Understanding Delta-Hedged Option Returns in Stochastic Volatility Environments

Abstract

In this paper, we provide a novel representation of delta-hedged option returns in a stochastic volatility environment. The representation of delta-hedged option returns provided in this paper consists of two terms: volatility risk premium and parameter estimation risk. In an empirical analysis, we examine delta-hedged option returns based on the result of a historical simulation with the USD-JPY currency option market data from October 2003 to June 2010. We find that the delta-hedged option returns for OTM put options are strongly affected by parameter estimation risk as well as the volatility risk premium, especially in the post-Lehman shock period.

Appendices

Appendix 1: Proof of Proposition 2

The following equation can be derived by applying Ito’s lemma to the market price $C_t^{M} = G(t, S_t, \sigma _t)$,

$$\begin{aligned} \begin{aligned} C_{t+\tau }^{M}&= C_t^{M} + \int _t^{t+\tau } \frac{\partial G}{\partial S}(u,S_u,\sigma _u)d S_u \\&\quad \,\, + \int _t^{t+\tau } \frac{\partial G}{\partial \sigma }(u,S_u,\sigma _u)d \sigma _u + \int _t^{t+\tau } \mathcal {D} G(u,S_u,\sigma _u)du, \end{aligned} \end{aligned}$$

(21)

where

$$\begin{aligned} \begin{aligned} \mathcal {D}G(t,S_t,\sigma _t)&= \frac{\partial G}{\partial t}(t,S_t,\sigma _t) + \frac{1}{2} \sigma _t^2 S_t^2 \frac{\partial ^2 G}{\partial S^2} (t,S_t,\sigma _t) \\&\quad \,\, + \frac{1}{2} \eta _t^2 \frac{\partial ^2 G}{\partial \sigma ^2}(t,S_t,\sigma _t) + \rho \eta _t \sigma _t S_t \frac{\partial ^2 G}{\partial S \partial \sigma }(t,S_t,\sigma _t), \end{aligned} \end{aligned}$$

and $C_t^{M} = G(t, S_t, \sigma _t)$ is also solves the Eq. (7). Under the two equations of (7) and (21), we can derive the following equation with the market price $C_t^{M}$,

$$\begin{aligned} C_{t+\tau }^{M}&= C_t^{M} + \int _t^{t+\tau } \frac{\partial C_u^{M}}{\partial S_u} d S_u + \int _t^{t+\tau } \frac{\partial C_u^{M}}{\partial \sigma _u} d \sigma _u \nonumber \\&\quad + \int _t^{t+\tau } \Big [ \Big ( r_d C_u^{M} - ( r_d - r_f ) S_u \frac{\partial C_u^{M}}{\partial S_u} \Big ) - (\tilde{\theta }_u - \lambda _u) \frac{\partial C_u^{M}}{\partial \sigma _u} - \frac{1}{2} {\tilde{\eta }_u}^2 \frac{\partial ^2 C_u^{M}}{\partial {\sigma _u}^2} \nonumber \\&\quad - \,\tilde{\rho } \tilde{\eta }_u \sigma _u S_u \frac{\partial ^2 C_u^{M}}{\partial S \partial \sigma } + \frac{1}{2} {\eta _u}^2 \frac{\partial ^2 C_u^{M}}{\partial {\sigma _u}^2} + \rho \eta _u \sigma _u S_u \frac{\partial ^2 C_u^{M}}{\partial S \partial \sigma } \Big ] du \nonumber \\&= C_t^M + \int _t^{t+\tau } r_d C_u^{M} du+ \int _t^{t+\tau } \lambda _u \frac{\partial C_u^{M}}{\partial \sigma _u} du \nonumber \\&\quad + \,\int _t^{t+\tau } \Big [ (\theta _u - \tilde{\theta }_u) \frac{\partial C_u^{M}}{\partial \sigma _u} + \frac{1}{2} ({\eta _u}^2 - {\tilde{\eta }_u}^2) \frac{\partial ^2 C_u^{M}}{\partial {\sigma _u}^2}\nonumber \\&\quad + \,(\rho \eta _u - \tilde{\rho } \tilde{\eta }_u) \sigma _u S_u \frac{\partial ^2 C_u^{M}}{\partial S \partial \sigma } \Big ] du \nonumber \\&\quad +\, \int _t^{t+\tau } \sigma _u S_u \frac{\partial C_u^{M}}{\partial S_u} (\sqrt{1-\rho ^2} d W_u^1 + \rho d W_u^2) + \int _t^{t+\tau } \eta _u \frac{\partial C_u^{M}}{\partial \sigma _u} d W_u^2.\nonumber \\ \end{aligned}$$

(22)

The last equation is valid because of the assumption of $\mu _u = r_d - r_f$ in this proposition.

$C_t^{M}[\lambda _u]$ denotes the time-$t$ call option price which is consistent with the underlying exchange rate process (2) and the equivalent martingale measure $\mathbb {Q}[\lambda _u]$. By replacing $\tau$ with $T-t$ in (22), the following equation can be derived because of the fact that $C_T^{M}[\lambda _u]=C_T^{M}[0]$ at the maturity date.

$$\begin{aligned} \begin{aligned}&C_t^M[\lambda _u] + \int _t^{T} r_d C_u^{M}[\lambda _u] du + \int _t^{T} \lambda _u \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} du \\&\quad + \int _t^{T} \Big [ (\theta _u - \tilde{\theta }_u) \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} + \frac{1}{2} ({\eta _u}^2 - {\tilde{\eta }_u}^2) \frac{\partial ^2 C_u^{M}[\lambda _u]}{\partial {\sigma _u}^2}\\&\quad + (\rho \eta _u - \tilde{\rho } \tilde{\eta }_u) \sigma _u S_u \frac{\partial ^2 C_u^{M}[\lambda _u]}{\partial S \partial \sigma } \Big ] du \\&\quad + \int _t^{T} \sigma _u S_u \frac{\partial C_u^{M}[\lambda _u]}{\partial S_u} (\sqrt{1-\rho ^2} d W_u^1 + \rho d W_u^2) + \int _t^{T} \eta _u \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} d W_u^2 \\&=\,C_t^M[0] + \int _t^{T} r_d C_u^{M}[0] du \\&\quad + \int _t^{T} \Big [ (\theta _u - \tilde{\theta }_u) \frac{\partial C_u^{M}[0]}{\partial \sigma _u} + \frac{1}{2} ({\eta _u}^2 - {\tilde{\eta }_u}^2) \frac{\partial ^2 C_u^{M}[0]}{\partial {\sigma _u}^2}\\&\quad + (\rho \eta _u - \tilde{\rho } \tilde{\eta }_u) \sigma _u S_u \frac{\partial ^2 C_u^{M}[0]}{\partial S \partial \sigma } \Big ] du \\&\quad + \int _t^{T} \sigma _u S_u \frac{\partial C_u^{M}[0]}{\partial S_u} (\sqrt{1-\rho ^2} d W_u^1 + \rho d W_u^2) + \int _t^{T} \eta _u \frac{\partial C_u^{M}[0]}{\partial \sigma _u} d W_u^2. \end{aligned} \end{aligned}$$

Rearranging the above equation and taking expectation to the rearranged equation under $\mathbb {P}$, we can obtain the following expression.

$$\begin{aligned} C_t^{M} [0] - C_t^{M}[\lambda _u]&= \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ \lambda _u \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} \Big ] du + r_d \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ C_u^{M}[\lambda _u]-C_u^{M}[0] \Big ] du \nonumber \\&\quad +\, \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ (\theta _u - \tilde{\theta }_u) \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} + \frac{1}{2} ({\eta _u}^2 - {\tilde{\eta }_u}^2) \frac{\partial ^2 C_u^{M}[\lambda _u]}{\partial {\sigma _u}^2}\nonumber \\&\quad +\,(\rho \eta _u - \tilde{\rho } \tilde{\eta }_u) \sigma _u S_u \frac{\partial ^2 C_u^{M}[\lambda _u]}{\partial S \partial \sigma }\Big ] du\nonumber \\&\quad -\,\int _t^{T} \mathbb {E}^\mathbb {P} \Big [ (\theta _u - \tilde{\theta }_u) \frac{\partial C_u^{M}[0]}{\partial \sigma _u} + \frac{1}{2} ({\eta _u}^2 - {\tilde{\eta }_u}^2) \frac{\partial ^2 C_u^{M}[0]}{\partial {\sigma _u}^2}\nonumber \\&\quad +\,(\rho \eta _u - \tilde{\rho } \tilde{\eta }_u) \sigma _u S_u \frac{\partial ^2 C_u^{M}[0]}{\partial S \partial \sigma } \Big ] du \nonumber \\&\!= \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ \lambda _u \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} \Big ] du + r_d \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ C_u^{M}[\lambda _u]-C_u^{M}[0] \Big ] du\nonumber \\&\quad +\,I_{t,T}(\lambda _u) - I_{t,T}(0). \end{aligned}$$

(23)

First, let us assume $\lambda _u \ge 0$. We have the following inequations on the option premium,

$$\begin{aligned} \begin{aligned} C_u^{M}[\lambda _u]-C_u^{M}[0]&\le 0 \quad (\forall u \in [t,T]) \\&\mathrm{and} \\ \frac{\partial }{\partial u} \mathbb {E}^\mathbb {P} \Big [ C_u^{M}[\lambda _u]-C_u^{M}[0] \Big ]&= \mathbb {E}^\mathbb {P} \Big [ \frac{\partial }{\partial u} \Big ( C_u^{M}[\lambda _u]-C_u^{M}[0] \Big ) \Big ] \ge 0. \end{aligned} \end{aligned}$$

Thus, from (23),

$$\begin{aligned} \begin{aligned}&C_t^{M}[0] - C_t^{M}[\lambda _u] \\&\quad = \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ \lambda _u \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} \Big ] du\\&\quad \quad + r_d \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ C_u^{M}[\lambda _u]-C_u^{M}[0] \Big ] du + I_{t,T}(\lambda _u) - I_{t,T}(0) \\&\quad \ge \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ \lambda _u \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} \Big ] du + r_d(T-t) \Big ( C_t^{M}[\lambda _u]-C_t^{M}[0] \Big )\\&\quad \quad + I_{t,T}(\lambda _u) - I_{t,T}(0). \\ \therefore&\quad \quad \Big (1+r_d(T-t) \Big ) \Big ( C_t^{M}[0]-C_t^{M}[\lambda _u] \Big ) + I_{t,T}(0) - I_{t,T}(\lambda _u) \\&\quad \ge \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ \lambda _u \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} \Big ] du. \end{aligned} \end{aligned}$$

We also have an inequation of $\mathbb {E}^\mathbb {P} \Big [ C_u^{M}[\lambda _u]-C_u^{M}[0] \Big ] \le 0$, so the following inequation can be derived with the equation of (23).

$$\begin{aligned} C_t^{M}[0] - C_t^{M}[\lambda _u] \le \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ \lambda _u \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} \Big ] du + I_{t,T}(\lambda _u) - I_{t,T}(0). \end{aligned}$$

Thus we can derive the following inequation with two inequations derived above,

$$\begin{aligned} \begin{aligned} C_t^{M}[0] - C_t^{M}[\lambda _u] + I_{t,T}(0) - I_{t,T}(\lambda _u)&\le \int _t^{T} \mathbb {E}^\mathbb {P} \Big [ \lambda _u \frac{\partial C_u^{M}[\lambda _u]}{\partial \sigma _u} \Big ] du \\&\le \Big (1+r_d(T-t) \Big ) \Big ( C_t^{M}[0]-C_t^{M}[\lambda _u] \Big ) \\&\quad + I_{t,T}(0) - I_{t,T}(\lambda _u). \end{aligned} \end{aligned}$$

Second, in the case of $\lambda _u < 0$, we have the following inequations,

$$\begin{aligned} \begin{aligned} C_u^{M}[\lambda _u]-C_u^{M}[0]&\ge 0 (\forall u \in [t,T]) \\&\mathrm{and} \\ \frac{\partial }{\partial u} \mathbb {E}^\mathbb {P} \Big [ C_u^{M}[\lambda _u]-C_u^{M}[0] \Big ]&= \mathbb {E}^\mathbb {P} \Big [ \frac{\partial }{\partial u} \Big ( C_u^{M}[\lambda _u]-C_u^{M}[0] \Big ) \Big ] \le 0. \end{aligned} \end{aligned}$$

Thus we can derive the inequation asserted in this Proposition with the similar approach to the discussion explored above. $\square$

Appendix 2: Time Series Data

See Table 7 and Figs. 3, 4, 5.

Table 7 Summary statistics for implied volatilities

Fig. 3

USD-JPY WM/Reuter closing spot rate. This figure shows the time-series data in the period from October 31, 2003 to June 30, 2010

Fig. 4

USD-JPY 1M ATM implied Volatility (Mid Price). This figure shows the time-series data in the period from October 31, 2003 to June 30, 2010

Fig. 5

USD-JPY 1M ATM mid-bid spread. This figure shows the time-series data in the period from October 31, 2003 to June 30, 2010

Appendix 3: Parameter Estimation Results for the Heston (1993) model

See Figs. 6, 7 and 8.

Fig. 6

The time-series of the estimated parameter: $\tilde{\kappa }$. This figure shows estimation results of $\tilde{\kappa }$ at each time point in the period from October 31, 2003 to June 30, 2010. These parameters are estimated with the maximum liklihood method proposed by Aït-Sahalia and Kimmel (2007) and we update the parameters daily based on historical 1,750 days daily data with a rolling estimation procedure

Fig. 7

The time-series of the estimated parameter: $\tilde{v}$. This figure shows estimation results of $\tilde{v}$ at each time point in the period from October 31, 2003 to June 30, 2010. These parameters are estimated with the maximum liklihood method proposed by Aït-Sahalia and Kimmel (2007) and we update the parameters daily based on historical 1,750 days daily data with a rolling estimation procedure

Fig. 8

The time-series of the estimated parameter: $\tilde{\rho }$. This figure shows estimation results of $\tilde{\rho }$ at each time point in the period from October 31, 2003 to June 30, 2010. These parameters are estimated with the maximum liklihood method proposed by Aït-Sahalia and Kimmel (2007) and we update the parameters daily based on historical 1,750 days daily data with a rolling estimation procedure