Volatility Management

The Equity Derivatives Pricing Expert

Volatility Management

The variance swap is an equity derivative with payoff the realized variance of the underlying equity or index. The Black-Scholes-Merton tradition of continuous delta hedging under diffusion confuses it with the log contract. As a consequence, it suggests the variance swap is redundant with the vanilla options. What the variance swap truly is, however, over and above the vanillas, is a play on the possible underlying jumps. We believe variance swaps mark a new age in volatility arbitrage. For this reason, we price them from scratch, independently of the diffusion assumption or even the idea that vanilla options may have ever been a play on variance:

  • We price the variance swaps under our generalized jump-diffusion model with stochastic volatility and stochastic jumps, also known as the “regime-switching model.”
  • We also price the log contract. This way, you can measure the difference due to the jumps.
  • It doesn’t matter whether jumps (in the equity or the index) have been known to occur or not to occur in the past. (A jump to default couldn’t have occurred in the past.) What matters is whether the market anticipates such jumps.
  • The empirical disconnect between the market price of the variance swap and the theoretical price of the log contract (a.k.a. the strip of vanillas), apparent even on the index, points in that direction.
  • We even calibrate the regime-switching model against the market prices of variance swaps of different starting dates and maturity dates, independently of the vanillas. Indeed, the variance swap is not redundant with the vanillas and its price carries additional information on the underlying process (as does the price of any path-dependent option, generally).
  • People should be suspicious, anyway, of any methodology that is incapable of valuing an instrument as natural and simple and homogeneous as the variance swap directly and says it requires a full strip of known vanilla options prices in order to do so!
  • On top of vanilla variance swaps, our all-numerical solving techniques enable us to price the following payoffs:
    • volatility swaps
    • capped volatility and variance swaps
    • gamma swaps
    • corridor variance swaps
    • up and down corridor variance swaps
    • conditional variance swaps
    • variance options
    • variance swaptions
  • All this is achieved in our general equity-to-credit framework, of which dividends and default risk are an integral part.

Equity-to-Credit is the new form of volatility arbitrage. Credit risk (through the probability of the underlying equity jumping to zero) adds a component to option premium that cannot be financed by the usual rebalancing of the delta hedge issuing from the Black-Scholes-Merton model. Another hedging instrument has to be held and continuously traded in order to hedge the jump to default. Jointly inferring the Brownian volatility and the hazard rate from the market data of instruments sensitive both to volatility and credit risk (equity options, CDS) and computing the composite dynamic hedging strategy are the new rule of volatility arbitrage.

Since Excel is probably the most popular front end among traders and hedge fund managers, we made sure that all the results of Opscore are published to Excel. There are two ways the user may view these results:
  • Using Microsoft Office Automation, the user can export all the terms and conditions of the convertible security from the Opscore data model to the Opscore Excel Analyzer. Several theoretical models can be defined and simulated on this worksheet. 3D surfaces of theoretical values and Greeks can be instantaneously plotted. More generally, every single output can be plotted as a function of every single input. This is achieved by the VBA routines of the Opscore Excel Analyzer, which, in turn, call the DLL of the pricing engine; moreover, the user can simulate terms and conditions different from the ones that are stored in the Opscore database, without affecting the stored data.
  • Alternatively, users can build their own Excel spreadsheet, laying out the results of the pricing engine any way they please thanks to the Opscore XLL functions. The Opscore XLLs accept the identification number of the given security (its internal database code, ISIN, CUSIP, Bloomberg number or SEDOL) as an argument. This allows the XLLs to retrieve the corresponding terms and conditions from the Opscore database. The remaining arguments are the theoretical parameters (such as Brownian volatility and hazard rate). They are produced on the spreadsheet, either by calibration routines or by direct user input.