Optimal strategies with option compensation under mean reverting returns or volatilities

Abstract

We study the problem of a fund manager whose contractual incentive is given by the sum of a constant and a variable term. The manager has a power utility function and the continuous time stochastic processes driving the dynamics of the market prices exhibit mean reversion either in the volatilities or in the expected returns. We provide an approximation for the optimal wealth and for the optimal strategy based on affine processes and the fast Fourier transform.

Appendix

1.1 A.1 Fourier transform of indicator functions

For the convenience of the reader here we recall how to express different combinations of indicator functions over an interval in terms of inverse Fourier transforms

$$\begin{aligned} x^{a}\mathbf {I}_{\{x < \bar{x}_1\}}= & {} \frac{1}{2\pi }\int _{-\infty }^{\infty }\frac{\bar{x}_1^{a+iu-R_1}}{a+iu-R_1} x^{-iu+R_1}du \end{aligned}$$

(44)

$$\begin{aligned} x^a\mathbf {I}_{\{x \ge \bar{x}_2\}}= & {} -\frac{1}{2\pi }\int _{-\infty }^{\infty }\frac{\bar{x}_2^{a+iu-R_2}}{a+iu-R_2} x^{-iu+R_2}du \end{aligned}$$

(45)

$$\begin{aligned} x^{a}\mathbf {I}_{\{\bar{x}_1\le x < \bar{x}_2\}}= & {} \frac{1}{2\pi }\int _{-\infty }^{\infty }\frac{\bar{x}_2^{a+iu-R_3}-\bar{x}_1^{a+iu-R_3}}{a+iu-R_3} x^{-iu+R_3}du \end{aligned}$$

(46)

where a is any real number, and $R_1 < a$, $R_2 > a$, and $R_3$ is any real number. The restrictions on $R_1$, $R_2$ and $R_3$ ensure that the dampened functions $x^{a-R_1}\mathbf {I}_{\{x\le \bar{x}_1\}}$, $x^{a-R_2}\mathbf {I}_{\{x \ge \bar{x}_2\}}$ and $x^{a-R_3}\mathbf {I}_{\{\bar{x}_1\le x < \bar{x}_2\}}$ are integrable.

1.2 A.2 Truncation error

In Sect. 3 we obtained an upper and a lower bound for the optimal portfolio value by truncating the integral (9). This gives a truncation error $W_t^{\epsilon }$ as defined in (12). Here we define an upper bound to $W_t^{\epsilon }$ by using that $\xi _T < h(\bar{y})$

$$\begin{aligned} W_t^{\epsilon }< \frac{1}{\xi _t}\sum _{j=1}^3 A_j E\Big [\xi _T^{c_j}Y_T^{d_j} \mathbf {I}_{\{\xi _T < h(\bar{y})\}}\mathbf {I}_{\{Y_T \ge \bar{y}\}}|\mathcal {F}_t\Big ] \end{aligned}$$

(47)

and we show how to evaluate it.

We use the same techniques already used to compute $W^{u}$ and $W^{l}$. By means of (44) and (45) in the Appendix A.1, we express the indicator functions in the right hand side of (47) in terms of inverse Fourier transforms for $j=1,2,3$. To have integrability for any j, we choose $R_{1} < 1-1/\gamma$ and $R_2 > 1$. Then by using Fubini’s theorem to exchange the order of integration, the upper bound for the truncation error reads

$$\begin{aligned} W_t^{\epsilon } < \frac{1}{\xi _t 4\pi ^2}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\Gamma (u_1,u_2)H_t(-iu_1+R_1,-iu_2+R_2)du_1du_2 \end{aligned}$$

where

$$\begin{aligned} \Gamma (u_1,u_2) = -\sum _{j=1}^3 A_j \frac{h(\bar{y})^{c_j+iu_1-R_1}}{c_j+iu_1-R_1}\frac{\bar{y}^{d_j+iu_2-R_2}}{d_j+iu_2-R_2}. \end{aligned}$$

Such an integral is then evaluated by means of a fast Fourier transform computation on a grid of values for $\xi _t$ and $Y_t$ as in (40)

$$\begin{aligned} W_{t,p,q}^{\epsilon } < \frac{\xi _{t,p}^{R_1-1}Y_{t,q}^{R_2}}{4\pi ^2}(-1)^{p+q}\sum _{m = 0}^{\bar{N}-1}\sum _{n = 0}^{\bar{N}-1}\varPhi (u_{1,m},u_{2,n})(-1)^{m+n} e^{-i\frac{2\pi }{\bar{N}}(pm+qn)}\varDelta _1\varDelta _2 \end{aligned}$$

where $u_{1,m}$ and $u_{2,n}$ are defined in (39), and

$$\begin{aligned} \varPhi (u_1,u_2) = \Gamma (u_1,u_2)F(T-t,\nu _t;-iu_1+R_1,-iu_2+R_2), \end{aligned}$$

where F is defined in (21).

In practical application we use Čebyšëv’s inequality to compute a bound to the probability that $Y_t \ge \mu _Y + \delta _Y \sigma _Y$, where $\mu _Y$ and $\sigma _Y$ are the expected value and the standard deviation of $Y_t$, and where $\delta _Y>0$ is a constant

$$\begin{aligned} P(|Y_t-\mu _Y|\ge \delta _Y \sigma _Y) \le \frac{1}{\delta _Y^2}. \end{aligned}$$

In particular we choose $\delta _Y$ in such a way that $\mu _Y-\delta _Y\sigma _Y < 0$, and hence $P(Y_t <\mu _Y-\delta _Y\sigma _Y) = 0$, and that $\frac{1}{\delta _Y^2} < \epsilon _{Y}$, where $\epsilon _Y > 0$ is a small constant. Then we have

$$\begin{aligned} P(Y_t \ge \mu _Y + \delta _Y \sigma _Y) = P(|Y_t-\mu _Y|\ge \delta _Y \sigma _Y)\le \frac{1}{\delta _Y^2} < \epsilon _Y. \end{aligned}$$

Using the same kind of argument we obtain that

$$\begin{aligned} P(\xi _t \ge \mu _{\xi } + \delta _{\xi } \sigma _{\xi }) = P(|\xi _t-\mu _{\xi }|\ge \delta _{\xi } \sigma _{\xi })\le \frac{1}{\delta _{\xi }^2} < \epsilon _{\xi }, \end{aligned}$$

where $\mu _{\xi }$ and $\sigma _{\xi }$ are the expected value and the standard deviation of $\xi _t$, and where $\delta _{\xi }$ and $\epsilon _{\xi }$ are positive constants.

As we already pointed out, the two-dimensional FFT algorithm gives results for a grid of values of $(\xi _t,Y_t)$ as in (40). However, some of these grid points are unlikely. Hence, from the original grid (40), we keep only the subset of points falling in a rectangular region $(0,\mu _{\xi } + \delta _{\xi } \sigma _{\xi })\times (0,\mu _Y + \delta _Y \sigma _Y)$, where it is most likely that the process takes values at time t, and we choose $\bar{y}$ in such a way that the truncation error corresponding to such points is lower than $\epsilon$, for a small $\epsilon > 0$.