# Pricing and hedging of vanilla options based on non-tradable underlying

Consider a non-tradable stock index $S$ which satisfies:

$dS_t=\mu S_tdt+\sigma S_tdW_t$

and a risk-free asset $B$.

I want to price an European Call option with the payoff $C_T=max(S_T-K,0)$. The No-Arbitrage condition tells us that if a portfolio consisting of underlyings $S$ and $B$ replicates the call option then the fair price of the call option is equal to the initial value of the portfolio. Now the problem is that the underlying $S$ is not tradable so that it's impossible to replicate (and hedge) the call option using $S$. We probably use futures based on the underlying $S$ to construct replicating portfolio. My question is how to price such index call option. I appreciate if you could share any reference. Thanks.

• If the stock index is not traded, you might look at its underlyings which probably will be traded. – Raskolnikov Feb 13 '18 at 6:44
• If $S$ is not traded you should look at what strategies allow you to synthesize a "delta one" exposure. For stock indices you can either trade futures that you roll until expriy or trade synthetic forwards (harder in practice since there is no listed options after 3Y for most indices). What matters is the average drift of these self financing delta one position (r-q-repo) and their volatility which are the same under BS framework. The BS formula can then be rewritten as a function of the futures/forward price as $C_0(T,K) = DF(0,T) ( F(0,T) N(d_1) - K N(d_2))$. – Quantuple Feb 13 '18 at 8:23
• @Quantuple Thanks for your comment. I think I understand your point. But I still don't understand how you derive the BS formula in the bottom line. I've found some articles about this topic, e.g. link. Basically they use utility-based pricing. I'm very curious that index option is very common but why i can't find textbooks talking about this topic. – Jack Wang Feb 13 '18 at 11:04
• I think the most straightforward approach to stick to the usual textbook theory would be to see the index as a weighted basket of individual stocks that can be traded. Then BS assumes that the future price of this basket is lognormally distributed and you apply the usual pricing theory. Would that help you? – Quantuple Feb 13 '18 at 11:18
• Note that the formula I wrote is not a new formula. It's simply a rewriting of the classic one accounting for the fact that $S_0 e^{(r-q-repo)T} = F(0,T)$. – Quantuple Feb 13 '18 at 11:29