Sato Process Research Articles

Simple Processes and the Pricing and Hedging of Cliquets

For data on market prices on 246 cliquets we consider the task of pricing these exotic options using a relatively simple path space subsequently stressed to market implied and then predicted stress levels. An additive process transitioning from a Sato process to a Levy process is formulated and estimated on vanilla options. Ask prices constructed from predicted stress levels are observed to have an in sample correlation of 92% with market prices. Interestingly, it is observed that capped cash flows have negative stress levels while uncapped products have positive stress levels. We illustrate the effect of hedging cliquet liabilities using call options as hedging assets permiting a 10% reduction in ask prices.

The S&P 500 Index as a Sato Process Travelling at the Speed of the VIX

The logarithm of SPX is modeled as a Sato process running at a speed proportional to the current level of the VIX. When the logarithm of the VIX is an exponential compound Poisson process with drift one may obtain exact expressions for the prices of equity options taken at an independent exponential maturity. The parameters for the Levy process are calibrated from VIX options while the parameters for the Sato process driving the stock may be calibrated from market option prices taken at an independent exponential maturity. Results confirm that both the option surface and the VIX time changed Sato process volatilities, skews and term volatility spreads are responsive to the VIX level and the VIX option surface.

Pricing and Hedging Basket Options to Prespecified Levels of Acceptability

The concept of stress levels embedded in S&P 500 options are defined and illustrated with explicit constructions. The particular example of a stress function used is MINMAXVAR. Seven joint laws for the top 50 stocks in the index are considered. The first time changes a Gaussian one factor copula. The remaining six employ correlated Brownian motion independently time changed in each coordinate. Four models use daily returns, either run as Levy processes or scaled, to the option maturity. The last two employ risk neutral marginals from the VGSSD and CGMYSSD Sato processes. The smallest stress function uses CGMYSSD risk neutral marginals and Levy correlation. Running the Levy process yields a lower stress surface than scaling to the option maturity. Static hedging of basket options to a particular level of acceptability is shown to substantially lower the price at which the basket option may be offered.

Sato processes and the valuation of structured products

We report on the adequacy of using Sato processes to value equity structured products. In models used to price options on realized variance, the latter must be a random variable with a positive variance. An analysis of this variance of realized variance for Sato processes shows that these processes may be suited to option contracts on realized volatility. Nonlinear pricing principles based on hedging to acceptability are outlined for the purpose of pricing structured transactions. It is shown that, typically, different products should be priced using different models. Pricing comparisons of Sato process prices with other standard models like Heston stochastic volatility, with and without jumps, VGSA, local volatility and local CGMY are also provided. Sato processes tend to overprice cliquets relative to other models. They also maintain the value of long dated out-of-the-money realized variance options.

Sato Processes and the Valuation of Structured Products

We report on the adequacy of using Sato processes to value equity structured products. An analysis of the variance of realized variance for Sato processes shows that these processes may be suited to option contracts on realized volatility. Nonlinear pricing principles based on hedging to acceptability are outlined for the purpose of pricing structured transactions. It is shown that typically different products should be priced using different models. Pricing comparisons of Sato process prices with other standard models like Heston stochastic volatility, with and without jumps, VGSA, local volatility and local CGMY are also provided. Sato processes tend to overprice cliquets relative to other models. They also maintain the value of long dated out-of-the-money realized variance options.