Probability density of lognormal fractional SABR model

Transcription

1 Probability density of lognormal fractional SABR model Tai-Ho Wang IAQF/Thalesians Seminar Series New York, January 24, 2017 (joint work with Jiro Akahori and Xiaoming Song)

2 Outline Review of SABR model and SABR formula Lognormal fractional SABR (fsabr) model A bridge representation for probability density of lognormal fsabr Edgeworth type of expansion Heuristic derivation of sample path large deviation principle from path-integral perspective Approximations of implied volatility in small time Conclusion

3 Stochastic αβρ (SABR) model Stochastic αβρ (SABR) model was suggested and investigated by Hagan-Lesniewski-Woodward as ds t = S β t α t (ρdb t + ρdw t ), S 0 = s; dα t = να t db t, α 0 = α where B t and W t are independent Brownian motions, ρ = 1 ρ 2. 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

4 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) D(ζ) {1 + O(τ)} as the time to expiry τ approaches 0. D and ζ are defined respectively as ( ) 1 2ρζ + ζ D(ζ) = log 2 + ζ ρ 1 ρ and ζ = ν s 1 β K 1 β α 1 β if β 1; ν α log ( ) s K if β = 1.

5 Plot of SABR formula for implied volatilities α = , ρ = , ν = , τ = 1. Implied Volatility s = s = s = backbone strike

6 Brief review on derivation of SABR formula Take ν = 1 for simplicity. Change of variable x = 1 ρ ( ) ds s 0 s β ρα = y = α 1 ρ s 1 β s 1 β 0 1 β ρ ρ α if β 1; ( ) 1 ρ log s s 0 ρ ρ α if β = 1,

7 Ito s formula implies Infinitestimal generator Y 2 t dx t = Y t dw t β 2 ρ (1 β)( ρx t ρy t ) dt dy t = Y t db t y 2 L = y 2 ( x 2 + y 2 ) β 2 ρ (1 β)( ρx ρy) x The principle part is the Laplace-Beltrami operator on 2-dimensional hyperbolic space. The corresponding diffusion process in called the hyperbolic Brownian motion. The transition density of 2-dimensional hyperbolic Brownian motion is given by the McKean kernel. Drift part does not play a role in the large deviation regime.

8 SABR density in small time The transition density of the process (X t, Y t ) from (x, y) to (ξ, η) in time τ is asymptotically given by p τ (ξ, η x, y) 1 d 2 (ξ,η x,y) 2πτ e 2τ, where d is defined as [ (ξ x) d(ξ, η x, y) = cosh η 2 + y 2 ] 2yη with ( cosh 1 z = log z + ) z 2 1. d is the geodesic distance between the initial point (x, y) and terminal point (ξ, η).

9 From density to option price For out-of-money calls, C(K, T ) = (s T K) + p T (s T, α T s 0, α 0 )ds T dα T 1 2πT 0 K (s T K)e d2 (s T,α T s 0,α 0 ) 2T ds T dα T By applying the Laplace asymptotic formula, we have, as T 0, C(K, T ) e d2 (s 0,α 0 ) 2T where d is the minimal distance from the initial point (s 0, α 0 ) to the half plane {(s, α) : s K}, i.e., d (s 0, α 0 ) = min {d(s, α s 0, α 0 ) : s K}.

10 Matching with Black-Scholes price The zeroth order SABR formula is thus obtained by matching the exponent with the corresponding term in Black-Scholes price. e d2 (s 0,α 0 ) We end up with 2T C(K, T ) = C BS (K, T ) e (log s 0 log K) 2 2σ BS 2 T d 2 (s 0, α 0 ) (log s 0 log K) 2 σbs 2. Thus, σ BS (K, T ) log s 0 log K. d (s 0, α 0 )

11 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 qth moments m(q, ) in various lags to E [ log σ t+ log σ t q ] = K q ζq proxied by daily realized variance estimates. K q denotes the qth moment of standard normal. At-the-money volatility skew is well approximated by a power law function of time to expiry

12 Gatheral-Jaisson-Rosenbaum Log-volatility behaves as a fractional Brownian Motion with Hurst exponent H of order 0.1 at any reasonable time scale

13 Gatheral-Jaisson-Rosenbaum Log-log plot of m(q, ) versus for various q.

14 Gatheral-Jaisson-Rosenbaum At-the-money volatility skew ψ(τ) = d dk k=0 σ BS (k, τ) is well approximated by a power law function of time to expiry τ

15 Fractional volatility process The observations suggest the following model for instantaneous volatility σ t = σ 0 e νw t H, where W 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 Xt where dx t = α(m X t )dt + νdw H t is a fractional Ornstein-Uhlenbeck process. Again, drift term plays no role in large deviation regime.

16 Review: fractional Brownian motion A mean-zero Gaussian process Bt H is called a fractional Brownian motion with Hurst exponent H [0, 1] if its autocovariance function R(t, s), for t, s > 0, satisfies [ R(t, s) := E Bt H Bs H ] = 1 2 ( t 2H + s 2H t s 2H). B H is self-similar, indeed, Bat H d = a H Bt H for a > 0 B H has stationary increments B H t is a standard Brownian motion when H = 1 2

17 Lognormal fsabr model Consider the following lognormal fsabr model ds t S t = α t (ρdb t + ρdw t ), α t = α 0 e νbh t, where B t and W t are independent Brownian motions, ρ = 1 ρ 2. Bt H is a fractional Brownian motion with Hurst exponent H driven by B t : B H t = t K H is the Molchan-Golosov kernel. 0 K H (t, s)db s. Goal: to obtain an easy to access expression for the joint density of (S t, α t ).

18 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 X t X 0 = Y 0 t Y t = Y 0 e νbh t. 0 e νbh s (ρdb s + ρdw s ) Y t 0 e 2νBH s ds, We derive a bridge representation for the joint density of (X t, Y t ) in a Fourier space.

19 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 ) e η 2 t 2ν 2 t 2H y t 2πν 2 t 1 2H 2π [ ( e i(xt x0)ξ E e i ρ ] t 0 y 0e νbh s db s+ y2 0 2 v t )ξ e ρ2 y 0 2 v t ξ 2 2 νbt H = η t dξ, where i = 1, v t = t 0 e2νbh s ds and ηt = log yt y 0.

20 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 ) e η 2 t 2ν 2 t 2H y t 2πν 2 t 1 2H 2π ] e i(xt x0)ξ E [e 1 2 (ξ i)ξy 0 2 vt νb H t = η t dξ, where i = 1, v t = t 0 e2νbh s ds and ηt = log yt y 0.

21 McKean kernel The McKean kernel p H 2(t, x t, y t x 0, y 0 ) reads p H 2(t, x t, y t x 0, y 0 ) = 2e t/8 (2πt) 3/2 d ξe ξ2 /2t cosh ξ cosh d dξ, 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 Riemannian volume form 1 dx yt 2 t dy t. 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.

22 Expanding around b s We expand the conditional expectation in the bridge representation around the deterministic path b s. Let E ηt [ ] = E [ νb t H ] = η t. First, define the deterministic path b s by ] b s = log E ηt [e 2νBH s. Indeed, b s = log E ηt [e 2νBH s ] = 2νE ηt [Bs H ] + 2ν 2 var ηt [Bs H { ] } = 2R(1, u)η t + 2ν 2 t 2H u 2H R 2 (1, u), where u = s t and R(t, s) = E [ Bt H Bs H ]. ] Note that e bs = E ηt [e 2νBH s. In other words, e bs is an unbiased estimator of e 2νBH s conditioned on νb H t = η t.

23 Now expand the conditional expectation in the bridge representation around the deterministic path b s as [ E ηt e 1 2 (ξ i)ξ ] t 0 y 0 2e2νBH s ds = e 1 2 (ξ i)ξ [ t 0 y 0 2ebs ds E ηt e 1 2 (ξ i)ξ ) ] t 0 y 0 (e 2 2νBH s e bs ds e 1 2 (ξ i)ξ t 0 y 0 2ebs ds ( 1 + n k=2 1 k! E η t [ { 1 (ξ i)ξ 2 t y0 2 0 ) } ]) k (e 2νBH s e bs ds. Note that, by definition of b s, the first order term in the last expansion is automatically zero.

24 In principle each term in the expansion can be computed in closed form since they are given by the integrals of moments of shifted lognormal distributions. Denoting these terms by H k, i.e., [ { t ) } ] k H k (t, η t ) := E ηt (e 2νBH s e bs ds For instance, = H 2 (t, η t ) = = where ˆv t = t 0 ebs ds. [0,t] k [0,t] 2 [0,t] 2 E ηt E ηt E ηt 0 [ k i=1 [ 2 i=1 ( e 2νBH s i e bs i ) ] ds 1 ds k ( e 2νBH s i e 2bs i ) ] ds 1 ds 2 [e 2νBH s 1 +2νB H s 2 ] ds 1 ds 2 ˆv 2 t,

25 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 ) 1 ηt 2 e 2ν 2 t 2H 2H y t 2πν 2 t 1 e i(xt x0)ξ e 1 2 (ξ i)ξˆvt 2π { n [ ] } k k! 2 (ξ i)ξ y0 2k H k (t, η t ) dξ, k=2 where ˆv t = t 0 y 2 0 ebs ds. Note that it gives a natural expansion in t via the H k functions.

26 Edgeworth type of expansion Finally, integrating term by term, we obtain a full Edgeworth type of expansion for the probability density p when ρ = 0 around the deterministic path b s. For example, we have up to second order p(t, x t, y t x 0, y 0 ) e η 2 t 2ν 2 t 2H 1 w2 t 2y e y t 2πν 2 t 2H 0 2πy 2 ˆv t 0 2 ˆvt y 2 0 ˆv t He 2 ( ) wt + 2y ( ) 0 wt He 3 + 1ˆv ( wt ˆvt ˆv t 3 ˆvt t 2 He 4 ) ˆvt y 4 0 H 2(t, η t ), where w t = x t x 0 + y 0 2 ˆvt 2 and He n ( ) denotes the nth Hermite polynomial. Recall that ˆv t = t 0 ebs ds and η t = log yt y 0.

27 Small time asymptotics - uncorrelated To the lowest order as t 0, the density p has the following small time asymptotic behaviour p(t, x t, y t x 0, y 0 ) = ηt 2 e 2ν 2 t 2H (x t x 0 ) 2 2y 2 0 ˆv t e y t 2πν 2 t 2H 2πy0 2ˆv t e x t x 0 2 {1 + o(1)}, where recall that ˆv t = t 0 ebs ds.

28 Probability density in small time - correlated case The Edgeworth type of expansion in the correlated case is more involved because of the appearance of the stochastic integral. However, the lowest order term is manageable. Define the functions C RK and C er by C RK (η) := 1 Then to the lowest order we have 0 p(t, x t, y t x 0, y 0 ) 1 2π 1 y t ν 2 t e R(1,u)η K H (1, u)du, C er (η) := ηt 2 e 2ν 2 t 2H 1 2H y 0 ṽt e 1 where ṽ t = tψ(η t ) := { (1 + ρ 2 )C er (η t ) ρ 2 C 2 RK (η t) } t. 0 e 2R(1,u)η du. ( ) 2 1 2y 0 2 x t x 0 ρy 0 C RK (η t) η t ṽt ν t 1 2 H

29 Approximate distance function Rewrite the joint density p as where 1 2π p(t, x t, y t x 0, y 0 ) 1 1 e y t ν 2 t 2H y 0 ṽt d 2 (xt,yt x 0,y 0 ) 2t 2H d(x t, y t x 0, y 0 ) := η2 t ν y 2 0 ψ(η t) ( xt x 0 t 1 2 H ρy 0C RK (η t ) η ) 2 t ν is regarded as the approximate distance function.

30 Convexity of approximate distance function Contour plot of approximate distance function Contour plot of approximate distance H = 0.75 η η x x Figure : The contour plots. Parameters ρ = 0.7, ν = 1, y 0 = 1, t = 0.5. H = 0.75 on the right; H = 0.25, on the left.

31 Implied volatility approximation by bridge representation By matching with the Black-Scholes price to the lowest order, we obtain a small time approximation of the implied volatility as follows. Let α = 1 2 H and k = log K s 0. Implied volatility approximation σ 2 BS { k2 η 2 ( T 2α ν k y0 2ψ(η ) T α ρy 0C RK (η ) η ) } 2 1 ν where η is the minimizer { η = argmin Note that η = η ( k T α ) η R : η2 ν y 2 0 ψ(η) ( k T α ρy 0C RK (η) η ) } 2. ν

32 Approximate implied volatility plots fsabr implied volatility, t = 0.01 fsabr implied volatility, t = 1 σ σ logmoneyness logmoneyness Figure : The implied volatility curves. t = 0.01 on the left, t = 1 on the right. Parameters are set as ρ = 0.4, ν = 0.58, α 0 = H = 0.1 in red, H = 0.3 in orange, H = 1 2 in green, H = 0.7 in blue, H = 0.9 in purple.

33 Recovery of SABR formula? Q: Does it recover the SABR formula to the lowest order when H = 1 2? SABR formula where σ BS (k) νk D(ζ), ζ = ν α 0 k, 1 2ρζ + ζ D(ζ) = log 2 + ζ ρ. 1 ρ

34 Recovery of SABR formula? Q: Does it recover the SABR formula to the lowest order when H = 1 2? A: NO! SABR formula where σ BS (k) νk D(ζ), ζ = ν α 0 k, 1 2ρζ + ζ D(ζ) = log 2 + ζ ρ. 1 ρ

35 Graphic comparison with SABR formula Comparison of implied volatilities Difference between implied volatilities σ fsabr with H = 0.5 SABR formula iv_sabr vol Logmoneyness Logmoneyness Figure : The implied volatility curves from SABR and fsabr formula. Parameters are set as τ = 1, ρ = , ν = 0.58, α 0 =

36 Recovery of SABR formula? Q: Does it recover the SABR formula to the lowest order when H = 1 2? A: NO! Q: Maybe a smarter choice of b s might work?

37 Recovery of SABR formula? Q: Does it recover the SABR formula to the lowest order when H = 1 2? A: NO! Q: Maybe a smarter choice of b s might work? A: Unfortunately, doesn t really work that way either.

38 Recovery of SABR formula? Q: Does it recover the SABR formula to the lowest order when H = 1 2? A: NO! Q: Maybe a smarter choice of b s might work? A: Unfortunately, doesn t really work that way either. Q: Is it even possible to recover the SABR formula from the bridge representation?

39 Recovery of SABR formula? Q: Does it recover the SABR formula to the lowest order when H = 1 2? A: NO! Q: Maybe a smarter choice of b s might work? A: Unfortunately, doesn t really work that way either. Q: Is it even possible to recover the SABR formula from the bridge representation? A: Most-likely-path from bridge representation

40 Bridge representation for multiperiod joint density Notations t = (t 1,, t n ), x t = (x t1,, x tn ), y t = (y t1,, y tn ), B H t = (B H t 1,, B H t n ), X t = (X t1,, X tn ), Y t = (Y t1,, Y tn ), ξ t = (ξ t1,, ξ tn ), η t = (η t1,, η tn ), ζ t = (ζ t1,, ζ tn ). For 0 < t 1 < < t n = T, we are interested in obtaining a bridge expression for the multiperiod joint density p(x t1, y t1,, x tn, y tn ) := P [(X t1, Y t1 ) = (x t1, y t1 ),, (X tn, Y tn ) = (x tn, y tn )].

41 The multiperiod joint density p has the following bridge representation p(x 1, y 1,, x n, y n ) n = E P k=1 e 1 2y 2 0 ρ2 vt k ( x tk y 0 ρ t k [y 0 e νbh t = y t ], 2πy 2 0 ρ2 v tk t k 1 e νbh s db s+ y2 0 2 v tk ) 2 νb H t = η t where v tk = v tk v tk 1 = t k t k 1 e 2νBH s ds.

42 Heuristic derivation of LDP Consider the log likelyhood function log p(x t1, y t1,, x tn, y tn ) n = log E k=1 e 1 2y 2 0 ρ2 vt k ( x tk y 0 ρ t k ] + log P [νb H t = η t 2πy 2 0 ρ2 v tk n log y tk. k=1 t k 1 e νbh s db s+ y2 0 2 v tk ) 2 νb H t = η We calculate the first two terms in the limit as n in the following.

43 Note that ] log P [νb H t = η t 1 2ν 2 η R 1 η, where R = [R(t i, t j )] is the covariance matrix of B H. We approximate R(t i, t j ) by a Riemann sum ] ti R(t i, t j ) = E [B H ti B H t j tj = K(t i, s)k(t j, s)ds i j K(t i, t k )K(t j, t k ) t = K K t, k=0 where K is the upper triangular matrix K(t i, t j ), if i j; K ij = 0, otherwise. 0

44 Therefore, R 1 = K 1 (K ) 1 1 t. Let b = (b t1, b tn ) be the unique solution to the linear system Hence, η ν = Kb t η t i = ν i K(t i, t k )b tk t. k=0 1 2ν 2 η R 1 η = 1 2 tb K R 1 Kb t = 1 2 b b t = 1 n bt 2 2 k t 1 2 T 0 k=1 b 2 t dt as n. Also, in the limit n, η t = ν t 0 K(t, s)b sds for t [0, T ].

45 As for the conditional expectation n log E k=1 e 1 2y 2 0 ρ2 vt k ( x tk y 0 ρ t k 2πy 2 0 ρ2 v tk ( n 1 E 2y0 2 x tk y 0 ρ ρ2 v tk k=1 t k 1 e νbh s db s+ y2 0 2 v tk ) 2 tk Note that conditioned on νb H = η, we have and v tk = = tk t k 1 e νbh e 2νBH s ds e 2ηt k 1 t = e 2ν k 1 t k 1 tk x tk y 0 ρ ( xtk t νb H t = η ) 2 s db s νb H = η j=0 K(t k 1,t j )b tj t t e νbh s db s x tk y 0 ρe ηt k 1 btk 1 t t k 1 y 0 ρe ν k 1 j=0 K(t k 1,t j )b tj t b tk 1 ) t

46 Thus, ( n 1 E 2y0 2 x tk y 0 ρ ρ2 v tk k=1 tk t k 1 e νbh ) 2 s db s νb H = η n 1 k=0 2y0 2 ρ2 e 2ν k 1 j=0 K(t k 1,t j )b tj t ( xtk y 0 ρe ν ) k 1 2 j=0 K(t k 1,t j )b tj t b tk 1 t t 1 T 1 ( 2 0 y0 2 ρ2 e 2ν t ẋ t y 0 ρe ν ) t 2 0 K(t,s)bsds 0 K(t,s)bsds b t dt = 1 2 as n. T 0 1 ρ 2 y 2 0 e2ηt (ẋ t ρy 0 e ηt b t ) 2 dt

47 Large deviations principle for fsabr We end up with log P [X t = x t, Y t = y t for t [0, T ]] = lim log p(x t n 1, y t1,, x tn, y tn ) = 1 2 = 1 2 T 0 T 0 1 ρ 2 y0 2 (ẋ t ρy 0 e ηt b t ) 2 dt + 1 e2ηt 2 1 ρ 2 y 2 t where b L 2 [0, T ] satisfying (ẋ t ρy t b t ) 2 dt η t = log y t log y 0 = ν t 0 T 0 T 0 b 2 t dt K H (t, s)b s ds for t [0, T ]. This should be the rate function for sample path LDP. b 2 t dt

48 Recovery of Freidlin-Wentzell when H = 1 2 Indeed, When H = 1 2, K 1 H b t = 1 ν K 1 H [η](t). is simply the usual differential operator, thus b t = η t ν = 1 ẏ t. ν y t Therefore, the rate function reduces to log P [X t = x t, Y t = y t for t [0, T ]] T ( ) 1 η 2 t ẋ t ρy t dt + 1 T ν 2 = ρ 2 yt 2 0 = 1 T 1 ( ν2ẋ ρ 2 ν 2 yt 2 t 2ρνẋ t ẏ t + ẏt 2 ) dt which recovers the classical large deviations principle of Freidlin-Wentzell ( ) 2 ηt dt ν

49 Implied volatility approximation by LDP Again, by matching with the Black-Scholes price, we obtain fsabr formula σ 2 BS k2 T ( T 0 ) 1 1 ρ 2 yt 2 (ẋ t ρyt bt ) 2 + bt 2 dt, where (x, b ) is the minimizer { T (x, b ) = argmin ẋ, b L 2 1 [0, T ] : 0 ρ 2 yt 2 with x T = k and y t is given by, for t [0, T ], log y t log y 0 = ν t 0 K H (t, s)b s ds This recovers the SABR formula when H = 1 2! } (ẋ t ρy t b t ) 2 + bt 2 dt

50 Conclusion We show a bridge representation for the (single and multi period) joint density of the lognormal SABR model. We obtained a full Edgeworth type of expansion in the uncorrelated case. Small time asymptotics to the lowest order are presented for both correlated and uncorrelated cases. A heuristic derivation of large deviations principle from multiperiod bridge representation for the fsabr model is shown, which recovers the classical Freidlin-Wentzell large deviations principle when H = 1 2. Approximations of implied volatility in small time are obtained accordingly by matching terms with the Black-Scholes price. We emphasize that the fsabr formula also holds for general fsabr model, not only for lognormal fsabr.

51 THANK YOU FOR YOUR ATTENTION.