Several factors can influence an option price that can either be favorable or adverse depending on the type of position a trader has taken. Option traders should understand the risk they are exposed to and the factors affecting their position(s).

Out of all greeks, options delta and gamma are the first partial derivative and the second partial derivative of the price of an option respectively that is most important for any trader to decide upon the directional risk, and gamma-delta hedging is a procedure that allows mitigating that risk maintaining a gamma-delta-neutral position/portfolio.

Please refer to our introductory article to understand the basics of Options Delta and Options Gamma where we have concluded that- options are a non-linear derivative meaning the delta is not constant and is changing due to change in factors such as moneyness of an option i.e., change in the underlying price, change in implied volatility, and time decay.

However, whether any change in the underlying price would lead to continuous rebalancing our portfolio? If that is the case then whether a trader has to monitor the market and the risk limits at all times? and what about the transaction costs, slippage costs, and regulatory constraints impacting the profitability? --Let's take a practical example to understand the procedure of gamma-delta hedging and how we can rebalance our portfolio to remain gamma-delta-neutral.

Suppose, an Investment Bank has entered into a contract on a $450 strike 1-Year European call option by taking a short position amounting to $42,479,129.18 on a non-dividend-paying stock that is presently trading at $500, lot size 445,000. The annualized volatility in the underlying price is 22.315% and the risk-free interest rate is 6.00% p.a compounded continuously.

Using Black-Scholes-Merton Model, the no-arbitrage price of this option is $89.84, and accordingly, the total contract price is $39,979,129.18 (i.e., 89.84 * 445,000) while the actual price at which the contract is entered is $42,479,129.18 (greater than the no-arbitrage/theoretical price of the contract).

The primary motive to enter into a short ITM call option contract is that the value to the short is $2,500,000.00 at the initiation of the contract. In other words, the bank is standing on a positive value at the initiation of the contract with a notional of $200,250,000.00 (i.e., $450.00 * 445,000).

It is not necessary that there is some arbitrage opportunity for the bank but one reason could be that- the implied volatility on the stock calculated by reverse engineering using the BSM model is higher than the historical volatility of the underlying. And since the actual volatility is not known and not constant, an option seller is suffering the risk, and therefore, an option buyer has to compensate it by paying a higher price.

Once the contract has been entered, risk managers are appointed to monitor the daily price movement in the underlying. If the underlying price changes, the call price changes too and that leads to a risk of price movement on the upside. Risk managers are responsible to develop the best techniques to curb the risk of any position taken by an Investment Bank and that best could be gamma-delta hedging and rebalancing technique in our case study.

### Gamma-Delta Hedging and Dynamic Rebalancing

Risk managers are concerned with more targeted and stable risk-taking. To manage and mitigate the risk efficiently on option products, they need to continuously monitor various risk attributes associated with the positions by calculating MTM PnL on a daily basis and check on the risk limits/flags associated with the use of option greeks.

Options Delta is the amount by which an option price is expected to change with respect to a $1 change in the underlying asset price, other factors being constant. Options Gamma is the amount by which an options delta is expected to change with respect to a $1 change in the underlying asset price, other factors being constant.

The delta of stock is 1 which means that the long position in 358,468 shares has a positive delta of $358,468 for every $1 change in the stock price while the short position in the 445,000 call option has a negative delta of $357,362 (i.e., 445,000*0.8) and a negative gamma of $1,106 (i.e., 445,000*0.002486). Henceforth, the options delta and gamma of the trader's overall position are neutralized.

To gamma-delta hedge the short options position, the equivalent number of stocks that need to be purchased would be 358,468 (i.e., (445,000 * 0.80) + (445,000 * 0.002486) = 358,468.12) [rounded off] resulting in offsetting positions, and the gain/loss on the long stock position would then tends to offset the loss/gain on the short option position.

Here the trader is aiming to set up a portfolio of two different positions that overall remain unaffected to daily market moves while enjoying the Options Theta (i.e., time decay) and Implied Volatility overstatement, not directional trading.

Let's suppose that the stock price on the very next day happens to be $505 and therefore, the call price changes to $93.89. The probability that the option will land ITM increases i.e., the options delta and gamma of the call option increases from 0.80 to 0.82 and from 0.002486 to 0.002367 respectively.

The MTM PnL would be as follows-

Long stock position = (505 - 500)*358,468 = $1,792,340.61

Short call option position = -(93.89 - 89.84)*445,000 = -$1,800,416.21

Excess hedge due to rounding off stocks = (505 - 500)*(358,468 - 358,468.12) = -$0.61

Net MTM PnL = ($1,792,340.00 + $0.61) - $1,800,416.21 = -$8,076.21

Since the overall position was gamma-delta hedged, the net MTM PnL of -$8,076.21 is due to the residual impact.

Theoratical price of an option due to change in delta and gamma = $89.84 + (505 - 500)*0.80 + (505 - 500)*0.002486 = $93.8686

Therefore, the residual premium = $93.8866 - $93.8685 = $0.0181

(amounting to $0.0181*445,000 = $8,075.59)

At this point, it is very important to understand that-

The overall position remains gamma-delta hedged only for a reasonable change in the asset price because for deep ITM options, delta plus gamma price is always an overestimate of the actual option price and for deep OTM options, delta plus gamma price is always an underestimate of the actual option price, and therefore, it still faces the residual risk for a large change in the option price.

The gamma is highest for ATM options and relatively lower for ITM and OTM options because ITM and OTM options delta will not change as quickly with the movement in the underlying. As the underlying price increases and move further ITM, the gamma for that option starts decreasing resulting in a reduction in the number of stocks that needs to be purchased in order to maintain gamma-delta-neutral position.

In our case study, the option delta and gamma both have changed from 0.80 to 0.82 and from 0.002486 to 0.002367 respectively, and we have to accordingly rebalance the portfolio by purchasing 5,346 stocks (i.e., 445,000*(0.82 - 0.80) + 445,000*(0.002367 - 0.002486)) [rounded off] to remain gamma-delta-neutral.

The gamma-delta-neutral portfolio will now contains 363,814 (i.e., 358,468 + 5,346) long stocks which are presently trading at $505 having a positive delta of $363,814.00 and 445,000 short call option priced at $93.89 with a negative delta of $3,62,760.55 (i.e., 445,000*0.82) and a negative gamma of $1,053.39 (i.e., 445,000*0.002367). It means, if the underlying price change by $1, the call option premium is expected to change by $0.82 in the same direction keeping other factors constant. However, this gamma-delta-neutral portfolio still faces the residual risk.

It is observed that, as implied volatility increases, the linearity of the delta also increases and that results in a directional risk that is more stable (i.e., gamma is relatively flat). It becomes easier and straightforward because the frequency of rebalancing could be a little bit less.

Call Delta | Volatility Ladder (Impact of Volatility on Delta)

Call Gamma | Volatility Ladder (Impact of Volatility on Gamma)

To conclude, gamma-delta hedging is a continuous process (as delta is changing due to change in underlying price and gamma is changing due to increased volatility in the options delta) and should be performed throughout the tenure of an option. And hence gamma-delta hedging can’t be done by taking a static position but it requires a dynamic rebalancing. Increasing the frequency of rebalancing reduces the overall volatility but at the expense of increased commissions which might not be as profitable as compared to holding till the expiration date for retail traders. Hence, the gamma-delta rebalancing is a great hedge against the directional risk at the same time the frequency of rebalancing is also an active decision to be taken properly.