# Forward starting options concepts

Consider $$t_0, with $$t_0=0$$ (today date) and the standard payoff of a vanilla forward starting call option,

$$F_{t,T} = (S_T - S_t\cdot K)^+$$, with strike $$K$$.

If the price of this option is quoted today at $$t_0$$, then we can infer some kind of Black-Scholes implied volatility $$\sigma_{imp}(t_0, K, t, T)$$ for which the corresponding BS-price agrees with the market price (at $$t_0$$).

Now, denote the BS-implied volatility at time $$t$$ of a call option with the above payoff by $$\hat{\sigma}(t,T,K,S_t)$$. Obviously, from the stand point of $$t_0$$ this is unknown as the market quotes for date $$t$$ do not yet exist.

My question is how does $$\sigma_{imp}(t_0, K, t, T)$$ relate to the unknown $$\hat{\sigma}_{imp}(t,T,K,S_t(\omega)$$? Is the first just a proxy of the second?

I'm aware the answer might be obvious but I'm trying to convince myself and better understand the concepts in bibliography. Any references/easy to read papers that clarifies all the above is appreciated.

## 1 Answer

Forward-Start Options are very interesting securities, you can find a lot about them on the internet. It turns out that there is an explicit pricing formula for them in Black-Scholes, the nicest derivation I can find is given in this paper - the pricing formula is given by:

$$S(t_0)\left(\textbf{N}(d_1)-e^{-r(T_1-T_0)}\theta\textbf{N}(d_2)\right)$$ where $$d_1 = \frac{\ln(1/\theta)+(r+\sigma^2/2)(T_1-T_0)}{\sigma\sqrt{T_1-T_0}}, \hspace{1em} d_2 = \frac{\ln(1/\theta)+(r-\sigma^2/2)(T_1-T_0)}{\sigma\sqrt{T_1-T_0}},$$ $$\textbf{N}$$ is the cumulative distribution function of the standard normal distribution, $$\theta S(T_0)$$ is the strike price set at $$t=T_0$$, and $$t=T_1$$ is the time when the option matures.

As for the forward implied volatility, it turns out there are a few ways to define it. In plain BS, volatility is deterministic at all times, so forward implied vol will just be the same as implied vol now. However, things get more interesting in Local Vol models (which can be thought of as a generalisation of BS), in which case deterministic vol models and stochastic vol models give VERY different forward vol surfaces - I wrote a bit about this (with graphs and code) in another answer.

In case it's of interest, it turns out that even in the Heston stochastic vol model we can find a semi-analytical formula for these options, for example the one given here: $$C_\text{FWS}(t,\nu(t),S(t)) = S(t)\hat{P}_1(t,\nu(t)) - ke^{-r(T-t^\ast)}S(t)\hat{P}_2(t,\nu(t))$$ with $$\hat{P}_j(t,\nu(t)) := \int_0^\infty P_j(t^\ast,1,\nu(t^\ast),k) \; f(\nu(t^\ast)|\nu(t))\text{ d}\nu(t^\ast)$$ where $$P_j$$ for $$j\in\{1,2\}$$ are the Heston probabilities given earlier in the paper and $$f(\nu(t^\ast)|\nu(t))=\frac{B}{2}\cdot e^{-(B\nu(t^\ast)+\Lambda)/2}\left(\frac{B\nu(t^\ast)}{\Lambda}\right)^{(R/2-1)/2}I_{R/2-1}\left(\sqrt{\Lambda B\nu(t^\ast)}\right).$$

If you want to experiment for yourself, both the Local Vol case and the Heston case have analytic (and also monte-carlo) pricing engines available via QuantLib.