# Dividend yield for an index

Let's say we want to price an option and so need a dividend yield to plug into Black-Scholes.

We can compute an implied dividend yield for a stock using:

$$F=S_0 e^{(r-d)T}$$

and by isolating for $d$. We can then use this implied dividend in pricing options.

But how do we find the implied dividend yield for an index? E.g. what if we want to price an option on an index, and there are no forwards for that index?

• I suppose you mean no futures contracts trading on that index? Even then, if you have listed (European) options, you can always apply call-put parity: $F(0,T) = K + (C(K,T)-P(K,T))/DF(0,T)$ where $K$ is the strike, $C(.)$ (resp. $P(.)$) European call (resp. put) price struck at $K$ with maturity $T$ and $DF$ the relevant discount factor, i.e. $DF = \exp(R(0,T)T)$. If options are American, things get a little more complicated. – Quantuple Sep 11 '17 at 12:15
• Right, but what if there are no futures/forwards on that index? That means in your equation the LHS ($$F(0,T)$$) is unknown. So how then do we find C(K,T)? Can we use a composite of the individual forwards on the components of the index. I.e. back out a weighted average of the dividend yields implied by the individual forwards on stocks in the index? – trade_the_basis Sep 11 '17 at 12:21
• To be clear, I'm assuming we want to price C(K,T) and so we don't yet have that. – trade_the_basis Sep 11 '17 at 12:22
• I see, maybe you should add that in your question to be clear. Yes, you could see the index as a basket of equities: $S_t = \sum_{i=1}^N w_i S_t^{(i)}$ (weighted sum of its constituents, the weights $w_i$ given by the index construction rules). You would then have that the index forward is the weighted sum of the individual forwards. This remains an approximation though because in practice the weights of an index are not constant (composition reshuffling). Also, you should be careful regarding 'compo' effects (e.g. constituents denominated in a different currency than that of the index). – Quantuple Sep 11 '17 at 12:29
• No I have no reference, sorry. But everything boils down to the modelling assumption: $S_t =\sum_{i=1}^N w_i S_t^{(i)}$. If you can write that, then you're fine. Indeed, the forward is an expectation under $\Bbb{Q}$ and since the expectation operator is linear you'll get: $F(0,T) = \sum_{i=1}^N w_i F(0,T)^{(i)}$. Our starting equation is certainly true for the spot price of an index. However, this may not be a good representation of the future price of an index given the reshuffling risk (e.g. how can you be sure component $k$ will still be in the index in $T=5Y$ for instance?). – Quantuple Sep 11 '17 at 12:39