Sensitivity Analysis on Long-term Cash flows Hyungbin Park Worcester Polytechnic Institute 19 March 2016 Eastern Conference on Mathematical Finance Worcester Polytechnic Institute, Worceseter, MA 1 / 49
Table of contents Introduction Martingale extraction Sensitivity Analysis Examples of Options Examples of Expected Utilities Examples of LEFTs Conclusion 2 / 49
Introduction 3 / 49
Introduction Related articles Borovicka, J., Hansen, L.P., Hendricks, M., Scheinkman, J.A.: Risk price dynamics. Journal of Financial Econometrics 9(1), 3-65 (2011) Fournie, E., Lasry J., Lebuchoux, J., Lions P., Touzi, N.: Applications of Mallivin calculus to Monte Carlo methods in finance. Finance Stoch. 3, 391-412 (1999) Hansen, L.P., Scheinkman, J.A.: Long-term risk: An operator approach. Econometrica 77, 177-234 (2009) Hansen, L.P., Scheinkman, J.A.: Pricing growth-rate risk. Finance Stoch. 16(1), 1-15 (2012) 4 / 49
Introduction Let W t = (W 1 (t), W 2 (t),, W d (t)) be a standard d-dimensional Brownian motion. Assumption An underlying process X t is a conservative d-dimensional time-homogeneous Markov diffusion process with the following form: dx t = b(x t ) dt + σ(x t ) dw t, X 0 = ξ. Here, b is a d-dimensional column vector and σ is a d d matrix with some technical conditions. 5 / 49
Introduction In finance, we often encounter the quantity of the form: p T := E Q [e T 0 r(xt) dt f (X T )]. Purpose: to study a sensitivity analysis for the quantity p T with respect to the perturbation of X t for large T. This sensitivity is useful for long-term static investors and for long-dated option prices. 6 / 49
Introduction Let Xt ɛ be a perturbed process of X t (with the same initial value ξ = X 0 = X0 ɛ ) of the form: dx ɛ t = b ɛ (X ɛ t ) dt + σ ɛ (X ɛ t ) dw t. (1.1) The perturbed quantity is given by p ɛ T := EQ [e T 0 r(x ɛ s ) ds f (X ɛ T ) ]. For the sensitivity analysis, we compute ɛ pt ɛ ɛ=0 and investigate the behavior of this quantity for large T. 7 / 49
Introduction For the sensitivity w.r.t. the initial value X 0 = ξ, we compute p T ξ and investigate the behavior of this quantity for large T. 8 / 49
Martingale extraction 9 / 49
Martingale extraction We denote the infinitesimal generator corresponding to the operator f p T = E Q [e T 0 r(xt) dt f (X T )] by L : L := 1 2 where a = σσ. d 2 a ij (x) + x i x j i,j=1 d i=1 b i (x) x i r(x) 10 / 49
Martingale extraction Let (λ, φ) be an eigenpair of Lφ = λφ with positive function φ. It is easily checked that is a local martingale. M t := e λt t 0 r(xs)ds φ(x t ) φ 1 (ξ) Definition When the local martingale M t is a martingale, we say that (X t, r) admits the martingale extraction with respect to (λ, φ). e t 0 r(xs)ds = M t e λt φ 1 (X t ) φ(ξ) : martingale M t is extracted from the discount factor 11 / 49
Martingale extraction When M t is a martingale, we can define a new measure P by P(A) := A M T dq = E Q [I A M T ] for A F T, that is, M T = dp dq The measure P is called the transformed measure from Q with respect to (λ, φ). p T can be expressed by FT p T = E Q [e T 0 r(xs)ds f (X T )] = φ(ξ) e λt E Q [M t (φ 1 f )(X t )] = φ(ξ) e λt E P [(φ 1 f )(X T )]. 12 / 49
Martingale extraction We have p T = E Q [e T 0 r(xs)ds f (X T )] = φ(ξ) e λt E P [(φ 1 f )(X T )]. This relationship implies that the quantity p T can be expressed in a relatively more manageable manner. The term E P [(φ 1 f )(X T )] depends on the final value of X T, whereas E Q [e T 0 r(xs)ds f (X T )] depends on the whole path of X t at 0 t T. This advantage makes it easier to analyze the sensitivity of long-term cash flows. 13 / 49
Martingale extraction In general, there are infinitely many ways to extract the martingale. We choose a special one. Definition Consider a martingale extraction such that E P [(φ 1 f )(X T )] converges to a nonzero constant as T. We say this is a martingale extraction stabilizing f. In this case, 1 T ln p T = λ. For example, if X t has an invariant distribution µ under P, then E P [(φ 1 f )(X T )] (φ 1 f )(z) dµ(z) for suitably nice f. 14 / 49
Sensitivity Analysis 15 / 49
Sensitivity Analysis The rho and the vaga Let Xt ɛ be a perturbed process of X t (with the same initial value ξ = X 0 = X0 ɛ ) of the form: dx ɛ t = b ɛ (X ɛ t ) dt + σ ɛ (X ɛ t ) dw t. (3.1) The perturbed quantity is given by p ɛ T := EQ [e T 0 r(x ɛ s ) ds f (X ɛ T ) ]. For the sensitivity analysis, we compute ɛ ln pt ɛ ɛ=0 and investigate the behavior of this quantity for large T. 16 / 49
Sensitivity Analysis Assume that (X ɛ t, r) also admits the martingale extraction that stabilizes f, then p ɛ T = φ ɛ(ξ) e λ(ɛ)t E Pɛ [(φ 1 ɛ f )(X ɛ T )]. Differentiate with respect to ɛ and evaluate at ɛ = 0, then ɛ ɛ=0 pt ɛ ɛ = ɛ=0 φ ɛ (ξ) λ ɛ (0) + ɛ=0 E P [(φ 1 ɛ f )(X T )] T p T T φ(ξ) T E P [(φ 1 f )(X T )] + ɛ ɛ=0 E Pɛ [(φ 1 f )(XT ɛ )] T E P [(φ 1 f )(X T )] Here, E P [(φ 1 f )(X T )] (a nonzero constant) as T.. 17 / 49
Sensitivity Analysis The long-term behavior of the rho and the vaga Under some conditions, the first, third and the last terms go to zero as T, thus we have 1 T ɛ ln pt ɛ = ɛ ɛ=0 pt ɛ = λ (0) ɛ=0 T p T 18 / 49
Sensitivity Analysis The dynamics under the transformed measure Let ϕ := σ φ φ (Girsanov kernel) then t B t := W t ϕ(x s ) ds 0 is a Brownian motion under P. The P-dynamics of X t are dx t = b(x t ) dt + σ(x t ) dw t = (b(x t ) + σ(x t )ϕ(x t )) dt + σ(x t ) db t. 19 / 49
Sensitivity Analysis The rho Want to control: 1 T The perturbed process X ɛ t expressed by ɛ ɛ=0 E Pɛ [(φ 1 f )(XT ɛ )] 0 as dx ɛ t = b ɛ (X ɛ t ) dt + σ(x ɛ t ) dw t = (σ 1 b ɛ + ϕ ɛ )(X t ) dt + σ(x ɛ t ) db ɛ t = k ɛ (X ɛ t ) dt + σ(x ɛ t ) db ɛ t. 20 / 49
Sensitivity Analysis The rho Assume that there exists a function g : R d R with k ɛ (x) ɛ g(x) on (ɛ, x) I R d for an open interval I containing 0 such that (i) there exists a positive number ɛ 0 such that T )] E [exp (ɛ P 0 g 2 (X t ) dt c e at 0 for some constants a and c on 0 < T <. (ii) for each T > 0, there is a positive number ɛ 1 such that E P T 0 g 2+ɛ 1 (X t ) dt is finite. (iii) 1 T EP [(φ 1 f ) 2 (X T )] 0 as T. 21 / 49
Sensitivity Analysis Then, E Pɛ [(φ 1 f )(XT ɛ )] is differentiable at ɛ = 0 and 1 T ɛ E Pɛ [(φ 1 f )(XT ɛ )] 0. ɛ=0 22 / 49
Sensitivity Analysis The vega One way: The method of Fournie et. al. with bounded-derivative coefficients Fournie et. al.: Applications of Mallivin calculus to Monte Carlo methods in finance. inance Stoch. 3, 391-412 (1999) The perturbed process X ɛ t : The P ɛ dynamics of X ɛ t are dx ɛ t = b(x ɛ t ) dt + (σ + ɛσ)(x ɛ t ) dw t dx ɛ t = (b + (σ + ɛσ)ϕ ɛ )(X ɛ t ) dt + (σ + ɛσ)(x ɛ t ) db ɛ t 23 / 49
Sensitivity Analysis The vega We take apart two perturbations by the chain rule. dx ρ t = (b + (σ + ρσ)ϕ ρ )(X ρ t ) dt + σ(x ρ t ) db ρ t, dx ν t = (b + σϕ)(x ν t ) dt + (σ + νσ)(x ν t ) db ν t. Then we have ɛ E Pɛ [(φ 1 f )(XT ɛ )] ɛ=0 = ρ E Pρ [(φ 1 f )(X ρ T )] + ρ=0 ν E Pν [(φ 1 f )(XT ν )]. ν=0 24 / 49
Sensitivity Analysis Let Z t be the variation process given by dz t = (b + σϕ) (X t )Z t dt + σ(x t )db t + d σ i(x t )Z t db i,t, Z 0 = 0 d i=1 where σ i is the i-th column vector of σ and 0 d is the d-dimensional zero column vector. Theorem Suppose that b + σϕ and φ 1 f are continuously differentiable with bounded derivatives. If 1 T EP [ Z T ] 0 as T, then 1 T ν E Pν [(φ 1 f )(XT ν )] = 0. ν=0 25 / 49
Sensitivity Analysis The vega One way : the Lamperti transform for univariate processes The perturbed process is expressed by dxt ɛ = b ɛ (Xt ɛ ) dt + σ ɛ (Xt ɛ ) dw t, X0 ɛ = X 0 = ξ, (3.2) Define a function u ɛ (x) := x ξ σ 1 ɛ (y) dy. (3.3) 26 / 49
Sensitivity Analysis The vega Then we have du ɛ (X ɛ t ) = (σ 1 ɛ b ɛ 1 2 σ ɛ)(x ɛ t ) dt + dw t, u ɛ (X ɛ 0) = u ɛ (ξ) = 0. Set U ɛ t := u ɛ (X ɛ t ), then du ɛ t = δ ɛ (U ɛ t ) dt + dw t, U ɛ 0 = 0, (3.4) where δ ɛ ( ) := (( σ 1 ɛ b ɛ 1 2 σ ɛ) vɛ ) ( ). 27 / 49
Sensitivity Analysis The delta Let p T := E Q ξ [e T 0 r(xs)ds f (X T )] The quantity of interest is for large T From the martingale extraction ξ p T. p T = E Q [e T 0 r(xs)ds f (X T )] = φ(ξ) e λt E P [(φ 1 f )(X T )], it follows that ξ p T p T = ξ φ φ(ξ) + ξ E P ξ [(φ 1 f )(X T )] E P. ξ [(φ 1 f )(X T )] 28 / 49
Sensitivity Analysis The long-term behavior of the delta We have if ξ p T p T = ξ φ φ(ξ) ξ E P ξ [(φ 1 f )(X T )] 0. 29 / 49
Sensitivity Analysis Let Y t be the first variation process defined by dy t = (b + σϕ) (X t )Y t dt + d σ i(x t )Y t db i,t, Y 0 = I d where σ i is the i-th column vector of σ and I d is the d d identity matrix. Corollary Assume that the functions b + σϕ and σ are continuously differentiable with bounded derivatives. If E P ξ (φ 1 f ) 2 (X T ) and E P ξ σ 1 (X T )Y T 2 are bounded on 0 < T <, then E P ξ [(φ 1 f )(X T )] is differentiable by ξ and ξ E P ξ (φ 1 f )(X T ) 0 as T. Here, is the matrix 2-norm. i=1 30 / 49
Examples of Options 31 / 49
Examples of Options The CIR short-interest rate model Under a risk-neutral measure Q, the interest rate r t follows with 2θ > σ 2. dr t = (θ ar t ) dt + σ r t dw t The short-interest rate option price p T := E Q [e T 0 rt dt f (r T )] Want to know the behavior for large T of p T θ, p T a, p T σ 32 / 49
Examples Assume f (r) is a nonnegative continuous function on r [0, ), which is not identically zero, and that the growth rate at infinity is equal to or less than e mr with m < a σ 2. The associated second-order equation is Lφ(r) = 1 2 σ2 rφ (r) + (θ ar)φ (r) rφ(r) = λφ(r). a 2 +2σ 2 a Set κ :=. It can be shown that the martingale σ 2 extraction with respect to (λ, φ(r)) := (θκ, e κr ) stabilizes f. 33 / 49
Examples For large T, we have that 1 T ln p T = θκ, 1 T ln p T = θ 1 T ln p T a 1 T ln p T σ r 0 ln p T = a 2 + 2σ 2 a σ 2, = θ( a 2 + 2σ 2 a) σ 2 a 2 + 2σ 2, = θ( a 2 + 2σ 2 a) 2 σ 3, a 2 + 2σ 2 a 2 + 2σ 2 a σ 2. 34 / 49
Examples The quadratic term structure model Let X t be a d-dimensional OU process under the risk-neutral measure Q dx t = (b + BX t ) dt + σ dw t where b is a d-dimensional vector, B is a d d matrix, and σ is a non-singular d d matrix. The short interest rate : r(x) = β + α, x + Γx, x where the constant β, vector α and symmetric positive definite Γ are taken to be such that r(x) is non-negative for all x R d. 35 / 49
Examples Interested in: p T = E Q [e T 0 r(xt) dt f (X T )] with suitably nice payoff function f. Let V be the stabilizing solution (the eigenvalues of B 2aV have negative real parts) of 2VaV B V VB Γ = 0, and let u = (2Va B ) 1 (2Vb + α). The martingale extraction stabilizes f is (λ, φ(x)) = (β 1 2 u au + tr(av ) + u b, e u,x Vx,x ). 36 / 49
Examples We have that for 1 i, j d, 1 T ln p T = β + 1 2 u au tr(av ) u b, 1 T ln p T = λ b i b i, 1 T ln p T B ij = λ, B ij 1 T ln p T σ i = λ, σ i ξ p T p T = u 2V ξ. 37 / 49
Examples of Expected Utilities 38 / 49
Examples The geometric Brownian motion Assume that S t satisfies ds t = µs t dt + σs t dw t with µ 1 2 σ2 > 0. Let Q be the objective measure. Interested in p T = E Q [e rt S α T ]. The corresponding infinitesimal generator: (Lφ)(s) = 1 2 σ2 s 2 φ (s) + µsφ (s) rφ(s). The stabilizing martingale extraction: (λ, φ(s)) := (r µα 1 2 σ2 α(α 1), s α ) 39 / 49
Examples The geometric Brownian motion With this (λ, φ), we conclude that 1 T ln p T = r + µα + 1 2 σ2 α(α 1), 1 T µ ln p T = λ µ = α, 1 T σ ln p T = λ = σα(α 1), σ ln p T = φ (S 0 ) S 0 φ(s 0 ) = α. S 0 40 / 49
Examples The Heston model An asset X t follows dx t = µx t dt + v t X t dz t, dv t = (γ βv t ) dt + δ v t dw t, where Z t and W t are two standard Brownian motions with Z, W t = ρt for the correlation 1 ρ 1. Interested in: p T := E Q [u(x T )] = E Q [X α T ] 41 / 49
Examples The Heston model 1 T µ ln p T = α 1 T γ ln p T = 1 (β ραδ) 2 α(1 α) 2 + δ 2 α(1 α) β + ραδ δ 2 1 (β ραδ) T β ln p T = 2 + δ 2 α(1 α) β + ραδ δ 2 (β ραδ) 2 + δ 2 α(1 α) 42 / 49
Examples The Heston model 1 T 1 T (β ραδ) δ ln p T = ρα 2 + δ 2 α(1 α) β + ραδ δ 2 (β ραδ) 2 + δ 2 α(1 α) ρ ln p T = α ln p T = α X 0 X 0 ln p T v 0 + ( (β ραδ) 2 + δ 2 α(1 α) β + ραδ) 2 δ 3 (β ραδ) 2 + δ 2 α(1 α) (β ραδ) 2 + δ 2 α(1 α) αβ + ρα 2 δ δ (β ραδ) 2 + δ 2 α(1 α) = 1 2 α(1 α) (β ραδ) 2 + δ 2 α(1 α) β + ραδ δ 2. 43 / 49
Examples of LEFTs 44 / 49
Examples of LEFTs The sensitivity analysis of the expected utility and the return of an exchange-traded fund (ETF) is explored. The leveraged ETF (LETF) L t can be written by ( ) L β t Xt β(β 1) r(β 1)t t = e 2 0 σ2 (X u)/xu 2 du. L 0 X 0 We consider a power utility function of the form u(c) = c α, 0 < α 1. Interested in the sensitivity analysis of p T := E Q [u(l T )] 45 / 49
Examples of LEFTs The 3/2 model The dynamics of X t follows the 3/2 model dx t = (θ ax t )X t dt + σx 3/2 t dw t with θ, a, σ > 0 under the objective measure Q. The expected utility and the return of LETF is p T := E Q [u(l T )] = E Q [e αβ(β 1)σ2 2 Interested in the behavior for large T of T 0 Xu du X αβ T ] e rα(β 1)T. p T θ, p T a, p T σ 46 / 49
Examples of LEFTs The 3/2 model The corresponding infinitesimal operator is 1 2 σ2 x 3 φ (x) + (θ ax)xφ (x) 1 2 αβ(β 1)σ2 xφ(x) = λφ(x). Set l := (1 2 + a σ 2 ) 2 ( 1 + αβ(β 1) 2 + a ) σ 2 It can be shown that the martingale extraction with respect to (λ, φ(x)) := (θl, x l) stabilizes f (x) := x αβ when β 3. 47 / 49
Examples of LEFTs The 3/2 model a If + 1 αβ > 0, then σ 2 (1 1 T ln p T = θ 2 + a σ 2 (1 1 T ln p T = θ 2 + a σ 2 ) 2 ( 1 + αβ(β 1) 2 + a ) σ 2, ) 2 ( 1 + αβ(β 1) 2 + a ) σ 2, 1 T ln p T a 1 T ln p T σ = = θ ( ( 1 2 + a σ 2 ) 2 + αβ(β 1) ( σ 2 + a)) σ 2 ( 1 2 + a σ 2 ) 2 + αβ(β 1), 2aθ ( ( 1 2 + a σ 2 ) 2 + αβ(β 1) ( 1 2 + a σ 2 ) ) σ 3 ( 1 2 + a σ 2 ) 2 + αβ(β 1). 48 / 49
Conclusion Thank you! 49 / 49