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Let $S_1$ and $S_2$ be two risky assets. The market also has a riskfree asset, and only one driving Wiener process. The parameters are as in the Black SCholes, with $\mu_1, \mu_2, \sigma_1, \sigma_2$.

$dS_i = \mu_1S_idt + \sigma_i S_idW^P$

I wish to price the option at time $t$ which gives us $1$ if $S_1 > S_2$ at time $T$, otherwise $0$.

My method: calculate $S_1(T) - S_2(T)$ explicitly. Then we need the $Q$-probability that this is strictly positive. However, assuming this is the correct method, I get stuck on calculating this $Q$-probability.

EDIT: I got an answer, but it depends on the cumulative distribution of a normal distribution. Does this look right guys?

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    $\begingroup$ Given that you worked out an answer - why don't you show it and sketch how you got there? Would make it make easier to provide feedback. $\endgroup$
    – LocalVolatility
    Jan 23, 2017 at 1:01

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You should work in the numeraire of $S_1$ (if it is a tradable which doesn't pay dividends). In this numeraire $S_1$ has no drift and $S_2/S_1$ is a martingale.

You should get a digital option on a log normal process which actually has a pricing formula in terms of the cumulative distribution of a normal distribution.

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  • $\begingroup$ Assuming correlated lognormals, this gives rise to the famous Margrabe's formula (en.wikipedia.org/wiki/Margrabe's_formula). Also, I do not completely agree, even if $S_1$ pays dividend it can still be used as a numéraire (as long as it is tradable and always postiive-valued) and you could also apply the same reasoning using $S_2$ as numéraire. $\endgroup$
    – Quantuple
    Jan 23, 2017 at 9:14
  • $\begingroup$ @Quantuple thanks for your comment. I was pretty sure that you need non paying dividends assets to have the martingale property. I will check it. (Let me explain why I believe the hypothesys is necessary: informally speaking, martingality, means that no investment is better than another, but the stochastic process of an instrument paying coupons is the process of an asset "sinking value"). Of course I need to check if my intuition is correct. $\endgroup$
    – jimifiki
    Jan 23, 2017 at 9:31
  • $\begingroup$ Well you are right: dividends ruin the "self-financing" property hence martingality. But a numeraire need not be a martingale. $\endgroup$
    – Quantuple
    Jan 23, 2017 at 9:37

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