# Pricing a claim dependent on two stock processes

QUESTION Consider two stock processes:

$$dS^1_t=S^1_t(r\,dt+\sigma_1\,dW^1_t)$$ $$dS^1_t=S^2_t(r\,dt+\sigma_2\,dW^2_t)$$ $$t,S^1_0,S^2_0\ge0$$ and $$W^1_t,W^2_t$$ are standard independent brownian motions under a risk neutral measure P. What is the price of the claim with payoff structure given by: $$H(S^1_t,S^2_t)=\frac {S^1_t}{S^2_t}$$ For a fixed maturity: $$T>0$$

MY ATTEMPT

I get that: $$V_0=\left(\frac {S^1_0}{S^2_0}\right)e^{(\sigma^2_2-r)T}$$

Is this correct, and further, if it is correct, what is the intuition behind having the price depending solely on the volatility of the second stock??

Cheers

The solutions of the stochastic differential equation are $$S_T^1=S_0^1e^{(r-\frac12\sigma_1^2)T+\sigma_1W_T^1},$$ and $$S_T^2=S_0^2e^{(r-\frac12\sigma_2^2)T+\sigma_2W_T^2},$$ thus $$\frac{S_T^1}{S_T^2}=\frac{S_0^1}{S_0^2}e^{(\sigma_2^2-\sigma_1^2)\frac T2+\sigma_1W_T^1-\sigma_2W_T^2}.$$ Therefore, $$e^{-rT}\mathbb E\left[\frac{S_T^1}{S_T^2}\right] =e^{-rT}e^{(\sigma_2^2-\sigma_1^2)\frac T2}\frac{S_0^1}{S_0^2}\mathbb E\left[e^{\sigma_1\sqrt{T}N}\right]\mathbb E\left[e^{-\sigma_2\sqrt{T}N}\right]=e^{-rT+\sigma_2^2T}.$$ As for a financial interpretation, I have none. I haven't done any mathematical finance in a while, and I find it strange to evaluate the ratio of stocks (and not interest rates). Nevertheless, I can give you a bit of mathematical intuition by remarking that there exists a Brownian motion $(W_t)$ such that $$\frac{S_T^1}{S_T^2}=e^{\sigma_2^2T}\frac{S_0^1}{S_0^2}\underbrace{e^{-(\sigma_2^2+\sigma_1^2)\frac T2+(\sigma_1^2+\sigma_2^2)W_T}}_{M_T},$$ and it is known that the stochastic process $(M_t)$ is a martingale (for any $\sigma_1,\sigma_2$), thus constant in expectation, with expectation $1$. Hopefully someone else can give you further insight on this result.