Hedging with Options in Excel

Hedging with options is a risk management strategy that involves using options contracts to reduce or eliminate the exposure to adverse price movements in an underlying asset. Options are derivatives that give the buyer the right, but not the obligation, to buy or sell the underlying asset at a predetermined price and time. By buying or selling options, an investor can create a hedge against the potential losses of their existing positions in the asset or a related asset.

There are two types of options: call options and put options. A call option gives the buyer the right to buy the underlying asset, while a put option gives the buyer the right to sell the underlying asset. The price at which the option can be exercised is called the strike price, and the date until which the option can be exercised is called the expiration date. The buyer of the option pays a premium to the seller of the option to acquire this right.

Hedging with options can be done in various ways, depending on the type of risk and the direction of the market. For example, an investor who owns a stock and is worried about a possible decline in its price can buy a put option on the same stock. This way, if the stock price falls below the strike price of the put option, the investor can exercise the option and sell the stock at the strike price, limiting their losses. However, if the stock price rises above the strike price, the investor can let the option expire worthless and keep the stock. The cost of this hedge is the premium paid for the put option.

Another example of hedging with options is to use a call option to protect a short position in an asset. A short position is when an investor sells an asset that they do not own, hoping to buy it back later at a lower price and profit from the difference. However, this strategy involves the risk of unlimited losses if the asset price rises instead of falling. To hedge this risk, the investor can buy a call option on the same asset. This way, if the asset price rises above the strike price of the call option, the investor can exercise the option and buy the asset at the strike price, covering their short position and limiting their losses. However, if the asset price falls below the strike price, the investor can let the option expire worthless and profit from their short position. The cost of this hedge is the premium paid for the call option.

Hedging with options has some advantages and disadvantages. Some of the advantages are:

Options are flexible and can be tailored to suit different risk profiles and market scenarios
Options have limited risk for buyers, as they only lose the premium paid if the option expires out of the money
Options can provide leverage, as they allow the buyer to control a large amount of the underlying asset with a small amount of capital

Some of the disadvantages are:

Options have time decay, which means that their value decreases as they approach their expiration date
Options have opportunity cost, which means that the buyer foregoes the potential gains of the underlying asset if the option expires in the money
Options can be expensive, especially for at-the-money or in-the-money options, or for options with high implied volatility

Basic Theory of Hedging with Options

Call Options

A call option gives the holder the right to buy an asset at a specified price within a specified period.

Put Options

A put option gives the holder the right to sell an asset at a specified price within a specified period.

Hedging Strategy

Long Call Option: Protects against a potential price increase.
Long Put Option: Protects against a potential price decrease.

Procedures for Hedging with Options

Determine the Risk Exposure: Identify the asset you want to hedge and assess the potential risk exposure.
Select the Appropriate Option: Choose between a call option and a put option based on whether you want to hedge against a price increase or decrease.
Determine Option Parameters: Decide on the option contract details, including strike price, expiration date, and the number of contracts.
Calculate the Cost of the Option: Use the Black-Scholes or other pricing models to determine the cost of the selected option.
Implement the Hedge: Execute the option trade to establish the hedge.

Excel Formulas for Hedging

Black-Scholes Formula for Option Pricing

The Black-Scholes formula for call option pricing is given by:

$C = S_0 \cdot N(d_1) - X \cdot e^{-rT} \cdot N(d_2)$

Where:

$C$ : Call option price
$S_0$ : Current stock price
$X$ : Option strike price
$r$ : Risk-free rate
$T$ : Time to expiration
$N(d_1)$ and $N(d_2)$ : Cumulative distribution functions of the standard normal distribution.

Similarly, the formula for put option pricing is given by:

$P = X \cdot e^{-rT} \cdot N(-d_2) - S_0 \cdot N(-d_1)$

Delta Formula

The Delta ( $\Delta$ ) of an option measures the sensitivity of its price to changes in the underlying asset price. For a call option:

$\Delta_{\text{Call}} = N(d_1)$

For a put option:

$\Delta_{\text{Put}} = -N(-d_1)$

Scenario: Hedging Against a Price Increase

Let’s consider a scenario where a business owns 1,000 shares of stock currently trading at $50. The business wants to hedge against a potential price increase over the next three months.

Current Stock Price ( $S_0$ ): $50
Strike Price ( $X$ ): $52
Time to Expiration ( $T$ ): 3 months
Risk-free Rate ( $r$ ): 3% per annum

Excel Calculation

Calculate $d_1$ and $d_2$ : Use the Black-Scholes formulas to calculate $d_1$ and $d_2$ .
Calculate Call Option Price ( $C$ ): Use the Black-Scholes call option pricing formula.
Determine Delta ( $\Delta_{\text{Call}})$ : Calculate Delta using the Delta formula.
Hedge Position: To hedge against a price increase, the business should buy call options equivalent to the delta of the stock position.

Excel Table

Parameter	Value
Current Stock Price ( $S_0$ )	$50
Strike Price ( $X$ )	$52
Time to Expiration ( $T$ )	3 months
Risk-free Rate ( $r$ )	3% per annum
Number of Shares	1,000

Calculation	Formula	Result
$d_1$	$\frac{\ln\left(\frac{S_0}{X}\right) + \left(r + \frac{\sigma^2}{2}\right)T}{\sigma \sqrt{T}}$	Result of $d_1$
$d_2$	$d_1 - \sigma \sqrt{T}$	Result of $d_2$
Call Option Price ( $C$ )	$S_0 \cdot N(d_1) - X \cdot e^{-rT} \cdot N(d_2)$	Result of $C$
Delta ( $\Delta_{\text{Call}})$	$N(d_1)$	Result of $\Delta_{\text{Call}}$
Number of Call Contracts	$\frac{\text{Number of Shares} \cdot \Delta_{\text{Call}}}{100}$	Result of Contracts

Result of the Scenario

Call Option Price ( $C$ ): Result of $C$ from the Excel table.
Number of Call Contracts: Result of Contracts from the Excel table.

The business can now execute the option trade based on the calculated values to hedge against a potential price increase in the stock.

Other Approaches

Hedging with Put Options: Follow the same steps as above but use put options if the business wants to hedge against a potential price decrease.
Dynamic Hedging: Adjust the hedge position over time by regularly recalculating the delta and adjusting the number of contracts.
Combination Strategies: Use a combination of call and put options to create more complex hedging strategies tailored to specific risk exposures.