Stochastic Volatility Models

The Black-Scholes assumption that future volatility is known and constant is clearly unrealistic. Stochastic volatility models adress the issue by including the volatility itself as a second source of uncertainty in the stochastic process (driven by the volatility of volatility), correlated with the spot movement. Most stochastic volatility models include also an element of volatility mean-reversion towards a long-term equilibrium level, in order to confine the volatility distribution to realistic levels.

Stochastic volatility models provide a scientific explanation of the volatility smile and its dynamic behaviour far superior to other models, as the expected skewness of the smile in these models is predicted to 'follow the spot direction', a behaviour mostly observed in the real marketplace. As far as the price of a number of option classes is sensitive to the dynamics of the smile, stochastic volatility models do provide option prices of superior quality. Futhermore, greeks based on stochastic volatility models typically provide a far better hedge strategy for traders, as they capture the full intricacy of the dynamics of the relevant variables.

Since stochastic volatility models are grounded in academic research, market participants have a solid foundation to justify their prices and their revaluations (mark-to-model) to external auditors and financial market authorities, a requirement further emphasized after the recent banking crisis.

Stochastic volatility models have become a 'must-have', due to their clear superiority. No matter the level of sophistication of the final user, from a simple vanilla to a complex exotic, the drive towards models solidly backed by academic research seems an irreversible trend.

 

Let Volmaster Handle the Complexity

 

Volmaster FX is a complete and ready-to-go software solution, natively built around stochastic volatility models and as such it handles all the complexity of the models on behalf of the final user. Focus on what really matters, the pricing and management of risk exposures, with a seamless user interface. Volmaster FX takes care of smile generation, calibration of the model, parametrization, without involving the user into complex settings.

Users will find an enormous benefit by interacting with Volmaster FX interface in an easy and direct way, as simple as an old-fashioned Black-Scholes pricer, while Volmaster FX manages all the technical complexity behind the curtains. 

 

Volmaster FX volatility surface (parametric smile)

 

Volmaster FX generates the volatility smile for each time bucket out of three market parameters: atm volatility, 25 delta butterfly, 25 delta risk-reversal, following broker's conventions. The smile surfaces are therefore generated out of the most liquid and meaningful strike/volatility pairs. For advanced users, the functional form of the parametric smile also allows a precise handling of the tails asymptotes.

The smile curves are implemented by mean of a proprietary variant of the SABR model. Volmaster FX smile curves in particular include an asymptotic correction in order to handle correctly the flattening of the smile for extreme strikes. The volatility surface is recalculated and re-validated istantaneously every time a market parameter is changed. Because smile curves are generated by a financial model and not an interpolation/extrapolation method (such as splines), the curves are financially meaningful and arbitrage-free by construction.

While the SABR model is used for the generation of a consistent and arbitrage-free volatility surface, Volmaster FX uses an advanced stochastic volatility model (Volmaster SLV Model) in order to price derivatives. The calibration process makes sure that Volmaster SLV Model fits the SABR-based volatility surface. The calibration process is seamlessly integrated into Volmaster FX workflow and it's implemented as a streamlined one-click experience for the user.

 

Volmaster SLV Model (Stochastic-Local Volatility)

 

Some stochastic volatility models (such as the Heston model) use a unique set of parameters for the whole volatility surface. However, this class of models is too rigid to capture the richness of the term structure of the volatility surface. Volmaster SLV Model uses instead a different set of parameters for each time-bucket, to ensure maximum flexibility. Furthermore, Volmaster SLV Model includes a parametric local volatility component. Users can define different weights for the stochastic volatility and local volatility components of the model.

The model includes the following parameters/features, per time-bucket:

• interest rates drift
• volatility drift
• volatility of volatility
• correlation spot-volatility
• equilibrium mean volatility
• volatility mean-reversion speed
• local volatility function

Volmaster SLV Model captures the full term structure of interest rates and the full shape of the volatility surface, by modeling a jointly consistent process of spot and volatility evolving over time with time-dependent parameters.

Although the typical calibration 'error' of Volmaster SLV Model is extremely small, Volmaster FX implements proprietary algorithms to make sure exotic options (such as barrier options) are priced relatively to the exact actual underlying vanilla option, as priced on the volatility smile.

The computation of actual option prices on Volmaster SLV Model is numerically extremely intense, due to the nature of the problem. Thanks to innovative and break-through calculus techniques, Volmaster FX can perform option price calculations orders of magnitude faster than the typical time required by standard techniques such as finite differences, lattices or monte-carlo simulations, effectively breaking the trade-off between speed and accuracy.

By combining an arbitrage-free smile generation, an advanced stochastic volatility model, a methodology for calibration error compensation and break-through calculus techniques, Volmaster FX delivers to all market participants the most cutting-edge technology in a rich, ready-to-go, fully integrated software solution deployed through the internet as a software-as-a-service business model.

Technical documentation is available on request to Volmaster FX registered users (terms and conditions apply).

 

Volmaster SLV+J Model (Stochastic-Local Volatility + Jumps)

 

In some circumstances, the stochasticity of the volatility and the local volatility are not a good explanation of an extreme convexity in the smile. This is particularly the case of pegged or semi-pegged currency pairs which tend to exhibit a very low level of absolute volatility, coupled with a very strong convexity of the smile and often a very pronounced skewness. A better explanation for the behaviour of such currency pairs is the chance of sudden jumps (changes of pegging, central bank interventions, etc.), which can be effectively modeled with Poisson stochastic processes. In most cases, such jumps are also asymmetric in size.

Volmaster FX implements a SLV+J model whereby a jump process is superimposed on top of the SLV model. Traders shall provide to the system two extra input parameters: the jumps density, expressed for convenience as the chance of an O/N jump and the jumps expected size, expressed as 'quota' of the atm volatility. A third paramter, the jumps asymmetry is automatically calibrated by the system.

Technical documentation is available on request to Volmaster FX registered users (terms and conditions apply).

 

Further Developments

 

In addition to the SLV and SLV+J models, available by default to all users, Volmaster is implementing even more advanced models, such as a three-factor stochastic volatility/stochastic skew model and models that include stochastic interest rates. Since the calculus techniques of Volmaster FX are largely model-agnostic, Volmaster can quickly and effectively evolve its pricing models into the latest cutting-edge advancements and follow closely the progress of academic research.