An integral representation of elasticity and sensitivity ...

An integral representation of elasticity and sensitivity for stochastic volatility models

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2

Zhenyu Cui

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∗

Duy Nguyen

†

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October 12, 2017

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Abstract This paper presents a generic probabilistic approach to study elasticities and sensitivities of financial quantities under stochastic volatility models. We describe the shock elasticity, the quantile sensitivity and the vaga value of cash flows with respect to perturbation of the volatility function of the model. The main contribution is to establish explicit formulae for these elasticities and sensitivities based on a novel application of the exponential measure change technique in Palmowski and Rolski (2002). We carry out explicit calculations for the Heston model and the 3/2 stochastic volatility model, and derive explicit expressions in terms of model parameters.

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Key-words: Sensitivity; elasticity; growth-rate risk; quantile; Greeks; exponential measure change; stochastic volatility models.

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Hyungbin Park‡

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Introduction

In financial mathematics and economics, a sensitivity analysis is the study of how changes in certain model parameters affect the financial quantities that depend on the parameters. Research in sensitivities is an important direction in the literature and has attracted the attention of researchers from both academia and industry. See, for example, Glasserman (2003) for methods, theory, and applications of sensitivities through simulation. As pointed out by Glasserman (2003, p.377): “whereas the (option) prices themselves can often be observed in the market, their sensitivities can not, so accurate calculation of sensitivities is arguably even more important than the calculation of prices”. In financial engineering, sensitivities of derivative security prices (i.e. Greeks) with respect to various parameters are relevant for the risk management of derivatives products. This paper performs the sensitivity analysis for stochastic volatility models. Stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed and are ∗

Corresponding author. School of Business, Stevens Institute of Technology, 1 Castle Point on Hudson, Hoboken, NJ. Email: [email protected]. † Department of Mathematics, Marist College, 3399 North Road, Poughkeepsie, NY. Email: [email protected] ‡ Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA. Email: [email protected], [email protected].

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used for modeling asset returns among practitioners and academics. They have been developed to model asset price returns to improve the pricing and hedging performance of the classical Black-Scholes model, and in particular, to account for certain imperfections in it, e.g. the inability to incorporate the volatility smile. An important and well-documented empirical feature of the stochastic volatility models can be found in Yu (2005). It is known that the correlation coefficient in the stochastic volatility model reflects the leverage effect, and is negative in practice when the model is used to model the equity market. In this paper, we consider a stochastic volatility model with a general correlation structure. The main purpose of the present paper is to explore the sensitivity of the volatility function in various circumstances. One of the most commonly used stochastic volatility component is of the form m(Vt ), which is a function of another stochastic process Vt . Volatility models of this type such as the Heston model are extensively used because of its analytical tractability. We are mainly interested in the sensitivity with respect to the perturbation on the volatility function m(·). When the volatility function is given by the form of m (·) := m(·) + γ (·) with perturbation parameter , we will study how the financial quantities are affected by a small change of . Three topics will be mainly discussed in the present paper: the shock elasticity, the quantile sensitivity and the vega value. First, we investigate the shock elasticity, which characterizes the riskreturn tradeoff. The shock elasticity explains how the expected rates of return over a small interval are altered in response to changes in the exposure to the underlying shocks that impinge on the economy. To account for realistic characterizations, several information structures (e.g. Black-Scholes model, stochastic volatility model) have been proposed and the characterization of the risk-return tradeoff has attracted a good amount of attention from researchers in academia and industry. For relevant literature, see for example, Hansen (2008), Hansen and Scheinkman (2009, 2012), and references therein. We will explore this topic in Section 3.1 in the same context of their papers with focus on the perturbation of the volatility function. Second, we carry out an analysis of the sensitivity of the quantile. The quantile or the value-at-risk is widely employed in the area of risk management and optimal investment. It is useful in measuring the probability that an asset price exceeds the prescribed threshold, and refer to Alexander et al. (2012), Pham (2003) and references therein. This paper studies how the quantile is sensitive to the perturbation of the volatility function. This issue will be discussed in Section 3.2. Third, we measure the vega value of an European call option. The Greeks are the sensitivities of the option prices with respect to market parameters, and they play an important role in financial risk management. The Greeks measure the market risk in an option position, and are managed by traders so that all risks are acceptable. They are also actively used in constructing hedging positions for financial derivatives. Popular Greeks include delta, gamma and vega, and for related literature, refer to Fourni´e et al. (1999), Hong et al. (2014) and references therein. The present paper deals with the vega value of a call option in order to investigate the susceptibility on the volatility function m(·) when the underlying process is given by a stochastic volatility model. The main idea of this paper is based on the likelihood ratio method (LRM) (see Glasserman (2003) for details) and is described as follows. Many financial quantities such as expected returns, quantiles and option prices are expressed as expectations of stochastic processes. If one can find the density function of the stochastic process, the expectation can be explicitly obtained, thus the sensitivity quantity can be directly analyzed. We will show that the expected R t 2return, the R t quantile and the option price can be expressed by the joint density function of (Vt , 0 m (Vs ) ds, 0 h(Vs ) ds) with a specific function h depending on each model. This is discussed in Section 4.1 with a method

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of exponential measure change. When the volatility function is perturbed by the form m (·) with a perturbation parameterR, the perturbed R t financial quantities can be expressed by the perturbed joint t density function of (Vt , 0 m2 (Vs ) ds, 0 h (Vs ) ds). For a moment, denote the perturbed joint density by q (v, y, z). Then the perturbed financial quantities can be expressed by the integral form Z ρ() := F (v, y, z)q (v, y, z) dvdydz with a function F (v, y, z) depending on each financial quantity. For the sensitivity analysis, we are interested in ρ0 (0), which is given by   Z ∂ 0 ρ (0) = F (v, y, z) (1) q (v, y, z) dvdydz, ∂ =0 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

89 90 91 92 93

94 95

assuming that the integration and differentiation orders can be interchanged. From this equality, we can directly obtain the sensitivity with respect to the perturbation of the volatility function m (·) by ∂ |=0 q (v, y, z). evaluating the derivative of the perturbed density ∂ This approach to the sensitivity analysis has many advantages. First, we can find the explicit formula for the sensitivity. It is because the sensitivity can be directly obtained from Eq.(1) when the density function q (v, y, z) is given. Second, our method can be applied to more general types of volatility functions. Compared to Fourni´e et al. (1999), which is based on the Malliavin calculus, our assumptions stated after Eq.(2) are weaker. RT RT Then we discuss how to find the joint density function of (VT , 0 m2 (Vs )ds, 0 h (Vs )ds) through an exponential measure change for time-homogeneous diffusions from Hurd and Kuznetsov (2008), which is a specialization of the more general form for c´adlag Markov processes that appears in Palmowski and Rolski (2002). Section 4.1 demonstrates how to calculate this joint density function. The main idea is to identify an auxiliary diffusion process (for example, the standard Bessel process) under which the joint density of the terminal value process and some integral functionals is known in closed-form, say, from Borodin and Salminen (2002). Then we recover the equivalent measure change linking this auxiliary diffusion and the original diffusion process by characterizing the associated Radon-Nikodym derivative. Then we translate the study of the joint density for the original diffusion process to that of the auxiliary diffusion process under the new measure, whose closed-form expressions can be determined. This exponential change of measure technique has been applied in Hurd and Kuznetsov (2008) to determine closed-form expressions for the joint Laplace transform of functionals of the Cox-Ingersoll-Ross (CIR) process. It has also been applied in Cui and Nguyen (2016) to obtain the density function of the generalized Verhulst process and the Bessel process with constant drift (see also Linetsky (2004)). Our paper makes the following contributions to the current literature. 1. We obtain a conditional representation of the probability density function of a general correlated stochastic volatility model, based on which we derive explicit conditional representation of the elasticities and sensitivities. We also propose a generic probabilistic approach to explicitly calculate the elasticities and sensitivities for the Heston and 3/2 models through an exponential measure change. 2. In Boroviˇcka et al. (2014), they investigated the elasticities by employing the Malliavin calculus techniques, which typically requires that the diffusion and volatility coefficients have bounded 3

derivatives. Our approach to the shock elasticity generalizes1 the result of Boroviˇcka et al. (2014) because we require weaker assumptions on the drift and volatility of the processes.

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The rest of this paper is structured as follows. Section 2 introduces the stochastic volatility model. Section 3 presents the main results about the sensitivity analysis for three cases: the shock elasticity, the quantile and the vega value. Section 4 demonstrates how the main result can be applied to specific models such as the Heston model and the 3/2 model with explicit integral expressions. Section 5 concludes the paper with some future research directions.

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Stochastic volatility models

Let (Ω, F, (Ft )t∈[0,∞) , Q) be a filtered probability space with a right continuous filtration (Ft )t∈[0,∞) (i.e. Ft = Ft+ for t ∈ [0, ∞)). At this stage, we do not specify whether the measure Q is a physical measure or a risk-neutral measure. Its role will be specified at each section depending on the purposes. (1) (2) Assume that there is a two-dimensional Brownian motion (Wt , Wt ) on the probability space with E[dWt(1) dWt(2) ] = ρ dt and the correlation coefficient satisfies −1 6 ρ 6 1. Consider the following stochastic volatility model dSt (1) = r dt + m(Vt ) dWt , St (2) dVt = µ(Vt ) dt + σ(Vt ) dWt ,

(2) V0 = v0 ,

where Vt is the state variable, and its state space is J = (`, u), −∞ 6 ` < u 6 ∞. The functions µ(·) and σ(·) are continuously differentiable and σ(·) > 0 on J. We assume that the second stochastic differential equation (SDE) of V has a strong solution and that the boundaries ` and u are unattainable. The process St represents the cash flow. The volatility function m(·) is assumed to be continuously differentiable on J. The first SDE has a solution given by   Z t Z 1 t 2 (1) m(Vs ) dWs . m (Vs ) ds + St = S0 exp rt − 2 0 0 From the Cholesky decomposition, we can write (1)

dWt

(2)

= ρ dWt

+

p

(3)

1 − ρ2 dWt ,

with E[dWt dWt ] = 0 for some standard Brownian motion Wt independent of Wt . Introduce the following two auxiliary functions: Z x m(u) σ 2 (x) 00 g(x) := du, h(x) := µ(x)g 0 (x) + g (x), (3) σ(u) 2 . (2)

(3)

(3)

(2)

then from Itˆo’s lemma, we have (2)

dg(Vt ) = h(Vt )dt + m(Vt )dWt 1

.

See Section 3.1 for the single volatility factor case, and see Appendix A for a discussion on the possible extension to the case of multivariate volatility factors.

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Integrating both sides from 0 to T , it follows that T

Z g(VT ) − g(V0 ) =

Z h(Vs )ds +

111 112

m(Vs )dWs(2) .

(4)

0

0

110

T

Rt Rt Many financial quantities can be expressed by the joint density function of (Vt , 0 m2 (Vs ) ds, 0 h(Vs ) ds). For the remainder of this section, we demonstrate how the distribution of ST is evaluated from this joint density function. From Eq.(2) and (4), we obtain Z T Z Z T  p 1 T 2 (3) (2) 2 ST = S0 exp rT − m(Vs )dWs m (Vs )ds + ρ m(Vs )dWs + 1 − ρ 2 0 0 0 Z Z T Z T     p 1 T 2 2 = S0 exp rT − m (Vs )ds + ρ g(VT ) − g(V0 ) − h(Vs )ds + 1 − ρ m(Vs )dWs(3) . 2 0 0 0 RT 2 RT (3) (2) Since W is independent of W , we have that conditioning on (VT , 0 m (Vs )ds, 0 h(Vs )ds), the log asset price St is normally distributed, i.e. Z T Z T   2 m (Vs )ds, h(Vs )ds (5) log(ST ) VT , 0 0 Z T Z T Z     1 T 2 2 2 ∼ N log(S0 ) + rT − m (Vs )ds . h(Vs )ds , (1 − ρ ) m (Vs )ds + ρ g(VT ) − g(V0 ) − 2 0 0 0 

This result is useful to represent probability quantities such as the conditional moment generating function, the conditional expectation and the conditional density function. Recall that for a normal random variable X ∼ N (µ, σ 2 ), we have the identity E[eθX ] = exp(µθ + σ 2 θ2 /2). We can find that the conditional moment generating function is Z T Z T  i h 2 θ log(ST ) m (Vs )ds, h(Vs )ds Ee VT , 0 0    Z T  (1 − ρ2 )θ2 − θ Z T  2 = exp θ log(S0 ) + rT + ρ g(VT ) − g(V0 ) − h(Vs )ds + m (Vs )ds . 2 0 0 The conditional expectation is given by letting θ = 1 in the above equality, which yields Z T Z T h  i 2 E ST VT , m (Vs )ds, h(Vs )ds 0 0 Z T    (1 − ρ2 ) − 1 Z T  = exp log(S0 ) + rT + ρ g(VT ) − g(V0 ) − h(Vs )ds + m2 (Vs )ds . 2 0 0 In addition, the cumulative distribution function (CDF) of ST can be expressed by the joint CDF of RT RT (VT , 0 m2 (Vs )ds, 0 h(Vs )ds). Let F (v, y, z) be its joint CDF, that is,  Z F (v, y, z) = Q VT 6 v,

T 2

Z

m (Vs )ds 6 y,

0

 h(Vs )ds 6 z

0

5

T

.

Rx 1 2 Consider the normal CDF N (x) := √12π −∞ e− 2 t dt. From Eq.(5), we know that Z T Z T   2 Q ST 6 x VT , m (Vs )ds, h(Vs )ds 0 0    RT RT log(x/S0 ) − rT + 21 0 m2 (Vs )ds − ρ g(VT ) − g(V0 ) − 0 h(Vs )ds  , q =N  RT (1 − ρ2 ) 0 m2 (Vs )ds and this yields the following representation of the CDF of ST Z ∞ Z ∞Z u  Z T Z 2 Q(ST 6 x) = Q ST 6 x VT = v, m (Vs ) = y, −∞

Z

∞

0

Z

`

∞

Z

u

=

N −∞

0

`

T

 h(Vs )ds = z dF (v, y, z) 0 0 ! log(x/S0 ) − rT + y/2 − ρ(g(v) − g(v0 ) − z) p dF (v, y, z) . (1 − ρ2 )y

Thus the joint density function of ST is Z ∞ Z ∞Z u (log(x/S0 )−rT +y/2−ρ(g(v)−g(v0 )−z))2 1 − 2(1−ρ2 )y p e dF (v, y, z) . Q(ST ∈ dx) = 2 )y x 2π(1 − ρ −∞ 0 ` 113 114 115

(6)

The above integration is the Riemann-Stieltjes integration. In particular, if there exists a joint density function p(v, y, z) such that p(v, y, z) dudydz = dF (v, y, z), then we can evaluate the integral with the usual Riemann integral.

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Remark 2.1. Here we consider a single volatility factor. In Appendix A, we illustrate the extension to multiple volatility factors.

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Sensitivity analysis

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The explicit expression of the conditional distribution of St is useful in finding the sensitivity of financial or economic quantities. In this section, we investigate three kinds of sensitivities and illuminate how the explicit expression can be obtained.

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3.1

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Shock Elasticities

In Hansen and Scheinkman (2012), the authors characterized the compensation for incremental exposure to growth-rate risk. We regard the measure Q in Section 2 as a physical measure, and let the processes Vt and St in Eq.(2) be the state variable and the cash flow, respectively. The constant r in the drift term of St is the return of cash flow St . The cash flow St is considered as a reference growth process, which corresponds to the process Gt in Hansen and Scheinkman (2012). We introduce a discount factor Dt defined by the following SDE dDt (1) = β dt + δ(Vt )dWt , Dt

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(7)

where β is a real number and δ(·) is continuously differentiable on J. Here, we assume a restrictive (1) condition that the Brownian uncertainty Wt in the discount factor is the same as that modeled in cash flow St . 6

The growth rate risk induced by a perturbed cash flow St is of interest to us. The shock elasticity is defined by 1 ∂ 1 ∂  ρt := log E [S ] − log E[St Dt ], t t ∂ =0 t ∂ =0 and note that here we do not need to assume that the perturbed reference growth process St is a martingale.2 . We investigate this elasticity induced by the volatility perturbation, thus assume that the perturbed cash flow is of the form dSt (1) = rdt + (m(Vt ) + γd (Vt )) dWt , St

(8)

where r is the return of the cash flow and γd is a continuously differentiable function on J. Note that the function m(·) corresponds to the notation γg (·) in Hansen and Scheinkman (2012). For convenience of comparison to the results in Hansen and Scheinkman (2012), we introduce the following process Gt defined as Z Z t 1 t  (1) (m(Vu ) + γd (Vu ))2 du , log Gt := log S0 + (m(Vu ) + γd (Vu ))dWu − 2 0 0 then it can be checked that the shock elasticity ρt can be expressed by 1 ∂ ρt = − log E[Gt Dt ] . t ∂ =0 For simplicity of notations, we denote m (·) := m(·) + γd (·) and Gt := G0t . Its SDE is given by (1)

dGt = m (Vt )Gt dWt , G0 = S0 . 132 133 134 135 136 137

This form of cash flow was also investigated in Hansen and Scheinkman (2012), and refer to the last equation on page 4 of their paper. Here γd (·) reflects the direction of the perturbation parameterized by the parameter . The conditional distribution of Gt Dt will be later used to calculate the shock elasticity ρt . The conditional distribution can be expressed in a similar way as Eq.(5). From direct calculation, it follows that Z Z t Z t   p 1 t 2 2 (2)  (δ (Vs )+m (Vs ))ds+ρ (δ(Vs )+m (Vs ))dWs + 1 − ρ2 (δ(Vs )+m (Vs ))dWs(3) . Gt Dt = S0 exp βt− 2 0 0 0 (9) Define k (x) := δ(x) + m (x) , Z x k (u) g (x) := du , σ(u) . σ 2 (x) 00 h (x) := µ(x)g0 (x) + g (x) . 2  2

(10)

 Hansen and Scheinkman (2012) start with a perturbed reference growth process St being a martingale, then in 1 ∂ their setting, we have ρt = − t ∂ log E[St Dt ], and refer to the first equation on page 2 of Hansen and Scheinkman =0 (2012) for this definition of the shock elasticity (or the price of growth-rate risk)

7

Then Gt Dt

Z

t

1 =S0 exp βt + δ(Vs )m (Vs )ds − 2 0 Z  t p + 1 − ρ2 k (Vs ) dWs(3) . 

Z

t

k2 (Vs ) ds



Z

+ ρ g (VT ) − g (V0 ) −

T

 h (Vs )ds

0

0

0

The conditional expectation of Gt Dt conditional on Vt and the three integral functionals is given by Z t Z t  Z t i h 2   k (Vs )ds, h (Vs )ds, δ(Vs )m (Vs ) ds (11) Kt : = E Gt Dt Vt , 0 0 0 Z t Z t     ρ2 Z t 2 = S0 exp βt + ρ g (Vt ) − g (V0 ) − h (Vs )ds − δ(Vs )m (Vs )ds , k (Vs )ds + 2 0 0 0 138 139 140

and put Kt := Kt0 . From these observations, we obtain the following proposition on how to evaluate ρt , which is valid for arbitrary correlations −1 6 ρ 6 1. Proposition 3.1. We have the following representation of the shock elasticity R R R E[(ρ 0t γd (Vs )dWs(2) − ρ2 0t (δ(Vs ) + m(Vs ))γd (Vs )ds + 0t δ(Vs )γd (Vs )ds) · Gt Dt ] ρt = − . t E[Gt Dt ]

(12)

∂ | E[Gt Dt ] Proof. From the equality ρt = − ∂ t=0 E[Gt Dt ] , we only need to calculate the numerator. From Eq.(11), it follows that ∂ ∂  E[Gt Dt ] = E[Kt ] ∂ =0 ∂ =0 Z t Z t Z t h ∂    i 2 =E ρ g (Vt ) − g (V0 ) − h (Vs )ds − ρ k0 (Vs )γd (Vs )ds + δ(Vs )γd (Vs )ds Kt ∂ =0 0 Z t 0 Z t0 h Z t  i γd (Vs )dWs(2) − ρ2 =E ρ k0 (Vs )γd (Vs )ds + δ(Vs )γd (Vs )ds Kt ,

0

0

0

and for the last equality, we have used the following identity due to Itˆo’s lemma Z t Z t g (Vt ) − g (V0 ) − h (Vs )ds = k (Vs ) dWs(2) . 0 141

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0

Finally, the law of iterated expectation yields the desired result. This completes the proof. Remark 3.1. The expression in Eq.(12) is exactly the same as a special case of Eq.(3.1) on page 5 of Hansen and Scheinkman (2012) when the dimension n = 1 and ρ = 1. This shows that our result is consistent with that of Hansen and Scheinkman (2012). Remark 3.2. In the above derivation, we have assumed that there is a perfect correlation between the shocks to the asset St and the stochastic discount factor Dt , i.e. they share the same one(1) dimensional Brownian motion Wt . For the general multi-dimensional framework, refer to Hansen and Scheinkman (2012). 8

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It is straightforward to extend our analysis here to the case of arbitrary correlations. For exam˜ t(1) , which satisfies ple, if we assume that the stochastic discount factor Dt is instead driven by W p ˜ t(1) dWt(1) ] = ρ˜dt. From Cholesky decomposition, there is dW ˜ t(1) = ρ˜dWt(1) + 1 − ρ˜2 dWt(4) = E[dW p p (2) (3) (4) (4) (2) (3) ρ˜ρdWt + ρ˜ 1 − ρ2 dWt + 1 − ρ˜2 dWt , where Wt is independent from Wt and Wt . Thus we can derive an analogous representation as in Eq.(9), and consequently a similar result as Proposition 3.1 can be obtained. Rt (2) Occasionally, the stochastic integral 0 γd (Vs )dWs in Eq.(12) of Proposition 3.1 is hard to work with. In this case, the following equality can be useful. Z t Z t (2) γd (Vs )dWs = a(Vt ) − a(V0 ) − b(Vs ) ds, (13) 0

0

where Z x γd (u) ∂ a(x) := g (x) = du , ∂ =0 σ(u) . ∂ γd0 (x)σ(x) − γd (x)σ 0 (x) µ(x) γd (x) + . b(x) := h (x) = ∂ =0 σ(x) 2 157

3.2

(14)

Quantile sensitivity

In finance, the quantile is an important quantity for risk management and optimal investment. Regard the measure Q in Section 2 as a physical measure. The pth-quantile of a random variable X is defined as χ(p) := inf{x ∈ R : p 6 Q(X 6 x)} , 0 < p < 1. This section studies the sensitivity of the quantile of the stochastic volatility model with respect to the perturbation of the volatility function. Consider the following dynamics dSt (1) = r dt + m (Vt ) dWt , St (2)

dVt = µ(Vt ) dt + σ(Vt ) dWt ,

V0 = v0 ,

with m (·) := m(·) + γd (·) under the measure Q. Let χt, (p) := inf{x : p 6 Q(St 6 x)} be the pth-quantile of St and let χt (p) := χt,0 (p). We are interested in computing the sensitivity qt (p) := 158

∂ χt, (p) . ∂ =0

In the following, we use notation qt instead of qt (p) without ambiguity. The sensitivity qt can be expressed by using the density function of St . Let Ft, (x) := Q(St 6 x) be the cumulative distribution function of St . Clearly, Ft, (χt, (p)) = p and by taking the partial derivative with respect to  on both sides, we have ∂ ∂ 0 Ft, (χt (p)) + Ft (χt (p)) χt, (p) = 0 . ∂ =0 ∂ =0 9

Thus, in particular, if St has a density function ft, (·), we get Rx ∂ F (x) ∂ t, = − −∞ qt = χt, (p) = − ∂ =00 x=χt (p) ∂ =0 Ft (x) 159 160



∂ f (u) du ∂ =0 t,

ft (x)

x=χt (p)

.

Define ft (·) := ft,0 (·). The sensitivity qt can be directly computed from the density function ft, (·) similar to Eq.(6). Proposition 3.2. The sensitivity qt of the pth-quantile with respect to the perturbation stated above is given by Rx ∂ f (u) du −∞ ∂ =0 t, qt = − . x=χt (p) ft (x)

161

Here the density ft, (x) is Z ∞ Z ∞Z ft, (x) = −∞

0

`

u

1

−

p e x 2π(1 − ρ2 )y

(log(x/S0 )−rt+y/2−ρ(g (v)−g (v0 )−z))2 2(1−ρ2 )y

162

where p (v, y, z) is the joint density function of (Vt ,

163

3.3

Rt 0

m2 (Vs )ds,

Rt 0

p (v, y, z) dvdydz,

h (Vs )ds).

Vega of call options

As an application of the explicit formula of the conditional distribution discussed in Section 2, we explore the sensitivity of the call option price with respect to the volatility function m(·) in Eq.(2). This sensitivity is a practically important quantity when the asset price is modeled by a stochastic volatility model. We regard the underlying measure Q as a risk-neutral measure. The short interest rate is assumed to be a constant r. Consider the following dynamics of the perturbed asset dSt (1) = r dt + m (Vt ) dWt , St (2)

dVt = µ(Vt ) dt + σ(Vt ) dWt , 164 165 166 167 168

169 170 171 172 173

(15) V0 = v0 ,

with m (·) = m(·)+γd (·). Here, we used the same notation r with Eq.(8) for the drift term. However, in this context, the constant r is the short interest rate whereas r was used as return of the cash flow in Eq.(8). Even though the same notation was used, the readers should not feel confused. This section does not need to introduce the discount factor Dt in Eq.(7) because the underlying meausre Q is a risk-neutral measure. As is well-known, the perturbed call price with strike K is E[e−rt (St − K)+ ]. The main purpose of this section is to study the sensitivity of the call option price with respect to the perturbation on the volatility function of the underlying process. The quantity ∂ νt := E[e−rt (St − K)+ ] ∂ =0 is of interest to us. This quantity is also referred to as the vega value of the call option price. The following representation R t 2of the call R t option price in a stochastic volatility model is based on the joint distribution of (Vt , 0 m (Vs )ds, 0 h (Vs )ds) and the function g (·). Here, the functions h (·) and g (·) are R t self-explanatory from R t Eq.(3) with m(·) replaced by m (·). For notational simplicity, we let Mt := 0 m2 (Vs )ds and Ht = 0 h (Vs )ds. 10

Proposition 3.3. The perturbed call option price can be expressed by

E[e−rt (St − K)+ ] = S0 E[eα (t)+ 2 β (t) N (d1 )] − Ke−rt E[N (d2 )], 1

where 1 α (t) := ρ(g (Vt ) − g (V0 )) − ρHt − Mt , 2 β (t) := (1 − ρ2 )Mt 174

and d1 =

p log(S0 /K) + rt + α (t) + β (t)  p , d2 = d1 − β (t) . β (t)

Proof. We recall from Eq.(5) that   1      2  log St (Vt , Mt , Ht ) ∼ N log S0 + rt − Mt + ρ(g (VT ) − g (V0 )) − ρHt , (1 − ρ )Mt . 2 The perturbed call price conditional on (Vt , Mt , Ht ) is

E[e−rt (St − K)+ | Vt , Mt , Ht ] = S0 eρ(g (VT )−g (V0 ))−ρHt − 2 ρ Mt N (d1 ) − Ke−rt N (d2 ) 

1 2



1

= S0 eα (t)+ 2 β (t) N (d1 ) − Ke−rt N (d2 ) . 175

We obtain the desired result from the law of iterated expectation. This completes the proof. Proposition 3.4. We have that h   ∂α (t) 1 ∂d  ∂d i ∂β (t)   νt = E S0 eα0 (t)+ 2 β0 (t) N (d01 ) + Φ(d01 ) 1 − Ke−rt Φ(d02 ) 2 + ∂ =0 ∂ =0 ∂ =0 ∂ =0 where Φ(x) =

1 2 √1 e− 2 x . 2π

The partial derivatives are

Z t Z t ∂α (t) = ρ(a(Vt ) − a(V0 )) − ρ b(Vs ) ds − m(Vs )γd (Vs ) ds, ∂ =0 0 0 Z t ∂β (t) 2 = 2(1 − ρ) m(Vs )γd (Vs ) ds, ∂ =0 0 and  ∂d1 ∂



=



+

∂β (t) ∂ =0





β0 (t) + 12 (log(S0 /K) + rt + α0 (t) + β0 (t)) ∂β∂ (t) =0 β0 (t)3/2

=0

 ∂d2 = ∂ =0 176

∂α (t) ∂ =0

∂α (t) ∂ =0



−

∂β (t) ∂ =0





β0 (t) + 12 (log(S0 /K) + rt + α0 (t) − β0 (t)) ∂β∂ (t) =0 β0 (t)3/2

The proof is straightforward from Proposition 3.3.

11

,

.

177

4

Explicit Examples

178

4.1

Change of measure

RT RT From Eq.(6), the next task is to find the joint density function of (VT , 0 m2 (Vs )ds, 0 h(Vs )ds) under the measure Q. We provide two approaches. One method is to express the terminal value process and the two integral functionals as a three-dimensional process   Z t Z t 2 h(Vs )ds , m (Vs )ds, (Vt , Yz , Zt ) := Vt , 0

0

and then we can find the joint density function q = q(v, y, z; T ) from the Kolmogorov forward equation ∂q 1 ∂ 2 (σ 2 (v)q) ∂(µ(v)q) ∂q ∂q = − m2 (v) − h2 (v) , − 2 ∂T 2 ∂v ∂v ∂y ∂z lim q(v, y, z; T ) = δ(v0 ,0,0) . T →0

179 180 181 182 183 184

This method is valid for general circumstances. However, the explicit solution to the above partial differential equation, i.e. the joint density function, is not easy to find. We will illustrate the second method of exponential measure change forR time-homogeneous diffuRT T 2 sions. This method will be used to find the joint Q-density function of (VT , 0 m (Vs )ds, 0 h (Vs )ds) in, for example, the Heston model and the 3/2 model in Proposition 4.2 and Proposition 4.4, respectively. The procedure is as follows. 1. Begin with a process V under a measure P such that the dynamics dVt = µ ˜(Vt )dt + σ(Vt ) dWtP ,

V0 = v0 ,

and the joint P -density  Z P VT ∈ dv,

T

Z

2

 h(Vs )ds ∈ dz

m (Vs )ds ∈ dy, 0

0 185

T

are known, e.g. available from Borodin and Salminen (2002). Here, W P is a P -Brownian motion. 2. Choose any positive function f such that there exists a function Γ of three variables satisfying   Z T Z T  Z t LP f (V )  f (Vt ) s 2 exp − ds = Γ VT , m (Vs )ds, h(Vs )ds , f (v0 ) f (Vs ) 0 0 0 where LP is the generator of V defined by LP f (x) = µ ˜(x)

186 187

∂f (x) 1 2 ∂ 2 f (x) + σ (x) . ∂x 2 ∂x2

Intuitively, the value R T of the Radon R T Nikodym derivative is fully determined if we know the value of the triplet (VT , 0 m2 (Vs )ds, 0 h(Vs )ds). 3. By Proposition 4.1, under some technical conditions, we can define a measure Q by   Z T Z T dQ 2 m (Vs )ds, h(Vs )ds . = Γ VT , dP FT 0 0 12

4. Under the measure Q, the dynamics of V is   f 0 (Vt ) dVt = µ ˜(Vt ) + σ(Vt ) dt + σ(Vt ) dWtP , f (Vt ) and the joint Q-density function is given by  Z Z T 2 m (Vs )ds ∈ dy, Γ(v, y, z) · P VT ∈ dv,

191 192 193

 h(Vs )ds ∈ dz

,

which can be obtained by the way we choose the measure P .

188

190

T

0

0

189

V0 = v0

Let us recall the following result on exponential measure change for time-homogeneous diffusions from Proposition 2.1 of Hurd and Kuznetsov (2008), which is a specialization of the more general form for cadlag Markov processes that appears in Palmowski and Rolski (2002). The link between the true martingale property and explosion behaviors of an auxiliary diffusion has been studied in Sin (1998), Ruf (2015) and Bernard et al. (2017). Proposition 4.1. Let f ∈ C 2 (J) with f (x) > 0 for all x ∈ J. Assume that the measure Q can be obtained from the measure P through the Radon-Nikodym derivative  Z t LP f (V )  dQ f (Vt ) s Zt := exp − ds , t ∈ (0, ∞) . = dP Ft f (v0 ) f (V ) s 0 If the process V is non-explosive (i.e. the two boundaries l and u are unattainable) under both measures P and Q, then Zt is a true martingale3 under P (thus, P and Q are equivalent). The drift functions µ(x) and µ ˜(x) have the relationship µ ˜(x) − µ(x) = σ 2 (x)

f 0 (x) . f (x)

199

Remark 4.1. The general guiding principle is to choose the dynamics of the process under P to be a well-known process, e.g. in the Heston model considered below, we choose the dynamics in Eq.(17), which is the squared Bessel process. The reason that we choose this process is because that its only difference from the original CIR process is in the drift. The difference between the two drifts is given by κ(θ − x) − κθ = −κx. Based on this difference, we can use the relation in Proposition 4.1 to solve for the corresponding function f (·), then define measure P as in Proposition 4.1.

200

4.2

The Heston model

201

4.2.1

The shock elasticity

194 195 196 197 198

In the context of Section 3.1, we calculate the shock elasticity for the Heston model. The growth process St , the stochastic discount factor Dt and the state variable Vt are assumed to satisfy p (1) dSt = rSt dt + c1 Vt St dWt , p (1) dDt = βDt dt + Vt Dt dWt , p (2) dVt = κ(θ − Vt )dt + σ Vt dWt , V0 = v0 3

In the proof of part (c) of Theorem 1 in Cheridito et al. (2007), even though the proof of it needs to use results of part (a) and part (b), the results of part (a) and part (b) are only used to show that the SDE (57) in Cheridito et al. (2007) has a unique solution. Because of this, for the proof that E P [ZT ] = 1 where Z is constructed from possibly non-affine processes Vt , one can just repeat the proof of Theorem 1 of Cheridito et al. (2007) verbatim. This is in the spirit of the discussion after Proposition 2.1 in Hurd and Kuznetsov (2008).

13

under a measure Q with positive constants r, c1 , κ, θ, σ and a real number β. Here, the Feller condition 2κθ > σ 2 is assumed not attain its boundaries. The functions m(·) and δ(·) in Section √ so that Vt does√ 3.1 are m(x) = c1 x and δ(x) = x, respectively. We √mainly consider the shock elasticity with respect to the volatility function of St , thus let γd (x) = c2 x. The perturbed cash flow is p (1) dSt = rSt dt + (c1 + c2 ) Vt St dWt . The process Gt defined in Section 3.1 is p (1) dGt = (c1 + c2 ) Vt Gt dWt . 202 203

This section illustrates how to calculate explicitly the shock elasticity R t ρt discussed in Proposition 3.1. For this purpose, we first seek to find the joint distribution of (Vt , 0 Vs ds). Proposition 4.2. Let q(v, y) be the joint density function such that Z T   Vs ds ∈ dy . q(v, y) dvdy = Q VT ∈ dv, 0

Then we have √  8  v  ν2 2(v0 + v) 2 v0 v  − κ2 (v−v0 )+ κ22θ T − κ22 y σ 2σ q(v, y) = 4 is 4y2 ν, T, 0, , , ·e σ σ σ v0 σ2 σ2 204 205

where the special function isy (·) is defined on page 644 of Borodin and Salminen (2002) and ν = 2κθ/σ 2 − 1 > 0. κ

Proof. Choose f (x) = e− σ2 x , and define a measure P as in Proposition 4.1, then consider the squared Bessel process p dVt = κθdt + σ Vt dWtP (17) under a measure P. It can be easily checked that the measures P and Q are linked by this function f in Proposition 4.1. More specifically, the Radon-Nikodym derivative is  Z T LP f (V )  RT κ κ2 θ κ2 dQ f (VT ) s exp − ds = e− σ2 (VT −v0 ) e σ2 T − 2σ2 0 Vs ds . = dP FT f (v0 ) f (Vs ) 0 It follows that Z T   q(v, y) dvdy = Q VT ∈ dv, Vs ds ∈ dy 0 h dQ i = EP 1{VT ∈dv,R T Vs ds∈dy} 0 dP F i h κ T RT κ2 θ κ2 = EP e− σ2 (VT −v0 ) e σ2 T − 2σ2 0 Vs ds 1{VT ∈dv,R T Vs ds∈dy} 0 h i κ κ2 θ κ2 = e− σ2 (v−v0 )+ σ2 T − 2σ2 y EP 1{VT ∈dv,R T Vs ds∈dy} 0 Z T   κ κ2 θ κ2 − 2 (v−v0 )+ 2 T − 2 y σ σ 2σ =e P VT ∈ dv, Vs ds ∈ dy . 0

14

(18)

From Eq.(18), to find the joint density q(v, y), we just need to calculate the joint density P (VT ∈ RT √ dv, 0 Vs ds ∈ dy) where V is the squared Bessel process defined in Eq.(17). Define Zt := 2 Vt /σ. From Itˆo’s lemma, we have dZt =

206 207

− 12 dt + dWtP , Zt

2κθ σ2

and the process Zt is a standard Bessel process with order ν = 2κθ/σ 2 − 1 > 0. From Vt = σ 2 Zt2 /4, it can be obtained that Z T    2√v  Z T   4y  Vs ds ∈ dy = P ZT ∈ d P VT ∈ dv, Zs2 ds ∈ d 2 , σ σ 0 0 √ ν    8 v 2 2(v0 + v) 2 v0 v  = 4 · is 4y2 ν, T, 0, dvdy . , σ σ v0 σ2 σ2 RT For the second equality, we used the joint density function of (ZT , 0 Zs2 ds) known from the formula (4.1.9.8) on page 378 of Borodin and Salminen (2002). This completes the proof. Proposition 4.3. The shock elasticity ρt of the Heston model with respect to the perturbation above is At ρt = − , Bt where Z ∞Z ∞     ρκc ρc2 2 (v − v0 − κθt) + − ρ2 c2 (1 + c1 ) + c2 y At = σ σ 0 0  ρ(1 + c )  ρκ(1 + c ) ρ2 (1 + c )2   1 1 1 · exp v+ − + c1 y q(v, y) dvdy , σ 2 Z ∞Z ∞ σ     ρ(1 + c1 ) ρκ(1 + c1 ) ρ2 (1 + c1 )2 Bt = t exp v+ − + c1 y q(v, y) dvdy . σ σ 2 0 0

208 209

Proof. Proposition 3.1 is applied to find the shock elasticity. By direct calculation, the functions in Eq.(10) are √ k (x) = (1 + c1 + c2 ) x, 1 + c1 + c2 g (x) = x, (19) σ (1 + c1 + c2 )κ h (x) = (θ − x) . σ Combining these functions with Eq.(13) and (14), we have Z t Z t Z t 2 (2) (δ(Vs ) + m(Vs ))γd (Vs )ds + δ(Vs )γd (Vs )ds ρ γd (Vs )dWs − ρ 0 0 0 Z t Z t Z t   2 = ρ a(Vt ) − a(V0 ) − b(Vs )ds − ρ (δ(Vs ) + m(Vs ))γd (Vs )ds + δ(Vs )γd (Vs )ds 0 0 0  ρκc Z t ρc2 2 2 (Vt − V0 − κθt) + − ρ c2 (1 + c1 ) + c2 Vs ds. = σ σ 0

15

212

The process Ht , which is the conditional expectation of Gt Dt , is given by    ρκ(1 + c ) ρ2 (1 + c )2 Z t ρκθ(1 + c1 )  ρ(1 + c1 ) 1 1 Vs ds . Ht = S0 exp β − t+ (Vt − V0 ) + − + c1 σ σ σ 2 0 Here, Eq.(11) with  = 0 is used to derive this. From these two equalities and the joint density Rt function of (Vt , 0 Vs ds), we obtain the desired result. We note that  ρκθ(1 + c1 )  ρ(1 + c1 )  S0 exp β − t− V0 σ σ Rt is the common factor, which is independent of the integrating variables v = Vt and y = 0 Vs ds, in ∂ | E[Gt Dt ] the numerator and the denominator of ρt = − ∂ t=0 E[Gt Dt ] , thus we can cancel out this term. This completes the proof.

213

4.2.2

210 211

Quantile sensitivity

This section demonstrates the explicit calculation of the sensitivity on the quantile for the Heston model. Assume that under Q, the stock price has the following dynamics p (1) (20) dSt = rSt dt + (c1 + c2 ) Vt St dWt , p (2) dVt = κ(θ − Vt )dt + σ Vt dWt . Recall that χt, (p) = inf{x : Q(St 6 x) = p} is the pth-quantile of St and let χt (p) = χt,0 (p). Our goal is to determine ∂ qt = qt (p) = χt, (p) . ∂ =0 By Proposition 3.2, we know the density function of St in this Heston model and the quantile sensitivity qt . The density function is y ρ Z ∞Z ∞ (log(x/S0 )− c rT + − (v−v0 −κθT +κy))2 1 +c2 2 σ 1 − 2 )y 2(1−ρ p ft, (x) = e q(v, y) dvdy , 2 )y x 2π(1 − ρ 0 0 where

√  8  v  ν2 2(v0 + v) 2 v0 v  , q(v, y) = e · is 4y2 ν, T, 0, , σ σ 4 v0 σ2 σ2 and the special function isy (·) is defined on page 644 of Borodin and Salminen (2002), and ν = 2κθ/σ 2 − 1. The sensitivity qt of the pth-quantile is Rx ∂ ft, (u) du qt = − 0 ∂ =0 ft (x) 2 rT y ρ R x R ∞R ∞ 1 − (log(u/S0 )− c1 + 2 − σ2(v−v0 −κθT +κy)) c2 rT 2(1−ρ )y √ e q(v, y) dvdydu 2c2 (1−ρ2 ) 0 0 0 uy y =− 1 , y ρ rT R ∞R ∞ 1 − (log(x/S0 )− c1 + 2 − σ2(v−v0 −κθT +κy))2 2(1−ρ )y √ e q(v, y) dvdy 0 0 x y −

214 215

2 2 κ (v−v0 )+ κ 2θ T − κ 2 y σ2 σ 2σ

where x = χt (p), that is, x is the number such that y ρ 2 Z x Z xZ ∞Z ∞ (log(u/S0 )− rT c1 + 2 − σ (v−v0 −κθT +κy)) 1 − 2 2(1−ρ )y p p= ft (u) du = e q(v, y) dvdydu . u 2π(1 − ρ2 )y 0 0 0 0 16

216

4.2.3

Vega of call options

We evaluate the sensitivity on call option prices ∂ νt := E[e−rt (St − K)+ ] ∂ =0 in the context of Section 3.3 for the Heston model stated in Eq.(20). To apply Proposition 3.4, we evaluate the components of νt in the formula there. By using Eq.(19), the functions Mt , Ht , α (t), β (t) defined in Section 3.3 can be computed directly. We have that Z t 2  Vs ds Mt = (c1 + c2 ) 0 Z (c1 + c2 )κθt (c1 + c2 )κ t  Ht = − Vs ds σ σ 0 and  (c1 + c2 )ρ  (c1 + c2 )σ  α (t) = Vt − v0 − κθt + κ − σ 2ρ Z t β (t) = (1 − ρ2 )(c1 + c2 )2 Vs ds .

Z

t

 Vs ds

0

0

For notational simplicity, let (t, v, y) ≡ (t, Vt ,

Rt

Vs ds) and define  (c1 + c2 )ρ  (c1 + c2 )σ   α(t, v, y) := α (t) = v − v0 − κθt + κ − y , σ 2ρ  c1 σ   c1 ρ  v − v0 − κθt + κ − y , α0 (t, v, y) := α(t, v, y)|=0 = σ 2ρ  ∂α(t, v, y) c2 ρκθt  c2 ρκ c2 ρ α b0 (t, v, y) := (v − v0 ) − + − c1 c2 y, = ∂ σ σ σ =0 0

and β(t, v, y) := β (t) = (1 − ρ2 )(c1 + c2 )2 y, β0 (t, v, y) := β=0 (t) = (1 − ρ2 )c21 y, ∂β (t) βb0 (t, v, y) := = 2(1 − ρ2 )c1 c2 y . ∂ =0 Similarly, we can easily compute d01 , d02 ,

∂d1 | ∂ =0 S0 K

∂d2 | ∂ =0

from




+ rt + α (t, v, y) + β (t, v, y) p , β (t, v, y) p d2 (t, v, y) := d1 (t, v, y) − β  (t, v, y) .

d1 (t, v, y)

:=

log

and

(21)

Put ∂d ∂d db01 (t, v, y) := 1 , db02 (t, v, y) := 2 . ∂ =0 ∂ =0 217 218

(22)

Now we apply Proposition 3.4 to obtain the following theorem. Recall that q(v, y) is the joint Rt density function of (Vt , 0 Vs ds) given in Proposition 4.2. 17

Theorem 4.1. The sensitivity νt in the Heston model with respect to the perturbation above is Z ∞Z ∞      1 S0 eα0 (t,v,y)+ 2 β0 (t,v,y) N (d01 (t, v, y)) α νt = b0 (t, v, y) + βb0 (t, v, y) + Φ(d01 (t, v, y))db01 (t, v, y) 0 0  − Ke−rt Φ(d02 (t, v, y))db02 (t, v, y) q(v, y) dvdy 219

with functions α0 , β0 , α b0 , βb0 , d01 , d02 , db01 , db02 given above.

220

4.3

3/2 Stochastic Volatility Model

Under measure Q, consider the 3/2 stochastic volatility model p (1) dSt = rSt dt + c Yt St dWt 3 2

(2)

dYt = (pYt − qYt2 )dt − Yt dWt ,

222 223

Y0 = y0 ,

where E[dWt dWt ] = ρ dt with positive constants r, c, p, q, , y0 and q < 2 /2. The state space of Yt is J = (0, ∞) and from Feller’s test of explosions, we have that the condition q < 2 /2 (see also Carr and Sun (2007)) guarantees that Yt does not attain its boundaries under the measure Q. We can translate the 3/2 stochastic volatility model to a form similar to the Heston model with a different volatility function. Define Vt := 1/Yt , then from the Itˆo’s lemma, we have p (2) dVt = (2 + q − pVt )dt +  Vt dWt , V0 = 1/y0 . √ (2) The process Vt is a CIR process that can be rewritten as dVt = κ(θ − Vt )dt + σ Vt dWt with κ = p, θ = (2 + q)/p, σ = . Then we shall rewrite the model in Eq.(23) as p (1) dSt = rSt dt + c/ Vt St dWt p (2) dVt = κ(θ − Vt )dt + σ Vt dWt , V0 = v0 , (1)

221

(23)

(2)

224

with v0 = 1/y0 . We will work with this form instead of the one given in Eq.(23).

225

4.3.1

The shock elasticity

In the context of Section 3.1, we calculate the shock elasticity under the 3/2 model. We are interested in the shock elasticity with respect to the volatility function of St . The perturbed cash flow process St , the stochastic discount factor Dt and the state variable Yt are assumed to be p (1) dSt = rSt dt + (c1 + c2 )/ Vt St dWt , (24) p (1) dDt = βDt dt + 1/ Vt Dt dWt , p (2) dVt = κ(θ − Vt )dt + σ Vt dWt , V0 = v0 √ under a physical measure Q. √The functions m(·), γd (·) and δ(·) in Section 3.1 are m(x) = c1 / x, √ γd (x) = c2 / x and δ(x) = 1/ x, respectively. The corresponding process Gt is p (1) dGt = (c1 + c2 )/ Vt Gt dWt . 226 227

This section illustrates how to explicitly calculate the shockRelasticity ρt discussed in Proposition t 3.1. For this purpose, we first find the joint distribution of (Vt , 0 1/Vs ds). 18

Proposition 4.4. Let q(v, y) be the joint density function such that Z T   q(v, y) dvdy = Q VT ∈ dv, 1/Vs ds ∈ dy . 0

Then we have

ν  2κ√v v  −(v0 +v) κ2 ch(κT /2) 2 2 2 σ κ(v/v0 ) 2 0 − κ2 (v−v0 )+ κ 2θ T + − ν σ8 y sh(κT /2) σ i σ2 y 2 ·e σ q(v, y) = , 8sh(κT /2) 8 σ sh(κT /2)

228 229

with ν = 2κθ/σ 2 − 1 where ch(·) and sh(·) are the hyperbolic cosine and hyperbolic sine functions, respectively. The special function iy (·) is defined on page 644 of Borodin and Salminen (2002). Proof. We recall the proof of Proposition 4.2. The P -dynamics of the squared Bessel process Vt in − κ2 x Eq.(17) and the Q-dynamics of Vt in Eq.(24) were linked by the function f (x) = e σ in Proposition 4.1. The Radon-Nikodym derivative is given by 2 2 RT dQ − κ (V −v ) κ θ T − κ V ds = e σ2 T 0 e σ2 2σ2 0 s . dP FT It follows that Z T   1 ds ∈ dy q(v, y) dvdy := Q VT ∈ dv, 0 Vs i h P dQ R T 1 =E 1 dP F {VT ∈dv, 0 Vs ds∈dy} h κ T i 2 2 R − 2 (VT −v0 ) κ 2θ T − κ 2 0T Vs ds P R σ σ 2σ =E e e 1{VT ∈dv, 0T V1 ds∈dy} s h i R 2 2 − κ2 (v−v0 )+ κ 2θ T − κ 2 0T Vs ds P σ =e σ E e 2σ 1{VT ∈dv,R T 1 ds∈dy} . 0

Vs

√ To evaluate the expectation in the last equality, we recall that Zt := 2 Vt /σ is a standard Bessel √ process with order ν = 2κθ/σ 2 −1 and Z0 = 2 v0 /σ. From the formula (4.1.21.7) part (1) on page 387 √ of Borodin and Salminen (2002) with the corresponding substitutions: x ← 2 v0 /σ, q ← κ/2, y ← √ σ 2 y/4, z ← 2 v/σ, we have i h κ2 R T 2 h κ2 R T i − 2 0 Vs ds P − 8 0 Zs ds P R 2σ R E e 1{VT ∈dv, 0T V1 ds∈dy} = E e 1{ZT ∈d( 2 √v), T 1 ds∈( σ2 y )} s 0 Z2 σ 4 s √ ν   −(v0 +v) κ2 ch(κT /2) 2 2 σ 2κ v0 v κ(v/v0 ) 2 − ν σ8 y sh(κT /2) = i σ2 y 2 ·e dvdy. 8sh(κT /2) 8 σ sh(κT /2) 230

This completes the proof. Theorem 4.2. The shock elasticity ρt of the 3/2 model for the given perturbation above is given by ρt = −

At Bt

where Z ∞Z

∞

c ρ  σρ κθρ   ρkc2 t 2 At = log(v/v0 ) − + c2 − + c1 + 2 y σ σ 2 σ 0 0  ρσ(1 + c ) ρκθ(1 + c ) ρ2 (1 + c )2   ρ(1+c1 ) 1 1 1 σ ·v − − + c1 y q(v, y) dvdy , exp 2 σ 2 Z ∞Z ∞  ρσ(1 + c ) ρκθ(1 + c ) ρ2 (1 + c )2   ρ(1+c1 ) 1 1 1 Bt = t v σ exp − − + c1 y q(v, y) dvdy . 2 σ 2 0 0 19

Proof. Similarly as the proof of Theorem 4.3, it follows that Z t Z t Z t 2 (2) (δ(Vs ) + m(Vs ))γd (Vs )ds + δ(Vs )γd (Vs )ds ρ γd (Vs )dWs − ρ 0 0 0 Z t Z t Z t   2 = ρ a(Vt ) − a(V0 ) − b(Vs )ds − ρ (δ(Vs ) + m(Vs ))γd (Vs )ds + δ(Vs )γd (Vs )ds 0 0 0  σρ κθρ Z t c2 ρ ρkc2 t = log(Vt /V0 ) − + c2 − + c1 + 2 1/Vs ds σ σ 2 σ 0 The process Kt , which is the conditional expectation of Gt Dt , is given by ρ(1+c1 )

233

Kt =S0 (Vt /v0 ) σ   ρσ(1 + c ) ρκθ(1 + c ) ρ2 (1 + c )2 Z t  ρκ(1 + c1 )  1 1 1 · exp β + t+ − − + c1 1/Vs ds . σ 2 σ 2 0 Rt From these two equalities and the joint density function of (Vt , 0 1/Vs ds), we obtain the desired result. We note that  ρ(1+c1 ) ρκ(1 + c1 )   σ exp β + t S0 (1/v0 ) σ Rt is the common factor, which is independent of the integrating variables v = Vt and y = 0 1/Vs ds, ∂ | E[Gt Dt ] in the numerator and the denominator of ρt = − ∂ t=0 E[Gt Dt ] , thus we can cancel out this term. This completes the proof.

234

4.3.2

231 232

Quantile sensitivity

This section demonstrates the sensitivity on the quantile for the 3/2 model. Assume that under Q, the stock price has the following dynamics p (1) (25) dSt = rSt dt + (c1 + c2 )/ Vt St dWt , p (2) dVt = κ(θ − Vt )dt + σ Vt dWt , V0 = v0 . By Proposition 3.2, we know the density function of St in this model and the quantile sensitivity qt . The density function is y 1 log(v/v )−( κθ − σ )y+ κ T ))2 Z ∞Z ∞ ((log(x/S0 )− c rT + −ρ( σ 0 σ σ 2 1 +c2 2 1 − 2(1−ρ2 )y p e ft, (x) = q(v, y) dvdy , x 2π(1 − ρ2 )y 0 0 where ν

1

κ

ch(κT /2) 2 2 σ2 κ (v0 v) 2 sh(κT /2) − κ2 (v−1/v0 )+ κ22θ T − ( v0 +v) − ν σ8 y sh(κT /2) σ q(v, y) = e σ · i σ2 y 8 8 235 236



 √ 2κ v . √ σ 2 v0 sh(κT /2)

Here ν = 2κθ − 1 and ch(·)/sh(·) are hyperbolic cosine/sine functions. The special function iy (·) is σ2 defined on page 644 of Borodin and Salminen (2002). The sensitivity qt on the pth-quantile is Rx ∂ ft, (u) du qt = − 0 ∂ =0 ft (x) 2 rT y 1 κθ σ κ 0 )−( σ − 2 )y+ σ T )) R x R ∞R ∞ 1 − (log(u/S0 )− c1 + 2 −ρ( σ log(v/v c2 rT 2 2(1−ρ )y √ e q(v, y) dvdydu 2c2 (1−ρ2 ) 0 0 0 uy y , =− 1 rT y 1 κθ σ κ 2 0 )−( σ − 2 )y+ σ T )) R ∞R ∞ 1 − (log(x/S0 )− c1 + 2 −ρ( σ log(v/v 2(1−ρ2 )y √ e q(v, y) dvdy 0 0 x y 20

where x = χt (p), that is, x is the number such that Z

Z xZ

x

ft (u) du =

p=

0

0

237

4.3.3

∞Z ∞

0

1

p e u 2π(1 − ρ2 )y

0

−

y 2 1 κθ σ κ (log(u/S0 )− rT c1 + 2 −ρ( σ log(v/v0 )−( σ − 2 )y+ σ T )) 2(1−ρ2 )y

q(v, y) dvdydu .

Vega of call options

We evaluate the sensitivity on call option prices ∂ νt := E[e−rt (St − K)+ ] ∂ =0 in the context of Section 3.3 for the 3/2 model stated in Eq.(25). Proposition 3.4 will be applied and in order to evaluate the components of νt in the formula there, we compute the functions Mt , Ht , α (t), β (t) defined in Section 3.3. By direct calculation, we have Z t 1  2 Mt = (c1 + c2 ) ds, 0 Vs  κθ σ  Z t 1 κ(c1 + c2 )  Ht = − t + (c1 + c2 ) − ds, σ σ 2 0 Vs and   κθ σ  1 Z t 1 ρκ(c1 + c2 ) ρ(c1 + c2 ) 2 log(Vt /v0 ) + t − ρ(c1 + c2 ) − − (c1 + c2 ) α (t) = ds, σ σ σ 2 2 0 Vs Z t 1 2 2 β (t) = (1 − ρ )(c1 + c2 ) ds . 0 Vs For notational simplicity, let (t, v, y) denote the values taken by (t, Vt , α(t, v, y) := α (t) =

Rt 0

1/Vs ds) and define

  κθ σ  1  ρ(c1 + c2 ) ρκ(c1 + c2 ) log(v/v0 ) + t − ρ(c1 + c2 ) − − (c1 + c2 )2 y, σ σ σ 2 2

  κθ σ  1  ρc1 ρκc1 α0 (t, v, y) := α(t, v, y)|=0 = log(v/v0 ) + t − ρc1 − − c21 y, σ σ σ 2 2   κθ σ   ∂α(t, v, y) ρc2 ρκc1 α b0 (t, v, y) := = log(v/v0 ) + t − ρc2 − − c1 c2 y, ∂ σ σ σ 2 =0 and β(t, v, y) := β (t) = (1 − ρ2 )(c1 + c2 )2 y, β0 (t, v, y) := β=0 (t) = (1 − ρ2 )c21 y, ∂β (t) βb0 (t, v, y) := = 2(1 − ρ2 )c1 c2 y . ∂ =0 238 239 240 241

From these, we can define the functions d1 (t, v, y), d2 (t, v, y) db01 (t, v, y) and db02 (t, v, y) in the same way as in Eq.(21) and (22). Now we apply Proposition 3.4 to obtain the following theorem. Recall that q(v, y) is the joint Rt density function of (Vt , 0 1/Vs ds) given in Proposition 4.4. 21

Theorem 4.3. The sensitivity νt in the 3/2 model with respect to the perturbation above is Z ∞Z ∞      1 b0 (t, v, y) + βb0 (t, v, y) + Φ(d01 (t, v, y))db01 (t, v, y) νt = S0 eα0 (t,v,y)+ 2 β0 (t,v,y) N (d01 (t, v, y)) α 0 0  − Ke−rt Φ(d02 (t, v, y))db02 (t, v, y) q(v, y) dvdy, 242

with intermediate functions α0 , β0 , α b0 , βb0 , d01 , d02 , db01 , db02 given above.

243

5

244 245 246 247 248 249 250 251 252 253 254 255 256 257 258

Conclusion and Future Research

In this paper, we characterize the elasticities and sensitivities of stochastic volatility models and derive explicit formulae in terms of model parameters. Three sensitivities were mainly discussed: the shock elasticity, the quantile sensitivity and the vega value. We used the exponential measure change technique in Palmowski and Rolski (2002) as a main tool. The shock elasticity expressed in this paper is consistent with that of Hansen and Scheinkman (2012). However, our results focus on the one-factor stochastic volatility model, which is restrictive compared to their results of multi-factor models. We illustrate our main results using two popular stochastic volatility models: the Heston model and the 3/2 model. The authors suggest the following two directions for future research. First, it is of interest to study the sensitivity with respect to the perturbation in the volatility of the volatility coefficient in the stochastic volatility model. In Eq.(15), the perturbation m (·) = m(·) + γ (·) is only related to the volatility coefficient of the reference process St . Future research lies in extending the perturbation to the form σ(·) + η (·) in the volatility of the volatility coefficient of Vt . Second, it would be interesting to explore the sensitivity of path-dependent functionals including the payoffs of barrier and American options in general stochastic volatility models.

259

260 261 262

Acknowledgments We would like to thank the anonymous referee for stimulating remarks, and comments which significantly help improve the paper. The usual disclaimer applies. The research of Duy Nguyen is partially supported by a Marist College summer research grant.

263

264

Compliance with ethical standards

265 266

Conflict of interest The authors declare that they have no conflict of interest.

267 268

Human/Animals participants This research does not involve human participants or animals.

269

22

270

271 272

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ˇka, J., L. P. Hansen, and J. A. Scheinkman (2014): “Shock elasticities and impulse responses,” MatheBorovic matics and Financial Economics, 8(4), 333–354.

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Hansen, L. (2008): “Modeling the long run: valuation in dynamic stochastic economies,” Fisher-Schultz Lecture at the European Meetings of the Econometric Society.

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A

302

Discussion of the Case of Multivariate Volatility Factors

In this section, we shall illustrate that similar representations like Eq.(5) also hold for the case of multivariate stochastic volatility factors, which is consistent with the model considered in Hansen and

23

Scheinkman (2012). In the following, we shall illustrate the details of this extension. Assume that the underlying state vector process V is n-dimensional, where each entry of V satisfies the following SDE (i)

dVt 307

(i)

(i)

(i)

= µ(Vt )dt + σ(Vt )dZt ,

V0i = v0i ,

i = 1, . . . , n.

Consider the following stochastic volatility model n

X dSt (i) (i) = r dt + mi (Vt ) dWt , St i=1 308

i = 1, . . . , n,

(26)

and this can be thought of as the multivariate generalization of the model in Eq.(2). (1)

(n)

In line with Hansen and Scheinkman (2012), we assume that Zt := (Zt , . . . , Zt ) is an n(i) dimensional Brownian motion, which means that the individual entries Zt are independent of each (1) (n) other. Similarly we assume that Wt := (Wt , . . . , Wt ) is an n-dimensional Brownian motion. Then (i) (j) (i) (j) we assume that there is a constant correlation between Wt and Zt , i.e. E[dWt dZt ] = ρi δij dt, (i) (j) i = 1, . . . , n, where δij is the Kronecker Delta notation. Note that Wt and Zt are independent of each other for i 6= j. From the Cholesky decomposition, we can write q (i) (i) (i) dWt = ρi dZt + 1 − ρ2i dZ¯t 309 310 311 312

(i) (i) (i) (i) with E[dZt dZ¯t ] = 0 for some standard Brownian motion Z¯t independent of Zt . Note that we (i) (1) (n) also have that Z¯t are independent of each other for i = 1, . . . , n, i.e. the vector Z¯t := (Z¯t , . . . , Z¯t ) is also an n-dimensional Brownian motion independent of Zt . Define two auxiliary functions gi (·) and hi (·) for i = 1, 2, · · · , n as in Eq.(3), then from Itˆo’s lemma, we have (i)

(i)

(i)

(i)

dgi (Vt ) = hi (Vt )dt + mi (Vt )dZt . Integrating both sides from 0 to T , it follows that Z T Z (i) (i) (i) gi (VT ) − gi (V0 ) = hi (Vs )ds + 0

T

mi (Vs(i) ) dZs(i) .

(27)

0

(i) From Eq.(26) and (27), and the fact that Z¯t are independent of each other for i = 1, . . . , n, we obtain

Z Z TX n n 1 T X 2 (i) ST = S0 exp rT − m (V )ds + ρi mi (Vs(i) )dZs(i) 2 0 i=1 i s 0 i=1 ! Z TX n q + 1 − ρ2i mi (Vs(i) )dZ¯s(i) 0

i=1

Z Z T n n   X 1 T X 2 (i) (i) (i) = S0 exp rT − ρi gi (VT ) − gi (V0 ) − hi (Vs(i) )ds mi (Vs )ds + 2 0 i=1 0 ! i=1 Z n q T X + 1 − ρ2i mi (Vs(i) )dZ¯s(i) . i=1

0

24

313 314

Since Z¯ (i) is independent of Z (i) and since Z¯ (i) (and Z (i) ) are independent of each other, we have that conditioning on the functionals of the vector stochastic processes   Z T Z T Z T Z T (1) (n) 2 (1) 2 (n) (n) (1) hn (Vs )ds , (28) VT , . . . , V T , m1 (Vs )ds, . . . , mn (Vs )ds, h1 (Vs )ds, . . . , 0

0

0

0

the log asset price St is normally distributed, i.e. Z Z T Z T  (1) (n) 2 (n) 2 (1) mn (Vs )ds, m1 (Vs )ds, . . . , log(ST ) VT , . . . , VT , ∼N

log(S0 ) + rT − n X

(1 − ρ2i )

i=1

Z

1 2

T

0

n X

h1 (Vs(1) )ds, . . . ,

Z n  X (i) (i) 2 (i) mi (Vs )ds + ρi gi (VT ) − gi (V0 ) −

i=1

T

320

321 322 323 324 325 326

 hi (Vs(i) )ds , (29)

s

s

i

0

i=1 n X i=1

319



0

i=1

=

318

T

m2i (Vs(i) )ds .

i

316

hn (Vs(n) )ds

0

i=1

317

T

!

We have also utilized the fact that conditioning on Eq.(28) ! q Z T Z n q n X X (i) (i) 2 2 Var 1−ρ mi (V ) dZ¯ = Var 1−ρ

315

Z 0

0

0

0

Z

T

T

mi (Vs(i) ) dZ¯s(i)



0

(1 − ρ2i )

Z

T

m2i (Vs(i) ) ds,

0

and this is due to the fact that Z¯ (i) are independent of each other. Remark A.1. Note that the conditional representation in Eq.(29) can be thought of as a multivariate generalization of Eq.(5). Thus we can similarly extend the conditional representation of the density of ST as given in Eq.(6), to the multivariate case. Also note that the main results in Section 3 and Section 4 also depend on the representation in Eq.(5), thus their statements can be similarly extended to the case of multivariate volatility factors. Remark A.2. The density of ST can be similarly obtained in this multivariate volatility beR T case, (i) R T (i) (i) 2 cause the volatility factors are independent from each other, i.e. (VT , 0 m1 (Vs )ds, 0 h1 (Vs )ds) RT (j) R T (j) (j) is independent from (VT , 0 m21 (Vs )ds, 0 h1 (Vs )ds) for i 6= j. Thus based on Eq.(29), the density of STR can be obtained the normal density against the products of density functions R T by integrating T (i) (i) (i) of (VT , 0 m21 (Vs )ds, 0 h1 (Vs )ds) for i = 1, . . . , n, each of which can be obtained through our exponential measure change technique as illustrated in Section 4.

25