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Pricing Power Options in the Black-Scholes Model

Power options are a class of exotic options in which the payoff at expiry is related to the power of the stock price, where . For a power option on a stock with price having strike price and time to expiry , the payoff is for a call, and for a put. Within the Black–Scholes model, closed-form solutions exist for the price of power options. In this Demonstration, prices as a function of the various parameters are explored.

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Given a stock with price , fixed dividend yield , and assumed fixed interest rate , the pricing formulas for power calls and puts with strike and time to expiry are
,
,
with and . Here is the cumulative distribution function for the standard normal distribution.
Snapshot 1: in the special case , power options reduce to regular European options
Snapshot 2: for , the payoff curve for call options becomes concave, and thus the option can have negative time value (i.e., current price < payoff)
Snapshot 3: for put options, the reverse is true: the payoff curve becomes concave for
Snapshot 4: for call options, the option value becomes very large as increases
E. G. Haug, The Complete Guide to Option Pricing Formulas, New York: McGraw–Hill, 2007.
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