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Greeks of the Black Scholes Model
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
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)
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
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:
25.
25.05.0
25).5.05.05.0()40/36ln( 2
1
d
5.025.05.025.02 d
The Black-Scholes Model
Step 2: Find the probabilities:
4013.0)25.0()1( NdN
3085.0)50.0()2( NdN
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
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?
• 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