Rainbow Option on the Minimum (Stulz)
Computes the price of a call on the minimum of two assets with the Stulz (1982) formula: the option pays based on the worse performer of two assets, minus the strike. It's the more conservative option of the rainbow pair, valuable when you want both assets to perform. An elegant identity holds: the sum of the call on the maximum and the call on the minimum equals the sum of two plain calls. It uses the bivariate normal. Enter the two prices, the strike, the two volatilities, the correlation, the rate and the term.
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Rainbow Option on the Minimum (Stulz)
Computes the price of a call on the minimum of two assets with the Stulz (1982) formula: the option pays based on the worse performer of two assets, minus the strike. It's the more conservative option of the rainbow pair, valuable when you want both assets to perform. An elegant identity holds: the sum of the call on the maximum and the call on the minimum equals the sum of two plain calls. It uses the bivariate normal. Enter the two prices, the strike, the two volatilities, the correlation, the rate and the term.
The option that needs both to perform
The rainbow on the minimum is the cautious cousin of the option on the maximum. Instead of following the winner, it pays based on the worse performer of two assets. That makes it cheaper and more conservative: it only rewards well if both assets rise, because the outcome is dragged by the weaker of the pair.
Why would anyone want this? Because there are situations where you need two things to go right at the same time, and you want protection or exposure tied precisely to the weakest link. The pricing, also from Stulz, mirrors the option on the maximum with flipped signs on the correlations, and satisfies a neat identity: maximum plus minimum equals two plain calls summed.
Enter the prices of the two assets, the strike, the two volatilities, the correlation, the interest rate and the term. The tool returns the call-on-the-minimum premium, using the bivariate normal. Since it depends on the weaker asset, correlation has the opposite effect to the option on the maximum: more correlation, more valuable.
Related Tools
Rainbow Option on the Maximum (Stulz)
Computes the price of a call on the maximum of two assets with the Stulz (1982) formula: the option pays based on the better performer of two correlated assets, minus the strike. It's a bet on the winner of a race between two assets, and its price depends heavily on the correlation between them — the less correlated, the more valuable, because there's a greater chance at least one takes off. It uses the bivariate normal. Enter the two prices, the strike, the two volatilities, the correlation, the rate and the term.
Margrabe Exchange Option
Computes the price of an exchange option with the Margrabe formula: the right to exchange one asset for another at expiry. It's the generalization of Black-Scholes to two risky assets, where the strike stops being fixed and becomes the price of a second asset. The relevant volatility is that of the ratio between the two, combining the individual volatilities and the correlation. It shows up in mergers, spread options and executive compensation. Enter the two prices, the two volatilities, the correlation and the term.
Two-Asset Correlation Binary
Computes the price of a two-asset binary option: it pays a fixed amount if and only if the first asset finishes above its strike AND the second asset finishes above its own. It's a conditional double bet whose price depends critically on the correlation between the assets — the more correlated, the more likely both conditions happen together. It uses the bivariate normal. Enter the two prices, the two strikes, the two volatilities, the correlation, the rate, the term and the payout.
Black-76 Call Price (Options on Futures)
Works out the premium of a European call option on futures with the Black-76 model, the variant of Black-Scholes used when the underlying is a futures or forward contract. The price is e^(−rT)·[F·N(d1) − K·N(d2)], with d1 and d2 built from the futures price, the strike, the volatility and the term. Because the future already embeds the cost of carry, the discount hits both terms and the interest rate never shows up inside d1. It is the standard for options on commodities, indices and rates. Enter the futures price, the strike, the risk-free rate, the term in years and the annual volatility.
Black-76 Put Price (Options on Futures)
Works out the premium of a European put option on futures with the Black-76 model, the Black-Scholes version for when the underlying is a future or forward contract. The price is e^(−rT)·[K·N(−d2) − F·N(−d1)], where d1 and d2 come from the futures price, the strike, the volatility and the term. The future already carries the cost of carry, so the discount factor multiplies both terms and interest does not enter d1. It applies to puts on commodities, indices and rates. Enter the futures price, the strike, the risk-free rate, the term in years and the annual volatility.
Compound Option (Call-on-Call, Geske)
Computes the price of a compound call-on-call option with the Geske (1979) model: a call option whose underlying is, itself, another call option. It's the structure behind many real-world contracts — an option to extend a project, for example, is an option on an option. The calculation requires finding the critical price at which exercising the first option is worthwhile and uses the bivariate normal. Enter the spot price, the two strikes, the two expiries, the rate and the volatility.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.