Optimal hedging of a perpetual American put with a single trade

TL;DR

This study considers the seller of a perpetual American put option who can hedge her portfolio once until the underlying stock price leaves a certain range of values, and determines optimal trading boundaries as functions of the initial stock holding, and an optimal hedging strategy for a bond/stock portfolio.

Abstract

It is well-known that using delta hedging to hedge financial options is not feasible in practice. Traders often rely on discrete-time hedging strategies based on fixed trading times or fixed trading prices (i.e., trades only occur if the underlying asset's price reaches some predetermined values). Motivated by this insight and with the aim of obtaining explicit solutions, we consider the seller of a perpetual American put option who can hedge her portfolio once until the underlying stock price leaves a certain range of values $(a,b)$. We determine optimal trading boundaries as functions of the initial stock holding, and an optimal hedging strategy for a bond/stock portfolio. Optimality here refers to the variance of the hedging error at the (random) time when the stock leaves the interval $(a,b)$. Our study leads to analytical expressions for both the optimal boundaries and the optimal stock holding, which can be evaluated numerically with no effort.

Figures (10)

• Figure 1: Plots of the functions $\Gamma(x)$ and $P'(x)$ using parameters $r=3\%$, $\sigma=30\%$, $K=100$ and $b=150$. Notice that $a=K/(1+d^{-1})=40$.
• Figure 2: Plots of the map $x\mapsto G(x,h)$ for different values of the initial stock holding $h$ using parameters $r=3\%$, $\sigma=30\%$, $K=100$, $b=150$, and $a = \hat{a} = K/(1+d^{-1})=40$.
• Figure 3: Plots of the map $x\mapsto V(x,h)$ and $M(x)$ for different values of the initial stock holding $h$ using parameters $r=3\%$, $\sigma=30\%$, $K=100$, $b=150$, and $a=\hat{a}=K/(1+d^{-1})=40$.
• Figure 4: Plots of the optimal stopping boundaries $x^*_{1,h},x^*_{2,h}$ as functions of $h$ using parameters $r=3\%$, $\sigma=30\%$, $K=100$, $b=150$ and $a=\hat{a}=K/(1+d^{-1})=40$.
• Figure 5: Left panel: 3-D plot of the value function $(x,h)\mapsto V(x,h)$. Right panel: plot of optimal stock holdings $x\mapsto h^*(x)$, $x\mapsto\Gamma(x)$ and the Black-Scholes Delta $x\mapsto P'(x)$ using parameters $r=3\%$, $\sigma=30\%$, $K=100$, $b=150$ and $a=\hat{a}=K/(1+d^{-1})=40$.

Theorems & Definitions (38)

• Definition 2.1: Trading strategy
• Remark 2.2
• Remark 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Lemma 3.4