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Valuation of Squared Power Options
Team Latte
August 10, 2011
Ever since their introduction more than two decades ago, Power options, as an exotic family of financial derivatives, have not been popular with most investors. The reason for this is the leverage built into their payoffs. One of the members of this family of options is the Squared Power option. In this section we will look into the governing stochastic differential equation (SDF) for the Squared Power option the closed form solution for the value of a Squared Power Call.
The payoff of a Squared Power Call option is given by:
The valuation of this payoff can be easily done either via Monte Carlo simulation or a closed form Black-Scholes formula.
The governing stochastic differential equation (SDF) of an  th power asset is given by
If we substitute  in the above equation we would get the SDF for an asset raised to the power of 2.
This shows that the squared power option can be easily valued in Black-Scholes model; the only difference would be that we need to replace the drift by  or  and the volatility needs to be replaced by 
Thus, for the above square power callpayoff, the closed form Black-Scholes option value would be given by:
It would be quite easy to find the price of the call option using the above closed form solution. The exaggerated payoff of the squared power call option is not simply due to the asset being raised to the power of 2; note that the volatility is also increased twofold.
Reference: The Handbook of Exotic Options, Edited by Israel Nelken (Irwin Publishing)
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