Pricing a Relative-Performance Barrier Option with a Stock Numeraire
Summary
The question considers an option that pays the difference between two stock prices at maturity only if stock A has remained above stock B throughout the contract. The accepted response reframes the condition by using stock B as the numeraire. In units of B, the relevant underlying is the ratio of the two stock prices, and the condition becomes a barrier at a ratio of one. The proposed price is obtained from a barrier option on that ratio, then converted back into currency by multiplying by the price of B.
Under the response’s assumptions, the ratio’s volatility depends on the volatilities of both stocks and their correlation. The exchange offers this as a simpler option-pricing route than directly calculating the probability that a modeled price difference avoids zero. A second answer suggests a stock-pair replication intuition, but does not establish that it captures the path-dependent barrier payoff. The discussion gives no complete model specification, market inputs, or numerical valuation, so implementation requires additional assumptions and validation.
Key ideas
- Using stock B as numeraire expresses stock A’s value as the ratio of their prices.
- The requirement that A stay above B becomes a barrier condition at a ratio of one.
- The relative price’s volatility incorporates both stocks’ volatilities and their correlation.
- A value calculated in units of B is converted to currency using B’s price.
- A simple long A and short B position does not by itself explain the path-dependent condition.
Tags
Full text
# How to price this option?
# 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?
## Answer by spaceisdarkgreen (score 7)
https://quant.stackexchange.com/a/36272
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.
## Answer by Adam N. (score 5)
https://quant.stackexchange.com/a/36282
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 $S_A-S_B$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.