greeks, bsm & binomial
TRANSCRIPT
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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
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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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