# Mark Joshi's book - quant interview questions

I am currently doing the question on pricing the option with payoff:

$$\max (S(S-K),0).$$

On the relevant question section, it's asked why would a bank be reluctant to sell such option? I can't really think of a convinced answer for this so would appreciate any thoughts here.

I suspect this is because, conditional on being in-the-money, the payoff of your option is convex in stock price $$-$$ whereas for a vanilla call, the payoff is linear. As a consequence, the delta $$\Delta$$ and gamma $$\Gamma$$ hedge ratios are larger, in particular gamma becomes much more sizeable.
Let us assume that rates are null to lighten notation. Then your payoff can be priced under the stock measure $$\mathcal{S}$$, see for example this answer, such that: \begin{align} V(t,S_t) &=E^\mathcal{Q}\left(S_T(S_T-K)^+|\mathscr{F}_t\right) \\ \tag{1} &=S_tE^\mathcal{S}\left((S_T-K)^+|\mathscr{F}_t\right) \end{align} As you can see in the linked question, under its own measure the stock price is still distributed like a Geometric Brownian Motion, but with drift $$r+\sigma^2=\sigma^2$$ due to the null rates assumption. Black-Scholes formula applies to $$(1)$$ and we get: $$V(t,S_t)=S_tf(t)$$ where $$f(t):=f(t,S_t,T,K)$$ is the Black-Scholes pricing formula for a vanilla call option but such that the stock price has drift $$\sigma^2$$. Therefore: \begin{align} \Delta_V(t,S_t)&=f(t)+S_t\Delta_{BS}(t,S_t) \\[6pt] \Gamma_V(t,S_t)&=2\Delta_{BS}(t,S_t)+S_t\Gamma_{BS}(t,S_t) \end{align} These hedge ratios should be much larger than for a plain vanilla call, in particular due to the $$S_t$$ factor: for example for a stock price of \\$10 that might yield hedge ratios more than 10 times larger than for the vanilla call.