Option implied riskiness and risk-taking incentives of executive compensation

Abstract

The riskiness developed by Aumann and Serrano (J Polit Econ 116:810–836, 2008) is a measure based on mean, standard deviation and higher order moments. Instead of relying on corporate policies as indirect measures of firm risk, we theoretically show a positive relation between the value of compensation contracts with convex payoff and the firm’s option implied riskiness through second-order stochastic dominance and provide supportive empirical evidence of this risk taking incentive. To address the endogeneity concern, we perform a difference-in-difference analysis using the implementation of FAS 123R in 2006, an accounting standard under which firms are required to recognize the fair value-based expense of stock option grants. Firms thereby are discouraged from granting executive stock options (ESO) because of the higher cost resulted from the strict expense recognition required by FAS 123R. Hence, the implementation of FAS 123R results in an exogenous negative shock to the use of ESO. Using this approach, we find a significant decrease in the option implied riskiness subsequent to FAS 123R, supportive of the risk-taking incentive associated with executive stock options.

Appendix

According to the spanning formula of Bakshi and Madan (2000) and Bakshi et al. (2003), any function of the form $H\left(S\right)$ with $E\left[H\left(S\right)\right]< \infty$ can spanned by a collection of call and put options:

$$H\left[ S \right] = H\left[ {\overline{S}} \right] + \left( {S - \overline{S}} \right)H_{S} \left[ {\overline{S}} \right] + \mathop \smallint \limits_{{\overline{S}}}^{\infty } H_{SS} \left[ K \right]\left( {S - K} \right)^{ + } dK + \mathop \smallint \limits_{0}^{{\overline{S}}} H_{SS} \left[ K \right]\left( {K - S} \right)^{ + } dK.$$

(8)

where ${H}_{S}\left(\bullet \right)$ and ${H}_{SS}\left(\bullet \right)$ represent the first and second derivative of H with respect to S and $\overline{S }$ is the initial stock price. We denote

$$g_{j,T} = \frac{{S_{j,T} - S_{j,0} }}{{S_{j,0} }}$$

(9)

In addition, S denotes ${S}_{j,T}$, $\overline{S }$ denotes ${S}_{j,0}$ and $H\left[{S}_{j,T}\right]={e}^{-\frac{{g}_{j,T}}{{{\varvec{R}}}_{j}}}$. Since $E\left({e}^{-\frac{{g}_{j,T}}{{{\varvec{R}}}_{j}}}\right)=1$ and by the spanning formula of Bakshi and Madan (2000) and Bakshi et al. (2003), we have

$$\begin{aligned} \frac{{E\left( {e^{{ - \frac{{g_{j,T} }}{{R_{j} }}}} } \right)}}{{\left( {1 + r_{T} } \right)}} & = \frac{1}{{\left( {1 + r_{T} } \right)}} \\ & = \frac{1}{{\left( {1 + r_{T} } \right)}} - \frac{{r_{0,T} S_{j,0} }}{{\left( {1 + r_{T} } \right)R_{j} }} + \mathop \smallint \limits_{0}^{{S_{j,0} }} v\left( {R_{j} \left[ {g_{j,T} } \right],K} \right)P_{j} \left( {K,T} \right)dK \\ & \quad + \mathop \smallint \limits_{{S_{j,0} }}^{\infty } v\left( {R_{j} \left[ {g_{j,T} } \right],K} \right)C_{j} \left( {K,T} \right)dK \\ \end{aligned}$$

(10)

where ${r}_{T}$ is the return on the risk-free security over a period T, ${P}_{j}\left(K,T\right)$ and ${C}_{j}\left(K,T\right)$ are put price and call price of stock j with strike price equal to K and time to maturity equal to T at the present time 0 respectively, and

$$v\left( {R_{j} \left[ {g_{j,T} } \right],K} \right) = \frac{1}{{R_{j} \left[ {g_{j,T} } \right]^{2} }}e^{{ - \frac{1}{{R_{j} \left[ {g_{j,T} } \right]}}\frac{{K - S_{j,0} }}{{S_{j,0} }}}}$$

(11)

Therefore, by Eqs. (10) and (11), we have (2).