Using Fourier Transforms for stock option pricing with stochastic interest rates

Question

Can Fourier transforms be used to derive the joint probability density function of stochastic interest rates and stock price Brownian motions of call options under stochastic interest rates?

So lets say we have for the interest rate the following process $$dr(t)=\lambda(\theta-r(t))dt+\sigma^{r}dW^{r}(t)$$

and for the stock process $$dS(t)=r(t)S(t)dt+\sigma^{S}S(t)dW^{S}(t)$$ A regular vanilla call option price is then given as $$\mathrm{Call}(t,K)=\mathbb{E}^{Q}\left[e^{-\int_{0}^{t}r(s) ds}(S(t)-K)^{+}\right]$$

Can the joint density $f(W^{r}(t),W^{S}(t))$ be derived from call option prices by use of the Fourier Transform?

Answer 1

You have there an incomplete market model, because the risky asset contains two sources of risk $W^S,W^r$ , which means that the option price cannot be hedged by a riskfree portfolio.

For such models, $Q$ has infinitely many solutions which means you cannot find the joint density in any way.

Answer 2

There are a couple ways I can think to approximate this from market data, but none use Fourier transforms.

1. Assuming that you have specified the processes under the risk neutral measure (since S is presumably under the risk neutral measure I am assuming that r is specified under the risk neutral measure) then you can calibrate the parameters to the market (for example, by minimizing the squared error over the unknown parameters). Given the parameters you can compute the (market implied) joint density, if not analytically than by solving the Focker-Plank equation numerically.

2. Augment the market with a risk free bond. Then you can price the option under the forward measure. Taking the second derivative with respect to the strike recovers the forward density of the stock (See the discussion here for an example of how to do this). This doesn't recover the joint density but does allow pricing of any european option with the same expiry date.

3. If you are after the risk-neutral covariance between the two process one can augment the market to include forward and future contracts. Let $f=\tilde{\mathbb{E}}[S_T]$ be the future price and $F=\frac{S_0}{B(0, T)}$ be the forward price where $B(0, T)=\tilde{\mathbb{E}}[e^{-\int_0 ^T r(s) ds}]=\tilde{\mathbb{E}}[D(0, T)]$ is the price of a zero-coupon default-free bond. Then clearly Cov(S, D)=$\tilde{\mathbb{E}}[S_T D(0, T)]-\tilde{\mathbb{E}}[S_T]\tilde{\mathbb{E}}[D(0, T)]=S_0-fB(0, T)$. This is greater than zero if and only if $S_0>fB(0, T)$ which is equivalent to $F>f$. So if the forward price is larger than the future price the correlation between S and D is positive, or equivalently, S and r are negatively correlated.