Small time asymptotics for fast mean-reverting stochastic volatility models Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque) March 11, 2011 Frontier Probability Days, March 10-12, 2011
1 Motivation - finance problem Stock price model 2 LDP Ultra-fast mean-reversion regime Slower mean-reversion regime 3 Bryc s lemma PDE Convergence of viscosity solutions 4 Option pricing Implied Volatility 5
Background Motivation - finance problem Stock price model Let S t denote stock price at time t. A European call option on this stock is a contract that gives its holder the right, but not the obligation to buy a unit of this stock at a certain price, called strike price, and at a given time called maturity time.
Background Motivation - finance problem Stock price model Let S t denote stock price at time t. A European call option on this stock is a contract that gives its holder the right, but not the obligation to buy a unit of this stock at a certain price, called strike price, and at a given time called maturity time. Let K = strike price, T = maturity time and suppose S 0 < K ( out-of-the- money).
Question Motivation - finance problem Stock price model What is the behavior of as time to maturity T 0? option price = E[e rt (S T K) + ]
Question Motivation - finance problem Stock price model What is the behavior of as time to maturity T 0? To estimate this quantity, we study option price = E[e rt (S T K) + ] P(S T > K) as T 0. This probability decays exponentially fast to 0. We get a large deviation estimate of the form lim T log P(S T > K) = I (K) T 0
Black-Scholes model Motivation - finance problem Stock price model A simple model for this stock price is the B-S model ds t = rs t dt + σs t dw t ; σ > 0 is called volatility. Under the B-S model, the price of a call option with strike price K and maturity time T is: is easy to calculate. option price = E[e rt (S T K) + ] However the assumption of constant volatility is unrealistic and we instead work with a more sophisticated model.
Motivation - finance problem Stock price model Stochastic volatility model for stock price Let S t denote stock price. ds t = rs t dt + S t σ(y t )dw (1) t, (1a) dy t = 1 δ (m Y t)dt + ν δ Y β t dw (2) t. (1b) where m R, r, ν > 0, W (1) and W (2) are standard Brownian motions with W (1), W (2) t = ρt, with ρ < 1 constant. -The process (Y t ) is a fast mean-reverting process with rate of mean reversion 1/δ (δ > 0).
Assumptions Motivation - finance problem Stock price model We assume that Assumption 1 β {0} [ 1 2, 1); 2 in the case of β = 1/2, we require m > ν 2 /2 and Y 0 > 0 a.s., in the case of 1/2 < β < 1, we require m > 0 and Y 0 > 0 a.s.; 3 σ(y) C(R; R + ) satisfies σ(y) C(1 + y σ ), for some constants C > 0 and σ with 0 σ < 1 β.
Examples of Y process Motivation - finance problem Stock price model Ornstein-Uhlenbeck process (take β = 0) CIR process (take β = 1/2) dy t = 1 δ (m Y t)dt + ν δ dw (2) t. dy t = 1 δ (m Y t)dt + ν δ Yt dw (2) t.
Rescaling time Motivation - finance problem Stock price model Let X t = log S t. Rescale time t ɛt. ( dx ɛ,t = ɛ r 1 ) 2 σ2 (Y ɛ,t ) dt + ɛ σ(y ɛ,t )dw (1) t (2a) dy ɛ,t = ɛ ɛ δ (m Y ɛ,t)dt + ν δ Y ɛ,tdw β (2) t. (2b) Our mean-reversion time δ is ɛ-dependent.
Rescaling time Motivation - finance problem Stock price model Let X t = log S t. Rescale time t ɛt. ( dx ɛ,t = ɛ r 1 ) 2 σ2 (Y ɛ,t ) dt + ɛ σ(y ɛ,t )dw (1) t (2a) dy ɛ,t = ɛ ɛ δ (m Y ɛ,t)dt + ν δ Y ɛ,tdw β (2) t. (2b) Our mean-reversion time δ is ɛ-dependent. Consider 2 regimes: δ = ɛ 4 (ultra-fast regime) δ = ɛ 2 (fast regime)
Rescaling time Motivation - finance problem Stock price model Let X t = log S t. Rescale time t ɛt. ( dx ɛ,t = ɛ r 1 ) 2 σ2 (Y ɛ,t ) dt + ɛ σ(y ɛ,t )dw (1) t (2a) dy ɛ,t = ɛ ɛ δ (m Y ɛ,t)dt + ν δ Y ɛ,tdw β (2) t. (2b) Our mean-reversion time δ is ɛ-dependent. Consider 2 regimes: δ = ɛ 4 (ultra-fast regime) δ = ɛ 2 (fast regime) As ɛ 0, we look at small time asymptotics of X process but large time asymptotics of the Y process. Y is mean-reverting and ergodic and approaches its invariant distribution in large time. The effect of Y gets averaged!
Large Deviation Principle (LDP) LDP Ultra-fast mean-reversion regime Slower mean-reversion regime Let X ɛ,0 = x. We prove the following large deviation estimates of the probabilities of {X ɛ,t > x } when x > x. THEOREM lim ɛ log P(X ɛ,t > x ) = I (x ; x, t) ɛ 0 with rate functions I (x ; x, t) as follows.
Rate function (δ = ɛ 4 case) LDP Ultra-fast mean-reversion regime Slower mean-reversion regime When δ = ɛ 4, I (x ; x, t) = x x 2 2 σ 2, t where σ 2 is the average of the volatility function σ(y) with respect to the invariant distribution of Y.
Rate function (δ = ɛ 2 case) LDP Ultra-fast mean-reversion regime Slower mean-reversion regime When δ = ɛ 2, where H 0 (p) = { I (x ; x, t) = t sup p p R ( x x t ) } H 0 (p) lim T 1 log E[e 1 R T 2 p2 0 σ2 (Y p s )ds Y p 0 T + = y]. Y p is the process with the perturbed Y process with generator B p B p g(y) = Bg(y) + ρσνy β p y g(y), (3) where B is the generator of the Y process.
Problem Bryc s lemma PDE Convergence of viscosity solutions Two aspects to this problem: It is a Large Deviation problem coupled with a homogenization problem.
Key Steps in Proof Bryc s lemma PDE Convergence of viscosity solutions Prove convergence of the following functionals u h ɛ (t, x, y) := ɛ log E[e ɛ 1 h(x ɛ,t) X ɛ,0 = x, Y ɛ,0 = y], h C b (R) to u0 h (t, x).
Key Steps in Proof Bryc s lemma PDE Convergence of viscosity solutions Prove convergence of the following functionals u h ɛ (t, x, y) := ɛ log E[e ɛ 1 h(x ɛ,t) X ɛ,0 = x, Y ɛ,0 = y], h C b (R) to u0 h (t, x). Prove exponential tightness of {X ɛ,t } ɛ>0, i.e. for any α > 0 there exists a compact set K α R such that lim ɛ log P(X ɛ,t / K α ) α. ɛ 0
Key Steps in Proof Bryc s lemma PDE Convergence of viscosity solutions Prove convergence of the following functionals u h ɛ (t, x, y) := ɛ log E[e ɛ 1 h(x ɛ,t) X ɛ,0 = x, Y ɛ,0 = y], h C b (R) to u0 h (t, x). Prove exponential tightness of {X ɛ,t } ɛ>0, i.e. for any α > 0 there exists a compact set K α R such that lim ɛ log P(X ɛ,t / K α ) α. ɛ 0 Then, by Bryc s inverse Varadhan lemma, {X ɛ,t } ɛ>0 satisfies a LDP with I (x ; x, t) := sup {h(x ) u0(t, h x)}. h C b (R)
Bryc s lemma PDE Convergence of viscosity solutions Convergence of u ɛ Fix h C b (R). How do we prove converges? u ɛ (t, x, y) := ɛ log E[e ɛ 1 h(x ɛ,t) X ɛ,0 = x, Y ɛ,0 = y]
Bryc s lemma PDE Convergence of viscosity solutions Convergence of u ɛ Possible ways: Compute u ɛ and take limit. Use PDE (viscosity solution) approach. Operator-theoretic approach (See Feng and Kurtz [2]): u h ɛ = S ɛ (t)h is a nonlinear contraction semigroup. Method: Let H ɛ denote nonlinear generator of S ɛ (t). Prove H ɛ H. Invoke Crandall-Liggett generation theorem to get the limit semigroup S(t) corresponding to H. A variational representation for S ɛ (t)h can be obtained. S ɛ (t) can be interpreted as a Nisio semigroup.
PDE Bryc s lemma PDE Convergence of viscosity solutions u ɛ satisfies the following nonlinear pde: t u = H ɛ u, in (0, T ] R E 0 ; (4a) u(0, x, y) = h(x), (x, y) R E 0. (4b) E 0 is the state space of Y.
Bryc s lemma PDE Convergence of viscosity solutions H ɛ Let A ɛ denote the generator of the markov process (X ɛ,, Y ɛ, ). Then H ɛ u(t, x, y) = ɛe ɛ 1u A ɛ e ɛ 1u (t, x, y) ( = ɛ (r 1 2 σ2 ) x u + 1 ) 2 σ2 xxu 2 + 1 2 σ xu 2 where, + ɛ2 1u δ e ɛ Be ɛ 1u + ρσ(y)νy β ( ɛ xy 2 u + 1 x u y u) δ δ ɛ 2 1u δ e ɛ Be ɛ 1u = ɛ δ Bu + 1 δ 1 2 νy β y u 2. (5)
u 0 for δ = ɛ 4 case Bryc s lemma PDE Convergence of viscosity solutions We prove that as ɛ 0, u ɛ u 0 where u 0 satisfies the HJB equation t u 0 = 1 2 σ xu 0 (x) 2 ; u 0 (0, x) = h(x), where σ 2 is the average of σ 2 ( ) with respect to the invariant distribution of Y.
u 0 for δ = ɛ 2 case Bryc s lemma PDE Convergence of viscosity solutions We prove that as ɛ 0, u ɛ u 0 where u 0 satisfies the HJB equation t u 0 = H 0 ( x u 0 ) u 0 (0, x) = h(x), where H 0 (p) = lim T 1 log E[e 1 R T 2 p2 0 σ2 (Y p s )ds Y p 0 T + = y].
Bryc s lemma PDE Convergence of viscosity solutions Rigorous proof of convergence of u ɛ We use viscosity solution techniques adapted from Feng and Kurtz [2]. Difficulties: In what sense do the operators H ɛ converge? How do we identify the limit operator? We are averaging over a non-compact space! -We need to carefully choose suitable perturbed test functions in our proof. -The perturbed test functions f ɛ (t, x, y) and H ɛ f ɛ (t, x, y) should have compact level sets.
Bryc s lemma PDE Convergence of viscosity solutions Rigorous proof of convergence of u ɛ We prove conditions for convergence of u ɛ, solutions (in the viscosity solution sense) of t u = H ɛ u, to a sub-solution of and a super-solution of t u(t, x) inf α Λ H 0(x, u(t, x), D 2 u(t, x); α), (8) t u(t, x) sup H 1 (x, u(t, x), D 2 u(t, x); α). (9) α Λ The method used is a generalization of Barles-Perthame s half-relaxed limit arguments first introduced in single scale, compact space setting (see Fleming and Soner [3]).
Asymptotics of option price Option pricing Implied Volatility Consider an out-of-the-money European call option i.e. S 0 < K or x = log S 0 < log K. Lemma lim ɛ log ɛ 0 E[e rɛt (S ɛ,t K) + ] = I (log K; x, t), + where I is the rate function for LDP of {X ɛ,t } ɛ>0.
Asymptotics of option price Option pricing Implied Volatility Proof. lim ɛ log ɛ 0 E[e rɛt (S ɛ,t K) + ] = lim ɛ log + ɛ 0 + (by Laplace principle) = lim ɛ log ɛ 0 + lim ɛ 0 + ɛ log K K K P(S ɛ,t > z)dz = inf I (log z; x, t) z (K, ) = I (log K; x, t) P(X ɛ,t > log z)dz { exp I (log z; x, t) ɛ as I is a continuous, increasing function in the interval (x, ). } dz
Option pricing Implied Volatility Details of this work can be found on http://arxiv.org/abs/1009.2782. Our paper titled Small time asymptotics for fast mean-reverting stochastic volatility models has been accepted (pending minor revisions) in Annals of Applied Probability.
Option pricing Implied Volatility References J. Feng, M. Forde and J.-P. Fouque, Short maturity asymptotics for a fast mean reverting Heston stochastic volatility model, SIAM Journal on Financial Mathematics, Vol. 1, 2010 (p. 126-141) J. Feng and T. G. Kurtz, Large Deviation for Stochastic Processes, Mathematical Surveys and Monographs, Vol 131, American Mathematical Society, 2006. W. Fleming, H.M. Soner, Controlled Markov Processes and Viscosity Solutions Second Edition, Springer, 2006. I. Kontoyiannis and S. P. Meyn, Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes, Electronic Journal of Probability, Vol. 10 (2005), (p. 61-123). D. Stroock, An introduction to the Theory of Large Deviations. Universitext, Springer-Verlag 1984, New York.
Option pricing Implied Volatility Thank You!
Black-Scholes model Option pricing Implied Volatility A simple model for this stock price is the B-S model ds t = rs t dt + σs t dw t ; σ > 0 is called volatility. Price of a call option with strike price K and maturity time T under this model can be easily calculated: E BS [ e rt (S T K) +] ( ) ( ) log(s0 /K) + rt + 1 2 = S 0 Φ σ2 T σ Ke rt log(s0 /K) + rt 1 2 Φ σ2 T T σ T (Black-Scholes Formula)
Implied Volatility Option pricing Implied Volatility Implied Volatility, Σ(T, K), is the volatility parameter value to be inputed in the Black-Scholes model to match a call option price. Implied volatility for Black-Scholes model is a constant for all T and K. However, implied volatilities of market prices are not constant and vary with T and K. Keeping T fixed, the graph of implied volatilities of market prices as a function of K is approximately U-shaped.
Implied volatility Option pricing Implied Volatility Let σ ɛ denote the implied volatility corresponding to strike price K of option price given by our stochastic volatility model. Then σ ɛ is obtained by solving: ( ) ( ) x log K + rɛt + 1 e rɛt 2 S 0 Φ σ2 ɛɛt x log K + rɛt 1 2 KΦ σ2 ɛɛt σ ɛ ɛt σ ɛ ɛt = E [ e rɛt (S ɛ,t K) +] I (log K;x,t) e ɛ
Option pricing Implied Volatility Taking lim ɛ 0 ɛ log on both sides, we get (log K lim ɛ 0 σ2 x)2 + ɛ = 2I (log K; x, t)t.
Option pricing Implied Volatility Taking lim ɛ 0 ɛ log on both sides, we get (log K lim ɛ 0 σ2 x)2 + ɛ = 2I (log K; x, t)t. In the regime δ = ɛ 4, we get lim ɛ 0 + σ 2 ɛ = σ 2. In the regime δ = ɛ 2, we need to first compute the quantity H 0 defined as the limit of a log moment. This can be computed for the Heston model i.e. when σ(y) = y and β = 1/2.
Option pricing Implied Volatility A LDP and applications to option pricing and implied volatility for the Heston Model are in Feng, Forde and Fouque [1]. Fig. 2.1. Here we have plotted Λ, Λ, and the implied volatility in the small-ǫ limit as a function Here x = log(k/s of the log-moneyness 0 ). Take x = log(k/s0). σ(y) The= parameters y and are t = the 1, ergodic parameters mean θ =.04, convexity are ν/κ = 1.74 (κ 1.15, ν =.2), and skew ρ =.4 (dashed blue), ρ = 0 (solid black), ρ = +.4 (dotted t = 1, β = 1/2, red). m =.04, ν = 1.74 and ρ =.4 (dashed blue), ρ = 0 (solid black), ρ = +.4 (dotted red). converges to zero (we referrohini to [10] Kumar for details).
Rate functions (δ = ɛ 4 ) I (x ; x, t) = x x 2 2 σ 2 t is the rate function for {Z ɛ,t } ɛ>0 where Z ɛ, satisfies Z ɛ,0 = x and (Z ɛ,t = log S BS ɛ,t ) t Z ɛ,t = x + ɛ (r 12 ) σ2 ds + t ɛ σdw s 0 0
Rate functions (δ = ɛ 4 ) I (x ; x, t) = x x 2 2 σ 2 t is the rate function for {Z ɛ,t } ɛ>0 where Z ɛ, satisfies Z ɛ,0 = x and (Z ɛ,t = log S BS ɛ,t ) t Z ɛ,t = x + ɛ (r 12 ) σ2 ds + t ɛ σdw s 0 0 i.e. in this regime, the mean-reversion of Y is so fast that σ(y ( )) gets averaged to σ and the stock price behaves effectively like the Black-scholes model with constant volatility σ.
Rate functions (δ = ɛ 2 ) { I (x ; x, t) = t sup p p R ( x x t ) } H 0 (p)
H 0 H 0 (p) = lim T 1 log E[e 1 R T 2 p2 0 σ2 (Y p s )ds Y p 0 T + = y]; Process Y p is multiplicative ergodic (a strong enough ergodic property that the above limit exists and is independent of Y p 0 = y), see Kontoyiannis and Meyn [4] for definition of multiplicative ergodicity.
H 0 (p) = lim T 1 log E[e 1 R T 2 p2 0 σ2 (Y p s )ds Y p 0 T + ( p 2 ) = sup σ 2 dµ J(µ; p). µ P(R +) 2 R + = y]} Where J( ; p) is the rate function for the LDP of the occupation measures {µ T ( )} T 0 : µ T (A) = 1 T T 0 1 {Y p s A}ds average amount of time Y p spends in set A. (See Stroock [5].)
Y p is a mean-reverting and ergodic process. As ɛ 0, the distribution of Yɛ,t p approaches its invariant distribution. J( ; p) measures the cost of deviation of Yɛ,t p from its invariant distribution.
Rate functions (δ = ɛ 2 ) I (x ; x, t) = t sup inf p R µ P(R +) { p ( x x t ) p 2 2 } σ 2 dµ + J(µ; p) R +
Rate functions (δ = ɛ 2 ) I (x ; x, t) = t sup inf p R µ P(R +) { p ( x x t ) p 2 2 } σ 2 dµ + J(µ; p) R + The deviation {X ɛ,t > x } is caused by a perturbation of Y ɛ, to Y p then Yɛ, p deviates from its invariant distribution. ɛ, and