Pricing Variance Swaps for Stochastic Volatility with Delay and Jumps

1. Pricing Variance Swaps for Stochastic Volatility with Delay and Jumps Li Xu PRMIA Presentation Department of Mathematics & Statistics University of Calgary

2. Outline Introduction Previous Results in Delay Model Pricing Model with Delay and Jumps Numerical Example and Comparison Conclusion

3. Introduction Variance swap: A forward contract on future realized asset variance, the square of the volatility, which can be used to trade variance. Payoff function: is the realized asset variance (quoted in annual terms) over the life of the contract, is the strike price for variance. is the notional amount of the swap in dollars per annualized volatility point square. The price of the variance swap in the risk neutral world is the expected present value of the payoff:

4. Who Can Use Variance Swaps Portfolio managers who wish to hedge vega ( ) exposure. Hedging implicit volatility exposure. Investors may require more frequent rebalancing and greater transactions expenses during volatile periods. Equity funds are probably short volatility because of the negative correlation between index and volatility. Trading the spread between realized and implied volatility. Clients who want to speculate on the future levels of volatility.

5. Stochastic Volatility with Delay The stochastic volatility depends on where is the delay factor and is the spot price of the underlying asset at time (see Swishchuk et al. (2002)). The initial data: deterministic function. Variance swaps can be priced under this model (see Swishchuk et al. (2002))

6. Main Features of this Model Continuous-time analogue of discrete-time GARCH model. Mean-reversion. Contains the same Wiener process as the asset. Market is complete. Incorporates the expectation of log-return. Delay as a measure of risk. Similar to Heston model, but easier to model and price the variance swaps.

7. Why Jumps in Stochastic Volatility By empirical study, it is more realistic to consider jumps in stochastic volatility in financial markets including energy market. Still keeps those good features of the previous model. An analytical pricing formula for variance swaps is still available and it is quick to implement.

8. Pricing Model The spot price of the asset satisfies the following SDDE: Consider the Poisson process in the stochastic volatility This is a continuous-time analogue of its discrete-time GARCH(1,1) model

9. Continue Change to risk neutral measure, and take expectation of the stochastic volatility under risk neutral probability, we have where The intensity of Poisson process does not change under risk neutral probability.

10. Continue Solving this Differential Equation, we get the approximate solution in general case: The analytical formula for variance swap with delay and Poisson jumps:

11. Compound Poisson Process Case Consider the following equation: Take expectation under risk neutral probability:

12. Continue In general case, the approximate solution: The analytical formula for variance swap with delay and compound Poisson jumps:

13. Numerical Example: S&P60 Canada Index (1997-2002)

14. Comparison

15. Dependence on Intensity

16. Conclusion Keeps all the advantages of the delay model without jumps. Incorporate jumps in stochastic volatility which is important in energy market. Easier to model and price variance swaps, no numerical approximation and time saving. Further extension: adding jumps in asset price, more complex form of jumps in stochastic volatility, pricing volatility swaps.

17. Thank you! lxu@math.ucalgary.ca