Greeks of the Black Scholes Model

Black-Scholes Model The Black-Scholes formula for valuing a call option where

The Black-Scholes Model Ps = the stock’s current market price X = the exercise price R = continuously compounded risk free rate T = the time remaining to expire s = risk (standard deviation of the stock’s annual return)

The Black-Scholes Model Further definitions: –X/e RT = the PV of the exercise price where continuous discount rate is used –N(d 1 ), N(d 2 ) = the probabilities

The Black-Scholes Model Example: Consider a call that expires in three months and has an exercise value of Rs40 (hence, T=0.25 and X=Rs40). The current price and volatility of the underlying stock are Rs36 and 50%, respectively. The risk free rate is 5% (hence, Ps=Rs36, R=0.05 and std. dev =0.5). What is the value of the call?

The Black-Scholes Model Step 1. Start by finding the value of d1 and d2:

The Black-Scholes Model Step 2: Find the probabilities:

The Black-Scholes Model Step 3: Use the Black-Scholes formula to estimate the value of the call option:

The Black-Scholes Model What happens to the fair value of an option when one input is changed while holding the other four constant? –The higher the stock price, the higher the option’s value –The higher the exercise price, the lower the option’s value –The longer the time to expiration, the higher the option’s value

The Black-Scholes Model What happens to the fair value of an option when one input is changed while holding the other four constant? –The higher the risk free rate, the higher the option’s value –The greater the risk, the higher the option’s value

Delta Measures change in the option value when the stock value changes Can be neutralized/hedged by taking selling/buying shares Delta –Positive –Negative

Delta Delta (for a call) –At the money call: 0.5 –Deep in the money: 1 –Deep out of the money: 0 Delta (for a put) –At the money put: -0.5 –Deep in the money: -1 –Deep out of the money: 0

Delta Delta is closer to one for longer maturities but tend towards 0, 0.5 or 1 near expiry Why should the delta decline over time??

Delta Hedging Conveniently done through buying/selling stocks How would you delta hedge your long positions in call? How would you delta hedge your long position in puts? How would you delta hedge a straddle!!

Delta Hedging Why is there a need for dynamic delta hedging? Can this strategy be profitable?

Gamma Arises on account of non linearity of options Very similar to the concept of ‘convexity’!! Gamma is the change in delta as the price of stock changes

Gamma Why would you love to have positive gamma in your portfolio? Gamma is maximum for at the money options It tends towards zero for out of the money and deep in the money options Think about creating a zero delta positive gamma portfolio!!

Gamma Gamma tends to explode as an at the money option nears maturity. Guess why??

Theta Theta is the change in the value of option due to passage of time Validates ‘Change is inevitable’. Why?

Theta Theta is negative for long positions in call or put. Theta is near zero for out of the money call and out of the money put Theta is significant for at the money options

Vega Change in the value of an option if the implied volatility changes What should the Vega of long positions in call or put be? How would Vega behave for an option marching towards expiry?

Your portfolio seems to react adversely to volatility but seems to be high on gamma. How do you neutralize the Greeks?

Clue: The portfolio has positive gamma but negative vega.

Your portfolio seems to react adversely to volatility but seems to be high on gamma. How do you neutralize the Greeks? Clue: The portfolio has positive gamma but negative vega. Ans: Buy long term option sell short term options