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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.