Does the Hurst index matter for option prices under fractional volatility?

Abstract

This study examines the effect of fractional volatility on option prices. To this end, we develop an approximation method for the pricing of European-style contingent claims when volatility follows a fractional Brownian motion. Through extensive numerical experiments, we confirm that the decrease in the smile amplitude under fractional volatility is much slower than that under the standard stochastic volatility model. We also show that the Hurst index under fractional volatility has a crucial impact on option prices when the maturity is short and speed of mean reversion is slow. On the contrary, the impact of the Hurst index on option prices reduces for long-dated options.

Appendix: A chaos expansion approach

In this appendix, we apply the Wiener–Ito chaos expansion approach to obtain an approximation formula for the pricing of European-style contingent claims.

First, note that (2.7) is a special case of Eq. (2.2) in Funahashi (2014). Therefore, according to Proposition 3.2 in Funahashi (2014), the following result can be obtained.

Lemma 1

Let $X_t:=\frac{S_t}{F(0,t)} - 1$. Then,

$$\begin{aligned} X_t = 1 + a_1(t) + a_2(t) + a_3(t) , \end{aligned}$$

where

$$\begin{aligned} a_1(t)= & {} \int _0^t \bar{V}(s) \mathrm{d}W_s, \\ a_2(t)= & {} \int _0^t f_0(s) \left( \int _0^s f_0(u) \mathrm{d}W_u \right) \mathrm{d}W_s + \int _0^t f^{(1)}_0(s) \left( \int _0^s \lambda _2(s,u) \mathrm{d}\bar{W}_u \right) \mathrm{d}W_s , \\ a_3(t)= & {} \int _0^t f_0(s) \left( \int _0^s f_0(u) \left( \int _0^u f_0(r) \mathrm{d}W_r \right) \mathrm{d}W_u \right) \mathrm{d}W_s \\&+\, \int _0^t f_0(s) \left( \int _0^s f^{(1)}_0(u) \left( \int _0^u \lambda _2(u,r) \mathrm{d}\bar{W}_r \right) \mathrm{d}W_u \right) \mathrm{d}W_s \\&+\, \int _0^t f^{(1)}_0(s) \left( \int _0^s f_0(u) \left( \int _0^u \lambda _2(s,r) \mathrm{d}\bar{W}_r \right) \mathrm{d}W_u \right) \mathrm{d}W_s \\&+\, \int _0^t f^{(1)}_0(s) \left( \int _0^s \lambda _2(s,u) \left( \int _0^u f_0(r) \mathrm{d}W_r \right) \mathrm{d}\bar{W}_u \right) \mathrm{d}W_s \\&+\, \int _0^t f^{(2)}_0(s) \left( \int _0^s \lambda _2(s,u) \left( \int _0^u \lambda _2(s,r) \mathrm{d}\bar{W}_r \right) \mathrm{d}\bar{W}_u \right) \mathrm{d}W_s . \\ \end{aligned}$$

Here, $f_0(t) = f(V(0,t))$, $f^{(n)}_0(t) = \partial ^{(n)}_x f(x) |_{x=V(0,t)}$, and

$$\begin{aligned} \bar{V}(s) = f_0(s) + \rho f^{(1)}_0(s) \int _0^s f_0(u) \lambda _2(s,u) \mathrm{d}u + \frac{1}{2} f^{(2)}_0(s) \int _0^s \lambda ^2_2(s,u) \mathrm{d}u . \end{aligned}$$

Note that $a_1(t)$ in Lemma 1 follows a normal distribution with zero mean and variance

$$\begin{aligned} \varSigma _t = \int _0^t (\bar{V}(s))^2 \mathrm{d}s . \end{aligned}$$

Then, by applying the following result, an approximation of the density function of $X_t$ can be obtained. The proof is found in Funahashi and Kijima (2015) under a general setting.

Lemma 2

Let us denote the density function of $X_t$ by $f_{X_t}(x)$. Then, the probability density function of $X_t$ is approximated as

$$\begin{aligned} f_{X_t}(x)= & {} n\left( x; 0, \varSigma _{t} \right) - \frac{\partial }{\partial {x}} \left\{ \mathbb {E}[ a_{2}(t) | a_{1}(t) = x ] n\left( x; 0, \varSigma _{t} \right) \right\} \nonumber \\&-\, \frac{\partial }{\partial {x}} \left\{ \mathbb {E}[ a_{3}(t) | a_{1}(t) = x ] n\left( x; 0, \varSigma _{t} \right) \right\} \nonumber \\&+\, \frac{1}{2} \frac{ \partial ^{2}}{\partial x^2} \left\{ \mathbb {E}[ a_{2}(t)^{2} | a_{1}(t) = x ] n\left( x; 0, \varSigma _{t} \right) \right\} + \cdots . \end{aligned}$$

(5.1)

where n(x; a, b) denotes the normal density function with mean a and variance b.

The conditional expectations in Lemma 2 can be evaluated explicitly by a standard argument. In other words, we obtain

$$\begin{aligned} \mathbb {E}[ a_{2}(t) | a_{1}(t) = x ]= & {} q_{1}(t) \left( \frac{x^{2}}{\varSigma _{t}^{2}}- \frac{1}{\varSigma _{t}} \right) , \\ \mathbb {E}[ a_{3}(t) | a_{1}(t) = x ]= & {} q_{2}(t) \left( \frac{x^{3}}{\varSigma _{t}^{3}}- \frac{3x}{\varSigma _{t}^{2}} \right) , \\ \mathbb {E}[ a^2_{2}(t) | a_{1}(t) = x ]= & {} q_{3}(t) \left( \frac{x^{4}}{\varSigma _{t}^{4}} - \frac{6x^{2}}{\varSigma _{t}^{3}} + \frac{3}{\varSigma _{t}^{2}} \right) + q_{4}(t) \left( \frac{x^{2}}{\varSigma _{t}^{2}}- \frac{1}{\varSigma _{t}} \right) + q_{5}(t) , \end{aligned}$$

where

$$\begin{aligned} q_{1}(t)= & {} \int _{0}^{t} f_0(s) \bar{V}(s) \left( \int _{0}^{s} f_0(u) \bar{V}(u) \mathrm{d}u \right) \mathrm{d}s \\&+\, \rho \int _{0}^{t} f^{(1)}(s) \bar{V}(s) \left( \int _{0}^{s} \lambda _2(s,u) \bar{V}(u) \mathrm{d}u \right) \mathrm{d}s , \\ q_{2}(t)= & {} \int _{0}^{t} f_0(s) \bar{V}(s) \left( \int _{0}^{s} f_0(u) \bar{V}(u) \left( \int _{0}^{u} f_0(r) \bar{V}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, \rho \int _{0}^{t} f_0(s) \bar{V}(s) \left( \int _{0}^{s} f^{(1)}(u) \bar{V}(u) \left( \int _{0}^{u} \lambda _2(u,r) \bar{V}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, \rho \int _{0}^{t} f^{(1)}(s) \bar{V}(s) \left( \int _{0}^{s} f_0(u) \bar{V}(u) \left( \int _{0}^{u} \lambda _2(s,r) \bar{V}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, \rho \int _{0}^{t} f^{(1)}(s) \bar{V}(s) \left( \int _{0}^{s} \lambda _2(s,u) \bar{V}(u) \left( \int _{0}^{u} f_0(r) \bar{V}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, \rho ^2 \int _0^t f^{(2)}_0(s) \bar{V}(s) \left( \int _0^s \lambda _2(s,u) \bar{V}(u) \left( \int _0^u \lambda _2(s,r) \bar{V}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s , \\ q_{3}(t)= & {} q^2_{1}(t) , \\ q_{4}(t)= & {} q_{4,1} (t) + q_{4,2} (t) + q_{4,3} (t), \end{aligned}$$

and

$$\begin{aligned} q_{5}(t)= & {} \int _{0}^{t} f_0(s)^2 \left( \int _{0}^{s} f_0(u)^2 \mathrm{d}u \right) \mathrm{d}s + \int _{0}^{t} (f^{(1)}_0(s))^2 \left( \int _{0}^{s} \lambda _2^2(s,u) \mathrm{d}u \right) \mathrm{d}s \\&+\, 2 \rho \int _{0}^{t} f_0(s) f^{(1)}_0(s) \left( \int _{0}^{s} \lambda _2(s,u) f_0(u) \mathrm{d}u \right) \mathrm{d}s . \end{aligned}$$

Here, we define

$$\begin{aligned} q_{4,1}(t)= & {} 2 \int _{0}^{t} f_0(s) \bar{V}(s) \left( \int _{0}^{s} f_0(u) \bar{V}(u) \left( \int _{0}^{u} f_0(r)^2 \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, 2 \int _{0}^{t} f_0(s) \bar{V}(s) \left( \int _{0}^{s} f_0(u)^2 \left( \int _{0}^{u} f_0(r) \bar{V}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, \int _{0}^{t} f_0(s)^2 \left( \int _{0}^{s} f_0(u) \bar{V}(u) \mathrm{d}u \right) ^2 \mathrm{d}s , \end{aligned}$$

$$\begin{aligned} q_{4,2}(t)= & {} 2 \int _{0}^{t} f^{(1)}_0(s) \bar{V}(s) \left( \int _{0}^{s} f^{(1)}_0(u) \bar{V}(u) \left( \int _{0}^{u} \lambda _2(s,r) \lambda _2(u,r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, 2 \rho ^2 \int _{0}^{t} f^{(1)}_0(s) \bar{V}(s) \left( \int _{0}^{s} f^{(1)}_0(u) \lambda _2(s,u) \left( \int _{0}^{u} \lambda _2(u,r) \bar{V}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, \rho ^2 \int _{0}^{t} (f^{(1)}_0(s))^2 \left( \int _{0}^{s} \lambda _2(s,u) \bar{V}(u) \mathrm{d}u \right) ^2 \mathrm{d}s , \end{aligned}$$

and

$$\begin{aligned} q_{4,3}(t)= & {} 2 \rho \int _{0}^{t} f_0(s) \bar{V}(s) \left( \int _{0}^{s} f^{(1)}_0(u) \bar{V}(u) \left( \int _{0}^{u} \lambda _2(u,r) f_0(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, 2 \rho \int _{0}^{t} f^{(1)}_0(s) \bar{V}(s) \left( \int _{0}^{s} f_0(u) \bar{V}(u) \left( \int _{0}^{u} \lambda _2(s,r) f_0(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, 2 \rho \int _{0}^{t} f_0(s) \bar{V}(s) \left( \int _{0}^{s} f_0(u) f^{(1)}_0(u) \left( \int _{0}^{u} \lambda _2(u,r) \bar{V}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, 2 \rho \int _{0}^{t} f^{(1)}_0(s) \bar{V}(s) \left( \int _{0}^{s} f_0(u) \lambda _2(s,u) \left( \int _{0}^{u} f_0(r) \bar{V}(r) \mathrm{d}r \right) \mathrm{d}u \right) \mathrm{d}s \\&+\, 2 \rho \int _{0}^{t} f_0(s) f^{(1)}_0(s) \left( \int _{0}^{s} f_0(u)\bar{V}(u) \mathrm{d}u \right) \left( \int _{0}^{s} \lambda (s,u) \bar{V}(u) \mathrm{d}u \right) \mathrm{d}s . \end{aligned}$$

By substituting the conditional expectations into (5.1), the approximate density function, denoted by $\tilde{f}_{X_t}(x)$, can be expressed as

$$\begin{aligned} \tilde{f}_{X_t}(x)= & {} \frac{1}{2} n\left( x; 0, \varSigma _{t} \right) \bigg [ \frac{q_{3}(t)}{\varSigma _{t}^{3}} h_{6} \left( \frac{x}{\sqrt{\varSigma _{t}}} \right) + \frac{\left( 2 q_{2}(t) + q_{4}(t) \right) }{\varSigma _{t}^{2}} h_{4} \left( \frac{x}{\sqrt{\varSigma _{t}}} \right) \nonumber \\&+\, \frac{2 q_{1}(t)}{\left( \sqrt{\varSigma _{t}} \right) ^{3}} h_{3} \left( \frac{x}{\sqrt{\varSigma _{t}}} \right) + \frac{q_{5}(t)}{\varSigma _{t}} h_{2} \left( \frac{x}{\sqrt{\varSigma _{t}}} \right) + 2 \bigg ], \end{aligned}$$

(5.2)

where $h_n(x)$ denotes the Hermite polynomial of order n defined by

$$\begin{aligned} h_{n}(x) = (-1)^{n} \mathrm{e}^{x^2/2} \frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}} \mathrm{e}^{-x^2/2}, \quad n=1,2, \dots , \end{aligned}$$

with $h_0(x)=1$.