###### Bootstrapping Spot Rates

Bootstrapping spot rates is a forward substitution method that allows investors to determine... **Read More**

The Greeks are a group of mathematical derivatives applied to help manage or understand portfolio risks. They include delta, gamma, Theta, Vega, and rho.

Delta is the rate of change of the option’s price with respect to a given change in the price of the underlying instrument, holding other parameters constant. The delta of long one stock share is +1 while that of short one share of stock is -1.

The option deltas of a call and put options are given as:

$$\text{Delta}_{c}=e^{-\delta T}N(d_{1})$$

$$\text{Delta}=-e^{-\delta T}N(-d_{1})$$

Where \(\delta\) is the continuously compounded dividend yield of the underlying stocks, note that \(\delta\) will be zero for non-dividend paying stocks.

Call options have positive deltas as the value of a call option increases with an increase in the underlying asset price. Conversely, put options have negative deltas as the value of a put decreases with an increase in the underlying price.

Gamma is the rate of change of portfolio delta with a change in the underlying price, holding all the other parameters constant.

Option gamma measures the convexity or curvature of the relationship between the price of the option and the price of the underlying asset.

A high value of gamma means that delta is more sensitive to the share price changes and vice versa. The gamma of a long or short position in one share of stock is zero. Recall that the delta of the underlying share is equal to one. The derivative of a constant is zero, and so the gamma of the underlying asset must be zero.

Gamma is always positive, and its value is highest when the option is near at the money and close to expiration. The portfolio gamma can be lowered by going short options and increased by going long options.

Both put and call options have equal gamma.

$$Gamma_{c}=Gamma_{p}=\frac{e^{-\delta T}}{S\sigma\sqrt{T}}n(d_{1})$$

Where:

\(n(d_{1})\) is the standard normal probability density function

Theta measures the sensitivity of the option value to a small change in calendar time, holding all else constant. Note that time is a variable that progresses with certainty. Therefore, it does not make sense to hedge against changes in the time since the start of the contract, similar to what we do for unexpected changes in the underlying asset price.

The greater the time to expiry, the higher the possibility that the share price will move in favor of the holder, given that the downside loss is capped. However, Theta is usually negative for both a call and a put option as the expiration date gets nearer. The speed of the decline in the option value increases as time goes by. The option will thus expire worthless. I.e., S=K. The change in option prices as time advances is known as **time decay. **The Theta of a stock is zero since stocks do not have an expiry.** **

Vega is the sensitivity of a portfolio to a given small change in the assumed level of volatility; all else held constant. The assumption of future volatility makes Vega a subjective risk management tool.

The Vega of both call and put options are equal and always positive. If the underlying security becomes more volatile, then there is a greater chance of the price moving in favor of the option holder.

Vega is high for at or near the money options and short-dated options.

Rho is defined as the change in a portfolio with respect to a small change in the risk-free rate of interest, holding everything else constant. Although the risk-free rate of interest can be determined with a good degree of certainty, it can vary by a small amount over the contract term.

Holding a call option can be viewed as having cash in the bank waiting to purchase the share. The holder of the call option will benefit in the meantime when interest rates rise.

Conversely, we can think of a holder of a put as already owning a share and is waiting to sell it for cash. The holder of the put will thus lose out on the interest in the meantime when interest rates rise.

Therefore, rho is positive for a call option and negative for a put option.

Additionally, rho changes over time. The option price will be less sensitive to the interest rate as the expiration date nears because interest rates lose much importance then.

## Question

The sensitivity of a portfolio value to a small change in calendar time, holding all else constant is most likely:

- Vega
- Theta
- Rho
## Solution

The correct answer is B:Theta measures the change in the value of a portfolio given a small change in calendar time, holding all else constant.

A is incorrect:Vega is the sensitivity of a portfolio to a given small change in the assumed level of volatility; all else held constant.

C is incorrect:Rho is defined as the change in a portfolio with respect to a small change in the risk-free rate of interest, holding everything else constant.

*Reading 38: Valuation of Contingent Claims *

*LOS 38(k) Interpret each of the option Greeks*