An integral representation of elasticity and sensitivity for stochastic volatility models

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 vega 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 (Bernoulli 8(6):767–785 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.

A 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 Scheinkman [12]. 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

$$\begin{aligned} dV^{(i)}_t&=\mu (V^{(i)}_t)dt+\sigma (V^{(i)}_t) dZ_t^{(i)}, \quad V_0^i=v_0^i, \quad i=1,\ldots ,n. \end{aligned}$$

Consider the following stochastic volatility model

$$\begin{aligned} \frac{{ dS}_t}{S_t}=r\, dt+ \sum _{i=1}^{n} m_i(V^{(i)}_t)\,{ dW}_t^{(i)}, \quad i=1,\ldots ,n, \end{aligned}$$

(25)

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

In line with Hansen and Scheinkman [12], we assume that $Z_t:=(Z_t^{(1)}, \ldots , Z_t^{(n)})$ is an n-dimensional Brownian motion, which means that the individual entries $Z_t^{(i)}$ are independent of each other. Similarly we assume that $W_t:=(W_t^{(1)}, \ldots , W_t^{(n)})$ is an n-dimensional Brownian motion. Then we assume that there is a constant correlation between $W_t^{(i)}$ and $Z_t^{(j)}$, i.e. $\mathbb {E}[{ dW}_t^{(i)} dZ_t^{(j)}]=\rho _i\delta _{ij} \,dt, i=1,\ldots , n$, where $\delta _{ij}$ is the Kronecker Delta notation. Note that $W_t^{(i)}$ and $Z_t^{(j)}$ are independent of each other for $i\ne j$. From the Cholesky decomposition, we can write

$$\begin{aligned} { dW}_t^{(i)}=\rho _i \,dZ_t^{(i)}+\sqrt{1-\rho _i^2}\,d\bar{Z}_t^{(i)} \end{aligned}$$

with $\mathbb {E}[dZ_t^{(i)}d\bar{Z}_t^{(i)}]=0$ for some standard Brownian motion $\bar{Z}_t^{(i)}$ independent of $Z_t^{(i)}$. Note that we also have that $\bar{Z}_t^{(i)}$ are independent of each other for $i=1,\ldots , n$, i.e. the vector $\bar{Z}_t:=(\bar{Z}_t^{(1)}, \ldots , \bar{Z}_t^{(n)})$ is also an n-dimensional Brownian motion independent of $Z_t$. Define two auxiliary functions $g_i(\cdot )$ and $h_i(\cdot )$ for $i=1,2,\ldots ,n$ as in Eq. (3), then from Itô’s lemma, we have

$$\begin{aligned} dg_i(V^{(i)}_t)=h_i(V^{(i)}_t) dt+ m_i(V^{(i)}_t) dZ_t^{(i)}. \end{aligned}$$

Integrating both sides from 0 to T, it follows that

$$\begin{aligned} g_i(V^{(i)}_T)-g_i(V^{(i)}_0)&=\int _0^T h_i(V^{(i)}_s) ds +\int _0^T m_i(V^{(i)}_s) \,dZ_s^{(i)}. \end{aligned}$$

(26)

From Eqs. (25) and (26), and the fact that $\bar{Z}_t^{(i)}$ are independent of each other for $i=1,\ldots , n$, we obtain

$$\begin{aligned} S_T&=S_0 \exp \left( rT-\frac{1}{2} \int _0^T \sum _{i=1}^{n} m_i^2(V^{(i)}_s) ds + \int _0^T \sum _{i=1}^{n} \rho _i m_i(V^{(i)}_s) dZ_s^{(i)}\right. \\&\quad \quad \left. +\int _0^T \sum _{i=1}^{n} \sqrt{1-\rho _i^2} m_i(V^{(i)}_s) d\bar{Z}_s^{(i)}\right) \\&=S_0 \exp \left( rT-\frac{1}{2} \int _0^T \sum _{i=1}^{n} m_i^2(V^{(i)}_s) ds + \sum _{i=1}^{n} \rho _i \Big (g_i(V^{(i)}_T)-g_i(V^{(i)}_0)-\int _0^T h_i(V^{(i)}_s) ds\Big ) \right. \\&\quad \quad \left. + \sum _{i=1}^{n} \sqrt{1-\rho _i^2} \int _0^T m_i(V^{(i)}_s) d\bar{Z}_s^{(i)}\right) . \end{aligned}$$

Since $\bar{Z}^{(i)}$ is independent of $Z^{(i)}$ and since $\bar{Z}^{(i)}$ (and $Z^{(i)}$) are independent of each other, we have that conditioning on the functionals of the vector stochastic processes

$$\begin{aligned} \left( V_T^{(1)},\ldots ,V_T^{(n)}, \int _0^T m_1^2(V^{(1)}_s) ds,\ldots ,\int _0^T m_n^2(V^{(n)}_s) ds, \int _0^T h_1(V^{(1)}_s) ds,\ldots ,\int _0^T h_n(V^{(n)}_s) ds\right) ,\nonumber \\ \end{aligned}$$

(27)

the log asset price $S_t$ is normally distributed, i.e.

$$\begin{aligned}&\log (S_T)\Big | \left( V_T^{(1)},\ldots ,V_T^{(n)}, \int _0^T m_1^2(V^{(1)}_s) ds,\ldots ,\int _0^T m_n^2(V^{(n)}_s) ds,\right. \nonumber \\&\quad \left. \int _0^T h_1(V^{(1)}_s) ds,\ldots ,\int _0^T h_n(V^{(n)}_s) ds\right) \nonumber \\&\sim \mathcal {N}\left( \log (S_0)+rT-\frac{1}{2} \int _0^T \sum _{i=1}^{n} m_i^2(V^{(i)}_s) ds+\sum _{i=1}^{n} \rho _i \Big (g_i(V^{(i)}_T)-g_i(V^{(i)}_0)\right. \nonumber \\&\left. -\int _0^T h_i(V^{(i)}_s) ds\Big ),\quad \quad \sum _{i=1}^{n} (1-\rho _i^2)\int _0^T m_i^2(V^{(i)}_s) ds \right) . \end{aligned}$$

(28)

We have also utilized the fact that conditioning on Eq. (27)

$$\begin{aligned} \mathrm {Var}\left( \sum _{i=1}^{n} \sqrt{1-\rho _i^2} \int _0^T m_i(V^{(i)}_s) \,d\bar{Z}_s^{(i)}\right)&=\sum _{i=1}^{n} \mathrm {Var}\left( \sqrt{1-\rho _i^2} \int _0^T m_i(V^{(i)}_s) \,d\bar{Z}_s^{(i)}\right) \\&=\sum _{i=1}^{n} (1-\rho _i^2)\int _0^T m_i^2(V^{(i)}_s)\, ds, \end{aligned}$$

and this is due to the fact that $\bar{Z}^{(i)}$ are independent of each other.

Remark A.1

Note that the conditional representation in Eq. (28) can be thought of as a multivariate generalization of Eq. (5). Thus we can similarly extend the conditional representation of the density of $S_T$ as given in Eq. (6), to the multivariate case. Also note that the main results in Sects. 3 and 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 $S_T$ can be similarly obtained in this multivariate volatility case, because the volatility factors are independent from each other, i.e. $(V_T^{(i)}, \int _0^T m_1^2(V^{(i)}_s) ds, \int _0^T h_1(V^{(i)}_s) ds)$ is independent from $(V_T^{(j)}, \int _0^T m_1^2(V^{(j)}_s) ds, \int _0^T h_1(V^{(j)}_s) ds)$ for $i\ne j$. Thus based on Eq. (28), the density of $S_T$ can be obtained by integrating the normal density against the products of density functions of $(V_T^{(i)}, \int _0^T m_1^2(V^{(i)}_s) ds, \int _0^T h_1(V^{(i)}_s) ds)$ for $i=1,\ldots ,n$, each of which can be obtained through our exponential measure change technique as illustrated in Sect. 4.