Stochastic Implied Volatility: A Factor-Based Model
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Since no new source of uncertainty was necessary, delta hedging was possible and the market was complete. However, there were also problems.
Several papers have tested the hedging performance of local volatility models and the general finding is that they could perform even worse than the Black-Scholes model. Hence, the usual conclusion is that the assumption of a deterministic spot volatility is too restrictive and that stochastic volatility models are more realistic.
Stochastic and local volatility models have been regarded as two alternative and competing approaches to the same unobservable quantity, the volatility of the underlying asset. But while these approaches are not inconsistent with each other, interestingly the few attempts to unify them into a single theory have not been much developed by further research.
The heart of the problem is the assumption of a deterministic spot volatility that is imposed by most local volatility models. However such assumption is not actually necessary for a local volatility model. See Rubinstein for some empirical evidence on the behaviour of implied volatilities before and after That is, the spot volatility is the diffusion coefficient in a geometric Brownian motion model for the underlying asset dynamics.
Forward volatility is a forecast of what the spot volatility will be at some future time. See Section II for more details on terminology and notation. Alexander and L. These authors define the local variance i. Hull and White, , and Heston, since x t can be any arbitrage-free set of processes consistent with options prices. So what is the problem with local volatility models?
It is precisely the residual uncertainty from x t after taking the expectation in 1 , and its influence on the spot volatility. This uncertainty is transferred to the local volatility surface itself. That is, although locally i.
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The residual uncertainty from x t does not just disappear from the model. In effect, the assumption that the spot volatility is deterministic is inconsistent with any dynamics for local volatility. This explains the poor empirical results on the hedging performance of local volatility models. In this paper we develop a general model for stochastic local volatility. At each trading date, we assume that a parameterised deterministic local volatility model is calibrated to a smile surface from market prices of Europen calls and puts. We introduce stochastic dynamics for these parameters over time, and hence additional uncertainty into the spot volatility.
We then show that the delta, gamma and theta of the stochastic local volatility model are equal to the equivalent deterministic local volatility hedge ratio plus and adjustment factor which depends on the degree of uncertainty in the local volatility parameters and on their correlation with the underlying price. Hence recent advances in both stochastic and local volatility have led to a unified model for the two approaches; the stochastic local volatility model that is introduced in this paper.
Later we relax this assumption. In this case the local volatility function is calibrated by changing parameters so that some distance metric 7 See Bouchouev and Isakov , and Avallaneda et. Of course, in 3 the local volatility will be sensitive to the calibration, at time t0. Thus the local volatility surface will be stochastic if the calibrated parameters v t are stochastic, a fact that has not been given much attention in the literature. That is, we assume that all uncertainty on the random variables x t in 1 is captured by the parameters v t of a local volatility model.
It is then only under the additional assumption that the parameters v t are constant and equal to v t0 that we have the a deterministic local volatility model such as in 3 above. In practice, for a variety of reasons it may be difficult to fit both the smile and the term structure of implied volatilities with just a few parameters. So it is common practice to restrict the calibration to near the money options and to a few maturities. However, restricting the calibration to very few maturities neglects the importance of the term structure and can lead to the wrong local volatility surface.
Thus it can evolve over time and there is an implicit dependence of this surface on time t, underlying asset price S t and other variables v t and their past histories. It is important to note that although the local volatility surface derived from a DLV model can fit the current smile, the assumption of a deterministic spot volatility is unrealistic. This can affect the price of exotics and other types of derivatives, and hedge ratios for all options. We shall discuss this in Section IV. Thus, in the 11 Note that, depending on the functional form assumed for the spot variance, some values for v t can introduce arbitrage opportunities if they violate at least one of the no-arbitrage conditions mentioned in Appendix B.
There is an extensive literature on the estimation of gt S — see Breeden and Litzenberger or Brunner and Hafner just to mention a few — but there is no easy way to calculate the joint density ht v, S unless we have a specific model for the spot variance. Now assume the dynamics for each parameter vi in v t under the risk-neutral measure follow 13 An analogy with the Heath-Jarrow-Morton model for interest rates is enlightening.
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The spot variance and local variance can be seen as analogous to the spot interest rate and the forward rate in the HJM model, so that the local volatility surface is the analogue to the forward yield curve. See Kani et al It does not matter how many Brownians we have in the dynamics since it is always possible to create new forward-like contracts to hedge all Brownians. We can always introduce new tradable hedging instruments as long as they can be priced under some martingale measure. Together 7 and 8 provide the full specification of the SLV model.
Note that here we actually model the stochastic parameters of a DLV model. The advantage of this approach is that it relates nicely to the extant literature on local volatility, addressing the typical parameter instability and its consequence for pricing and hedging of derivatives. Note that 9 holds locally, i. However, since the calibrated parameters are likely to be different at each re-calibration, we assume v t is stochastic and defined such as in 8 above.
Hence, that uncertainty will affect the actual dynamics of the claim price fL, given in the following theorem.
volatility - Local vol, stochastic vol, implied vol - Quantitative Finance Stack Exchange
The proof of the theorem as well as of most results in this paper is available in Appendix A. Therefore, when the parameters v t of a deterministic local volatility model are stochastic, the claim price has a multi-factor dynamics including one Brownian motion from the underlying asset price dynamics 7 and another Brownian motion for each stochastic parameter vi in the model.
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This result has important implications for hedging, the focus of the next section. Hedging with Local Volatility revisited Since the stochastic local volatility SLV model 7 and 8 introduces new sources of randomness, perfect hedging is complex and would require a combination of several traded options. Then, by combining these gadgets a multitude of hedging possibilities are available to the volatility trader.
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In this paper, however, we do not focus on perfect hedging. Instead, here we show that the uncertainty about the local volatility surface in the future — in terms of its parameters — can explain the poor hedging performance that has been reported in the literature on local volatility models. Dumas et al and McIntyre find that delta hedges derived from local volatility models perform worse than even the simple Black-Scholes delta hedge, but Coleman et al find the opposite for long hedging periods. More importantly, Hagan et al claim that local volatility deltas will be inaccurate since these models fail to capture proper dynamics of the implied volatility.
However readers should be very cautious of these findings. In common with the majority of research literature in this field, the local volatility models studied were incomplete in this respect. For instance, Dumas et al assume and test several different parametric or semi-parametric forms for the local volatility function.
maisonducalvet.com/donde-conocer-chicos-de-navaconcejo.php Their conclusion is that Black-Scholes hedge ratios appear to be more reliable than those obtained from the local volatility models they tested. However, they do not explore the impact of the instability of the local volatility surface as implied from the instability of the calibrated parameters on the hedge ratios.
This could be the main reason for their disappointing conclusion. Indeed we now show that the hedge ratios derived from such a view of volatility are biased, and the intuition behind that bias is the implicit and unrealistic assumption of a static local volatility surface. The next theorem shows that the SLV model requires an adjustment for the deterministic local volatility DLV hedge ratios. The intuition behind Theorem 2 is as follows: because the vector v t is stochastic the local volatility surface also evolves stochastically; thus a correction term must be added to hedge ratios to account for correlation between movements of each vi and the asset price S.
Hence, whilst the delta hedge strategy can be unbiased zero expected hedging error , it cannot be perfect when the spot volatility is stochastic. However, there is an important distinction to be made here. Finally, all results above are based on the assumption that 7 and 8 are a good approximation of reality.
In particular, the underlying asset price is assumed to follow a continuous process as in 7. Hence, if for instance the price process is discontinuous with jumps , the expressions for the hedging error above may not hold and the delta hedge strategy may be even biased with non-zero expected value for the hedging error. Model Estimation This section considers how the model parameters in 8 can be estimated in practice.
Proper estimation of the model will entail advanced econometric and optimisation techniques: it involves a calibration over a time series of cross-sectional option prices for several strikes and maturities simultaneously. This is beyond the scope of the present paper. Nevertheless, we would like to illustrate a practical example of the model in Section VII. Proof: See Appendix A. Here we assume that sample moments approximate population moments, and this is perhaps rather strong.