Several factors can influence an option's price which can either be favorable or unfavorable depending on the type of trade/position a trader has entered into. Option traders should understand the risks that they are exposed to and the factors affecting their position(s). Risk managers are appointed to help them out.

Out of all greeks, the option's delta is crucial because it helps, (1). traders estimate the directional movement in the price of an option for any change in the price of the underlying asset. (2). risk managers to gauge the directional risk and it is the most crucial risk attribute for analysis.

In the end, the ultimate objective of using the option's delta may not only capture the directional movement /risk but also hedge the delta to mitigate the directional /delta risk by maintaining delta-neutral positions.

Refer to our introductory article to understand the basics of the option's delta.

Here we have concluded that the delta of an option does not remain constant. It changes due to changes in several factors, and one of the most important factors is the moneyness of an option because a significant portion of the option's delta is driven due to a change in the price of the underlying asset.

Does that mean that any change in the price of the underlying asset would lead to a continuous change /rebalance in the position quantity? If that is the case then the traders have to monitor the market and the risk attributes at all times because traded instruments follow a stochastic process.

And what about the transaction costs, slippage costs, and regulatory constraints impacting the profitability while rebalancing the position(s)?

Let's take a case study and understand the procedure of static hedging and delta hedging & rebalancing the position(s) to remain delta-neutral.

Suppose,

an Investment Bank has entered a short $1460 strike European call option contract with a lot size of 5,000 at a premium of $161,250.00, 2 days to expire. The underlying is a non-dividend paying stock asset, presently trading at $1488.05.

The historical volatility in the underlying asset was observed to be 23.8640% p.a and the risk-adjusted interest rate derived from the futures turns out to be 22.6743% p.a continuous.

As per the Black-Scholes-Merton model, the theoretical /no-arbitrage price of this option is $31.4866, and accordingly, the total contract price should be $157,433.12 (i.e. 31.4866 * 5,000) while the actual price at which the contract has been entered is $161,250.00 (higher than the theoretical /no-arbitrage price of the contract).

It seems like, the primary motive to enter into a short in-the-money call option contract could be the value to the short of $3,816.88 at the initiation of the contract. In other words, the bank is sitting on a positive value at the initiation of the contract i.e. the actual premium of $161,250.00 minus the no-arbitrage premium of $157,433.12.

There doesn't need to be some sort of arbitrage opportunity available to the bank, but one reason could be that the implied volatility calculated by reverse engineering using the Black-Scholes-Merton model turns out to be higher than the observed /historical volatility in the underlying asset. And since the actual volatility is not known and not constant, the option writer suffers the risk, and therefore, the option buyer has to compensate it by paying a higher premium.

Once the contract is entered, risk managers are appointed to monitor the movement in the underlying risk factors like the price of the underlying asset. If the price of the underlying asset changes, the price of a call option changes too, and that brings the risk of the underlying's price moving on the upside. The risk managers are responsible to suggest the best risk management strategy to curb the risk from this position taken by the bank. And here, one of the risk management strategies could be delta-hedging and rebalancing in our case study.

### Static Hedging

To hedge the directional risk in a short call option position, the trader can purchase an equal number of underlying units (i.e. 5,000) from the secondary market at the best ask price of around $1488.05 trading currently and hold both positions till the expiration of the options contract.

The exposure from the underlying asset would amount to $7,440,250.00 (i.e. $1488.05 * 5,000) and the net investment would be $7,279,000.00 (i.e. 7,440,250.00 - $161,250.00) due to offsetting positions.

This risk management strategy involves static hedging by taking long-short positions on the same underlying but via different instruments. However, this strategy is efficient only if both positions are held until the option's contract expires. On the expiry date, the in-the-money call option behaves almost like its underlying asset as there will be a 1-to-1 movement in the price of the call option with respect to the price of the underlying asset.

Scenario-1: If the option lands /expires in-the-money, the bank has to deliver 5,000 units of the underlying asset at the asking price equal to the strike price of $1460.00. Here, the bank can deliver the units purchased under the hedging strategy to fulfill its obligation in the short call option's contract (or) can sell the underlying asset in the secondary market at the prevailing price and settle the short call option's contract in cash.

Case-1: If the bank chooses to settle by using purchased units of the underlying asset, the total realization from the hedging strategy will amount to $7,461,250.00 (i.e. from the underlying asset of $7,300,000.00 ($1460.00 * 5,000) + the premium received on a short call option at the initiation of $161,250.00). Hence, the overall profit realization from the hedging strategy will be $21,000 (i.e. total realization of $7,461,250.00 - the net investment amount of $7,440,250.00).

Case-2: If the bank chooses to settle by using cash,

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Scenario-2: If the option lands /expires worthless, the short call option's contract lapses, and the overall profit realization from the hedging strategy will be $21,000 (i.e. unrealized loss in the underlying asset of -$140,250.00 (i.e. ($1460.00 - $1488.05) * 5,000), assuming that the price of the underlying asset turns out to be $1460.00 on the expiry date + the premium received on a short call option at the initiation of $161,250.00).

At the outset, this hedge strategy looks great because this strategy will bring in gains of $21,000 under both scenarios (i.e. maximum the bank could lose in the underlying asset is $140,250.00 (i.e. (1488.05 - 1460.00) * 5,000) and that can be recovered from the premium received at the initiation of the option's contract of $161,250.00).

However, if the price of the underlying asset falls significantly from 1460.00 to 1440.00 to 1400.00 levels, then the impact /unrealized loss /negative mark-to-market will be significantly higher than the premium received in the short call option's contract.

The bank can further improve on this hedge strategy by buying an equal number of units in the underlying asset at the $1460.00 level as soon as the price of the underlying asset rises above that level and square-off the same at $1460.00 as soon as the price falls below that level. In this way, the bank will be hedged on the upside to fulfill its obligation in the short call option's contract, and on the downside, the bank will have no position in the underlying asset to lose. However, this strategy will involve transaction costs, financing costs, and slippage costs in buying and selling the underlying asset which can hurt the profitability on $21,000 from the overall strategy.

### Delta Hedging & Rebalancing

the risk managers are always concerned with more targeted and dynamic risk hedging. And to accomplish that, they need to continuously monitor various risk attributes associated with the positions by computing mark-to-market profit and loss regularly and checking whether the bank is trading within the acceptable risk limits or not.

As we know that an option's delta is the amount by which the price of an option is expected to change with respect to a $1 change in the price of the underlying asset, other factors being constant. It means that the delta of the underlying asset will always be 1, and hence, a long position of 1 unit in the underlying asset will have a delta of $1 for every $1 change in the price of the underlying asset - the sensitivity to itself.

However, the delta of a 1460 strike call option = [ (LN(St/K) + (r-q+σ^2/2)*T) / σ*√T ] from the standard normal distribution = $0.8471 for every $1 change in the price of the underlying asset, and hence, the delta of 5,000 short call option position will have a negative delta of -$4,235.7 (i.e. 0.8471 * -5,000).

In order to hedge the risk i.e. negative delta in a short call option position, the trader has to book an equivalent amount of positive delta resulting in long-short delta /offsetting positions, and the loss/gain on the short call option position would then tend to offset the gain/loss on the long underlying asset position.

In this financial book, the discussion involves the underlying asset as a hedge position, and therefore, to create a delta-hedged position, the trader has to purchase 4,235 units in the underlying asset to have a positive delta of $4,235 for every $1 change in the price of the underlying asset.

"delta-hedging is a procedure to mitigate the linear risk associated with a trade/position."

Henceforth, the short delta risk of -$4,235.7 in a short call option position can be hedged via a long delta of $4,235 in an underlying asset position resulting in an overall delta-neutral position.

Here, the trader is aiming to setup a position of two different instruments that overall remains unaffected by daily market moves (ignoring higher-order movements) in the underlying asset while enjoying the option's theta (i.e. time-decay) and implied volatility overstatement, not directional trading.

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