Utility-Based Pricing, Timing and Hedging of an American Call Option Under an Incomplete Market with Partial Information

Abstract

This paper studies the pricing, timing and hedging of an American call option written on a non-tradable asset whose mean appreciation rate is not observable but is known to be a Gaussian random variable. Our goal is to analyze the effects of the partial information on investment in the American option under an incomplete market. The objective of the option holder is to maximize the expected discounted utility of consumption over an infinite lifetime. Thanks to consumption utility-based indifference pricing principal, stochastic control and filtering theory, under CARA utility, we derive the value and the exercise time of the American call option, which are determined by a semi-closed-form solution of a free-boundary PDE problem with a finite time horizon. We provide numerical results by finite difference methods and compare the results with those under a fully observable case. Numerical calculations demonstrate that partial information leads to a significant loss of the implied value of the American call option. This loss increases with the uncertainty of the mean appreciation rate. If the option holder is risk-averse enough, a growth of the systematic/idiosyncratic risk will increase/decrease the implied option value and the option is exercised later/sooner. Whether a stronger positive correlation between the tradable asset and the non-tradable underlying asset increases the option value and the information value depends on the risk attitude of the option holder. On the contrary, a stronger negative correlation will definitely make the option and the information more valuable. In addition, explicit expressions of the utility-based pricing for a perpetual American call are presented if the tradable risky asset is perfectly correlated with the underlying asset.

Appendix

1.1 Computation Method

For the comparative static analysis, we apply a finite-difference method, which is efficient and powerful to numerically solve our problem. We focus on solving the implied value $g$ of the American option.

First, we take the domain of the PDE (28) as a finite domain $[0,x_{M}]~\times [-s_{L},s_{M}]\times [0,T]$, where $x_{M},s_{L}$ and $s_{M}$ need to be large enough to keep the accuracy of the numerical solutions and also appropriately satisfy $g(x_{M},s,t)=x_{M}-I$. Then, we divide the domain to a $[N_{x},N_{s},N_{t}]$ grid, or mesh. A special note of the grid is $\{((i-1)\Delta _{x},(j-1)\Delta _{s},(k-1)\Delta _{t}):i=1,\ldots , N_{x}+1;j=1,\ldots ,N_{s}+1;k=1,\ldots ,N_{t}+1\}$, where the grid spacings are $\Delta _{x}=\frac{x_{M}}{N_{x}},\,\Delta _{s}=\frac{s_{M}+s_{L}}{N_{s}}$, and $\Delta _{t}=\frac{T}{N_{t}}$ respectively. Next, we define a representative point of the function $g_{i,j}^{k}= g(x_{i},s_{j},t_{k})$ with $x_{i}=(i-1)\Delta _{x},s_{j}=(j-1)\Delta _{s}$ and $t_{k}=(k-1)\Delta _{t}$.

In the end, we begin to solve the free-boundary problem by the following finite-difference scheme which solves for all $g_{i,j}^{k}$ iteratively backward in time starting at $t_{N_{t}}=T$. Especially, when $k=1$, we get the implied option value at the initial time. For this end, in our paper, we discretize the partial differential equality in (28) using central differences approximation for the first-order and second-order partial derivatives with respect to $x$ and $s$:

$$\begin{aligned} \frac{\partial g}{\partial x}(x_{i},s_{j},t_{k})&= \frac{g_{i+1,j}^{k}-f_{i-1,j}^{k}}{2\Delta x},\\ \frac{\partial ^{2}g}{\partial x^{2}}(x_{i},s_{j},t_{k})&= \frac{g_{i+1,j}^{k}-2g_{i,j}^{k}+g_{i-1,j}^{k}}{\Delta x^{2}},\\ \frac{\partial g}{\partial s}(x_{i},s_{j},t_{k})&= \frac{g_{i,j+1}^{k}-g_{i,j-1}^{k}}{2\Delta s},\\ \frac{\partial ^{2}g}{\partial s^{2}}(x_{i},s_{j},t_{k})&= \frac{g_{i,j+1}^{k}-2g_{i,j}^{k}+g_{i,j-1}^{k}}{\Delta s^{2}},\\ \frac{\partial ^{2}g}{\partial x\partial s}(x_{i},s_{j},t_{k})&= \frac{g_{i+1,j+1}^{k}-g_{i+1,j-1}^{k}+g_{i-1,j-1}^{k}-g_{i-1,j+1}^{k}}{4\Delta x\Delta s}. \end{aligned}$$

And the derivative with regard to $t$ is approximated applying the backward difference

$$\begin{aligned} \frac{\partial g}{\partial t}(x_{i},s_{j},t_{k})=\frac{g_{i,j}^{k+1}-g_{i,j}^{k}}{\Delta t}. \end{aligned}$$

In addition to the fixed boundary conditions, the free boundary conditions given by (29) must be satisfied at each step. At any state $(s,t)$, by comparing the implied option value and the intrinsic value, we can derive the function $x^{*}(s,t)$ which separates the holding domain from the exercising domain. When the underlying asset price $X_{t}$ hits the level $x^{*}$, it is optimal for the option holder to exercise the option.