Risk-adjusted option-implied moments

Abstract

This paper provides a new way of converting risk-neutral moments into the corresponding physical moments, which are required for many applications. The main theoretical result is a new analytical representation of the expected payoffs of put and call options under the physical measure in terms of current option prices and a representative investor’s preferences. This representation is then used to derive analytical expressions for a variety of ex-ante physical return moments, showing explicitly how moment premiums depend on current option prices and preferences. As an empirical application of our theoretical results, we provide option-implied estimates of the representative stock market investor’s disappointment aversion using S&P 500 index option prices. We find that disappointment aversion has a procyclical pattern. It is high in times of high index levels and declines when the index falls. We confirm the view that investors with high risk aversion and disappointment aversion leave the stock market during times of turbulence and reenter it after a period of high returns.

Appendix

To prove Proposition 1, we use Eq. (1) and the following result from measure theory¹⁹:

Let $(\Omega ,\mathcal {A},\mu )$ be a finite measure space, f a non-negative, real-valued measurable function, and $\varphi :[0,\infty ) \rightarrow [0,\infty )$ a continuously differentiable and monotonically increasing function with $\varphi (0)=0$. Then,

$$\begin{aligned} \int \varphi \circ f \, d\mu \; = \; \int \limits _{0}^{\infty } \varphi '(x) \, \mu (f>x) \, dx. \end{aligned}$$

(16)

Since U is twice continuously differentiable, the function $\frac{1}{U'(x)}$, $x\in [0,\infty )$, has the following properties:

• (i) $\frac{1}{U'(x)}$ is continuously differentiable, since it is a composition of continuously differentiable functions;

• (ii) $\frac{1}{U'(x)}$ increases monotonically for all $x>0$, since $\left( \frac{1}{U'(x)}\right) '=\frac{-U''(x)}{U'(x)^2}>0$;

• (iii) for $x \rightarrow 0$, $\frac{1}{U'(x)}$ reaches its minimum and converges to a non-negative value. Therefore, $\frac{1}{U'(x)}$ is a non-negative function.

It follows that $\frac{1}{U'(x)}-\frac{1}{U'(0)}$ satisfies all conditions required for $\varphi$, where $\frac{1}{U'(0)}$ stands for $\lim \limits _{x \rightarrow 0} \,\, \frac{1}{U'(x)}$.

The discounted expected payoff of a call option under the physical measure equals

$$\begin{aligned} C^{P}(t,\tau ,K)&= E^P \left[ e^{-r\tau } (S_{t+\tau } -K)^+ \right] \\&= \int \limits _{0}^{\infty } e^{-r\tau } (S_{t+\tau }-K)^+ P(dS_{t+\tau }). \end{aligned}$$

Using the relation between physical and risk-neutral measures from Eq. (1) yields

$$\begin{aligned}&\frac{1}{c} \cdot \int \limits _{0}^{\infty } e^{-r\tau } (S_{t+\tau }-K)^+ \frac{1}{U'(S_{t+\tau })} Q(dS_{t+\tau }) \\&\quad = \frac{1}{c} \cdot \int \limits _{0}^{\infty } \left\{ \frac{1}{U'(S_{t+\tau })} - \frac{1}{U'(0)} + \frac{1}{U'(0)}\right\} \underbrace{e^{-r\tau } (S_{t+\tau }-K)^+ Q(dS_{t+\tau })}_{\equiv \mu _C(dS_{t+\tau })}, \end{aligned}$$

where $\mu _C$ defines a measure. We can now apply the above result from measure theory, which leads to

$$\begin{aligned} C^{P}(t,\tau ,K)&= \frac{1}{c} \cdot \int \limits _{0}^{\infty } \frac{-U''(x)}{U'(x)^2} \mu _{C}\{S_{t+\tau }>x\} dx + \frac{C(t,\tau ,K)}{c \cdot U'(0)} \\&= \frac{1}{c} \cdot \int \limits _{0}^{\infty } \frac{-U''(x)}{U'(x)^2} \left\{ \int \limits _{x}^{\infty } e^{-r\tau } (S_{t+\tau }-K)^+ Q(dS_{t+\tau })\right\} dx \\&\quad +\frac{C(t,\tau ,K)}{c \cdot U'(0)} . \end{aligned}$$

For $x<K$, the inner integral $\int \limits _{x}^{\infty } e^{-r\tau } (S_{t+\tau }-K)^+ Q(dS_{t+\tau })$ equals the value of a plain-vanilla call option with strike price K. For $x > K$, it follows that

$$\begin{aligned}&\int \limits _{x}^{\infty } e^{-r\tau } \underbrace{(S_{t+\tau }-K)^+}_{> 0} Q(dS_{t+\tau }) \\&\quad = \int \limits _{x}^{\infty } e^{-r\tau } (S_{t+\tau }-x) Q(dS_{t+\tau }) +\int \limits _{x}^{\infty } e^{-r\tau } (x-K) Q(dS_{t+\tau })\\&\quad = C(t,\tau ,x) + (x-K) e^{-r\tau } Q\{S_{t+\tau }>x\} \\&\quad = C(t,\tau ,x) + (x-K) D(t,\tau ,x), \end{aligned}$$

where $D(t,\tau ,x)$ denotes the price of a digital option that pays $1 if $S_{t+\tau }>x$.

Finally, we obtain the following expression:

$$\begin{aligned} C^P(t,\tau ,K)= & {} \frac{1}{c} \cdot \int \limits _{0}^{\infty } \frac{-U''(x)}{U'(x)^2} C(t,\tau ,K)1_{\{x < K \}} dx + \frac{C(t,\tau ,K)}{c \cdot U'(0)} \\&+ \frac{1}{c} \cdot \int \limits _{0}^{\infty } \frac{-U''(x)}{U'(x)^2} \left\{ C(t,\tau ,x) + (x-K) D(t,\tau ,x) \right\} 1_{\{x > K \}} dx \\= & {} \frac{1}{c} \cdot C(t,\tau ,K) \left\{ \frac{1}{U'(K)}-\frac{1}{U'(0)}\right\} + \frac{C(t,\tau ,K)}{c \cdot U'(0)} \\&+ \frac{1}{c} \cdot \int \limits _{K}^{\infty } \frac{-U''(x)}{U'(x)^2} \left\{ C(t,\tau ,x) + (x-K) D(t,\tau ,x) \right\} dx . \end{aligned}$$

The expression for the constant c can be derived similarly:

$$\begin{aligned} c = \int \limits _{0}^{\infty } \left\{ \frac{1}{U'(S_{t+\tau })} - \frac{1}{U'(0)} + \frac{1}{U'(0)}\right\} Q(dS_{t+\tau }). \end{aligned}$$

Applying the above result from measure theory to the measure Q yields

$$\begin{aligned} c&= \int \limits _{0}^{\infty } \frac{-U''(x)}{U'(x)^2} \Bigg \{ \int \limits _{x}^{\infty } Q(dS_{t+\tau }) \Bigg \} dx + \frac{1}{U'(0)}\\&= \int \limits _{0}^{\infty } \frac{-U''(x)}{U'(x)^2} e^{r\tau } D(t,\tau ,x) dx + \frac{1}{U'(0)}. \end{aligned}$$

The proof for the expected discounted payoff of a put option proceeds similarly. $\square$