Pricing a Contract That Pays the Higher of Two Stock Prices

Summary

The document considers a one-year contract on two stocks, each initially priced at 100, that pays the higher stock price at maturity. It explains that the issue price cannot be determined from the initial prices and maturity alone: asset volatilities and their correlation are also needed, with rates and dividends affecting the valuation assumptions.

Under zero rates and dividends, the payoff can be decomposed into one stock plus an exchange option on the difference between the two stocks. The stock component is worth its current price, while the exchange option has positive value; a Black–Scholes framework can value that component using the Margrabe formula. The document therefore establishes that the contract is worth more than 100 under those assumptions, but provides no numeric price because key market inputs are missing. A separate answer suggests a risk-free discounted value, illustrating the ambiguity when the option component is not modeled.

Key ideas

• The maximum of two stock prices can be expressed as one stock price plus an option on their difference.
• The option component has positive value, so under zero rates and dividends the contract is worth more than either initial price.
• A numerical price requires the assets' volatilities and correlation.
• The exchange option component can be valued with the Margrabe formula within a Black–Scholes framework.
• Rates and dividends also matter when they are not assumed to be zero.

Tags

# What would be the issue price of the following contract?


# What would be the issue price of the following contract?












- Stock currently A has a price equal to 100.

- Stock B also currently has a price of 100.

- The contract has a maturity $\mu$ of one year.

- At maturity the payout is the max price of either A or B.

What is the contract worth at issue?

## Answer by Ivan (score 1, accepted)

https://quant.stackexchange.com/a/38471

The actual answer depends on the volatility of both assets and their correlation, assuming for simplicity that rates and dividends are zero.

With what you’re given you should be able to at least answer that it’s worth more than 100 (rates are 0), and to show this you can simply express max(A,B) as B + Max(A-B,0). Take the expected value in the risk-neutral world to price it. The second term is a so-called exchange option obviously whose value is > 0. This is straightforwardly valued in a BS framework (see Margrabe formula). The first term is 100 if rates and dividends are zero.

## Answer by abb (score 0)

https://quant.stackexchange.com/a/38466

This is probably a homework question... that being said this should be valued similar to how you value an option. Except the payout would be: max(A,B) instead of max(St-k,0) for a call. Because it doesn’t give any volatility parameters there’s not a whole lot you can say other than assume the contract is worth 100 discounted for a year at the risk free rate.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.