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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

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1 Answer 1

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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.

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