Question

What actions are required to both delta-hedge and gamma-hedge a written option position?

Market makers buy and sell just like anyone else with different products. Obviously buying low selling high is the goal. This process is selected and done by demand and supply of the instrument or product not by personal interest. One way that market makers in the derivatives market that they can control this risk is by Delta Hedging. The market maker calculates the purchase of the option delta and takes an offsetting position. This delta-hedged strategy is commonly used in the markets especially these days. That investor usually invests own capital to do this. The Black-Scholes model is a key part of this.

Answer 1

Delta hedging is an options trading strategy that aims to reduce, or hedge, the directional risk associated with price movements in the underlying asset. The approach uses options to offset the risk to either a single other option holding or an entire portfolio of holdings. The investor tries to reach a delta neutral state and not have a directional bias on the hedge

The most basic type of delta hedging involves an investor who buys or sells options, and then offsets the delta risk by buying or selling an equivalent amount of stock or ETF shares. Investors may want to offset their risk of move in the option or the underlying stock by using delta hedging strategies. More advanced option strategies seek to trade volatility through the use of delta neutral trading strategies. Since delta hedging attempts to neutralize or reduce the extent of the move in an option's price relative to the asset's price, it requires a constant rebalancing of the hedge. Delta hedging is a complex strategy mainly used by institutional traders and investment banks.

The delta represents the change in the value of an option in relation to the movement in the market price of the underlying asset. Hedges are investments—usually options—taken to offset risk exposure of an asset.

Gamma hedging is an options hedging strategy used to reduce the risk created when the underlying security makes strong up or down moves, particularly during the last day or so before expiration.

Gamma hedging consists of adding additional option contracts to an investors portfolio, usually in contrast to the current position. For example, if a large number of calls were being held in a position, then a trader might add a small put-option position to offset an unexpected drop in price during the next 24-48 hours, or sell a carefully chosen number of call options at a different strike price. Gamma hedging is a sophisticated activity that requires careful calculation to be done correctly.

Delta tells a trader how much an option's price is expected to change because of a small change in the underlying stock or asset--specifically a one-dollar change in price.

Gamma refers to the rate of change of an option's delta with respect to the change in price of an underlying stock or other asset's price. Essentially, gamma is the rate of change of the rate of change of the price of an option.

Gamma is the Greek-alphabet inspired name of a standard variable from the Black-Scholes Model, the first formula recognized as a standard for pricing options. Within this formula are two particular variables that help traders understand the way option prices change in relation to the price moves of the underlying security: Delta and Gamma.

Delta-Gamma Hedging

Delta-gamma hedging is an options strategy that combines both delta and gamma hedges to mitigate the risk of changes in the underlying asset and in delta itself.

In options trading, delta refers to a change in the price of an option contract per change in the price of the underlying asset. Gamma refers to the rate of change of delta.

Using a Delta-Gamma Hedge

With delta hedging alone, a position has protection from small changes in the underlying asset. However, large changes will change the hedge (change delta) leaving the position vulnerable. By adding a gamma hedge, the delta hedge remains intact.

Using a gamma hedge in conjunction with a delta hedge requires an investor to create new hedges when the underlying asset’s delta changes. The number of underlying shares that are bought or sold under a delta-gamma hedge depends on whether the underlying asset price is increasing or decreasing, and by how much.

Large hedges that involve buying or selling significant quantities of shares and options may have the effect of changing the price of the underlying asset on the market, requiring the investor to constantly and dynamically create hedges for a portfolio to take into account greater fluctuations in prices.

Example of Delta-Gamma Hedging Using the Underlying Stock

Assume a trader is long one call of a stock, and the option has a delta of 0.6. That means that for each $1 the stock price moves up or down, the option premium will increase or decrease $0.60, respectively. To hedge the delta, the trader needs to short 60 shares of stock (one contract x 100 shares x 0.6 delta). Being short 60 shares neutralizes the effect of the positive 0.6 delta.

As the price of the stock changes, so will the delta. At-the-money options have a delta near 0.5. The deeper in-the-money an option goes, the closer delta gets to one. The deeper out-of-the-money an option goes, the closer it gets to zero.

Assume that the gamma on this position is 0.2. That means that for each dollar change in the stock, the delta changes by 0.2. To offest the change in delta (gamma) the prior delta hedge needs to be adjusted. If delta increases by 0.2, then delta is now 0.8. That means the trader needs 80 short shares to offset delta. They already shorted 60, so they need to short 20 more. If delta decreased by 0.2, the delta is now 0.4, so the trader only needs 40 shares short. They have 60, so they can buy 20 shares back.

Gamma hedging is essentially constantly readjusting the delta hedge as delta changes (gamma).

The Black Scholes model, also known as the Black-Scholes-Merton (BSM) model, is a mathematical model for pricing an options contract. In particular, the model estimates the variation over time of financial instruments. It assumes these instruments (such as stocks or futures) will have a lognormal distribution of prices. Using this assumption and factoring in other important variables, the equation derives the price of a call option.

The model assumes the price of heavily traded assets follows a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option's strike price, and the time to the option's expiry.

Assumptions of The Black Scholes model:

1. The option is European and can only be exercised at expiration.

2. No dividends are paid out during the life of the option.

3. Markets are efficient (i.e., market movements cannot be predicted).

4. There are no transaction costs in buying the option.

5. The risk-free rate and volatility of the underlying are known and constant.

6. The returns on the underlying are normally distributed.

Black Scholes Formula

C= StN(d1)-Ke-rtN(d2)

d1=

d2= d1-σ*√t

where:

C=Call option price

S=Current stock (or other underlying) price

K=Strike price

r=Risk-free interest rate

t=Time to maturity

N=A normal distribution​

The model assumes stock prices follow a lognormal distribution because asset prices cannot be negative (they are bounded by zero). This is also known as a Gaussian distribution. Often, asset prices are observed to have significant right skewness and some degree of kurtosis (fat tails). This means high-risk downward moves often happen more often in the market than a normal distribution predicts.

The assumption of lognormal underlying asset prices should thus show that implied volatilities are similar for each strike price according to the Black-Scholes model. However, since the market crash of 1987, implied volatilities for at the money options have been lower than those further out of the money or far in the money. The reason for this phenomena is the market is pricing in a greater likelihood of a high volatility move to the downside in the markets.

This has led to the presence of the volatility skew. When the implied volatilities for options with the same expiration date are mapped out on a graph, a smile or skew shape can be seen. Thus, the Black-Scholes model is not efficient for calculating implied volatility.