## Delta

When financial institutions sell tailored options in over-the-counter markets, they face unique challenges in managing risk, particularly when these options don’t align with standardized products traded on exchanges. This chapter delves into various approaches to hedge such risks, focusing on the “Greek letters” or “Greeks,” which measure different dimensions of risk in an option position. The aim for traders is to manage these Greeks to keep all risks within acceptable limits. The concepts discussed apply to both market makers in exchange-traded options and traders in over-the-counter markets.

For instance, consider a financial institution that has sold a European call option for $300,000 on 100,000 shares of a non-dividend-paying stock. The stock price is $49, the strike price is $50, and the option’s theoretical Black–Scholes–Merton price is about $240,000. The institution has made a profit over the theoretical value but needs to manage the associated risks.

There are several strategies to hedge such risks:

**Naked and Covered Positions**: The institution might choose to do nothing, a strategy known as a naked position. This works well if the stock price remains below the strike price. Alternatively, it could adopt a covered position, buying 100,000 shares as soon as the option is sold. This strategy has its own risks, especially if the stock price drops significantly.**Stop-Loss Strategy**: This involves buying the stock when its price rises above the strike price and selling it when it falls below. The goal is to hold a naked position when the stock price is less than the strike price and a covered position when it’s more. However, this strategy has its limitations due to the difficulty of executing trades at the exact strike price and the associated costs.**Delta Hedging**: Most traders use sophisticated procedures involving Greek measures like delta. Delta is the rate of change of the option price with respect to the price of the underlying asset. By maintaining a delta-neutral position and rebalancing regularly (dynamic hedging), traders can effectively hedge their positions. This strategy involves calculating the delta of the option and adjusting the stock position to offset the option’s delta, ensuring the overall position remains neutral to price changes in the underlying asset.

Each of these strategies has its pros and cons. Naked and covered positions can be simple but carry significant risks. Stop-loss strategies, while attractive in theory, can be costly due to frequent trades and price discrepancies. Delta hedging offers a more refined approach but requires regular rebalancing and careful monitoring of the option’s delta.

## Rest of the Greeks

Managing the risks associated with options trading involves understanding various factors, commonly referred to as the “Greeks.” These include the previously described Delta, Theta, Gamma, Vega, and Rho. Each of these parameters measures a different aspect of the risk in an option position, helping traders to balance their portfolios effectively.

**Delta (**\(\Delta\)**)**: This measures the sensitivity of an option’s price to changes in the price of the underlying asset. A high Delta implies that the option’s price will move significantly for a relatively small change in the asset’s price. Traders often adjust their portfolios to maintain Delta neutrality.**Theta (**\(\Theta\)**)**: Theta indicates the rate at which an option’s value decreases as time passes, all other factors being equal. This ‘time decay’ tends to accelerate as the option approaches its expiration date.**Gamma (**\(\Gamma\)**)**: Gamma measures the rate of change of Delta with respect to changes in the underlying asset’s price. A high Gamma suggests that Delta is very sensitive to price changes, necessitating more frequent portfolio adjustments.**Vega (**\(\mathcal{V}\)**)**: Vega shows the sensitivity of an option’s price to changes in the volatility of the underlying asset. A portfolio with high Vega is significantly affected by changes in market volatility.**Rho (**\(\rho\)**)**: Rho assesses the sensitivity of an option’s price to changes in interest rates. It’s particularly relevant for options with longer maturities.

In practical scenarios, achieving perfect balance in these Greeks is challenging. Traders often focus on Delta neutrality, adjusting their portfolios regularly to respond to market changes. Gamma and Vega neutrality are more complex to maintain due to the difficulty in finding suitable trading options. These Greeks play a crucial role in the dynamic hedging strategies employed by financial institutions, aiding them in navigating the complexities of options trading and risk management.