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# The Greeks

## What are the Greeks?

Broadly, the Greeks measure the sensitivity of an option’s premium to changes in the underlying variables. They are necessary for determining how to properly hedge a portfolio and are therefore important for risk management.

• In this presentation we’ll cover Greeks in the Black-Scholes world. This means we are assuming options are European on a non-dividend paying stock. It also assumes the underlying stock follows a geometric Brownian motion process.

• This means the underlying variables are the follwing: the stock price, volatility, the risk-free rate, and time.

• Note, each Greek (being a partial derivative of the Black-Scholes equation) assumes all other variables remain constant. The Black-Scholes equation for the premium of a European call option is shown on the next slide.

## Black-Scholes Formula:

• Call_0 = S_0N(d_1) - Xe^{-rT}N(d_2)
• Put_0 = N(-d_2)K\exp{-r(T-t)} - N(-d_1)S_0

where

d_1 = \frac{ln(\frac{S_0}{X}) + (r+\frac{\sigma^2}{2})T}{\sigma\sqrt(T)}

d_2 = d_1 - \sigma\sqrt(T)

• S_0: the value of the call option at time 0.

• N(): the cumulative standard normal density function (NORMSDIST() in Excel)

• X: the exercise or strike price.

• r: the risk-free interest rate (annualized).

• T: the time until option expiration in years.

• \sigma: the annualized standard deviations of log returns.

• e and ln are the exponential and natural log functions respectively (EXP() and LN() in Excel).

## Greeks

Let P refer to the equation for either a call or put option premium. Then the greeks are defined as:

Delta (\Delta = \frac{\partial P}{\partial S}): Where Sis the stock price.

Gamma (\Gamma = \frac{\partial^2 P}{\partial S^2}): Where S is the stock price.

Theta (\Theta = \frac{\partial P}{\partial t}): Where t is time.

Rho (\rho = \frac{\partial P}{\partial r_f}): Where r_f is the risk-free rate.

Vega (v = \frac{\partial P}{\partial \sigma}) (Not Greek): Where \sigma is volatility.

## Delta: \frac{\partial P}{\partial S}

Delta is the rate of change on the option’s price with respect to changes in the price of the underlying asset (stock). For a call option the Delta is: \Delta = N(d_1)

where N() is the standard cumulative normal density function. The Delta for a put is: \Delta = N(d_1) - 1

Delta is very useful, because it is the number of shares to buy (or sell) to hedge out the risk of changes in the underlying stock’s price when short a call (or put) option.

• In other words, if you have a portfolio short 1 call option and long Delta shares of stock, then your portfolio is riskless (over very short time periods). This is referred to as delta hedging.

• Similarly, a portfolio short one put and short Detla shares of stock is riskless.

• Call Deltas range from 0 to 1, and put Deltas range from -1 to 0.

## Interactive Apps

This presentation contains interactive apps for each Greek – on the following slide is the app for an option’s Delta.

• Each app will allow you to graph the variation of a Greek, where you can choose the variable on the horizontal axis. You can also change the other inputs into the option pricing model and see how this affects the relationship.

• Many of the relationships are greatly affected by the moneyness of the option, so first try changing the stock or strike price.

## Gamma: \frac{\partial^2 P}{\partial S^2}

Gamma is the rate of change of the option’s Delta with respect to changes in the underlying stock. Gamma for a both a call and put is:

\Gamma = \frac{N'(D_1)}{S_0 \sigma \sqrt(T)}

• The higher the Gamma (in absolute value) the more often you’ll need to rebalance a delta-neutral portfolio.

Suppose the Gamma of a call option on a stock is 0.03.

• This means that a $1 increase in the stock’s price will increase the Delta of the option by 0.03. To create a Gamma-neutral portfolio, you’ll have to trade in an option on the underlying stock – or some derivative which is not linearly related to the underlying stock. ## Theta: \Theta = \frac{\partial P}{\partial t} Theta is the rate of change of the option premium with respect to time. It is also known as the option’s time decay. Theta for a call is: \Theta = -\frac{S_0 N'(d_1) \sigma}{2\sqrt(T)}-rXe^{-rT} N(d_2) and for a put is: \Theta = -\frac{S_0 N'(d_1) \sigma}{2\sqrt(T)} + rXe^{-rT} N(-d_2) Theta is usually negative, which means as time passes the option’s premium declines (all other variables held constant). • Theta is calculated in years, but if we divide theta by 252, we get the daily decline in the option premium solely due to time decay. • For example, say Theta is -25, then in days Theta is \frac{-25}{252} = -0.09921 which means all else constant, the option’s price will decline by$0.09921 per day.

• Theta is often quoted as this time decay per day (-0.9921 above).

• Use the following app to find the situation for which theta is positive? Hint: when does an option’s value decline with increased time to expiration?

## Rho: \rho = \frac{\partial P}{\partial r_f}

Rho is the rate of change of the option premium with respect to the risk-free rate. Rho for a call is:

\rho = XTe^{-rT}N(d_2)

and for a put is:

\rho = -XTe^{-rT}N(-d_2)

Rho measures the sensitivity of the option value to interest rates.

• For example, say for a put option Rho is -30.

• Then, if the risk-free rate increases by 1%, the option’s value will decline by 1\% \times 30 = \$0.30 • Often Rho is simply quoted as the change in the option’s price for each 1% change in the risk-free rate (the 0.30 above). ## Vega: \frac{\partial P}{\partial \sigma} Vega is the rate of change of the option premium with respect to the volatility of the underlying asset. Note, the Black-Scholes model assumes volatility is constant–so there is a contradiction in deriving Vega from the Black-Scholes model. • More appropriately, we should calculate Vega from a stochastic volatility model, but in practice Vega from Black-Scholes is very similar to Vega from stochastic volatility models. Vega for both a call and put is: V = S_0\sqrt(T)N'(D_1) • The higher Vega is in absolute value, the more sensitive the option’s price is to changes in volatility. • The Vega of a long position in a call or put option is positive. • The Vega of an at-the-money option decreases as expiration approaches. • If the Vega of a put option is 40, then a 1% increase in volatility increases the value of the option by 1\% \times 40 = \$0.40.

• Often Vega is simply quoted as the change in the option’s price for each 1% change in volatility (the 0.40 above).

## Credits and Collaboration

Click the following links to see the codeline-by-line contributions to this presentation, and all the collaborators who have contributed to 5-Minute Finance via GitHub.