1. Introduction to Derivatives
Dr. Islam Azzam Assistant Professor of Finance Department of Management American University in Cairo

2. Examples of Derivatives
Futures Contracts
Forward Contracts
Options 3

3. Terminology Call Put Long Right to buy Pay premium Can Cancel
Right to sell
Short
Obligation to sell
Receive premium
Obligation to buy 10

4. Terminology Future/Forward Spot Long Obligation to buy
Buying the stock
Short
Obligation to sell
Short Selling 10

5. Long Call on eBay
Profit from buying one eBay European call option: option price = $5, strike price = $100, option life = 2 months 30 20 10 -5 70 80 90
100
110
120
130
Profit ($)
Terminal
stock price ($)

6. Short Call on eBay
Profit from writing one eBay European call option: option price = $5, strike price = $100
-30
-20
-10 5 70 80 90
100
110
120
130
Profit ($)
Terminal
stock price ($)

7. Long Put on IBM
Profit from buying an IBM European put option: option price = $7, strike price = $70 30 20 10 -7 70 60 50 40 80 90
100
Profit ($)
Terminal
stock price ($)

8. Short Put on IBM
Profit from writing an IBM European put option: option price = $7, strike price = $70
-30
-20
-10 7 70 60 50 40 80 90
100
Profit ($)
Terminal
stock price ($)

9. American vs European Options
An American option can be exercised at any time during its life
A European option can be exercised only at maturity

10. Intel Option Prices (May 29, 2003; Stock Price=20.83)
Strike Price
June Call
July Call
Oct Call
June Put
July Put
Oct Put
20.00
1.25
1.60
2.40
0.45
0.85
1.50
22.50
0.20
1.15
1.85
2.20
2.85

11. Options vs Futures/Forwards
A futures/forward contract gives the holder the obligation to buy or sell at a certain price
An option gives the holder the right to buy or sell at a certain price

12. Derivatives Markets Exchange traded Over-the-counter (OTC)
Traditionally exchanges have used the open-outcry system, but increasingly they are switching to electronic trading
Contracts are standard there is virtually no credit risk
Over-the-counter (OTC)
A computer- and telephone-linked network of dealers at financial institutions, corporations, and fund managers
Contracts can be non-standard and there is some small amount of credit risk

13. Forward Contracts vs Futures Contracts
FORWARDS
FUTURES
Private contract between 2 parties
Exchange traded
Non-standard contract
Standard contract
Usually 1 specified delivery date
Range of delivery dates
Settled at end of contract
Settled daily
Delivery or final cash
Contract usually closed out
settlement usually occurs
prior to maturity
Some credit risk
Virtually no credit risk 16

14. Futures Contracts Available on a wide range of underlyings
Exchange traded
Specifications need to be defined:
What can be delivered,
Where it can be delivered, &
When it can be delivered
Settled daily

15. Margins
A margin is cash or marketable securities deposited by an investor with his or her broker
The balance in the margin account is adjusted to reflect daily settlement
Margins minimize the possibility of a loss through a default on a contract

16. Ways Derivatives are Used
To hedge risks
To speculate (take a view on the future direction of the market)
To lock in an arbitrage profit 4

17. Optimal Hedge Ratio
Proportion of the exposure that should optimally be hedged is
where
sS is the standard deviation of DS, the change in the spot price during the hedging period,
sF is the standard deviation of DF, the change in the futures price during the hedging period
r is the coefficient of correlation between DS and DF.

18. Hedging Using Index Futures
To hedge the risk in a portfolio the number of contracts that should be shorted is
where P is the value of the portfolio, b is its beta, and A is the value of the assets underlying one futures contract

19. Reasons for Hedging an Equity Portfolio
Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.)
Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will outpeform the market.)

20. Example Value of S&P 500 is 1,000 Value of Portfolio is $5 million
Beta of portfolio is 1.5
What position in futures contracts on the S&P 500 is necessary to hedge the portfolio?

21. Determination of Forward and Futures Prices

22. Pricing Forward/Futures when Interest Rates are Measured with Continuous Compounding
F0 = S0erT
This equation relates the forward price and the spot price for any investment asset that provides no income and has no storage costs

23. When an Investment Asset Provides a Known Dollar Income
F0 = (S0 – I )erT
where I is the present value of the income during life of forward contract

24. Futures and Forwards on Currencies
A foreign currency is analogous to a security providing a dividend yield
The continuous dividend yield is the foreign risk-free interest rate
It follows that if rf is the foreign risk-free interest rate

25. Options Strategies

26. Types of Strategies Take a position in the option and the underlying
Take a position in 2 or more options of the same type (A spread)
Combination: Take a position in a mixture of calls & puts (A combination)

27. Positions in an Option & the Underlying
Long Stock
Profit
Profit
Long Call K K ST ST
(a)
Short Call
(b)
Short Stock
Profit
Profit
Long Stock
Short Put K ST K ST
Long Put
(c)
(d)
Short Stock

28. Bull Spread Using Calls
Long Call at K1
Profit ST K1 K2
Short Call at K2

29. Bull Spread Using Puts K1 K2
Profit ST
Short Put at K2
Long Put at K1

30. Bear Spread Using Calls
Profit
Long Call at K2 K1 K2 ST
Short Call at K1

31. Bear Spread Using Puts K1 K2
Profit ST
Short Put at K1
Long Put at K2

32. Butterfly Spread Using Calls
Long Call at K1
Profit
Long Call at K3 K1 K2 K3 ST
Short 2 Calls at K2

33. Butterfly Spread Using Puts
Profit K1 K2 K3 ST
Long put at K1
Long put at K3
Short 2 puts at K2

34. A Straddle Combination
Long Call at K
Profit K ST
Long put at K

35. A Strangle Combination
Profit
Long a Call at K2 with same T K1 K2 ST
Long a put at K1 with same T

36. Binomial Trees

37. Generalization
A derivative lasts for time T and is dependent on a stock
S0u ƒu
S0d ƒd S0 ƒ

38. Generalization
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
where

39. Choosing u and d One way of matching the volatility is to set
where s is the volatility of the stock price and Dt is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein

40. p as a Probability S0u ƒu p S0 ƒ S0d (1 – p ) ƒd
It is natural to interpret p and 1-p as probabilities of up and down movements
The value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate
S0u ƒu
S0d ƒd S0 ƒ p
(1 – p )

41. Example S0u = 22 ƒu = 1 p S0 ƒ S0d = 18 (1 – p ) ƒd = 0
A stock price is currently $20 and it is known that at the end of 3 months it will be either $22 or $18. We are interested in valuing an option to buy the stock for K= $21 in 3 months. (u = 1.1 & d = 0.9 & r = 0.12)
Alternatively, we can use the formula p S0 ƒ
S0d = 18
ƒd = 0
(1 – p )

42. Valuing the Option Using Risk-Neutral Valuation
The value of the option is
e–0.12´0.25 (0.6523´ ´0)
= 0.633
S0u = 22
ƒu = 1
S0d = 18
ƒd = 0 S0 ƒ
0.6523
0.3477

43. Valuing a Call Option
Value at node B = e–0.12´0.25(0.6523´ ´0) =
Value at node A = e–0.12´0.25(0.6523´ ´0) =
24.2
3.2 D 22 B 20
1.2823
19.8
0.0
2.0257 A E 18 C
0.0
16.2
0.0 F

44. A Put Option Example; K=52
K = 52, a 2 year European Option with time step = 1yr
r = 5%, u = 1.2, d = 0.8
Market price = 50 50
4.1923 60 40 72 48 4 32 20
1.4147
9.4636 A B C D E F

45. What Happens When an Option is American50
5.0894 60 40 72 48 4 32 20
1.4147
12.0 A B C D E F

46. V = P[N(d1)] - Xe -rRFt[N(d2)]. d1 = .  t d2 = d1 -  t.
Black-Scholes Model
Call Option
P: Spot Price
X: Strike Price
r = Risk-Free Rate
t = Time to Maturity
σ = Risk of the stock
V = P[N(d1)] - Xe -rRFt[N(d2)].
d1 =
 t
d2 = d1 -  t.
ln(P/X) + [rRF + (2/2)]t

47. V = Xe -rRFt[N(-d2)] - P[N(-d1)] d1 = .  t d2 = d1 -  t.
Put Option
Black-Scholes Model
V = Xe -rRFt[N(-d2)] - P[N(-d1)]
d1 =
 t
d2 = d1 -  t.
ln(P/X) + [rRF + (2/2)]t

48. What is the value of the following
call option? Assume: P = $27; X = $25; rRF = 6%; t = 0.5 years: 2 = 0.11
V = $27[N(d1)] - $25e-(0.06)(0.5)[N(d2)].
ln($27/$25) + [( /2)](0.5)
(0.3317)(0.7071) =
d2 = d1 - (0.3317)(0.7071) = d
= =
d1 =

49. N(d1) = N(0.5736) = =
N(d2) = N(0.3391) = =
Note: Values obtained from Excel using NORMSDIST function.
V = $27(0.7168) - $25e-0.03(0.6327)
= $ $25( )(0.6327)
= $ 16