How to derive a pricing PDE for an asset that follows a mean-reverting process?

Question

I want to derive a Black-Scholes type partial differential equation to price options on an asset that follows a mean-reverting process (Schwartz model).

My attempt follows the methodology of deriving the Black-Scholes PDE but using a mean-reverting process to describe the asset instead of a geometric Brownian motion:

Let $S$ follow a mean-reverting stochastic process $$ S = \kappa(\mu-\ln S)S dt + \sigma SdW $$ and let $V=V(S,t)$ denote the value of the option. From Itô's lemma we have $$ dV=\left(\frac{\partial V}{\partial t}+\kappa(\mu-\ln S)S\frac{\partial V}{\partial S}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right)dt+ \sigma S \frac{\partial V}{\partial S} dW_t. $$

Let's perform a delta hedge, i.e. construct a portfolio $\Pi=-V+\frac{\partial V}{\partial S} S$. We see that $$ d\Pi = \left(\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right)dt, $$ and since the portfolio $\Pi$ does not involve any risk, it must earn the risk-free interest rate, i.e. $$ d\Pi = r\Pi dt= r\left(-V+\frac{\partial V}{\partial S} \right)dt. $$ Thus, we will have a PDE of the form $$ \frac{\partial V}{\partial t} +rS\frac{\partial V}{\partial S} +\frac{1}{2}\sigma^2 S^2 \frac{\partial ^2V}{\partial S^2}-rV=0, $$ which is the regular Black-Scholes PDE.

Is this correct, or where do I go wrong here?

I believe that the PDE should be $$ \frac{\partial V}{\partial t} +\kappa\left(\mu - \lambda-\ln S\right)S\frac{\partial V}{\partial S} +\frac{1}{2}\sigma^2 S^2 \frac{\partial ^2V}{\partial S^2}-rV=0, $$ where $\lambda$ is the market price of risk. This form of the PDE can be found in this post, for example.

Answer 1

In a Black-Scholes-Merton-style hedge portfolio, we'd get:

$$dS_t=\kappa\left(\mu-\ln S_t\right)S_tdt+\sigma S_t dW_t $$

with a hedged portfolio

$$\Pi_t\equiv V_t-\Delta_tS_t$$

and

$$ d\Pi_t=\frac{\partial V}{\partial t}dt+\frac{\partial V}{\partial S}dS+\frac{1}{2}\frac{\partial^2 V}{\partial S^2}dS^2-\Delta_tdS_t$$

as usual, the portfolio is hedged iff $\Delta_t=\frac{\partial V}{\partial S}$ at all times. Then:

$$\begin{align} d\Pi_t&=\frac{\partial V}{\partial t}dt+\frac{1}{2}\frac{\partial^2 V}{\partial S^2}dS^2=r\left(V_t-\Delta_tS_t\right)dt\\ \Rightarrow rV_t&=\frac{\partial V}{\partial t}+rS_t\frac{\partial V}{\partial S}+\frac{1}{2}\frac{\partial^2 V}{\partial S^2}dS^2 \end{align} $$

...and $\mu,\kappa$ do not show up as the physical-world drift component does not have any say in this perfectly hedged (i.e. risk-neutral) world. HTH?

Answer 2

Let $\{r_t, \, t\ge 0\}$ be the interest rate process. For maturity $T$ and $0\le t \le T$, note that \begin{align*} V(S_t, t) = e^{\int_0^t r_s ds}\,\mathbb{E}\left(e^{-\int_0^T r_s ds}V(S_T, T) \mid \mathscr{F}_t\right), \end{align*} where $\mathbb{E}$ is the expectation under the risk-neutral probability measure. Then $M_t = e^{-\int_0^t r_s ds}V(S_t, t)$ is a martingale, for $0\le t \le T$. Moreover, note that \begin{align*} dM_t &= e^{-\int_0^t r_s ds}dV - r_t e^{-\int_0^t r_s ds}V dt\\ &=e^{-\int_0^t r_s ds}\left[\bigg(-r_t V_t + \frac{\partial V}{\partial t}+\kappa(\mu-\ln S)S\frac{\partial V}{\partial S}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \bigg)dt + \sigma S \frac{\partial V}{\partial S} dW_t \right] \end{align*} Then, \begin{align*} \frac{\partial V}{\partial t}+\kappa(\mu-\ln S)S\frac{\partial V}{\partial S}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}-r_t V_t=0. \end{align*}

Regarding the application to futures option valuation, see, for example, this question. For application to a bond price, see here.

Comments

The issue in your derivation is that $\Pi=-V+\frac{\partial V}{\partial S} S$ is not a self-financing portfolio. See discussions in this question.

Answer 3

(Just a complementary note)

A variable change $X = \ln S$ ($S =\exp X$) gives:

$$ dS/S = dX + \sigma^2/2 dt,$$

making $X$ an Ornstein-Uhlenbeck process

$$ dX = \kappa[(\mu - \sigma^2/(2\kappa)) - X] dt + \sigma dW, $$

which allows direct calculations of the time-$0$ price of the commodity futures with last trading date $T$, $F(S_0,T)$:

$$F(S_0,T)=E_0[S_T]=\exp \left( E_0[X_T] +V_0[X_T]/2 \right)$$

$$ = \exp \left( e^{-\kappa T} \ln S_0 + (1- e^{-\kappa T})(\mu - \sigma^2/(2\kappa)) + \sigma^2/(2\kappa) (1- e^{-2\kappa T})\right) $$

One can then check that $F(S_0,T)$ solves:

\begin{align*} -\frac{\partial F}{\partial T}+\kappa(\mu-\ln S)S\frac{\partial F}{\partial S}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 F}{\partial S^2}=0, \; \; F(S,0)=S \end{align*}

This model was first applied in the commodity space, and enhanced to two and three dimensions, by Eduardo Schwartz here.

Also, the market price of risk $\lambda$ indeed simply adjusts $\mu$ to $\mu-\lambda$ both in the underlying's SDE and derivative's PDE.