# How to price this option?

I was asked this question in an interview.

There is an option as follows. It monitors the prices of two stocks A and B, and pays the difference in their prices at time $T$, if stock A has been higher than stock B all through till $T$. There is nothing paid if stock A has fallen lesser than stock B at any time before $T$.

How do we price this option?

I gave an answer by modeling the difference as a Brownian motion, and computing the probability that the zero-hitting time for the BM to be greater than $T$. However the interviewer said there was a simpler method based on option pricing.

Can anyone help me?

• What was the interview for (ie what level)? The answer they're looking for will depend on this. – will Oct 3 '17 at 7:19
• @will: quant associate position... top investment bank. – helloman Oct 3 '17 at 7:42

Well "based on option pricing" is a little vague, but the desired solution is probably to use one of the stocks (say stock $B$) as your numeraire. If you're unfamiliar the intuitive idea is that imagine instead of money, people used stock $B$ as currency / measure of wealth, etc. Then the "value" of stock $A$ (i.e. the number of shares of stock $B$ it is worth) is $S_A/S_B$ and you have a fixed strike / Barrier of $1.$ So you price a barrier option with initial price $S_{A,0}/S_{B,0}$, barrier / strike $1$ and volatility $$\sqrt{\sigma_A^2 + \sigma_B^2-2\rho\sigma_A\sigma_B}$$ (i.e. the volatility of $S_A/S_B$) and that gives the price of the option in units of $S_B,$ so just multiply by $S_B$ to get the price in dollars.
I suspect you could replicate this trade by buying $A$ and selling short $B$ (additionally, when the price of $A$ touches $B$ at any time before $T$, you liquidate the position for $0$ payoff). If so, today's price of this option is just $S_A-S_B$.
• @dm63 That's not what I get from reading the question (it just says it pays the value of the spread at time $T$.) That said, it's entirely possible that it was American in the interviewer's actual question... would explain the 'simple solution'. – spaceisdarkgreen Oct 4 '17 at 12:18
• @dm6 (you gotta @ me) I don't know what you mean by the knockout being American. The solution above would apply to a derivative that can be cashed in for the value of the spread at any time before expiration (i.e. American exercise) and becomes worthless if it ever hits zero. If you only get to cash out at $T$ you lose the option value of cashing out before that simply buying the spread would provide. It's easy to see such a derivative's value goes to down as vol or $T$ get large since prob of spread hitting zero before $T$ increases – spaceisdarkgreen Oct 4 '17 at 15:22