FINANCIAL MATHEMATICS ACTIVITIES

Financial Mathematics Seminars - April 25, 2001

4:30 - 5:30 p.m.

Optimal Contracting, Incentive Effects and the Valuation of Executive Stock Options

Yisong Tian,Schulich School of Business, York University

In this paper, I develop a new approach to the valuation of non-traded stock options for undiversified executives. The concept of certainty equivalent value, originally proposed by Lambert et al. (1991), is modified with two new features. First of all, the executive's personal investment opportunity set is expanded to include risky assets as well as the risk-free asset. Risky investments are not considered in previous studies. Furthermore, the value of non-traded stock options is determined within a standard principal-agent model with information asymmetry. The principal (shareholder) optimally selects a compensation contract in order to induce the desired level of effort from the agent (executive), fully aware that the executive take action to maximize his own expected utility. Unlike previous research, option value determined this way is consistent with the executive's utility maximization behavior.

Using parameters consistent with the average pay of CEOs from S&P 500 companies and typical stock market conditions over the last two decades, I find that the executive value of non-traded stock options depends largely on the executive's degree of risk aversion and effort aversion, his investment opportunities, and the compensation package. This value can be higher than, lover than or close to the Black-Scholes/Merton value of the option. In addition, the optimal contract between the shareholder and the executive is a combination of incentive pay and cash pay. However, option is the optimal form of incentive pay only if the executive has a relatively low degree of risk aversionand effort aversion. Restriced stock is optimal for other executives. The optimal incentive pay cannot always overcome the executive's risk and effort aversion because the optimal proportion of incentive pay is actully negatively related to the executive's degree of risk aversion and effort aversion. The shareholder is always better off if the executive is less risk averse and effort averse. These findings are robust with respect to various parameters concerning option exercise price, stock volatility, outside investment opportinities, and the executive's initial wealth, utility function and disutility of effort.

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6:00 - 7:00 p.m.

News Arrival, Jump Dynamics and Volatility Components in Individual Stock Returns

Tom McCurdy, University of Toronto

This paper allows different types of news to have different impacts on returns and expected volatility. The conditional variance is a mixture of smoothly changing (augmented GARCH) and discontinuous (jump) components. This mixture captures occasional large changes in price, due to news events such as earnings surprises, as well as smoother changes in prices which may be due to liquidity trading or strategic trading as information disseminates. The parameterization allows asymmetric effects from good and bad news and asymmetric feedback from previous jump innovations to future volatility.
A heterogeneous Poisson process governs the likelihood of jumps and is summarized by a time-varying conditional intensity parameter. The model is applied to returns from IBM, Intel and Texaco and demonstrates that both the conditional jump intensity and the jump size dis-tribution vary significantly over time. Some of the empirical features we investigate are the proportion of volatility explained by jumps, changes in jump likelihood over time, and the differential impact and continuation effects of jumps versus normal news events.

The full version of Tom McCurdy's paper is available by emailing him at: tmccurdy@rotman.utoronto.ca