Stochastic Volatility Models

Within stochastic volatility models (sometimes shortened to “stoch vol models”), volatility has its own process. There are many different stochastic volatility models but one of the original and best known is the Heston model from 1993 because it has intuitive parameters that mirror the risk reversal and butterfly instruments and a closed-form expression for vanilla options. The stochastic differential equations (SDE) of the Heston model are:

The first line of the model is identical to Black-Scholes except that volatility has been replaced with the square root of a variance term . The process for variance includes parameters for long-run variance , the speed of mean reversion , volatility of instantaneous variance , although this parameter is often referred to as the “vol-of-vol,” and the correlation is applied to the two independent Wiener processes . The first Wiener process drives spot and the two together drive variance. The Ornstein-Uhlenbeck (OU) model and Exponential Ornstein-Uhlenbeck and SABR models are other commonly used stochastic volatility models. Within the Exponential Ornstein-Uhlenbeck model, volatility cannot go to zero, which is an issue within the Heston model.

Pure stochastic volatility models do not depend on the level of spot; an equivalent volatility surface will be generated no matter the initial spot. Therefore, stochastic volatility models can be thought of as sticky delta models (see Chapter 14).

The parameters within the model are calibrated such that the vanilla volatility surface is (approximately) reproduced within the model. For example, if the initial and long run volatility is set to 10%, with zero vol-of-vol and correlation, the volatility smile produced by the model is a flat 10% as shown in Exhibit 19.2.

If the vol-of-vol parameter is increased, the wings of the volatility smile move higher as shown in Exhibit 19.3. Note that in this instance, the ATM level increases too, unlike the standard volatility smile construction.

If the correlation parameter is increased, the volatility smile tilts such that topside strikes have higher implied volatility than same delta downside strikes as shown in Exhibit 19.4. This is equivalent to a larger topside skew within the volatility smile.

Parameter calibration within the stochastic volatility model is an automatic process that uses vanilla option prices. This is why closed-form expressions or accurate approximations for vanilla option prices under a given pricing model are so important; it enables the model to be calibrated quickly. In practice, there are often multiple different parameter sets that generate near-identical volatility surfaces. When using any model with calibrated parameters it is important that traders observe the parameters over time and ensure they are stable and respond sensibly to changes in the volatility surface; that is, changes in the skew of the volatility surface should be mainly reflected by changes in the correlation parameter .

These Heston parameters can attempt to match a volatility smile at a single tenor but in order to match the entire volatility surface, parameters must change over time. In other words, the variance parameters within the SDE become functions of time:

When calibrating models with parameter sets that are functions of time, the parameters should evolve smoothly rather than jumping around.

In practice, in order to successfully calibrate to vanilla contracts, stochastic volatility models often have a higher volatility of implied volatility than is observed in the market. Put another way, volatility convexity is overvalued by stochastic volatility models. For this reason, stochastic volatility models do not consistently match prices in the interbank broker exotics market, particularly for contracts with significant convexity, like double-no-touch options, for which the stochastic volatility model price will be too high. The same effect also causes forward smiles to be overvalued.

In addition, under stochastic volatility models the skew within the volatility smile can get overwhelmed by the volatility mean reversion, plus some stochastic volatility models can struggle to have enough vol-of-vol at short maturities to produce the correct volatility smile.

Local Volatility Models

The local volatility model was developed in the mid-1990s by superstar quant Bruno Dupire. The key to the model is that volatility, rather than being constant, is a deterministic function of spot and time. Therefore, the Black-Scholes stochastic differential equation is extended to:

A local volatility surface is generated within the model using the full volatility surface. The local volatility surface can be thought of as a grid of forward volatilities spanning out in time as shown in Exhibit 19.5. Implied volatility and local volatility smiles at one tenor in USD/JPY (a currency pair with a small downside risk reversal currently) and USD/BRL (a currency pair with a large topside risk reversal currently) are shown in Exhibits 19.6 and 19.7 respectively.

The local volatility model is neither a sticky strike nor a sticky delta model, although it is more sticky strike than sticky delta. The local volatility depends on the level of spot but that does not necessarily imply that the implied volatility for a specific strike is fixed as spot moves.

The Dupire local volatility model has no parameters to calibrate and the model is quick to setup and price. Plus, if the vanilla volatility surface contains either strike spread arbitrage (call options with increasing value at higher strikes) or calendar spread arbitrage (options with decreasing value at longer maturity), the model will be unstable due to undefined local volatility. This feature can be used to identify potential problems with the volatility surface.

Another possible approach is to build the local volatility surface using a defined functional form. This approach has the advantage of guaranteeing an arbitrage-free volatility surface but it is slower because the parameters within the functional form must be calibrated.

Because vanilla options are correctly priced within the model and no time consuming calibration is required, the Dupire local volatility model is a good pricing model to use for exotic contracts with no/minimal path dependence.

The main problem with local volatility models is that the volatility of implied volatility generated from the local volatility function is lower than reality. Put another way, volatility convexity is undervalued by local volatility models. For this reason, local volatility models do not consistently match prices in the interbank broker exotics market, particularly for contracts with significant convexity, like double-no-touch options, for which the local volatility model will be too low.

Another feature of the model is that forward smile can be undervalued, again, caused by low volatility of implied volatility. This can be checked by querying for the 1yr in 1yr forward volatility smile from the model. In most currency pairs this forward volatility smile under local volatility will have significantly lower wings and skew than the current 1yr volatility smile. Also, since volatility is a deterministic function within local volatility models it should not be used to price forward volatility agreements or forward start options (see Chapter 31), where option value depends primarily on the random nature of implied volatility.

In summary, local volatility models are stable, quick, and give fairly accurate valuations and exposures, particularly for options with minimal path dependence. However, for contracts with significant convexity or forward skew exposures, local volatility often gives inaccurate prices.

Mixed Volatility Models

Mixed volatility models (sometimes called stochastic local volatility models) are, as it sounds, a combination of a stochastic volatility model (which overvalues volatility convexity) and a local volatility model (which undervalues volatility convexity).

There are many different ways in which the stochastic and local volatility models can be combined. One possible approach would be to extend a Heston stochastic volatility model with a local volatility component plus a mixing weight :

Mixing weights define how much stochastic volatility is applied at each tenor. Conceptually, the stochastic volatility model can be calibrated first and the local volatility component can then be used to ensure that the correct vanilla surface is generated after the stochastic volatility component is weighted by the mixing weight. Within this formulation, a mixing weight of 0% implies full local volatility and a mixing weight of 100% implies full stochastic volatility.

The behavior of the model can be driven by a single mixing weight or a term structure of mixing weights set by traders to match exotic contracts in the interbank market. Alternatively, the model could be calibrated directly to prices of exotic contracts. In general, if convexity is being more highly valued in the market than the model, the model price will be lower than the market price for exotic contracts with significant volga exposures. In this case, mixing weights should be moved higher to increase the stochastic volatility element within the mixed volatility model. In obscure pairs with few quotes on interbank broker exotics, mixing weights in similar, more liquid currency pairs can be used.

Variations on the mixed volatility model are used by the vast majority of FX derivatives trading desks to price exotic contracts out to two-year maturities in most currency pairs. It is so successful that development of new single asset pricing models on bank FX derivatives trading desks has virtually stopped.

Jump Diffusion Models

There is a lot of evidence of jumps in the history of financial markets. One of the original jump diffusion models is the Merton model from 1976, which extends the Black-Scholes SDE by adding a jump term:

where is a poisson process with intensity and is log-normally distributed with mean . Note that the introduction of jumps also requires an additional correction to the drift.

Spot paths like Exhibit 19.8 are generated from a Merton model, with the probability and size of up or down jumps generating the wings and skew in the volatility surface.

Merton is a popular model because it has a semi-closed-form expression for vanilla options, hence allowing for quick calibration, and like Heston, it has intuitive parameters that mirror the skew and wings of the volatility surface. Jump diffusion models can generate a wide range of volatility smiles but their pricing doesn't consistently match exotics prices observed in the interbank market for liquid currency pairs. Jump models work best in managed or pegged currency pairs where the model dynamic best matches the real market dynamic. Jump models can also be used to, for example, imply an expected probability and size of spot jump from the volatility smile. By comparing this with their intuition traders can identify relative value in the volatility surface.

Stochastic Interest Rate Models

The smile models reviewed in this chapter all have interest rates that are static or deterministic (i.e., they move in a predetermined manner). In practice, interest rates have a volatility of their own and they can move in a correlated manner with spot. Stochastic interest rate models extend the Black-Scholes SDE by introducing processes for the two interest rates (or perhaps their spread). For example, if the so-called “short rates” are being modeled:

The effect of stochastic interest rates is particularly important on long-dated (approximately past two-year) exotic contracts. When a contract has a large sensitivity to interest rates, the effect of stochastic interest rates must be quantified and included within the TV adjustment.

Finally, it is important to note that generating TV adjustments with a smile model and an interest rate model separately and then summing the adjustments is not always a valid approach. The smile and interest rates may well interact with each other. Local volatility is often added into stochastic interest rate models in order to value the volatility surface in a manner which requires no additional parameters to be calibrated.