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The Greek Letters
Finance (Derivative Securities) 312
Tuesday, 17 October 2006
Readings: Chapter 15
Hedging
Suppose that:
• A bank has sold for $300,000 a European call
option on 100,000 shares of a non-dividend
paying stock
• S0 = 49, K = 50, r = 5%, s = 20%, m = 13%,
T = 20 weeks
• Black-Scholes value of option is $240,000
How does the bank hedge its risk?
Naked and Covered Positions
Naked position
• If stock price < $50 at expiry, bank profits
• If stock price > $50, bank must pay prevailing
stock price x 100,000, no limit to loss
Covered position
• If stock price > $50 at expiry, bank profits
• If stock price < $50, bank loses on stock
position
Stop-Loss Strategy
 Ensures covered position if option closes in-themoney, naked position if it closes out-of-themoney
• Buy 100,000 shares if price rises over $50
• Sell 100,000 shares if price falls below $50
 Cost of hedge would always be less than BlackScholes price, leading to riskless profit
• Ignores time value of money
• Purchases and sales will not be made at K exactly
Delta Hedging
 Delta (D) is the rate of change of the option price
with respect to the underlying asset price
Option
price
Slope = D
B
A
Stock price
Delta Hedging
Suppose that:
• Stock price is $100, option price is $10,
D = 0.6
• Trader sells 20 calls on 2,000 shares
How can the trader employ delta hedging?
• Buy 0.6 x 2,000 = 1,200 shares
• Option D is 0.6 x –2,000 = –1,200
• Overall D is 0 (delta neutral)
European Stock Options
Call on non-dividend paying stock:

D = N(d1)
Put on non-dividend paying stock:

D = N(d1) – 1
Call on stock paying dividends at rate q

D = N(d1)e–qT
Put on stock paying dividends at rate q

D = e–qT [N(d1) – 1]
Effect of Dividends
Suppose that:
• A bank has sold six-month put options on £1m
with strike price of 1.6000
• Current exchange rate is 1.6200, UK r = 13%,
US r = 10%, volatility of sterling is 15%
How can the bank construct a delta neutral
hedge?

P
A
Effect of Dividends
Put option D = -0.458
• Exchange rate rises by DS, price of put falls by
45.8% of DS
• Bank must add £458,000 to its position to
make it delta neutral
Note that deltas on a portfolio are a
weighted average of individual derivative
deltas
Delta of Futures
Futures price on non-dividend paying
stock is S0erT
When stock price changes by DS, futures
price changes by DSerT
• Marking-to-market ensures investor realises
profit/loss immediately, thus D = erT
• With dividends, D = e(r-q)T
• Not the case with forwards
Delta of Futures
To achieve delta neutrality
• HF = e–rT HA
• HF = e–(r–q)T HA (with dividends)
• HF = e–(r–rf)T HA (with currency futures)
From earlier example, hedging using ninemonth futures requires short position of:
• e–(0.10–0.13)9/12 x 458,000 = £468,442
• Each contract = £62,500, no. of contracts = 7
Theta
Theta (Q) is the rate of change of value
with respect to time
Usually negative for an option
For a call option, theta is
• Close to zero when the stock price is very low
• Large and negative when at-the-money, and
approaches –rKe-rT as stock prices gets larger
Useful as a proxy for gamma
Gamma
Gamma (G) is the rate of change of delta
(D) with respect to the price of the
underlying asset
Small gamma implies less frequent
rebalancing required
Sensitivity of value of portfolio to DS
Delta, Theta, Gamma are connected
Gamma
Call
price
C′′
C′
C
Stock price
S
S′
Gamma Neutrality
Suppose that:
• Delta neutral portfolio has a gamma of –3,000
• Delta and gamma of an option are 0.62 and
1.50 respectively
How can this portfolio be made gamma
neutral?
Gamma Neutrality
Adding wT options with gamma GT to a
portfolio with gamma G gives wT GT + G
wT must therefore be –G/ GT
Include long position of 3,000/1.5 = 2,000
call options
Delta will change from zero to 2,000 x 0.62
= 1,240
Sell 1,240 units of the underlying asset
Theta as a Proxy for Gamma
For a delta neutral portfolio,
DP  QDt + ½GDS 2
DP
DP
DS
DS
Positive Gamma
Negative Gamma
Vega
Vega (n) is the rate of change of the value
of a derivatives portfolio with respect to
volatility
High vega implies high portfolio sensitivity
to small changes in volatility
To ensure both gamma and vega
neutrality, at least two derivatives must be
used
Vega Neutrality
Suppose that:
• A delta neutral portfolio has gamma of –5,000
and vega of –8,000
• Option 1 has gamma of 0.5, vega of 2.0, delta
of 0.6
• Option 2 has gamma of 0.8, vega of 1.2, delta
0.5
How can this portfolio be made gamma,
vega and delta neutral?
Vega Neutrality
Simultaneous equations
• –5,000 + 0.5w1 + 0.8w2 = 0
• –8,000 + 2.0w1 + 1.2w2 = 0
w1 = 400, w2 = 6,000
Delta = 400 x 0.6 + 6,000 x 0.5 = 3,240
Vega is always positive for a long position
in either European or American options
Rho
 Rho is the rate of change of the value of a
derivative with respect to the interest rate
 Long calls and short puts have positive Rhos
(increase in interest rate would mean increase in
call premium)
 Rho becomes more significant the longer the
time remaining to expiry of the options
 For currency options there are two rhos
corresponding to the two different interest rates
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
Synthetic Positions
Strategy I: investing part of portfolio into
the risk-free asset
Recall, for a put, D = e– qT [N(d1) – 1]
• Ensure that at any time, a proportion of e– qT
[N(d1) – 1] stocks in the portfolio has been
sold and invested into riskless assets
Synthetic Positions
Suppose that:
• A portfolio is worth $90m
• Six-month European put is required with strike
price of $87m
• rf is 9%, dividend yield is 3%, volatility is 25%
p.a.
How can the option be synthetically
created?
Synthetic Positions
D = –0.3215
32.15% of portfolio should be sold initially
and invested into riskless assets
If portfolio value falls to $88m after one
day, delta becomes –0.3679 and further
4.64% should be sold
Synthetic Positions
Strategy II: use index futures
Using previous example:



D = eq(T–T*)e–rT* [N(d1) – 1] A1/A2
T = 0.5, T* = 0.75, A1 = 100,000, A2 = 250,
d1 = 0.4499
D = 122.95, ≈ 123
123 futures contracts should be shorted