Notes: Kristensen and Mele (2011)

1. Introduction

In this post, I work through Adding and Subtracting Black Scholes, JFE (2011) by Antonio Mele and Dennis Kristensen. This paper develops a method for approximating the price of an asset where no closed form expression exists. In the analysis below, I focus my attention on pricing a European option on a stock when the stock price displays stochastic volatility.

In Section $2$ , I describe the approximation method which hinges on the wedge between the infinitesimal generator of the asset price under the true specification (e.g., with stochastic volatility) and a simpler reference specification (e.g., with constant volatility). Then, in Section $3$ I conduct some simple numerical exercises to see how the approximation behaves in practice.

2. The Method

Consider a stock with a price $X_t$ in units of $\$$ ‘s at time $t$ in units of $\mathtt{yr}$ ‘s whose evolution under the risk neutral measure is described by the equations below:

(1) $\begin{align*} \mathtt{d} X_t &= r \cdot X_t \cdot \mathtt{d}t + V_t^{\scriptscriptstyle \frac{1}{2}} \cdot X_t \cdot \mathtt{d}B_{x,t} \\ \mathtt{d} V_t &= \kappa \cdot \left( \alpha - V_t \right) \cdot \mathtt{d}t + \omega \cdot | V_t |^\xi \cdot \mathtt{d}B_{v,t} \end{align*}$

Here, $r$ is the risk free rate in units of $1/\mathtt{yr}$ , $V_t$ is the instantaneous variance of the stock price in units of $1/\mathtt{yr}$ , $\kappa$ is the mean reversion rate of the variance of the stock price in units of $1/\mathtt{yr}$ , $\alpha$ is the mean of the variance of the stock price in units of $1/\mathtt{yr}$ and both $\omega$ and $\xi$ are constants parameterizing the volatility of the variance of the stock price where $\omega$ has units of $\mathtt{yr}^{\xi - {\scriptscriptstyle \frac{3}{2}}}$ and $\xi$ is unitless.

Variance and price of the underlying stock.

Let $\phi(X) = \max\left\{ X - K,0 \right\}$ be the payout of a European option on this stock with strike price $K$ (in $\$$ ). Then, define the price of this option $P$ as follows:

(2) $\begin{align*} P(X,V;t) &= \mathtt{E} \left[ \ \phi(X_T) \ \middle| \ X_t, V_t \ \right] \end{align*}$

As shown in Ait-Sahalia and Kimmel (2007), when $\xi \neq 1/2$ there is no closed form solution for $P$ . The authors approximate $P(X,V;t)$ by computing the difference between the infinitesimal generators of $P(X,V;t)$ and $\overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t)$ where $\overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t)$ is the price of a European option on a stock whose volatility is constant and set to $\sigma$ .

Definition (Infinitesimal Generator): Let $\{Y_t\}$ be a time homogeneous Ito diffusion in $\mathcal{R}^N$ . The infinitesimal generator $\mathtt{A}$ of $Y_t$ is defined as:

(3) $\begin{align*} \mathtt{A}[f(y)] &= \lim_{t \searrow 0} \left\{ \frac{\mathtt{E}[f(Y_t)] - f(y)}{t} \right\} \end{align*}$

Source: Oksendal (2003)

The infinitesimal generator characterizes how a stochastic process $\{Y_t\}$ would look different if we peered an instant $\mathtt{d}t$ into the future. The theorem below gives a method for computing the infinitesimal generator.

Theorem (Infinitesimal Generator of an Ito Process): Let $\{Y_t\}$ be the Ito diffusion in $\mathcal{R}^N$ :

(4) $\begin{align*} \mathtt{d}Y_t &= m(Y_t) \cdot \mathtt{d}t + s(Y_t) \cdot \mathtt{d}B_t \end{align*}$

If $f \in \mathcal{C}^2$ with a compact support then:

(5) $\begin{align*} \mathtt{A}[f(y)] &= \sum_{n=1}^N m_n(y) \cdot \partial_n[f(y)] + \frac{1}{2} \cdot \sum_{n=1}^N \sum_{n'=1}^N \left( s_n(y) s_{n'}(y)^{\top} \right) \cdot \partial_{n,n'}[f(y)] \end{align*}$

Source: Oksendal (2003)

To tie this mathematical construction back into the financial application of options pricing, note that in a world with no financial frictions a la Black and Scholes (1973) the instantaneous change in the price of an asset must exactly equal the interest payment earned by depositing the current price of the asset in a bank for the next instant. Thus, in generator form the second order partial differential equation pinning down the price of a European option on a stock whose price has constant volatility can be written as follows:

(6) $\begin{align*} 0 &= \overset{\mathtt{\scriptscriptstyle BS}}{\mathtt{A}}[\overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t)] - r \cdot \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t) \end{align*}$

where the generator can be characterized as:

Proposition (Infinitesimal Generator given Constant Variance): The generator for the price of a European option on a stock with constant volatility $\sigma$ is given by:

(7) $\begin{align*} \overset{\mathtt{\scriptscriptstyle BS}}{\mathtt{A}} &= \partial_t + r \cdot X_t \cdot \partial_x + \frac{\sigma^2 \cdot X_t^2}{2} \cdot \partial_{x,x} \end{align*}$

Proof: If a stock price displays constant volatility $\sigma$ , I can write $dX_t$ as follows:

(8) $\begin{align*} \mathtt{d}X_t &= r \cdot X_t \cdot \mathtt{d}t + \sigma \cdot X_t \cdot \mathtt{d}B_{x,t} \end{align*}$

where $\sigma$ is the instantaneous volatility of the stock price in units of $1/\sqrt{\mathtt{yr}}$ . Plugging this differential equation into the formula above yield:

(9) $\begin{align*} m(X_t) &= r \cdot X_t \\ s(X_t) &= \sigma \cdot X_t \end{align*}$

I can conduct a similar exercise to characterize the price of a European option on a stock with stochastic volatility. For this asset, I know that:

(10) $\begin{align*} 0 &= \mathtt{A}[P(X,V;t)] - r \cdot P(X,V;t) \end{align*}$

What’s more, I can use the same theorem to characterize the infinitesimal generator as:

Proposition (Infinitesimal Generator given Stochastic Variance): The generator for the price of a European option on a stock with stochastic volatility $V_t$ is given by:

(11) $\begin{align*} \mathtt{A} &= \partial_t + r \cdot X_t \cdot \partial_x + \frac{V_t \cdot X_t^2}{2} \cdot \partial_{x,x} \\ &\qquad \qquad + \kappa \cdot \left( \alpha - V_t \right) \cdot \partial_v + \frac{\omega^2 \cdot V_t^{2 \cdot \xi}}{2} \cdot \partial_{v,v} \\ &\qquad \qquad \qquad \qquad + \rho \cdot \omega \cdot V_t^{\xi + 1/2} \cdot X_t \cdot \partial_{x,v} \end{align*}$

Proof: Applying Ito’s lemma to $f(X,V;t)$ and defining $y = (X,V,t)$ for shorthand yields an expression for $\mathtt{d}f(y)$ as:

(12) $\begin{align*} \mathtt{d}f(y) &= \partial_t [f(y)] \cdot \mathtt{d}t + \partial_x [f(y)] \cdot \mathtt{d}X + \partial_v [f(y)] \cdot \mathtt{d}V \\ &\qquad \qquad + \frac{1}{2} \cdot \left\{ \ \partial_{x,x} [f(y)] \cdot (\mathtt{d}X)^2 + \partial_{x,v} [f(y)] \cdot (\mathtt{d}X \cdot \mathtt{d}V) + \partial_{v,v}[f(y)] \cdot (\mathtt{d}V)^2 \ \right\} \\ &= \partial_t [f(y)] \cdot \mathtt{d}t + \partial_x [f(y)] \cdot \left\{ r \cdot X \cdot \mathtt{d}t + V^{\scriptscriptstyle \frac{1}{2}} \cdot X \cdot \mathtt{d}B_x \right\} \\ &\qquad + \partial_v [f(y)] \cdot \left\{ \kappa \cdot (\alpha - V) \cdot \mathtt{d}t + \omega \cdot |V|^\xi \cdot \mathtt{d}B_v \right\} \\ &\qquad \qquad + \frac{1}{2} \cdot \left\{ \ \partial_{x,x} [f(y)] \cdot \left( V \cdot X^2 \right) \cdot \mathtt{d}t \right. \\ &\qquad \qquad \qquad \left. + \partial_{x,v} [f(y)] \cdot \left( \rho \cdot \omega \cdot V^{\xi + {\scriptscriptstyle \frac{1}{2}}} \cdot X \right) \cdot \mathtt{d}t \right. \\ &\qquad \qquad \qquad \qquad \left. + \partial _{v,v} [f(y)] \cdot \left( \omega^2 \cdot V^{2 \cdot \xi} \right) \cdot \mathtt{d}t \ \right\} \\ &= \Big\{ \ \partial_t [f(y)] + \partial_x [f(y)] \cdot r \cdot X + \partial_v [f(y)] \cdot \kappa \cdot (\alpha - V) \\ &\qquad \qquad + \frac{1}{2} \cdot \left( \partial_{x,x} [f(y)] \cdot \left( V \cdot X^2 \right) \right. \\ &\qquad \qquad \qquad \left. + \partial_{x,v} [f(y)] \cdot \left( \rho \cdot \omega \cdot V^{\xi + {\scriptscriptstyle \frac{1}{2}}} \cdot X \right) \right. \\ &\qquad \qquad \qquad \qquad \left. + \partial _{v,v} [f(y)] \cdot \left( \omega^2 \cdot V^{2 \cdot \xi} \right) \right) \ \Big\} \cdot \mathtt{d}t \\ &\qquad \qquad + \left\{ \partial_x [f(y)] \cdot V^{\scriptscriptstyle \frac{1}{2}} \cdot X \right\} \cdot \mathtt{d}B_{x,t} \\ &\qquad \qquad \qquad + \left\{ \partial_v [f(y)] \cdot \omega \cdot |V|^\xi \right\} \cdot \mathtt{d}B_{v,t} \end{align*}$

Collecting terms yields expressions for $m$ and $s$ which given $\mathtt{A}$ .

The authors’ key insight is that applying the infinitesimal generator $\mathtt{A}$ to the difference $P(X,V;t) - \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t)$ yields the expression:

(13) $\begin{align*} \delta(X,V,\sigma;t) &= \mathtt{A}\left[ P(X,V;t) - \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t) \right] - r \cdot \left(P(X,V;t) - \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t) \right) \\ &= \frac{(\sigma^2 - V_t) \cdot X_t^2}{2} \cdot \partial_{x,x}[ \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t)] \end{align*}$

given the boundary condition that $P(X,V;T) - \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;T) = 0$ for all $V$ and $X$ . This wedge given by $\delta(X,V,\sigma;t)$ has the natural interpretation of the hedging cost incurred by a trader continuously hedging the risk of the European option with stochastic volatility using the incorrect constant volatility model. Thus, we can write the true price as the Black and Scholes price plus an error correction term:

(14) $\begin{align*} P(X,V;t) &= \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t) + \mathtt{E}\left[ \ \int_t^T e^{-r \cdot (u-t)} \cdot \delta(X,V;u) \cdot \mathtt{d}u \ \middle| \ X_t, V_t \ \right] \end{align*}$

This is a particularly nice formulation as each of the error terms is a function of only the Black and Scholes (1973) formulation with constant volatility.

3. Numerical Results

To put this approximation method to work, use a Taylor expansion around $\overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t)$ :

(15) $\begin{align*} P(X,V;t) &= \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t) + \sum_{n=0}^\infty \frac{(T-t)^{n+1}}{(n+1)!} \cdot \delta_n(X,V;t) \end{align*}$

Each of the expansion terms can be expressed recursively as follows:

(16) $\begin{align*} \delta_n(X,V;t) &= \mathtt{A}[\delta_{n-1}(X,V;t)] - r \cdot \delta_{n-1}(X,V;t) \\ \delta_0(X,V;t) &= \frac{(\sigma^2 - V_t) \cdot X_t^2}{2} \cdot \partial_{x,x}[ \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t)] \end{align*}$

The Black and Scholes (1973) options price given constant volatility $\sigma$ can be expressed as:

(17) $\begin{align*} \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t) &= N(q_1) \cdot X - N(q_2) \cdot K \cdot e^{-r} \\ q_1 &= \frac{\log \left( \frac{X}{K} \right) + \left( r + \frac{\sigma^2}{2} \right)}{\sigma} \\ q_2 &= q_1 - \sigma \end{align*}$

Thus, expanding out the first $2$ approximating terms yields:

(18) $\begin{align*} P_0(X,V;t) &= \overset{\mathtt{\scriptscriptstyle BS}}{P}(X;t) + \frac{(\sigma^2 - V_t) \cdot X_t^2}{2} \cdot \left( \frac{N(d_1)}{X \cdot \sigma} \right) \\ P_1(X,V;t) &= P_0(X,V;t) + \frac{1}{2} \cdot \left[ \frac{\kappa \cdot (\alpha - V_t) \cdot X_t^2}{2} \cdot \left( \frac{N(d_1)}{X \cdot \sigma} \right) + \omega \cdot \rho \cdot V_t^{\xi + {\scriptscriptstyle \frac{1}{2}}} \cdot X_t \cdot \left\{ X_t \cdot \right\} \right] \end{align*}$

In order to simplify calculations, I take $V_t=0.05$ to kill off the $P_0(X,V;t)$ term in my numerical analysis. Below I replicate Figure $3$ from the Kristensen and Mele (2011) for $N \in \{0,1\}$ and plot the mispricing as a fraction of the Black and Scholes (1973) price with $\xi = 1/2$ for $N = 0, 1$ . The strike price is $K = 100$ , the time to maturity is $1$ year, the remaining parameter values are $\kappa = 2$ , $\alpha = 0.04$ , $\omega = 0.10$ , $\rho = -0.50$ and $r = 0.10$ .

Mispricing as a fraction of the Black and Scholes (1973) price with xi = 1/2 for N = 0, 1. The strike price is K = 100, the time to maturity is 1 year, the remaining parameter values are kappa = 2, alpha = 0.04, omega = 0.10, rho = -0.50 and r = 0.10.

The code for the simulations can be found on www.SageMath.org. The simulation represents estimates from $500$ iterations at $40$ price points between $X_0 = 80$ and $X_0 = 120$ .