Black–Scholes PDE for Options on Powers of a Lognormal Asset
Summary
The document clarifies a common terminology question in option pricing: “Black–Scholes equation” usually means the Black–Scholes partial differential equation, while “Black–Scholes formula” refers to its closed-form call or put solution under specific assumptions. It discusses a call whose payoff depends on the square of a lognormal asset as an exercise in applying the Black–Scholes framework, rather than treating the standard call formula as directly interchangeable with the PDE.
The response also cautions that extending Black–Scholes reasoning to unusual payoffs can require careful justification, including self-financing replication, risk-neutral valuation, or a change of numeraire. The document does not derive a price or PDE for the squared-asset payoff, and its brief treatment of path dependence is only a pointer to another discussion. Its main value is clarifying terminology and flagging the reasoning needed to apply the framework beyond a vanilla European option.
Key ideas
- “Black–Scholes equation” commonly refers to the pricing PDE, whereas the formula is its analytical solution for standard call and put payoffs.
- A payoff based on a power of an asset can require extending the usual Black–Scholes arguments.
- Replication, risk-neutral valuation, and changes of numeraire may be relevant to justifying that extension.
- The standard formula and the pricing PDE have distinct roles in option valuation.
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Full text
# Call Option on the Square of a Log-Nomral Asset
# Call Option on the Square of a Log-Nomral Asset
I'm working on a quant interview question from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors).I cannot understand its answer well and really appreciate your advice:
Here is the question: suppose you have a call option on the square of a log-normal asset. What equation does the price satisfy?
The answer says "the price still satisfy the Black-Scholes equation", I'm confused with Black Scholes formula {ex. call option premium = SN(d1)-Kexp(-rt)N(d2)} and Black Scholes PDE {ex. dC/dt+rS*dC/dS+(1/2)*sigma^2*S^2*d^2C/dS^2 = r*C}. So what is "Black Scholes equation" (from the answer) referring to? The Black Scholes PDE or Black Scholes formula? My understanding about the difference between the two is: PDE has boundary condition that can be used to price almost all options(ex.European,American,Asian), but Black Scholes formula can be used to price European option, is that right? In other words, PDE can be used to price an option whose value is dependent on past prices since we can solve PDE backwards and take those past prices into account, but for Black Scholes formula, it can only price an option whose value is dependent on maturity, is that right?
## Answer by Magic is in the chain (score 4, accepted)
https://quant.stackexchange.com/a/49720
Generally the Black Scholes equation is used to refer to the Black Scholes PDE (PD equation). And the formula refers to the analytical formula, usually cover both call and put versions.
The extension of the BS to the square or power of S is frequently covered in the textbooks and tests; however, it could be tricky in an interview situation. When using Black Scholes logic, they can question you on self financing portfolio or martingale measure or change of numeraire. As you can see, justifying the extension of Black Scholes arguments to these payoffs is by no means trivial. Hence these pay offs are a good exercise to learn about the machinery behind the Black Scholes, and when to use or not to use the BS PDE.
For the path dependent options, please see the discussion here: Can we use Black-Scholes to price path dependent options?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.