framework Heuristic sample path large deviation principle Target volatility option (TVOs) pricing in lognormal fSABR Decomposition formula Approximations of the price of a TV call Conclusion

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Stochastic αβρ (SABR) model Stochastic αβρ (SABR) model was suggested and investigated by Hagan-Lesniewski-Woodward as dS t = S β t α t ( ρ dB t + ¯ ρ dW t ) , S 0 = s ; d α t = να t dB t , α 0 = α where B t and W t are independent Brownian motions, � 1 − ρ 2 . ρ = ¯ SABR model is market standard for quoting cap and swaption volatilities using the SABR formula for implied volatility. Nowadays also used in FX and equity markets. β = 0 is referred to as normal SABR β = 1 is referred to as lognormal SABR

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR SABR formula The SABR formula is a small time asymptotic expansion up to first order for the implied volatilities of call/put option induced by the SABR model. σ BS ( K , τ ) = ν log( s / K ) { 1 + O ( τ ) } D ( ζ ) as the time to expiry τ approaches 0. D and ζ are defined respectively as �� 1 − 2 ρζ + ζ 2 + ζ − ρ � D ( ζ ) = log 1 − ρ and  s 1 − β − K 1 − β ν if β � = 1;  α 1 − β  ζ = � s ν �  α log if β = 1 .  K

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR SABR formula - zeroth order The zeroth order SABR formula is obtained by matching the exponents − (log s 0 − log K )2 e − d 2 ∗ ( s 0 ,α 0) 2 σ 2 BS T ≈ C ( K , T ) = C BS ( K , T ) ≈ e 2 T thus, σ BS ( K , T ) ≈ | log s 0 − log K | . d ∗ ( s 0 , α 0 ) where d ∗ is the minimal distance from the initial point ( s 0 , α 0 ) to the half plane { ( s , α ) : s ≥ K } .

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Why fractional process? Gatheral-Jaisson-Rosenbaum observed from empirical data that Log-volatility behaves as a fractional Brownian Motion with Hurst exponent H of order 0.1 at any reasonable time scale. Indeed, they fitted the empirical q th moments m ( q , ∆) in various lags ∆ to E [ | log σ t +∆ − log σ t | q ] = K q ∆ ζ q proxied by daily realized variance estimates. K q denotes the q th moment of standard normal. At-the-money volatility skew is well approximated by a power law function of time to expiry

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Gatheral-Jaisson-Rosenbaum Log-volatility behaves as a fractional Brownian Motion with Hurst exponent H of order 0.1 at any reasonable time scale

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Gatheral-Jaisson-Rosenbaum Log-log plot of m ( q , ∆) versus ∆ for various q .

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Gatheral-Jaisson-Rosenbaum � is well � � d � � At-the-money volatility skew ψ ( τ ) = k =0 σ BS ( k , τ ) � dk approximated by a power law function of time to expiry τ

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Fractional volatility process The observations suggest the following model for instantaneous volatility σ t = σ 0 e ν B H t , where B H is a fractional Brownian motion with Hurst exponent H . As stationarity of σ t is concerned, GJR suggested the model for instantaenous volatility as σ t = σ 0 e X t where dX t = α ( m − X t ) dt + ν dB H t is a fractional Ornstein-Uhlenbeck process. Again, drift term plays no role in large deviation regime.

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Review: fractional Brownian motion A mean-zero Gaussian process B H t is called a fractional Brownian motion with Hurst exponent H ∈ [0 , 1] if its autocovariance function R ( t , s ), for t , s > 0, satisfies = 1 � � � t 2 H + s 2 H − | t − s | 2 H � B H t B H R ( t , s ) := E . s 2 B H is self-similar, indeed, B H d = a H B H t for a > 0 at B H has stationary increments B H t is a standard Brownian motion when H = 1 2 t is neither a semimartingale nor Markovian unless H = 1 B H 2 B H t is H¨ older of order β for any β < H almost surely

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Lognormal fSABR model Consider the following lognormal fSABR model dS t = α t ( ρ dB t + ¯ ρ dW t ) , S t α t = α 0 e ν B H t , where B t and W t are independent Brownian motions, � 1 − ρ 2 . B H ρ = ¯ t is a fractional Brownian motion with Hurst exponent H driven by B t : � t B H t = K H ( t , s ) dB s . 0 K H is the Molchan-Golosov kernel. Goal: to obtain an easy to access expression for the joint density of ( S t , α t ).

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Slightly more explicit form Defining the new variables X t = log S t and Y t = α t , we may rewrite the lognormal fSABR model in a slightly more explicit form as � t � t ρ dW s ) − Y 2 e ν B H e 2 ν B H s ( ρ dB s + ¯ 0 s ds , X t − X 0 = Y 0 2 0 0 Y t = Y 0 e ν B H t . We derive a bridge representation for the joint density of ( X t , Y t ) in a “Fourier space”.

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Bridge representation for joint density The joint density of ( X t , Y t ) has the following bridge representation p ( t , x t , y t | x 0 , y 0 ) η 2 t e − 2 πν 2 t 2 H × 1 2 ν 2 t 2 H = √ 2 π × y t y 2 � � � � t � � 0 y 0 e ν BH 0 ρ 2 y 2 � − ρ s dB s + ξ i 2 v t ¯ 0 vt � ξ 2 e i ( x t − x 0 ) ξ E e − � ν B H e t = η t d ξ, 2 � � where i = √− 1, v t = � t 0 e 2 ν B H s ds and η t = log y t y 0 .

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Bridge representation in uncorrelated case The bridge representation for the joint density of ( X t , Y t ) reads simpler when ρ = 0: p ( t , x t , y t | x 0 , y 0 ) η 2 t e − 2 πν 2 t 2 H × 1 � 2 ν 2 t 2 H � e − 1 � � 2 ( ξ − i ) ξ y 2 e i ( x t − x 0 ) ξ E 0 v t � ν B H = √ t = η t d ξ, � 2 π y t where i = √− 1, v t = � t 0 e 2 ν B H s ds and η t = log y t y 0 .

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR McKean kernel The McKean kernel p H 2 ( t , x t , y t | x 0 , y 0 ) reads √ � ∞ ξ e − ξ 2 / 2 t 2 e − t / 8 √ cosh ξ − cosh d d ξ, p H 2 ( t , x t , y t | x 0 , y 0 ) = (2 π t ) 3 / 2 d where d = d ( x t , y t ; x 0 , y 0 ) is the geodesic distance from ( x t , y t ) to ( x 0 , y 0 ). Note that the McKean kernel is a density with respect to the 1 Riemannian volume form t dx t dy t . y 2 The bridge representation can be regarded as a generalization of the McKean kernel. Indeed, in the case where H = 1 2 , ν = 1 and ρ = 0, Ikeda-Matsumoto showed how to recover the McKean kernel.

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Expanding around b s We expand the conditional expectation in the bridge representation ·| ν B H � � around the deterministic path b s . Let E η t [ · ] = E t = η t . First, define the deterministic path b s by � � e 2 ν B H b s = log E η t . s Indeed, b s = log E η t [ e 2 ν B H s ] = 2 ν E η t [ B H s ] + 2 ν 2 var η t [ B H s ] u 2 H − R 2 (1 , u ) 2 R (1 , u ) η t + 2 ν 2 t 2 H � � = , where u = s � B H t B H � t and R ( t , s ) = E . s � � Note that e b s = E η t e 2 ν B H . In other words, e b s is an s unbiased estimator of e 2 ν B H s conditioned on ν B H t = η t .

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Now expand the conditional expectation in the bridge representation around the deterministic path b s as � � t � 0 e 2 ν BH e − 1 0 y 2 2 ( ξ − i ) ξ s ds E η t � t � � e 2 ν BH � � � t e − 1 0 y 2 s − e bs e − 1 0 y 2 0 e bs ds E η t 2 ( ξ − i ) ξ ds 2 ( ξ − i ) ξ = 0 � t 0 e bs ds × { 1 + o (1) } . e − 1 0 y 2 2 ( ξ − i ) ξ ≈

SABR Bridge representation Small time approximations Heuristic LDP TVO pricing in fSABR Substituting the last expansion into bridge representation we obtain the following expansion (in the Fourier space) in terms of the H k functions as p ( t , x t , y t | x 0 , y 0 ) η 2 1 t 2 πν 2 t 2 H e − 2 ν 2 t 2 H × ≈ √ y t 1 � e i ( x t − x 0 ) ξ e − 1 v t { 1 + o (1) } d ξ, 2 ( ξ − i ) ξ ˆ 2 π � t 0 y 2 0 e b s ds . where ˆ v t =