derivative introduction

1. 1
Introduction to
Derivatives

2. What are derivatives?
Derivatives are financial instruments
whose prices depend on, or are
derived from, the prices of other
assets.

3. What are the assets on which
their prices depend?
 Underlying is financial in nature
 Stock prices
 Credit rating
 Interest rates
 Exchange rates
 Also..
 Electricity
 The weather
 Insurance
 Cattle prices
In sum, if you can price the underlying asset, there can
always be a derivative on it.

4. Why are derivatives useful?
 Hedging
 Speculation
 Arbitrage profit-making
 Balance sheet changes
 To change the nature of a liability
 To change the nature of an
investment/assets

5. The various kinds of derivatives
There are three principal classes of
derivative securities:
 Options
 Futures and Forwards
 Swaps
In addition, it is possible to have options
on futures, futures on options,
swaptions.. Infinite complexity.

6. Forward Contracts
 An agreement to buy or sell an asset
at a certain time in the future for a
certain price
 No daily settlement. When the
contract expires, one party buys the
asset for the agreed price from the
other party.
 The contract is an over-the-counter
(OTC) agreement between 2
institutions.

7. 7
Profit from Forward Positions
Profit
Price of Underlying
at Maturity
Profit
Price of Underlying
at Maturity
Long position Short position

8. 8
Working of a forward contract
Exchange Rate
(INR/USD)
Bid Ask
Spot 39.94 40.11
1-month 39.78 40.02
3-month 39.55 39.94

9. 9
Working of a forward contract
 Suppose Reliance has to make a
payment of $25 M to Total 3 months
from now and wish to lock in an
exchange rate.
 They would enter a 3-month forward
contract to buy $ at the 3-month
forward exchange rate.
 What happens if the exchange rate 3-
months from now is different from
39.55?

10. Futures Contracts
 An agreement to buy or sell an
asset at a certain time in the future
for a certain price.
 They are exchange-traded, and
hence, are standardized contracts.
 Important futures exchanges:
CBOT, CME, NYMEX etc.
 In India: BSE, NSE, MCX, NCDEX.

11. Examples of Futures Contracts
 Agreement to:
 Buy 1000 shares of Reliance at
Rs. 2500 in May (BSE)
 Sell $1 million at 1.5000 US$/£
in april (CME)

12. Futures Arbitrage
Suppose that:
 The spot price of Reliance is
Rs. 1500
 The quoted 1-year futures
price is Rs. 2000
 1-year interest rate is 10%
 Is there an arbitrage
opportunity?

13. Futures Arbitrage
Solution:
 Borrow Rs. 1500 at 10%
 Go long spot
 Short futures
 End of year profit
 = 2000 – 1500*(1+10%)
 What if the futures price
is Rs. 1600?

14. Non-arbitrage futures price
If the spot price is S and futures price is F for a
contract deliverable in T years, then
F = S (1+r )T
If F > S (1+r )T, go short on futures and long
on spot and vice versa.
 What is the value of the futures
contract?

15. Daily settlement and margins
 Margin is cash or marketable
securities deposited by an investor
with the broker
 Marking to market: Balance in the
margin account is adjusted to reflect
daily settlement
 Margins guard against default
 How are margins set?

16. Forward Contracts vs Futures
Contracts
Private contract between 2 parties Exchange traded
Non-standard contract Standard contract
Usually 1 specified delivery date Range of delivery dates
Settled at maturity Settled daily
Delivery or final cash
settlement usually occurs
Contract usually closed out
prior to maturity
FORWARDS FUTURES

17. Options
 An option is a security that gives the holder the
right but not the obligation to buy or sell a
security for a specified price at a specified
date.
 Basic classification of options:
 Call options/Put options
 American options/European options
 How are options different from
futures/forwards?

18. Intrinsic and Time Value
 Option premium = Intrinsic Value + Time Value
 Intrinsic value: payoff if option is exercised
immediately, always greater than or equal to zero.
 Usually the price of an option in the marketplace
will be greater than its intrinsic value. The
difference between the market value of an option
and its intrinsic value is called the time value of an
option.
 What are in-the-money, out-of-the-money and
at-the-money options?

19. Long Call
Profit from buying European call option: option price =
$5, strike price = $100, option life = 2 months
30
20
10
0
-5
70 80 90 100
110 120 130
Profit ($)
Terminal
stock price ($)

20. Short Call
Profit from writing a European call option: option price =
$5, strike price = $100
-30
-20
-10
0
5
70 80 90 100
110 120 130
Profit ($)
Terminal
stock price ($)

21. Long Put
Profit from buying an European put option: option price
= $7, strike price = $70
30
20
10
0
-7
70
60
50
40 80 90 100
Profit ($)
Terminal
stock price ($)

22. Short Put
Profit from writing an European put option: option price
= $7, strike price = $70
-30
-20
-10
7
0
70
60
50
40
80 90 100
Profit ($)
Terminal
stock price ($)

23. Effect of Variables on Option
Pricing
c p C P
Variable
S0
X
T

r
D
+ + –
+
? ? + +
+ + + +
+ – + –
–
– – +
– + – +
• What is the relation between price of an American
option and a European option?

24. Put-Call Parity; No Dividends
 Consider the following 2 portfolios:
 Portfolio A: European call on a stock + PV of
the strike price in cash
 Portfolio C: European put on the stock + the
stock
 Both are worth Max (ST , X) at the maturity of the
options
 They must therefore be worth the same today
 This means that
c + Xe -rT = p + S0

25. Valuation using Black
Scholes equation
c S N d X e N d
p X e N d S N d
d
S X r T
T
d
S X r T
T
d T
rT
rT
 
   

 

 
 


0 1 2
2 0 1
1
0
2
0
1
2 2
2 2
where
( ) ( )
( ) ( )
ln( / ) ( / )
ln( / ) ( / )





•What happens if S is a) very large? b) very small?

26. What is Risk-Neutral
Valuation?
1. Assume that the expected
return from an asset is the
risk-free rate
2. Calculate the expected
payoff from the derivative
3. Discount at the risk-free
rate

27. Naked and Covered Positions
Naked position
Take no action
Covered position
Buy 100,000 shares today
 Both strategies leave the bank
exposed to significant risk.
How?

28. Stop-Loss Strategy
This involves:
 Buying 100,000 shares as soon
as price reaches $50
 Selling 100,000 shares as soon
as price falls below $50
 What is the problem with
this strategy?

29. The Greeks: Delta
 Delta (D) is the rate of change of the
option price with respect to the underlying
Option
price
A
B
Slope = D
Stock price

30. Delta…
 Delta = sensitivity of an option's theoretical
value to a change in the price of the underlying
contract.
delta = change in the option price
change in the stock price
 What is the range of deltas for calls and
puts?
 Why is delta also called the hedge ratio?

31. Theta
 Theta (Q) is the rate of change of the value with respect
to time.
 Gamma (G) is the rate of change of delta (D) with
respect to price of the underlying asset.
 Vega (n) is the rate of change of the value of a
derivatives portfolio with respect to volatility.
 Rho is the rate of change of the value of a derivative
with respect to the interest rate.

32. Managing Delta, Gamma, &
Vega
 Delta, D, can be changed by
taking a position in the
underlying asset
 To adjust gamma, G, and vega, n,
it is necessary to take a position
in an option or other derivative

33. 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

34. Exotic Options
 Bermudan option - non-standard American option in which
early exercise is limited to certain dates during the life of
the option. Also referred to as "hybrid-style" exercise.
 Forward start option is an option that is paid for now, but
does not begin until some later date.
 Compound option is an option on an option. Compound
options have two strike prices and two expiration dates. For
example, a call on a call is purchased. At some specified
date in the future, a person will have the right but not the
obligation of purchasing a call option.

35. Exotic Options
 Chooser option, also called an "as you like it" option, allows
the holder to choose after a specified period of time
whether the option is a call or a put.
 Barrier option is an option in which the payoff depends on
whether the underlying asset's price reaches a certain level
during the life of the option.
 Up-and-out option becomes worthless once the underlying asset price
reaches a specified boundary price.
 Up-and-in option requires the underlying asset price to reach the
boundary price before the option can be activated.
 Rainbow option is an option involving two or more risky
assets.

36. Exotic Options
 Lookback option - payoffs depend on the maximum or
minimum the stock price reaches over the life of the option.
 Asian option (average price option) - payoff depends on the
average price of the asset (not the stock price itself) over a
specified amount of time during the life of the option.
 Spread option - strike price is the spread between two
underlying assets. For example, crack spreads on the
spread between the price of crude and its by-products.
 Basket option - payoff depends upon a portfolio of assets.

37. Terms to watch out for
 Volatility trading, correlation trading
 Delta hedging, Gamma Hedging
 Long, short, spread
 Basis risk
 Libor, yield curve

38. Butterfly Spread Using Calls
 Butterfly Spread: buying a call option with a
relative low strike price, K1, buying a call option
with a relative high strike price. K3, and selling
two call options with a strike price halfway in
between, K2.
Stock price
Range
Payoff
from First
Long Call
Option
Payoff from
Second
Long Call
Option
Payoff from
Short Calls
Total Payoff
ST ≥ K3
K2 < ST < K3
K2 < ST < K3
ST ≤ K1
ST - K1
ST - K1
ST - K1
0
ST - K3
0
0
0
-2(ST - K2)
-2(ST - K2)
0
0
0
K3 - ST
ST - K1
0

39. Butterfly Spread Using Calls
 Example: Call option prices on a $61 stock are: $10 for a $55
strike, $7 for a $60 strike, and $5 for a $65 strike. The investor
could create a butterfly spread by buying one call with $55 strike
price, buying a call with a $65 strike price, and selling two calls
with a $60 strike price.
Stock price
Range
Payoff
from First
Long Call
Option
Payoff from
Second
Long Call
Option
Payoff from
Short Calls
Total Payoff
ST ≥ $65
$60 < ST
<$65
$55 < ST
<$60
ST ≤ $55
ST - $55
ST - $55
ST - $55
0
ST - $65
0
0
0
-2(ST - $60)
-2(ST - $60)
0
0
0
$65 - ST
ST -$55
0

40. Butterfly Spread Using Calls
K1 K3
Profit
ST
K2

41. Butterfly Spread Using Puts
K1 K3
Profit
ST
K2

42. Introduction to financial
swaps
 An asset-liability management technique which permits
a borrower (investor) to access one market and then
exchange the liability (asset) for another type of
liability (asset).
 Swaps are not a funding instrument; they are a device
to obtain the desired form of financing indirectly which
otherwise might be inaccessible or too expensive.
 Swaps may also be used purely for hedging purposes.

43. Major Types of Swap
Structures
 Swaps involve exchange of a series of periodic
payments between two parties, usually through an
intermediary which is normally a large international
financial institution which runs a “swap book”.
 The two major types are Interest Rate Swaps (also
known as Coupon Swaps) and Currency Swaps.
 Major classification of swaps:
 Liability swaps
 Asset swaps

44. A Three Year Fixed-to-Floating Interest
Rate Swap
Notional principal P = $50 million
Trade Date : August 30, 2004
Effective Date : September 1, 2004
Fixed Rate : 9.5% p.a. payable semiannually, Actual/360
Floating Rate : 6 Month LIBOR

45. The fixed payments are as follows :
__________________________________________________
Payment Date Day Count Fraction Amount
__________________________________________________
1/3/2005 181/360 $2388194.40
1/9/2005 184/360 $2427777.80
1/3/2006 181/360 $2388194.40
1/9/2006 184/360 $2427777.80
1/3/2007 181/360 $2388194.40
1/9/2007 184/360 $2427777.80

46. Suppose the floating rates evolve as
follows :
Reset Date LIBOR (% p.a.)
30/8/2004 9.80
28/2/2005 9.20
30/8/2005 9.50
27/2/2006 8.90
30/8/2006 9.70
27/2/2007 10.20

47. This will give rise to the following
floating payments :
Payment Date Amount ($)
1/3/2005 2477222.20
1/9/2005 2351111.10
1/3/2006 2388194.40
1/9/2006 2274444.40
1/3/2007 2438472.20
1/9/2007 2606666.70
Normally, the payments would be netted out with only the net
payment being transferred from the deficit to the surplus party.

48. A TYPICAL USD IRS
A FIXED-TO-FLOATING INTEREST RATE SWAP
6.75% Fixed 6.5% fixed
Prime-25bp Prime-25bp
Prime+75bp 6.5% Fixed
To Floating to Fixed
Rate Lenders Rate Lenders
SWAP BANK
XYZ CORP. ABC BANK

49. Major Types of Swap
Structures
 A number of variants of the standard structure are
found in practice
• Varying notional principal – Amortizing,
Accreting and Roller-Coaster Swaps
 A Zero-Coupon Swap has only one fixed
payment at maturity
 A Basis Swap involves an exchange of two
floating payments, each tied to a different
market index
 In an Extendable Swap, one of the parties has
the option to extend the swap beyond the
scheduled termination date
 Index maturity not equal to reset frequency

50. Major Types of Swap Structures
 In a Forward Start Swap, the effective date is
several months even years after the trade date
so that a borrower with a future funding need
can take advantage of prevailing favourable
swap rates to lock in the terms of a swap to be
entered into at a later date
An Indexed Principal Swap is a variant in which the
principal is not fixed for the life of the swap but
tied to the level of interest rates - as rates
decline, the notional principal rises according to
some formula
 In a Callable Swap the fixed rate payer has the
option to terminate the agreement prior to
scheduled maturity while in a Putable Swap the
fixed rate receiver has such an option

51. Credit Default Swaps
Default protection
buyer
Default protection
seller
Protection
Premium

52. Credit Default Swaps
Bond 1
Bond 2
Bond 3
…
Bond n
Trust
Tranche 1
1st 5% of loss
35% return
Tranche 2
2nd 10% of loss
15% return
Tranche 3
Residual loss
6% return