The Greek Letters
Chapter 17
1
Goals



OTC risk management by option market
makers may be problematic due to
unique features of the options that are
not available on exchanges.
Many dimensions of risk (greeks) must
be managed so that all risks are
acceptable.
Synthetic options and portfolio insurance
2
Delta

Delta (D) is the rate of change of the option price
with respect to the underlying:
D=f/S

Remember, this creates a hedge (replication of
option payoff):
D*S - f = 0

Two ways to think about it:
 If S =$1 => f = $D
 If short 1 option, then need to buy D shares to
hedge it
3
Example

Let Dc=0.7. What does this mean?
-c + 0.7S = 0
 Short call can be hedged by buying
0.7 shares or
 For $1 change in S, C changes by
70 cents.

4
Example: Cost of hedging

Assume the following conditions:






r = 10%,
T = 6 months;
S0 = 20; u=1.1;
d = 0.9;
K = 20.
Compute the cost of hedging a short call
position on a two-period binomial tree
(each period = 3 months).
5
Example: Cost of hedging





Suu=24.2;
Sud=Sdu=19.8;
Sdd=16.2
Cuu=4.2; Cud=Cdu=Cdd=0
Risk-neutral probability:


p = (e0.1/4 – 0.9)/(1.1-0.9)=0.62658
Cu=e-0.1/4*p*4.2 = 2.56667; C0 = 1.5685
6
Example (cont’d)

The delta is changing over time:




Du = (4.2-0)/(24.2-19.8) = 0.95455
D0 = (2.56667-0)/(22-18) = 0.64167
Dd = 0
Assume that the stock ends up going up
twice in a row: uu move
7
Example (cont’d)

Hedging cost can be computed as follows:






t=0 : the risk-free portfolio is - C + 0.64167*S
Cost(t=0) = 0.64267*20 = 12.8334 (borrow it)
t=1: Du = 0.95455 => buy (Du - D0) =
0.31288 more shares
Cost(t=1) = 0.31288*22 = 6.88336
t=2 : option is exercised by the long position
holder
You get:-C + Du*S = 0.95455*24.2 – 4.2
=18.90011 (inflow)
8
Example (cont’d)

Present Value of the costs:
PVt=0 = 12.8334 + 6.88336e-0.1/4
– 18.90011e-0.1/2 = 1.5685 = call premium


In practice, transaction costs and
discreteness of hedging lead to imperfect
hedging
9
Example (cont’d)

What if the path is du?



Dd = 0
No shares needed to hedge the short call as
there is no uncertainty about its value any
more (it is OTM).
At home confirm that in this case, i.e., du
path, the cost of hedging is also equal to
call premium and that it does not depend
on the path of the underlying asset.
10
Example (cont’d)
What if the stock pays a dividend yield
q = 4%? Tree is the same. What are
the D’s now?
 Need to take into account both the
cap. gains and the dividends now:


- Cu + Du Su
-Cuu + Du *Suueqt
Cud + Du *Sudeqt
11
Example (cont’d)

Thus,
Du  e
 qt
C
0.04 / 4
e
0.95455 
S
 0.94505

Delta is smaller. Is option price
bigger or lower in this case? What
about costs of hedging?
12
Deltas for Currency and Futures
Options
Let f be the price of an option.
 Currency options:

Du  e

 r f t
f
S
Futures options (F is the futures
price):
Du  e
 rt
f
F
13
Two Special Cases

Forwards:
S0
Su
Su – Ke-r(T-t)=fu
Sd
Sd – Ke-r(T-t)=fd


For DS – f to be a hedge, D = 1(perfect
short hedge)
It is all cap gains here, no income.
14
Two Special Cases cont’d

Futures:
S0
Su
Fu – F0
Sd
Fd – F0


For DS – f to be a hedge, D = er(T-t)
(perfect short hedge); can ignore t, if
small.
It is all income here, no cap gains.
15
Using Futures for Delta
Hedging


Futures can be used to make an option position Dneutral.
The delta of an option contract on a dividend-paying
asset is e-(r-q)T times the delta of a spot contract:
c c S c ( r  q )T


e
f
S f
S

The position required in futures for delta hedging is
therefore e-(r-q)T times the position required in the
corresponding spot contract
16
Using Futures for Delta
Hedging

Example: hedging a currency option on £ requires a
short position in £458,000. If r=10% and rf = 13%,
how many 9-month futures contracts on £ would
achieve the same objective?
c
(.1.13) 9 / 12
 458,000e
 468,442
f

One contract is for £62,500 => need to sell about 7
contracts
17
Theta

Theta (Q) of a derivative (or portfolio of
derivatives) is the rate of change of the
value with respect to the passage of time
c
Q
0
t
18
Gamma


Gamma (G) is the rate of change of
delta (D) with respect to the price of
the underlying asset
In our previous example:
D D u  D d 0.95455  0
G


 0.23864
S Su  Sd
22  18

Assets linear in S (futures, stock) do
not affect Gamma.
19
Gamma Addresses Delta Hedging
Errors Caused By Curvature
Call
price
C’’
C’
C
Stock price
S
S
’
20
Interpretation of Gamma

For a delta neutral portfolio,
P  Q t + ½GS
P
2
P
S
S
Positive Gamma
Negative Gamma
21
Vega


Vega (n) is the rate of change of the
value of a derivatives portfolio with
respect to volatility
See Figure 17.11 for the variation of n
with respect to the stock price for a call
or put option
22
Managing Delta, Gamma, &
Vega

D can be changed by taking a position in the underlying
or the futures on it

To adjust G & n it is necessary to take a position in an
option or other derivative

The Greeks of a portfolio are computed as follows:
D
N
n D
i 1
i
i
23
Delta and Gamma of a Protective
Put


Protective put = long in the put and the
stock. The put has a delta of -0.4. Let a call
on the stock have a delta of 0.6 and a
gamma of 2.
Delta of the protective put:



D = Dp + DS = -0.4+1 = 0.6 (why is it equal to
call delta?)
Gamma: G = 2+0=2
You cannot change gamma buy trading shares
24
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
rho’s
25
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
26
Hedging vs Creation of an
Option Synthetically


When we are hedging we take
positions that offset D, G, n, etc.
When we create an option
synthetically we take positions that
match D, G, & n
27
Portfolio 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
D of the put option
In our earlier example: at t=0 we need to
sell -Dp = 1 – 0.64167 = 0.35833 =
35.83% of the portfolio and invest the
proceeds into riskless assets.
28
Portfolio Insurance


Can achieve the same with index
futures. One benefit is lower
transactions costs.
Recall
p
( r  q ) T p
e
f
S

Remember, T is the maturity of the
futures.
29
Portfolio Insurance
continued


As the value of the portfolio increases, the
D of the put becomes less negative and
some of the original portfolio is
repurchased
As the value of the portfolio decreases,
the D of the put becomes more negative
and more of the portfolio must be sold
30
Portfolio Insurance
continued
The strategy did not work well on October
19, 1987...
31