Derivatives
Financial products that depend on
another, generally more basic,
product such as a stock
Examples
•
•
•
•
Forward contracts
Futures
Options
Swaps
Forward contracts
• A agrees to buy and B agrees to sell an asset
– at specific price, K (forward price)
– On specific date, T (delivery date)
• A takes ‘long position’
• B takes ‘short position’
• Agricultural crops and commodities
But price fluctuates in the future
time!
• Actual price at time T:Y(T) and K can differ
• Payoff can be both positive and negative
– Whatever is gained by A is lost by B
– And vice versa
Y(T)
Y(T)
Short position
of seller
K
Long position of buyer
K
Options
• Gives holder right to exercise a given action
– Buy (or sell) underlying asset
– At time T
• exercise date; expiration date; date of maturity
– for price K
• Strike price; exercise price
• European option
– can only be exercised at maturity (t= T)
• American option
– can be exercised anytime between t=0 & t=T
• Exotic options
– Even more complicated boundary conditions!
European Calls and Puts
– Call option
• A has right (but not obligation) to buy underlying asset at
strike price at maturity
• B is obliged to sell
• Secured by paying fee C(Y,t) to B (via option dealer)
– Unlike future: No symmetry between buyer and
seller
• Buyer pays C at time t=0 and acquires right to buy at t=T
• Seller receives cash but faces potential liabilities at t=T
Payoff for European call options
Y(T)
C(Y,0)
K
Payoff for buyer of call option
C(Y,0)
Y(T)
K
Payoff for seller of call option
Put options
• Buyer has right to sell
underlying asset at strike
price, K at maturity time, T
Y(T)
C(Y,0)
K
Payoff for buyer of put option C(Y,0)
Y(T)
K
Payoff for seller of put option
?
• How much would one pay for the option?
• How can the seller minimise the risk associated
with his obligation?
Simple example
• Suppose in 3 months time using a call option,
may purchase 1 share in Acme Ltd for 2.50
• Scenario 1
– In 3 months Acme Ltd trades at 2.70
• Exercise option: buy for 2.50; sell at 2.70; profit is .20
• Scenario 2
– In 3 months Acme Ltd trades at 2.30
• Let option lapse
Assume 2 equal probability scenarios
• Expected profit is
– ½*0 +½*20 = 10
– Ignore interest rate effects reasonable to assume
value of option is 10
• Scenario 1
– Profit on exercise:
– cost of option:
– Net profit:
20
(10)
10
• gain100%
• Scenario 2
– Cost of option:
– Net loss
• Loss 100%
(10)
(10)
But suppose buy shares
• At T = 0 share price 250
– Buy 1 share
• Scenario 1
– Sell at 270;
– profit 20;
• Gain 20/250*100 = +8%
• Scenario
– Sell at 230;
– loss 20;
• Loss -8%
• Options respond in more exaggerated way
• More highly geared
• Used for speculating/ gambling and insurance
Put options
• Allows holder to sell asset at prescribed
price
• strike or exercise price
• Holder of ‘calls’ hopes asset price will rise
• Holder of ‘puts’ hopes price to fall
– Can also use as insurance against fall of prices
in portfolio
Hedging - a form of insurance
• Say UK company must pay S = 10000 euro to
Irish firm in 180 days
• 1 can write forward contract at present exchange rate for S
• 2 Buy call option for given strike price at 180 days maturity
– Eliminates risk associated with exchange rate
fluctuations
– Risk is
• exposure to losses in forward contract
• Or cost of option contract
Equity options
(Financial Times 22 November 2003)
Option
Br Air
214
200
220
………….
Jan
21.25
11.75
Centrica
190
180
200
14.4
4.5
20
10
22
11
3.5
13.5
7.5
17
10.5
19.5
Option
ARM
109
100
110
Dec
12
6
Mar
18
12
Jun
22
17
Dec
2.5
6.5
Mar
8
12
Jun
11
16
Unilever
509
500
550
14.5
0.5
30
8
38.5
15
4
41
15.5
44.5
26.5
55
Calls
Apr
32.25
22
…………. ………….
Jun
Jan
36.26
8.25
25.75
16.75
Puts
Apr
14.5
24
…………
Jun
20.25
30.25
Extent of trade in calls and puts
(vanilla options)
• ~ $10,000 billion worldwide
• In late 1992 Citicorp alone had contracts
totalling ~ $1426 billion
• May in some markets have a value greater
than the underlying asset
• In some cases, options are more liquid than
actual asset
FTSE 100 index option (Euronext.liffe)
£10 per full index point
4025
4125
4225
C
P
C
Nov
287
0.25
187
Dec
310
17
220
27
139
Jan
333
30.5
250
46.5
Feb
362
50.5
282
Mar
361
76
286
4625
4725
P
C
P
C
P
0.25 13.5 0.25
114
0.25
214
0.25
314
0.25
414
45
73.5
80
31
137
11.5
218
4
310
1
407
174
70
109
105
60.5
155
31.5
226
15.5
310
6.5
400
69
209
95
145
131
94
179
56.5
241
31.5
314
17.5
400
100
219
132
159
171
111
221
71.5
281
45
353
26.5
433
0.25 86.5 0.25
P
Y(21 November)=4319
C
4525
C
P
C
4425
P
P
C
4325
November 21
FTSE Index Call Options v Exercise Price
(November 2003)
At expiry
C = -(K-4325) K<Y(T)
C = 0 K>Y(T)
400
350
Option Price
300
November
250
December
200
January
150
February
100
March
50
0
-503800
4000
4200
4400
4600
4800
Exercise Price
Y(T)=4325
Option values at expiry
(FTSE = 4319 at 21 Nov 2003 - 3rd Friday in month)
‘Risk-less’ portfolio
(Binomial model)
Hold  h shares for option of price C
Y; C 
Y; C
Value of portfolio at t  0 :   Y  h  C
Riskless investment requires for t  t1:   
t=0
t=t1
time
Y; C
Y  h  C  Y  h  C
C  C  C 
h 
 
Y  Y
 Y t
• Y changes with time, hence h must also be changed to
maximise hedging process and minimise risk
Rational and fair price
• Holder of shares selling a derivative of stock at time, t
C
Value of portfolio  
Y C
Y
C
 
Y  C
Y
• Change must equal gain obtained by investing in
riskless security (eg cash)
  rt
Differential v stochastic calculus
C
C
 y  a( y, t ) t  dC ( y, t ) 
dy 
dt
y
t
dY  a (Y , t )dt  b(Y , t )dW
C
C
1  2C
2
 C 
Y 
t 
Y   .......
2 
Y
t
2 Y
For Gaussian processes:  W  ~ t
2
Ito's theorem:
C
C
 2C 2
C 
Y 
t 
b (Y , t )t
2
Y
t
2Y
Assumptions of Black Scholes 1973
Brownian / Weiner random walk
Gaussian fluctuations
dY  Ydt   YdW
Ito

C C ( Y ) 2  2C 
C
dC   Y


dt 
 YdW
2
Y t
2 Y 
Y

C
Y C
Y
C
 
Y  C
Y
  rt

Black Scholes SPDE
Y  Y t   Y W

C C ( Y ) 2  2C 
C
C   Y



t

 Y W
2
Y t
2 Y 
Y

C
C ( Y )  C
rC 
 rY

2
t
Y
2 Y
2
2
• Valid for all types of options
• Choice of solution determined by boundary
conditions
Boundary conditions: Call option
C(Y, K, T) = Y(T)-K if Y>K
C(Y, K, T) = 0 if Y<K
C = max{Y-K, 0}
C(Y,K,T)
K
Y(T)
Relation to heat transfer equation
C (Y , t )  e r (t T ) y ( x, t ')

 2 
2   2 
x  2 r 
ln(
Y
/
K
)

r

(
t

T
)




 
2 
2



2  2 
t'   2 r 
 (t  T )
 
2 
y ( x, t ')  y ( x, t ')

2 2
t '
 x
2
C (Y , t )  YN (d1 )  Ke r (t T ) N (d 2 )
ln(Y / K )  (r   2 / 2)(T  t )
d1 
 T t
d 2  d1   T  t
1
N ( x) 
2
x
e

x '2

2
dx '
20
Call Price
15
10
T=.001Yr
5
T=.25Yr
0
-5
0
50
100
Strike Price
(Risk free rate 3%; volatility 80%)
150
Solution for puts
C  P  Y  Ke
 P  Ke
 r ( T t )
 r ( T t )
N (d 2 )  YN (d1 )
Problems
• Interest rates, r may vary
• Volatility, σ is not constant
– Fat tails, not a Gaussian
• Historical volatility
– Over what time period?
• Implied volatility
– Use Black Scholes formula in inverse sense to compute
volatility given set of C values (For BS it would be
constant
– Gives indication of level of volatility expected by
market traders