I have a question regarding the general approach to questions like the following and the solution in this special case.
I'm referring to the question 2.6 from the book "Quant Job Interview Questions And Answers" from Mark Joshi, i.e.
"In the Black Scholes world, price a European option with payoff of $ \max(S_T^2 – K, 0)$ at time $T$."
His solution and the application of the Black formula (using the forward price) aren't lucid to me.
The methodology applied by Joshi is basically to translate the problem first into the standard assumptions in the Black-Scholes theory, i.e. to assume the stock moves lognormally, $dS_t = \mu S_t dt +\sigma S_t dW_t$ with $W_t$ denoting Brownian motion. He goes on to translate the equation into the risk-neutral measure, i.e. replacing $\mu$ with $r$ in the equation above.
By stating the solution to said equation with $S_t = S_0 \cdot \exp \{(r – 0.5\sigma^2)t + \sigma W_t\}$ he simply squares this for our problem to achieve $S_t^2 = S_0^2 \cdot \exp \{(2r – \sigma^2)t + 2\sigma W_t\}$ (first question: why can we just do that?).
He then goes on to say that since we are dealing with a geometric Brownian motion here, we're able to apply the Black formula. He finishes his explanation by stating that "we simply have to write the $S_T^2$ as $ F_T (0)e^{-\nu^2T/2 + \nu W_T}$ and compute the values for $F_T(0)$ and $\nu$ (second question: why and how can we do that?).
There has to be a general approach and recipe regarding pricing and translating such simple problems into the Black-Scholes framework, maybe outside the proposed solution by Joshi.
Answers are highly appreciated.
Best Answer
First I believe you mean you want a European option with payoff of $\max(S^2_T- K, 0)$, not $\max(S^2_T, K)$. As to why the solution seems so specific and not very general, that book is for job interview questions, so it assumes the reader is very comfortable with its contents already and wants solutions that are fast, easy to remember, and do not need a lot of complex calculations.
For the first question, $S_t = S_0 \cdot \exp((r - \frac{\sigma^2}{2})t + \sigma W_t)$ is the solution to Geometric Brownian motion for any time $t$. Nothing says if we want $S_t^2$ we can't just square both sides giving $S_t^2 = S_0^2 \exp((2r -\sigma^2)t + 2\sigma W_t))$. We could also find this using Ito's lemma with $f(t, x) = x^2$. To see this let $S_t$ follow $$dS_t = rS_tdt + \sigma S_tdW_t.$$
Using $f(t, x) = x^2$ by Ito's lemma we have
$$df = (2rf + \sigma^2 f)dt + 2 \sigma fdW_t.$$
Then if we let $x = S_t$ we have
$$dS_t^2 = (2rS_t^2 + \sigma^2 S_t^2)dt + 2 \sigma S_t^2dW_t.$$
This then has the solution for $S_t^2$ given above.
For the second question $F_T(0)$ is the forward price of $S_t$ at time $t=0$ and $\nu$ would be equal to $2\sigma$. The use of the Black model is not very intuitive and most derivations for this do not use that approach. I think one of the more intuitive approaches would be to see that $S_t^2$ is a Geometric Brownian motion with drift $2r + \sigma^2$ and diffusion $2\sigma$ and use the Black-Scholes formula with the proper adjustments. The adjustments would be use the Black-Scholes formula with dividends and in it let $\hat{\sigma} = 2\sigma$, $\hat{r} = r$, and $\hat{q} = -(\sigma^2 + r)$.
As far as a more general framework for option pricing your best bet is to consult a book on it. Two that come to mind are Martingale Methods in Financial Modelling by Musiela and Rutkowski or Arbitrage Theory in Continuous Time by Bjork, but almost any will suffice they tend to just vary in how deep mathematically they go.