Why is it useless to model stochastic volatility when pricing Vanilla style derivatives?

Question

With respect to the answer by user AFK in Ideas about Stochastic volatility models.

I am specifically interested in interest rate options (IR Caps/Floors and Swaptions).

Answer 1

I think that the main advantage of using a stochastic volatility model is to produce a consistent volatility smile. Let's consider the pricing formulas for the normal and lognormal volatilities: $$dS_t=\sigma dW_t\Rightarrow \mathbb{E}[(S_T-K)^+]=(S_t-K)\Phi\left(\frac{s-S_t}{\sigma\sqrt{\Delta t}}\right)+\sigma\sqrt{\Delta t}\phi\left(\frac{s-S_t}{\sigma\sqrt{\Delta t}}\right)$$ $$dS_t=\sigma S_tdW_t\Rightarrow \mathbb{E}[(S_T-K)^+]=S_t\Phi\left(\frac{\ln(S_t/K)+\sigma^2\Delta t/2}{\sigma\sqrt{\Delta t}}\right)-K\Phi\left(\frac{\ln(S_t/K)-\sigma^2\Delta t/2}{\sigma\sqrt{\Delta t}}\right)$$ Those two models can't produce the prices of options for different strike values, and there is no consistent way to aggregate the deltas for options for different $K$'s. Therefore, we can't properly hedge a European vanilla options because the smile changes every day and both Bachelier and Black model don't give any information about its dynamics. Moreover, in the case of IR caps and floors, those formulas only hold for one specific forward rate, i.e. only one forward rate is a martingale under the risk-neutral pricing measure, not for the entire forward curve.

The SABR model for IR vanillas was introduced to generate the skew and smile in a quite flexible manner, since it is homogeneous in both forward price and volatility and given the dynamics $$dS_t=\alpha_t S^\beta_tdW_t$$ $$d\alpha_t=\alpha_t \nu dZ_t, \quad d\langle W,Z\rangle_t=\rho dt$$ it has the approximation $$\alpha S^\beta+\frac{\rho\nu+\alpha S^\beta\frac{\beta}{S}}{2}(K-S)+\frac{(2-3\rho^2)\nu^2+(\alpha S^\beta)^2\frac{\beta(\beta-2)}{S^2}}{12\sigma}(K-S)^2$$

where $\alpha S^\beta=\sigma^{ATM}$. Interpolating accurately across tenors and expiries we can construct the smile. Note that now we would need to hedge the delta and the vega, since both $S$ and $\alpha$ are stochastic. However, since they are correlated (usually negatively), by delta-hedging one can also reduce the vega exposure because $$d\alpha_t=\alpha_t\nu(\rho \underbrace{dW_t}_{=\frac{1}{\alpha_t S_t^\beta}dS_t}+\sqrt{1-\rho^2}dZ_t^{\perp})$$. leaving a smaller residual vega to be hedged.

Answer 2

Because vanilla derivatives with European exercise depend only on total variance , not on it's dynamics in time.

If you have a simpler model (like interpolation of these total variances from your volatility surface) you don't have as much of unobservable parameters stochastic volatility models have.

Having more parameters (which many times would need to be calibrated/guessed from other instruments than vanillas) introduces unnecessary model risk/mispricings.

Answer 3

Both @alexprice and @FunnyBuzer have some good points, and I have upvoted them. I think I have enough to add here that I'll make another answer entry.

First off, @AFK was fairly correct that you do not need stochastic volatility for vanilla (European exercise) option pricing, since (as he says and alexprice elaborates) you just interpolate the surface of cumulated variance.

FunnyBuzer makes an important point, though, which may be somewhat buried by all the SABR math he included: a stochastic vol model can help you hedge better even if it does not improve pricing. Without stochastic vol, your vanilla hedge parameters will (depending on the strike) assume too much or too little delta.

Whether the hedging errors are important enough to make the effort and model risk of using stochastic vol depends on the portfolio and the hedging practices, so there's no "right" answer. For swaptions, SABR is a pretty good way of keeping the stochastic vol parameter count and computational effort low.

I would like to make an additional point, which is that stochastic vol can allow you better extrapolation. As AFK said, you interpolate the vanilla surface. That's fine until the underlying moves a great distance under your portfolio, or you need to price at a long tenor/far-from-the-money. In these cases you find yourself wanting to extrapolate that surface, which is a considerably less reliable operation. In these cases, good stochastic vol models can help.

(Note: SABR does not necessarily extrapolate well, at least if you are using the convenient approximations)