Asymptotic behavior of distribution densities in stochastic volatility models

Transcription

1 Asymptotic behavior of distribution densities in stochastic volatility models Archil Gulisashvili Ohio University joint work with E. M. Stein) Seminar talk at Stanford University 09/8/007 1

2 Samuelson s model of the stock price In their celebrated work on pricing of options, Black and Scholes used Samuelson model of the stock price. Samuelson suggested to describe the random behavior of the stock price by a diffusion process X t satisfying the following stochastic differential equation: dx t = µx t dt + σx t dw t, X 0 = x 0, where µ is a real constant, σ is a positive constant, and W t is a standard Brownian motion. The constants µ R 1 and σ > 0 are called the drift and the volatility of the stock, respectively. Explicit formula The following formula holds for the stock price process in Samuelson s model: X t = x 0 exp µ 1 ) } σ t + σw t. The process X t is called a geometric Brownian motion.

3 Stock price distribution density The distribution function of the stock price X t is given by r P X t < r), 0 r <, where P stands for the Wiener measure. The distribution density of X t if it exists) is a function D t on [0, ) such that for all r > 0. P X t < r) = r 0 D t u)du Stock price distribution in Samuelson s model The distribution density of the stock price process X t in Samuelson s model can be computed explicitly. We have 1 log x D t x) = exp πtσx x 0 ) µ 1 σ) t tσ. Such densities are called log-normal. 3

4 Black-Scholes model The Black-Scholes formula for the price of a European call option at t = 0 is the following: where C BS = x 0 N d 1 ) Ke rt N d ), d 1 = log x 0 log K + r + 1 σ) T σ T, d = d 1 σ T, and Nz) = 1 π z exp y } dy. Here T stands for the expiration date, K is the strike price, and r denotes the interest rate. Constant volatility assumption is erroneous 4

5 It is assumed in the Black-Scholes model that the volatility is constant. However, it has been observed that the implied volatility for a European call option depends on the strike price and the expiration date. This fact contradicts the constant volatility assumption. The implied volatility considered as a function of the strike price is locally convex near its minimum. This is the so-called volatility smile effect. Stochastic volatility models To improve the Black-Scholes model, various stochastic volatility models were suggested. Some of the wellknown stochastic volatility models are the Hull-White model where the volatility is a geometric Brownian motion; the Stein-Stein model where the absolute value of an Ornstein-Uhlenbeck process is used as the volatility process; and the Heston model where the volatility is a Cox-Ingersoll-Ross process. Let us first consider the following general stochastic volatility model: dxt = µx t dt + f Y t ) X t dw t dy t = b t, Y t ) dt + σ t, Y t ) dz t Here, µ R 1 ; b and σ are continuous functions on [0, T] R 1 ; W t and Z t are independent one-dimensional 5

6 standard Brownian motions; and f is a nonnegative function on R 1. The process X t plays the role of the stock price process, while f Y t ) is the volatility process. The initial conditions for the processes X t and Y t will be denoted by x 0 and y 0, respectively. We also assume that the second equation in in the model above has a unique strong solution Y t. Explicit formula Under appropriate conditions, X t = x 0 exp µt 1 t f Y s ) ds + 0 t 0 } f Y s ) dw s. Distribution densities We will denote by D t the distribution density of the stock price X t, and by m t the mixing distribution density associated with the volatility process f Y t ), that is, the distribution density of the random variable α t = 1 t t 0 f Y s ) ds}1. 6

7 The Hull-White model The stock price process X t and the volatility process Y t in the Hull-White model satisfy the following system of stochastic differential equations: dxt = µx t dt + Y t X t dw t dy t = νy t dt + ξy t dz t. The Stein-Stein model In the Stein-Stein model, the stock price process X t and the volaility process Y t satisfy the following system of stochastic differential equations: dxt = µx t dt + Y t X t dw t dy t = q m Y t ) dt + σdz t. We will only consider the case where µ R, q 0, m = 0, and σ > 0. Then the Stein-Stein model becomes dxt = µx t dt + Y t X t dw t dy t = qy t dt + σdz t. The solution to the second stochastic differential equation in??) is an Ornstein-Uhlenbeck process, for which the 7

8 long run mean equals zero. The reason why we restrict ourselves to the case where m = 0 is that CIR-processes and Ornstein-Uhlenbeck processes with m = 0 can be dealt with in a similar way, using Bessel processes. The case where m 0 is more complicated, and new methods may be needed. The Heston model In the Heston model, the stock price process X t and the volaility process Y t satisfy the following system of stochastic differential equations: dxt = µx t dt + Y t X t dw t dy t = a + by t ) dt + c 1) Y t dz t. We will consider the case where µ R, a 0, b 0, and c > 0. It is often assumed that the volatility equation in 1) is written in a mean-reverting form. Then the Heston model becomes dxt = µx t dt + Y t X t dw t dy t = r m Y t ) dt + c Y t dz t, where r 0, m 0, and c > 0. Explicit formula 8

9 The following formula holds for the stock price process: X t = x 0 exp µt 1 t t } Ys ds + Y s dw s. 0 0 Integral representation The following equality holds: D t x) = 1 x 0 e µt 0 L t, y, ) x m x 0 e µt t y)dy, where L is the log-normal density defined by 1 log v + ty Lt, y, v) = exp πtyv ty It follows that D t x0 e µt x ) = 0 y 1 m t y) exp 1 x 0 e µt πt x 3 [ 1 ty log x + ty 8 ). ]} dy. 9

10 Symmetry of the stock price distribution density D t x0 e µt x ) ) 1 = x 3 D t x ; 0, ν, ξ, 1, y 0. Asymptotic behavior of the mixing distribution The Hull-White model. Special case: m t y; 1 ), 1, 1 = c 1 y c log y) c 3 exp 1 log yy0 + 1 ) } tξ log log yy0 )) 1 + O log y) 1, y. Formulas for the constants c 1 = 1 t 1 exp πt } log ) t, c = 1 1 log, t 10

11 and c 3 = 1 + log. 4t The law of the time integral m 0) T denotes the law of the time integral A ρ) T = T of a geometric Brownian motion. 0 exp ρu + Z u )} du The following formula holds: m 0) T y) = c 1 1 c 3 T c +1 y c 1 log y ) c3 T exp 1 y log T T + 1 ) } y log log T )) 1 + O log y) 1, y, where c 1, c, and c 3 are as above with t replaced by T. The Hull-White model. General case: 11

12 m t y; ν,ξ,y 0 ) = c 1 y c log y) c 3 exp 1 tξ )) 1 + O log y) 1 as y. log yy0 + 1 log log yy0 ) } Formulas for the constants: c 1 = 1 ξ πt 1 and α 0 exp log 1 tξ tξ y c = α log tξ, c 3 = α 1 + log 4tξ, α = ν ξ ξ. The law of the time integral } log ) exp α ξ } t, tξ The following formula holds for the law of the time inte- 1

13 gral of a geometric Brownian motion: m ρ) T y) = C 1 1 C 3 T C +1 y C 1 y )) C3 log T exp 1 y log T T + 1 ) } y log log T )) 1 + O log y) 1 as y. Formulas for the constants: C 1 = 1 } T 1 log ) exp exp πt T C = ρ log T } ρ T,, and C 3 = ρ 1 + log. 4T Asymptotic behavior of the mixing distribution The Hull-White model. Case where x 0. Let < ν <, ξ > 0, y 0 > 0, and t > 0. Then there exists a positive constant b = b ν,ξ,y 0, t) such that } m t y; ν,ξ, y 0 ) = by α 1 exp y 0 1 )) + O y tξ y as y 0. Asymptotic behavior of the stock price density 13

14 Asymptotics near infinity. The Hull-White model. Special case: D t x; 0, 1, 1, 1, 1 ) log x) c 1 exp 1 t 1 + O log = c 1 c 1 c 3 t c +1 x [ ]) log x c3 t log x log t )) log log x) ) log x log log t as x, where the constants c 1, c, and c 3 are as above. Asymptotics near infinity and zero. The Hull- White model. General case: Let < µ <, < ν <, ξ > 0, x 0 > 0, y 0 > 0, and t > 0. Then 14

15 ) D t x0 e µt c 1 x;µ, ν,ξ, x 0, y 0 = µtc 1 c3 [ ]) x 0 e c3 log x) c 1 log x log t [ ] [ 1 log x exp log + 1 log log 1 tξ 1 + O as x, and y 0 log log x) 1 t )) ) D t x0 e µt c 1 x;µ, ν,ξ, x 0, y 0 = µtc 1 c3 [ ]) x 0 e log 1 c3 1 log 1 x )c x log t exp 1 tξ 1 + O as x 0. log 1 log 1 x y 0 t log log 1 )) x ) 1 t c +1 x 1 y 0 t c log log ]) log x t x 1 1 y 0 log 1 x t Asymptotic behavior of the mixing distribution The Heston model. General case: 15

16 For a 0, b 0, and c > 0, there exist A > 0, B > 0, and C > 0 such that )) m t y; a, b,c, y 0 ) = Ay 1 +a c e By e Cy 1 + O y 1 as y. Asymptotic behavior of the mixing distribution The Heston model. Special case: The following formula holds: m t y, a, 0, c, y 0 ) = Ay 1 +a c e By e Cy 1 + O as y. Formulas for the constants )) y 1 A = a 1 c 4 y a c 0 c t exp } 4y0, c t B = y 0 π, and C = π c t c t. Asymptotic behavior of the stock price distribution The Heston model. 16

17 For a 0, b 0, and c > 0, there exist A 1 > 0, A > 0, and A 3 > 0 such that for every δ with 0 < δ < 1, as x. D t x0 e µt x ) = A 1 log x) a c e A log x x A 3 )) 1 + O log x) δ Asymptotic behavior of the stock price distribution The Stein-Stein model. For q 0, m = 0, and σ > 0, there exist B 1 > 0, B > 0, and B 3 > 0 such that for every δ with 0 < δ < 1, D t x0 e µt x ) = B 1 log x) 1 e B log x x B 3 )) 1 + O log x) δ as x, and D t x0 e µt x ) = B 1 log 1 ) 1 e B log x 1 x B 3 3 x 1 + O log 1 ) )) δ x as x 0. Conclusions. Fat tails. 17

18 The distribution density D t of the stock price in the Hull- White model decays extremely slowly. This is the socalled fat tail effect. The function x D t x) behaves near infinity as 1 times logarithmic factors. In addition, x the function x D t x) behaves near zero as 1 x multiplied by a logarithmic factor making the density integrable near zero. In a sense, the stock price distribution density in the Hull-White model has the slowest decay possible in comparison with stock price distribution densities in similar stochastic volatility models. It follows from Theorem?? that the stock price distribution density D t x) in the Heston model behaves at infinity roughly as the function x A 3 and at zero as the function x A3 3. Since C = π and A c 3 = 3 t + 4π + c t ct for b = 0, we see that in this case the density D t x) behaves as the function x 3 4π +c t ct near infinity and as the function x 3 + 4π +c t ct near zero. For the Stein-Stein model, Theorem?? implies that the function D t x) behaves at infinity roughly as the function x B 3 and at zero as the function x B3 3, where B 3 = 3 + 8C + t t and C = 1 ) tq + t 1 r σ tq. ) 18

19 Market price of risk and martingale measures Since the volatility in the Hull-White model is a stochastic process, there is an additional source of uncertainty in the random behavior of the stock price. This uncertainty is controlled by a stochastic process γ t which is called the market price of the volatility risk. Assumption: the market price of risk is constant We assume that γ t = a, where a is a number. Then, under the corresponding martingale measure P,a, the Hull-White equations can be rewritten in the following form: dxt = rx t dt + Y t X t dw t dy t = ν ξa)y t dt + ξy t dz t, where W t and Z t are independent standard Brownian motions, and r > 0 is the interest rate. The pricing function 19

20 By the martingale property of the discounted stock price with respect to the martingale measure P,a, the price of a Hull-White European call option at t = 0 is given by the following formula: V 0 K) = E,a [ e rt X T K) + ]. where K is the strike price, and T is the expiration date. Explicit formula for the pricing function V 0 K) = e rt xd T x)dx e rt K K K D T x)dx. Implied volatility in the Hull-White model The implied volatility in a general option pricing model is the volatility in the Black-Scholes model for which the 0

21 corresponding Black-Scholes price of the option is equal to its price in the model under consideration. Therefore, for K > 0 the implied volatility IK) in the Hull-White model is determined from the equality C BS K, IK)) = V 0 K). 1

22 Implied volatility smile is a mathematical reality Consider a stochastic volatility model such that the volatility process is independent of the standard Brownian motion driving the stock price equation. Then the volatility function K IK) is convex in a neighborhood of its global minimum K min = x 0 e rt. This is a celebrated result of Renault and Touzi. Asymptotic behavior of the implied volatility Large values of the strike price. IK) = log K T log log K + log log log K ξt + O1) as K.

23 Small values of the strike price. IK) = log 1 K T log log 1 K + log log log 1 K Tξ + O1) as K 0. Asymptotic behavior of I K) and I K) Large values of the strike price. I K) = ) 1 log log K + O. TK log K K log K ) I 1 log log K K) = TK log K + O. K log K as K. 3

24 Small values of the strike price. I 1 K) = TK log 1 K + O ) log log 1 K K log 1. K I K) = 1 TK log 1 K + O ) log log 1 K K log 1 K as K 0. Structural properties of the implied volatility The implied volatility as a function of the strike price is concave near infinity and convex near zero. These properties combined with the volatility smile effect show that the global shape of the implied volatility function K IK) resembles an infinite dipper. 4