15.1 the greek letters chapter 15. 15.2 example a bank has sold for $300,000 a european call option...
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15.1
The Greek Letters
Chapter 15
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15.2
Example
• A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock
• S0 = 49, X = 50, r = 5%, = 20%,
T = 20 weeks, = 13%
• The Black-Scholes value of the option is $240,000
• How does the bank hedge its risk?
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15.3
Naked & Covered Positions
Naked position
Take no action
Covered position
Buy 100,000 shares today
Both strategies leave the bank exposed to significant risk
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15.4
Stop-Loss Strategy
This involves:
• Buying 100,000 shares as soon as price reaches $50
• Selling 100,000 shares as soon as price falls below $50
This deceptively simple hedging strategy does not work well
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15.5Delta (See Figure 15.2)
• Delta () is the rate of change of the option price with respect to the underlying
Option
price
A
BSlope =
Stock price
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15.6
Call Delta
• Call Delta = N(d1)
• Always positive
• Always less than one
• Increases with the stock price
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15.7Delta Hedging
• This involves maintaining a delta neutral portfolio
• The delta of a European call on a stock is N (d 1)
• The delta of a European put is
N (d 1) – 1
Put delta is always less than one and increases with the stock price
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15.8
Example
• Suppose c = 10, S = 100, and delta = .75• If stock price goes up by $1, by approximately
how much will the call increase?• Answer: $.75• If you write one call, how many shares of
stock must you own so that a small change in the stock price is offset by the change in the short call?
• Answer: .75 shares
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15.9
Check
• Suppose stock goes to $101. Then the change in the position equals
.75x$1 - .75x$1 = 0• Suppose the stock goes to $99. Then the
change in the position equals(.75)(-1) – (.75)(-1) =-.75 + .75 = 0
• This strategy is called Delta neutral hedging
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15.10
Delta Hedgingcontinued
• The hedge position must be frequently rebalanced
• Delta hedging a written option involves a “buy high, sell low” trading rule
• See Tables 15.3 (page 307) and 15.4 (page 308) for examples of delta hedging
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15.11
Gamma
• Gamma () is the rate of change of delta () with respect to the price of the underlying asset
• See Figure 15.9 for the variation of with respect to the stock price for a call or put option
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15.12Gamma Addresses Delta Hedging Errors Caused By Curvature
(Figure 15.7, page 313)
S
CStock price
S’
Callprice
C’C’’
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15.13Interpretation of Gamma• For a delta neutral portfolio,
t + ½S 2
S
Negative Gamma
S
Positive Gamma
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15.14
Vega
• Vega () is the rate of change of the value of a derivatives portfolio with respect to volatility
• See Figure 15.11 for the variation of with respect to the stock price for a call or put option
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15.15
Managing Delta, Gamma, & Vega
• Delta, , can be changed by taking a position in the underlying asset
• To adjust gamma, and vega, it is necessary to take a position in an option or other derivative
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15.16
Rho
• Rho is the rate of change of the value of a derivative with respect to the interest rate
• For currency options there are 2 rhos
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15.17
Hedging in Practice
• Traders usually ensure that their portfolios are delta-neutral at least once a day
• Whenever the opportunity arises, they improve gamma and vega
• As portfolio becomes larger hedging becomes less expensive
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15.18
Scenario Analysis
A scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities
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15.19
Hedging vs Creation of an Option Synthetically
• When we are hedging we take
positions that offset , , , etc.
• When we create an option synthetically we take positions
that match &
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15.20Portfolio Insurance
• In October of 1987 many portfolio managers attempted to create a put option on a portfolio synthetically
• This involves initially selling enough of the portfolio (or of index futures) to match the of the put option
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15.21
Portfolio Insurancecontinued
• As the value of the portfolio increases, the of the put becomes less negative and some of the original portfolio is repurchased
• As the value of the portfolio decreases, the of the put becomes more negative and more of the portfolio must be sold
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15.22
Portfolio Insurancecontinued
The strategy did not work well on October 19, 1987...