### Greeks- Theory and Illustrations

```Greeks : Theory and Illustrations
By A.V. Vedpuriswar
June 14, 2014
Introduction
Greeks help us to measure the risk associated with derivative
positions.
Greeks also come in handy when we do local valuation of
instruments.
But Greeks are not useful to get an aggregated view of risk.
1
Delta
 Delta is the rate of change in option price with respect to the
price of the underlying asset.
 It is the slope of the curve that relates the option price to the
underlying asset price.
A position with Delta of zero is called Delta neutral.
Since Delta keeps changing, the investor’s position may remain
delta neutral for only a relatively short period of time.
 The hedge has to be adjusted periodically.
 This is known as rebalancing.
The delta of European call option is N(d1) in the Black Scholes
equation.
The delta of a European put option is N(d1) – 1 in the Black
Scholes equation.
2
Gamma
 The gamma is the rate of change of the portfolio’s
respect to the price of the underlying asset.
delta with
It is the second partial derivative of the portfolio price with
respect to the asset price.
If gamma is small, it means delta is changing slowly.
relatively infrequently.
However, if gamma is large, the delta is highly sensitive to the
price of the underlying asst.
It is then quite risky to leave a delta neutral portfolio
unchanged for any length of time.
3
Theta
Theta of a portfolio is the rate of change of value of the
portfolio with respect to change of time.
Theta is also called the time decay of the portfolio.
Theta is usually negative for an option.
As time to maturity decreases with all else remaining the
same, the option loses value.
4
Vega
 Vega is the rate of change of the value of the portfolio with
respect to the volatility of the underlying asset.
 High Vega means high sensitivity to small changes in
volatility.
 A position in the underlying asset has zero Vega.
 The Vega can be changed by adding options.
To make the portfolio Gamma and Vega neutral, two traded
derivatives dependent on the underlying asset are needed.
5
Rho
 Rho of a portfolio of options is the rate of change of value of
the portfolio with respect to the interest rate.
If interest rate increases, value of call increases. Why?
6
Problem
A bank has a \$ 25 million par position in a 5 year zero coupon
bond with a market value of \$ 19,059,948. What is the
modified duration of the bond?
19,059,948=25,000,000/[(1+r)^5]
r = .0558
Modified duration = 5/{1 + .0558/2} = 4.86 years
7
Problem
 An investor holds the following bonds in her portfolio. Calculate the duration.
 \$ 2,000,000 par value of 10 year bonds, duration of 6.95 ,price 95.5
 \$3,000,000 par value of 15 year bonds, duration of 9.77, price 88.6275
 \$ 5,000,000 par value of 30 year bonds, duration of 14.81, price 115.875
 Market value of Bond 1 = 2,000,000 x .955
= 1,910,000, weight = .19
 Market value of Bond 2 = 3,000,000 x .886275 = 2, 658,825, weight = .26
 Market value of Bond 3 = 5,000,000 x 1.15875 = 5,793,750, weight = .56
 Portfolio duration = 6.95x.19 + 9.77x .26 + 14.81x.56 = 12.15
8
Problem
If all the spot interest rates are increased by one basis point,
the value of a portfolio of swaps will increase by \$ 1100. How
many Euro dollar futures contracts are needed to hedge the
portfolio?
A Eurodollar contract has a face value of \$ 1 million and a
maturity of 3 months. If rates change by 1 basis point, the
value changes by (1,000,000) (.0001)/4= \$ 25.
So the number of futures contracts needed = 1100/25=44
9
Problem
A bank has sold USD 300,000 of call options; with strike price
of 50 on 100,000 shares currently trading at 49.5.How should
the bank do delta hedging?
Current delta = -.5x 300,000 + 100,000 = - 50,000
So she must buy 50,000 shares.
10
Problem
Suppose an existing short option position is delta neutral and
has a gamma of －6000. Here, gamma is negative because
we have sold options. Assume there exists a traded option
with a delta of 0.6 and gamma of 1.25. Create a gamma and
delta neutral position.
Solution
To gamma hedge, we must buy 6000/1.25 = 4800 options.
Then we must sell (4800) (.6) = 2880 shares to maintain a
gamma neutral and original delta neutral position.
11
Problem
A delta neutral position has a gamma of －3200. There is
an option trading with a delta of 0.5 and gamma of 1.5.
How can we generate a gamma neutral position for the
existing portfolio while maintaining a delta neutral
hedge?
Solution
=
2133 options
Sell (2133) (.5)
=
1067 shares
12
Problem
 Suppose
a
portfolio
is
delta
= - 5000 and vega = - 8000. A traded option has
delta = 0.6. How do we achieve vega neutrality?
neutral,
with
gamma
gamma = .5, vega = 2.0 and
To achieve Vega neutrality we can add 4000 options.
 Delta increases by (.6) (4000) = 2400
So we sell 2400 units of asset to maintain delta neutrality.
As the same time, Gamma changes from – 5000 to
(.5) (4000) – 5000 = - 3000.
 If there is a second traded option with gamma = 0.8, vega = 1.2, delta = 0.5.
 if w1 and w2 are the weights in the portfolio,

 w1 = 400
- 5000 + 0.5w1 + 0.8w2 = 0
- 8000 + 2.0w1 + 1.2w2 = 0
w2 = 6000.
 This makes the portfolio gamma and vega neutral.
 But delta = (400) (.6) + (6000) (.5) = 3240
 3240 units of the underlying asset will have to be sold to maintain delta neutrality.
Ref : John C Hull, Options, Futures and Other Derivatives,
13
The Black Scholes Model and the Greeks
For a European call option on a non dividend paying
stock,
Delta =
N(d1)
For Put,
Delta =
N(d1) -1
For a dividend paying stock,
For Call,
Delta =
e-qt N(d1)
For Put,
Delta =
e-qt [N(d1) – 1]
14
The Black Scholes Model and the Greeks
For a European call or put option on a non dividend
paying stock,

e d1 2 2 

S 0 T 2 
1
Gamma
=
 For a European call or put option on a dividend paying
2
stock,
 qT  d1 2
e e

Gamma
= S 0 T 2
15
Problem
Stock price = 49
Strike price = 50; Volatility = 20%
Risk free rate = 5%; Time to exercise = 20 weeks
Using Deriva Gem spreadsheet, we get :
Call option price
= 2.40
Delta
= .522/\$
Gamma
= .066/\$/\$
Vega
= .121/%
Theta
= -.012/day
Rho
= .089/%
16
Problem
 Strike price = 25; Risk free rate of interest = 6%
 Time to maturity = 0.5 years; Stock volatility = 30%
 Establish the relationship between option price, delta, gamma and underlying
price.
Stock price
Call price
Intrinsic
value
Delta
Gamma
10
.00001
0
0
0
15
.01626
0
0.0153
.0121
20
.45875
0
0.2106
.0680
25
2.47066
0
.5977
.0730
27.5
4.17428
2.5
.757
.0536
30
6.21317
5
.8658
.0340
32.5
8.46707
7.5
.9311
.0192
35
10.844409
10.0
.9666
.0100
40
15.75
15
.9931
.0023
45
20.7425
20
.9987
.0004
17
Problem
 Calculate the delta of an at-the-money 6-month European call option on a nondividend-paying stock when the risk-free interest rate is 10% per annum and the
stock price volatility is 25% per annum.
In this case S0 = K, r = 0.1, σ = 0.25, and T = 0.5. Also,
ln(S0 / K )  (0.1  0.252 / 2)0.5
 0.3712
 d1 
0.25 0.5
The delta of the option is N(d1) or 0.6450.
We can also calculate using Deriva Gem.
Ref : John C Hull, Options, Futures and Other Derivatives,
18
Problem
 What is the delta of a short position in 1,000 European call options on silver
futures? The options mature in 8 months, and the futures contract underlying the
option matures in 9 months. The current 9-month futures price is \$8 per ounce,
the exercise price of the option is \$8, the risk-free interest rate is 12% per
annum, and the volatility of silver is 18% per annum.
 The delta of a European futures option is usually defined as the rate of change
of the option price with respect to the futures price (not the spot price).
 It is
e-rT N(d1)
 In this case F0 = 8, K = 8, r = 0.12, σ = 0.18, T = 0.6667
ln(8 / 8)  (0.182 / 2)  0.6667
d1 
 0.0735
0.18 0.6667
 N(d1) = 0.5293 and the delta is e-0.12x0.6667 x 0.5293 = 0.4886
The delta of a short position in 1000 futures options is therefore -488.6.
Ref : John C Hull, Options, Futures and Other Derivatives,
20
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