Pricing European Calls on a Power of the Underlying
Summary
The document shows how to price a European option with payoff equal to the positive part of the underlying raised to a constant power, less the strike. Under Black–Scholes, the underlying follows geometric Brownian motion. Applying Itô’s lemma to its power shows that the transformed variable is also geometric Brownian motion, with an adjusted drift and volatility. The payoff can therefore be viewed as a call on that transformed process and priced with a Black–Scholes formula using the corresponding parameters.
A second answer expresses the price as an integral over terminal underlying values above the threshold implied by the power and strike. These approaches explain the basic transformation and payoff region. The discussion does not spell out all parameter and domain conditions, such as handling a zero power or ensuring the strike threshold is meaningful for the chosen exponent, so those details need care in implementation.
Key ideas
- Raising a geometric Brownian motion to a constant power produces another geometric Brownian motion.
- Itô’s lemma gives the transformed process’s adjusted drift and volatility.
- The power-option payoff can be treated as a call payoff on the transformed underlying.
- The payoff is positive where the original underlying exceeds the threshold implied by the strike and exponent.
- The transformation requires attention to the exponent and payoff domain.
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Full text
# What is the price of the European option with the payoff of $\max(S^a-K,0)$?
# What is the price of the European option with the payoff of $\max(S^a-K,0)$?
I interpret such an option as a power option but I do not find any literatures or existing methods to price it. Can it be priced with Black-Scholes with simple changes?
## Answer by Sanjay (score 6, accepted)
https://quant.stackexchange.com/a/44651
Under the Black-Scholes framework the dynamics of $S$ is a GBM ($dS_t = \alpha S_t dt + \sigma S_t dW_t $).
Introduce a new variable $Y_t:= S_t^a$ for $a$ being a real valued constant. Then by Ito the dynamics of $Y$ is given by: $$dY=aS^{a-1}dS+\frac{1}{2}a(a-1)S^{a-2}(dS)^2 \\ = (a\alpha+\frac{1}{2}a(a-1)\sigma^2)S^adt+a\sigma S^a dW \\ = (a\alpha+\frac{1}{2}a(a-1)\sigma^2)Ydt+a\sigma Y dW $$ Let $\mu :=a\alpha+\frac{1}{2}a(a-1)\sigma^2$ and $\gamma:=a\sigma$ then $Y_t$ is a GBM ($dY_t = \mu Y_t dt + \gamma Y_tdW_t$) with drift $\mu$ and volatility $\gamma$.
Can it be priced with Black-Scholes with simple changes?
Yes. $\max (S^a_T-K,0)=\max (Y_T-K,0)$. $X:=\max (Y_T-K,0)$ is a payout equivalent to a European Call option with the underlying having a price process $Y_t$.You can use Black-Scholes formula to find the value of X at any time $t \in [0,T]$
## Answer by ZRH (score 1)
https://quant.stackexchange.com/a/44640
Yes indeed. Bearing in mind that for the option to be in the money, the underlying needs to fulfill the following inequality to be in-the-money: $S^a\geq K$, i.e. $S\geq K^{1/a}$. Calling the terminal pdf of underlying prices $\rho(\xi)$, the price can be computed by evaluating the following integral:
$C_a(K)=\int_{K^{1/a}}^{\infty}(\xi^a-K)\rho(\xi)d\xi$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.