Covered Call Writing and Framing: A Cumulative Prospect Theory Approach

Abstract

The covered call writing, which entails selling a call option on one’s underlying stock holdings, is perceived by investors as a strategy with limited risk. It is a very popular strategy used by individual, professional and institutional investors. Previous studies analyze behavioral aspects of the covered call strategy, indicating that hedonic framing and risk aversion may explain the preference of such a strategy with respect to other designs. In this contribution, following this line of research, we extend the analysis and apply Cumulative Prospect Theory in its continuous version to the evaluation of the covered call strategy and study the effects of alternative framing.

Appendix: European Options Valuation

Versluis et al. [12] provide the prospect value of writing call options, considering different intertemporal frames. The option premium represents a sure gain for the writer, whereas the negative payoff is a potential loss; the writer may aggregate or segregate such results in different ways. Nardon and Pianca [8] extend the model of [12] to the case of put options, considering the problem both from the writer’s and holder’s perspectives, and use alternative weighting functions.

Let S _t be the price at time t ∈ [0, T] of the underlying asset of a European option with maturity T. Let c be the call option premium with strike price X. At time t = 0, the option’s writer receives c and can invest the premium at the risk-free rate r, obtaining c e ^rT. At maturity, the amount S _T − X is paid to the holder if the option expires in-the-money.

In the time segregated case the option premium is evaluated separately (through the value function) from the option payoff. Considering zero as a reference point (status quo), the prospect value of the writer’s position is

$$\displaystyle{ V ^{sc} = v^{+}\left (c\,e^{rT}\right ) +\int _{ X}^{+\infty }\psi ^{-}\left (1 - F(x)\right )f(x)v^{-}\left (X - x\right )\,dx\,, }$$

(15)

with f and F being, respectively, the probability density function and the cumulative distribution function of the future underlying price S _T , and v is defined as in (2). One equates V ^sc at zero and solves for the price c.

In the time aggregated frame, gains and losses are integrated in a unique mental account, then one obtains the prospect value

$$\displaystyle{ \begin{array}{ll} V ^{ac}& = w^{+}\left (F(X)\right )v^{+}\left (c\,e^{rT}\right ) \\ & +\int _{ X}^{X+c\,e^{rT} }\psi ^{+}\left (F(x)\right )f(x)v^{+}\left (c\,e^{rT} - (x - X)\right )\,dx \\ & +\int _{ X+c\,e^{rT}}^{+\infty }\psi ^{-}\left (1 - F(x)\right )f(x)v^{-}\left (c\,e^{rT} - (x - X)\right )\,dx.\end{array} }$$

(16)

In this case, the option price evaluated by a PT investor is implicitly defined by the equation V _a = 0 and has to be determined numerically.