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.

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Notes

  1. 1.

    See Mandelbrot (1997) for the introduction of fBMs in finance.

  2. 2.

    See, e.g., Sottinen (2001) and Cheridito (2003) for related problems. Hu and Öksendal (2003) develop a no-arbitrage model by introducing Wick integrals, but this model cannot add a natural economic interpretation.

  3. 3.

    Alos and Yang (2014) derive an approximation formula of European option prices by using a different methodology when volatility follows a fractional Heston model.

  4. 4.

    To consider the long-memory feature of volatility, we restrict ourselves to the case that \(0.5 \le H <1\) under the physical measure \(\mathbb {P}\) until Sect. 3.2.

  5. 5.

    It seems reasonable to assume a mean-reverting process for the evolution of volatility over a long period of time under the physical measure.

  6. 6.

    To be precise, this formulation is a truncated version of the Mandelbrot–Van Ness representation of fBMs. In the next section, we consider its full version. See Comte and Renault (1998) for details.

  7. 7.

    Alternatively, as a market practice, assuming that \(\bar{\eta }_t\) is a deterministic function (e.g., piecewise constant) of time t, \(\bar{\eta }_t\) can be used to fit the option prices observed in the market.

  8. 8.

    They consider the mean-reverting volatility process as \(\mathrm{d}\sigma _t = \kappa (\widetilde{\theta }- \sigma _t ) \mathrm{d}t + \gamma \sigma _t \mathrm{d}w_t\). Hence, the parameter \(\theta \) in our model corresponds to \(\kappa \widetilde{\theta }\) in their model.

  9. 9.

    However, because the convergence speed is very slow in the fractional Monte Carlo simulation, we stop our simulations with 1,000,000 trials. The Monte Carlo simulation for fBMs is difficult to perform because of the non-Markovian nature (see, e.g., Kijima and Tam 2013).

  10. 10.

    This observation suggests that the fractional volatility model may have a strong impact on the prices of path-dependent options such as Asian and barrier options.

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Acknowledgements

The authors thank anonymous referees for careful reading our manuscript and for giving helpful comments. Kijima is grateful for the research grant funded by the Grant-in-Aid (A) (#21241040) from Japan’s Ministry of Education, Culture, Sports, Science and Technology.

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Correspondence to Masaaki Kijima.

Appendix: A chaos expansion approach

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(xab) 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 \(h_0(x)=1\).

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Funahashi, H., Kijima, M. Does the Hurst index matter for option prices under fractional volatility?. Ann Finance 13, 55–74 (2017). https://doi.org/10.1007/s10436-016-0289-1

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Keywords

  • Fractional Brownian motion
  • Hurst index
  • Stochastic volatility
  • Mean-reverting process
  • Implied volatility

JEL Classification

  • G12
  • G13
  • G17