greeks, bsm & binomial

7/30/2019 Greeks, BSM & Binomial

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A stock price is currently $20 In 3 months it will be either $22 or $18

Stock Price = $18

Stock Price = $22

Stock price = $20


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Stock Price = $18Option Price = $0

Stock Price = $22Option Price = $1

Stock price = $20Option Price=?

A 3-month call option on the stock has astrike price of .


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Consider the Portfolio: long D shares short1 call option

Portfolio is riskless when 22D 1 =18D orD = 0.25

22D 1

18D


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The riskless portfolio is:

long 0.25 sharesshort 1 call option

The value of the portfolio in 3months is 22 0.25 1 = 4.50

The value of the portfolio today is

4.5e 0.12

0.25

= 4.3670


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The portfolio that islong 0.25 sharesshort 1 option

is worth 4.367 The value of the shares is

5.000 (= 0.25 20 )

The value of the option is therefore

0.633 (= 5.000 4.367 )


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TdT

TrKSd

T

TrKSd

dNSdNeKp

dNeKdNSc

rT

rT

1

0

2

0

1

102

210

)2/2()/ln(

)2/2

()/ln(

)()(

)()(

where


9/26

A bank has sold for `300,000 aEuropean call option on 100,000 sharesof a non-dividend paying stock

S0

= 49, K = 50, r = 5%, = 20%,T = 20 weeks, m = 13%

The Black-Scholes value of the optionis ` 240,000

How does the bank hedge its risk to

lock in a ` 60,000 profit?


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Naked position

Take no action

Covered position

Buy 100,000 shares today

Both strategies leave the bankexposed to significant risk


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This involves: Buying 100,000 shares as soon as

price reaches ` 50 Selling 100,000 shares as soon as

price falls below ` 50This deceptively simple hedgingstrategy does not work well


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Delta (D) is the rate of change of theoption price with respect to theunderlying

Optionprice

A

B Slope = D

Stock price


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This involves maintaining a delta neutralportfolio

The delta of a European call on a non-

dividend paying stock isN(d1) The delta of a European put on the

stock is

N(d1)1


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The hedge position must be frequentlyrebalanced

Delta hedging a written option involves abuy high, sell low trading rule


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Theta (Q) of a derivative (or portfolio ofderivatives) is the rate of change of thevalue with respect to the passage of time

The theta of a call or put is usuallynegative. This means that, if time passeswith the price of the underlying assetand its volatility remaining the same, the

value of a long option declines


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When we are hedging we take

positions that offset D, G, n, etc.

When we create an optionsynthetically we take positions that

match D, G, &n


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Gamma (G) is the rate of change of delta(D) with respect to the price of theunderlying asset

Gamma is greatest for options that are

close to the money (At the Money)


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S

C

Stock price

S'

Callprice

C''C'


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Vega (n) is the rate of change of thevalue of a derivatives portfolio withrespect to volatility

Vega tends to be greatest for optionsthat are close to the money (At theMoney)


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D can be changed by taking a positionin the underlying

To adjust G & n it is necessary to take aposition in an option or other derivative


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Rho is the rate of change of the valueof a derivative with respect to theinterest rate

For currency options there are 2 rhos


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(Greek)

Asset price (S) Positively related Negatively related

(Delta> 0) ( Delta< 0)

Volatility (s) Positively related Positively related(Vega> 0) (Vega> 0)

Risk-free rare (r) Positively related Negatively related

(Rho> 0) (Rho< 0)

Time to expiration Value 0 Value0as call maturity as put maturity

(Theta< 0) (Theta


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greeks, bsm & binomial

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