We investigate the effect of discrete sampling and asset price jumps on fair variance and volatility swap strikes. Fair discrete volatility strikes and fair discrete variance strikes are derived in different models of the underlying evolution of the asset price: the Black-Scholes model, the Heston stochastic volatility model, the Merton jump-diffusion model and the Bates and Scott stochastic volatility and jump model. We determine fair discrete and continuous variance strikes analytically and fair discrete and continuous volatility strikes using simulation and variance reduction techniques and numerical integration techniques in all models. Numerical results show that the well-known convexity correction formula may not provide a good approximation of fair volatility strikes in models with jumps in the underlying asset. For realistic contract specifications and model parameters, we find that the effect of discrete sampling is typically small while the effect of jumps can be significant.

The first column shows the fair variance strike computed using the PDE method, simulation, and analytical value in the SV model. The second column shows the fair volatility strikes computed with the same methods. The Journal of Derivatives 2008.15.3:7-24. Downloaded from www.iijournals.com by PRINCETON UNIVERSITY on 09/18/13.It is illegal to make unauthorized copies of this article, forward to an unauthorized user or to post electronically without Publisher permission. 2 2 2 d dt rGdt = SPRING 2008 THE JOURNAL OF DERIVATIVES 13 E X H I B I T 3The left plot shows the square root of the fair variance strike and the fair volatility strike versus initial volatility. The right plot shows the convexity value (66) versus initial volatility.These prices are for one-year maturity options corresponding to the SV model parameters given in Exhibit 1.

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We propose a multiperiod model to value employee options allowing for the possibility that a risk-averse employee strategically exercises her options over time rather than at a single date. Our results describing the representative employee's option exercise behavior are broadly consistent with existing empirical evidence. The value of options to the employee and their effective cost to the firm are significantly different from the predictions of a constrained model that assumes “single date” strategic option exercise. The constrained model substantially underestimates the cost of options to the firm when, ceteris paribus, the employee's relative risk aversion and/or the time to maturity and/or the stock volatility exceed respective thresholds. Hence, the incorporation of “multiple-date” exercise has important economic and accounting consequences.