Valuing a Chooser Option with a Discrete Rate Distribution

Summary

The example values a one-year payoff that rewards either a low or a high average short-term interest rate. The rate has four stated possible outcomes with assigned probabilities, so the direct method is to calculate the payoff in each case, weight each payoff by its probability, and add the results. The document reports an expected payoff of 0.037.

It also rewrites the payoff as a put and call struck at the midpoint of the two thresholds, plus a constant payment. This gives an equivalent decomposition into a put value, a call value, and the fixed amount; the stated component values also sum to 0.037. The calculation is a simple illustration of payoff decomposition and expectation under a discrete distribution. It does not discuss discounting, whether the probabilities are risk-neutral, or how to infer the distribution from market data, so the stated figure should not be treated as a general market pricing formula.

Key ideas

• A discrete set of possible rate outcomes can be valued by weighting each payoff by its probability.
• The example payoff can be decomposed into a put, a call at the midpoint strike, and a constant payment.
• The reported expected payoff and the option decomposition both produce a value of 0.037.
• The example does not specify discounting or whether the probabilities are suitable for market pricing.

Tags

# Can you please give an extremely simple formula for valuing a so-called "chooser option"?


# Can you please give an extremely simple formula for valuing a so-called "chooser option"?












Can you please give a simple, high-level formula for valuing a so-called "chooser option"?

For example, say X represents the average short-term rate over the next year (time 0 to time 1). There's a 20% chance X will be 0.05, a 30% that X will be 0.07, a 20% chance that X will be 0.09, and a 30% chance that X will be 0.15. Say that there's a "chooser option" on the average interest rate where a cash flow equal to max(0.08 - X, X - 0.07) is paid at time one. How would one go about finding the price of this "chooser option" at time 0?

## Answer by Attack68 (score 2)

https://quant.stackexchange.com/a/82125

Your $X$ value only has 4 distinct outcomes so you easily model this probabilistically and take the expected value, which is 0.037.

Alternatively, you can observe that your payoff function is the same as the following:

$$ payoff = max(0.08-X, X-0.07) = max(0.075-X, 0) + max(X-0.075, 0) + 0.005 $$

And therefore the more general formula for the cost of this chooser, expressed in terms of puts and calls is:

$$ cost = Put(K=0.075) + Call(K=0.075) + 0.005 $$

Probabilistically, the value of the put is 0.0065, and the value of the call is 0.0255 and adding 0.005 get to the 0.037 value.

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