forward implied volatility skew

Question

I would like to calculate implied forward volatility skew. I have stochastic volatility monte carlo. What kind of payoff do I need to price and how to use Black() formula to calculate the implied volatility.

Answer 1

There are two approaches.

1. Price call and put options with various strikes. Plot their BS implied volatilities. Find the slope of the graph.

2. Price a call and digital call with the requisite strike. Compute the implied volatility of the call. Use the fact that

$ DC(model) = DC(BS) - skew \times callvega,$

to solve for the skew. (See eg Section 7.7 of my book "concepts and practice of mathematical finance")

This would give today's skew. To get the forward skew, you would need to do this conditional on future value of spot which basically means running MC and getting the implied future prices as a function of spot via bucketing.

Answer 2

Forward implied volatility smile is implied from forward start options. For example call options have payoff $$ g_{T+\theta} = \left( \frac{S_{T+\theta}}{S_T} -K\right)_+ $$ If you are in a stochastic volatility model this can be rewritten
$$ g_{T+\theta} = \left( e^{ \int_T^{T+\theta} r - \frac{1}{2}\sigma_t^2 dt + \int_T^{T+\theta}\sigma_tdW^S_t } -K\right)_+ $$ You can compute the price of the forward start call using MC: $$ C_t(T\to T+\theta,K) = \mathbb{E}^{\mathbb{Q}}_t[e^{-r(T+\theta - t)}\left( e^{ \int_T^{T+\theta} r - \frac{1}{2}\sigma_t^2 dt + \int_T^{T+\theta}\sigma_tdW^S_t } -K\right)_+ ] $$ It is worth noting that conditionning wrt the fixing time $T$, $$ C_t(T\to T+\theta,K) = \mathbb{E}^{\mathbb{Q}}_t[e^{-r(T - t)} C_T(S_T = 1,T+\theta,K) ] $$ So forward start calls are average over all scenarios of the future model prices of calls.

The forward implied volatility $\Sigma_t(T\to T+\theta,K)$ (this is $\theta$ years in $T$ years as seen from time $t$) is characterized by $$ C_t(T\to T+\theta,K) = C_{BS}(S=1,\theta,K,r,\Sigma_t(T\to T+\theta,K)) $$ The formula is set so that, in the BS model with time dependant volatility, $\Sigma_t(T\to T+\theta,K)^2 = \frac{1}{\theta}\int_T^{T+\theta} \sigma(t)^2 \, dt$ which is what we expect as a forward volatility.

This way you can get your prices by MC simulation, find the corresponding forward implied vol for different strikes and plot your forward smile. You can approximate the skew by finite difference if you have enough strikes.

The method based on digitals suggested by Mark Joshi works similarly.

Answer 3

There is no skew/smile for forward contracts, but there is for options based on it (caps, floors, swaptions, options on futures). Then it would be the simple Black Formula that should be used in theory (using futures price).

The mere existence of the smile is an indicator that the model is fundamentally flawed and it is important to apply a correction.