Gamma Scalping and Dynamic Hedging in Convertible Bond Arbitrage

The Essence of Gamma in Convertible Bonds

Convertible bonds exhibit convexity, a feature that is central to their appeal in arbitrage strategies. This convexity arises from the embedded option to convert the bond into a predetermined number of shares of the underlying equity. The price of the convertible bond is therefore sensitive to changes in the price of the underlying stock, and this sensitivity is not linear. The delta of a convertible bond, which measures the rate of change of the bond's price with respect to the stock's price, is not constant. The rate of change of the delta is known as gamma.

A high gamma means that the delta of the convertible bond will change significantly in response to a small change in the price of the underlying stock. This is a desirable characteristic for a convertible bond arbitrageur. The strategy typically involves being long the convertible bond and short a delta-neutral amount of the underlying stock. As the stock price fluctuates, the delta of the position changes. To maintain a delta-neutral hedge, the arbitrageur must dynamically adjust the short stock position. This process of re-hedging is where the opportunity for profit arises.

Gamma Scalping in Practice

Gamma scalping, also known as dynamic hedging, is the process of adjusting the hedge ratio of a convertible bond arbitrage position to profit from the volatility of the underlying stock. When the stock price rises, the delta of the convertible bond increases. To maintain a delta-neutral position, the arbitrageur must short more shares of the stock. Conversely, when the stock price falls, the delta of the convertible bond decreases, and the arbitrageur must buy back some of the shorted shares.

The profit from gamma scalping comes from the fact that the arbitrageur is systematically buying low and selling high. When the stock price falls, the arbitrageur buys back shares at a lower price than they were shorted at. When the stock price rises, the arbitrageur shorts more shares at a higher price. This process generates a steady stream of small profits, which can accumulate over time to a significant amount.

The profitability of gamma scalping is directly related to the realized volatility of the underlying stock. The more the stock price fluctuates, the more opportunities there are to re-hedge the position and generate profits. Therefore, convertible bond arbitrageurs actively seek out convertible bonds with high gamma and underlying stocks with high historical and implied volatility.

The Vega Component

Vega is the measure of a portfolio's sensitivity to changes in the volatility of the underlying asset. In convertible bond arbitrage, the arbitrageur is long vega, meaning that the position profits from an increase in the implied volatility of the underlying stock. This is because the value of the embedded call option in the convertible bond increases as volatility rises.

When implied volatility is low, the market is underpricing the potential for future stock price fluctuations. A convertible bond arbitrageur can take advantage of this by establishing a long convertible bond position and a short stock position. If the implied volatility subsequently increases, the value of the convertible bond will rise, and the arbitrageur will profit. This is another key source of return for the strategy, in addition to gamma scalping.