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Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call...

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Greeks of the Black Scholes Model
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Page 1: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

Greeks of the Black Scholes Model

Page 2: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

Black-Scholes Model

The Black-Scholes formula for valuing a call option

where

)()( 21 dNe

XPdNV

RTsc

T

TRXPd s

)5.()/ln( 2

1

Td

T

TRXsP

d

1

)25.()/ln(

2

Page 3: 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 priceX = the exercise priceR = continuously compounded risk

free rateT = the time remaining to expires = risk (standard deviation of the

stock’s annual return)

Page 4: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

The Black-Scholes Model

• Further definitions:– X/eRT = the PV of the exercise price where

continuous discount rate is used

– N(d1 ), N(d2 ) = the probabilities

Page 5: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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?

Page 6: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

The Black-Scholes Model

Step 1. Start by finding the value of d1 and d2:

25.

25.05.0

25).5.05.05.0()40/36ln( 2

1

d

5.025.05.025.02 d

Page 7: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

The Black-Scholes Model

Step 2: Find the probabilities:

4013.0)25.0()1( NdN

3085.0)50.0()2( NdN

Page 8: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

The Black-Scholes Model

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

26.2$19.12$45.14$

3085.025.005.0

40$36$4013.0

e

cV

Page 9: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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

Page 10: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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

Page 11: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

Delta

• Measures change in the option value when the stock value changes

• Can be neutralized/hedged by taking selling/buying shares

• Delta– Positive – Negative

Page 12: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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

Page 13: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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??

Page 14: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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!!

Page 15: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

Delta Hedging

• Why is there a need for dynamic delta hedging?

• Can this strategy be profitable?

Page 16: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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

Page 17: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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!!

Page 18: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

Gamma

• Gamma tends to explode as an at the money option nears maturity.

Guess why??

Page 19: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

Theta

• Theta is the change in the value of option due to passage of time

• Validates ‘Change is inevitable’. Why?

Page 20: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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

Page 21: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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?

Page 22: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

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

Page 23: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

• 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.

Page 24: Greeks of the Black Scholes Model. Black-Scholes Model The Black-Scholes formula for valuing a call option where.

• 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


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