Corso di Matematica Finanziaria 3 Introduzione ai Derivati
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Corso di Matematica Finanziaria 3
Introduzione ai Derivati
Docente: Arturo Leccadito
Università della Calabria
Slides taken from Hull’s web-site
http://www.rotman.utoronto.ca/∼hull/ofodslides/
Arturo Leccadito (Unical) Introduzione ai Derivati 1 / 36Derivatives Markets
A derivative is an instrument whose value depends on the values of other
more basic underlying variables.
Examples of Derivatives:
Futures Contracts
Forward Contracts
Options
Swaps
Derivatives Markets:
Exchange traded:
Contracts are standard there is virtually no credit risk
Over-the-counter (OTC):
Contracts can be non-standard and there is some small amount of
credit risk
Arturo Leccadito (Unical) Introduzione ai Derivati 2 / 36Types of Derivatives
Forward/Futures Contract:
Agreement to buy or sell an asset for a certain price at a certain time.
A forward contract is traded OTC, whereas a futures contract is
traded on an exchange. It costs nothing to take either a long or a
short position.
Swaps:
Agreement to exchange cash flows at specified future times according
to certain specified rules
Options:
◮ A call (put) option is an option to buy (sell) a certain asset by a
certain date for a certain price (the strike price)
◮ An American option can be exercised at any time during its life
whereas a European option can be exercised only at maturity
◮ If a futures/forward contract gives the holder the obligation to buy or
sell the asset, an option gives the holder the right to buy or sell the
asset.
Arturo Leccadito (Unical) Introduzione ai Derivati 3 / 36Payoffs from Forward Contracts
The payoff from a long position in a forward contract on one unit of an
asset is ST − K , where K is the delivery price and ST is the spot price of
the asset at maturity T .
Similarly the payoff from a short position in a forward contract on one unit
of an asset is K − ST .
Profit Profit
K
K ST ST
Long Position Short Position
Arturo Leccadito (Unical) Introduzione ai Derivati 4 / 36Forward Prices
The forward price for a contract is the delivery price that would be
applicable to the contract if it were negotiated today (i.e., it is the delivery
price that would make the contract worth exactly zero)
Some notation:
S0 : Spot price today
T: Time until delivery date
F0,T : Futures or forward price today
r: Risk-free interest rate for maturity T
For any investment asset, i.e. an assets held by significant numbers of
people purely for investment purposes (Examples: stocks, bonds, gold,
silver), that provides no income (for instance, no dividends) and has no
storage costs
F0,T = S0 erT
assuming interest rates are measured with continuous compounding.
Arturo Leccadito (Unical) Introduzione ai Derivati 5 / 36If F0,T > S0 erT an arbitrage opportunity arises from the following strategy
(Remember the rule: “Buy the cheap, sell the expensive”!!):
1. Sell the forward;
2. Borrow the amount S0 at the rate r ;
3. Buy the stock.
The result is a riskless profit of F0,T − S0 erT at time T .
0 T
1. 0 F0,T − ST
2. S0 −S0 erT
3. −S0 ST
0 F0,T − S0 erT
Arturo Leccadito (Unical) Introduzione ai Derivati 6 / 36Similarly, if F0,T < S0 erT an arbitrage opportunity arises from the
following strategy:
1. Buy the forward;
2. Short sell the stock;
3. Invest the proceeds of the short sale at the rate r .
The strategy results in a riskless profit of S0 erT − F0,T at time T .
0 T
1. 0 ST − F0,T
2. S0 −ST
3. −S0 S0 erT
0 S0 erT − F0,T
Arturo Leccadito (Unical) Introduzione ai Derivati 7 / 36Payoffs from Options
The payoff from a long position in a call on one unit of an asset is
max{ST − K , 0}, where K is the exercise price and ST is the spot price of
the asset at maturity T .
Similarly the payoff from a short position in a call on one unit of an asset
is − max{ST − K , 0} = min{K − ST , 0}.
Payoff Payoff
K
K ST ST
Long Call Short Call
Arturo Leccadito (Unical) Introduzione ai Derivati 8 / 36The payoff from a long position in a put on one unit of an asset is
max{K − ST , 0}, where K is the exercise price and ST is the spot price of
the asset at maturity T .
Similarly the payoff from a short position in a put on one unit of an asset
is − max{K − ST , 0} = min{ST − K , 0}.
Payoff Payoff
K
K ST ST
Long Put Short Put
Arturo Leccadito (Unical) Introduzione ai Derivati 9 / 36Terminology
Moneyness:
At-the-money option: zero cash flow if immediately exercised
(S0 = K )
In-the-money option: positive cash flow if immediately exercised
(S0 > K for a call and S0 < K for a put)
Out-of-the-money option negative cash flow if immediately exercised
(S0 < K for a call and S0 > K for a put)
Intrinsic value:
the maximum of zero and the value the option would have if it were
exercised immediately
Time value:
the difference between option value and intrinsic value. Time value is the
part of the option’s value that derives from the possibility of future
favorable movements in the stock price.
Arturo Leccadito (Unical) Introduzione ai Derivati 10 / 36American vs European Options
Notation:
c : European call option price C : American Call option price
p : European put option price P : American Put option price
S0 : Stock price today ST : Stock price at option maturity
K : Strike price T : Life of option
r : Risk-free rate for maturity T σ : Volatility of stock price
An American option is worth at least as much as the corresponding
European option:
c ≤C
p ≤ P.
Arturo Leccadito (Unical) Introduzione ai Derivati 11 / 36Upper and Lower Bounds for Option Prices
For options on non-dividend-paying stocks the upper bounds are:
c ≤ S0 and C ≤ S0
−rT
p ≤ Ke and P ≤ K
and the lower bounds:
c ≥ max{S0 − K e−rT , 0}
p ≥ max{K e−rT − S0 , 0}
Arturo Leccadito (Unical) Introduzione ai Derivati 12 / 36Put-Call Parity
The relationship between p and c is known as Put-Call Parity. Suppose we
combine a long position in a call and a short position in a put (same
stock, strike and maturity). Both options are European. The final payoff is
max{ST − K , 0} − max{K − ST , 0} = ST − K .
Note that this is the payoff of a forward contract. It follows that the
present value of the payoff is simply S0 − K e−rT . Thus for options on
non-dividend-paying stocks
c − p = S0 − K e−rT .
Arturo Leccadito (Unical) Introduzione ai Derivati 13 / 36Formally, we first assume c − p > S0 − K e−rT and consider the strategy
1. Sell the call;
2. Buy the put;
3. Buy the stock;
4. Borrow the amount K e−rT at the rate r .
The result is a riskless profit of c − p − S0 + K e−rT at time 0.
0 T
1. c − max{ST − K , 0}
2. −p max{K − ST , 0}
3. −S0 ST
4. K e−rT −K
c − p − S0 + K e−rT −[ST − K ] + ST − K = 0
Arturo Leccadito (Unical) Introduzione ai Derivati 14 / 36Similarly, one could build an arbitrage strategy when c − p < S0 − K e−rT .
The put-call parity justifies the lower bounds for option prices. Since p > 0
and c > 0 we have
c = p + S0 − K e−rT ≥ S0 − K e−rT
p = c + K e−rT − S0 ≥ K e−rT − S0 .
Arturo Leccadito (Unical) Introduzione ai Derivati 15 / 36Early Exercise
Usually there is some chance that an American option will be exercised
early. An exception is an American call on a non-dividend paying stock
that should never be exercised early. Recall that c ≥ S0 − K e−rT . For
r > 0 and T > 0 this implies c > S0 − K and as C ≥ c, we have
C > S0 − K .
Now, if it were optimal to exercise early, C would be equal to the intrinsic
value S0 − K . We deduce that it can never be optimal to to exercise early.
Arturo Leccadito (Unical) Introduzione ai Derivati 16 / 36The intuition behind the result is as follows. Suppose the owner of the
American call wants to hold the stock after the call’s maturity. He should
not exercise early because
No income is sacrificed
Payment of the strike price is delayed (investor would lost the interest
paid on the strike price in the case of an early exercise)
Holding the call provides insurance against stock price falling below
strike price
On the other hand, if the owner plans to exercise and sell the stock, he
would be better off selling the option than exercising it.
Arturo Leccadito (Unical) Introduzione ai Derivati 17 / 36Swaps A swap is an agreement to exchange cash flows at specified future times. Usually the calculation of the cash flows involves the future values of one or more market variables. A forward can be viewed as a simple example of a swap. At maturity the buyer pays K and receives the market value of the asset S. Whereas a forward leads to the exchange of cash flows on just one future date, swaps typically involve exchanges on several future dates. The most common type of swap is a “Plain Vanilla” Interest Rate Swap: B agrees to pay A cash flows equal to interest at a fixed rate on a notional principal and A agrees to pay B cash flows equal to interest at a floating rate on the same notional principal. Arturo Leccadito (Unical) Introduzione ai Derivati 18 / 36
Example
Consider an agreement initiated on March 5, 2004 by Microsoft to receive
6-month LIBOR (rate of interest offered by banks on deposits from other
banks in Eurocurrency markets) and pay a fixed rate of 5% per annum
every 6 months for 2 years on a notional principal of $100 million. Cash
flows (Millions of Dollars) are reported in the following table. Note that
the first exchange of payments takes place on September 5, 2004 and
every payment is based on the LIBOR of the previous period.
LIBOR Floating Fixed Net Cash Flow
05 Mar 04 4.20%
05 Sept 04 4.80% 2.1 -2.5 -0.4
05 Mar 05 5.30% 2.4 -2.5 -0.1
05 Sept 05 5.50% 2.65 -2.5 0.15
05 Mar 06 5.60% 2.75 -2.5 0.25
Arturo Leccadito (Unical) Introduzione ai Derivati 19 / 36Typical Uses of an Interest Rate Swap
Converting a liability from
◮ fixed rate to floating rate
◮ floating rate to fixed rate
Some companies may have a comparative advantage when borrowing in
fixed-rate markets, other companies may have a comparative advantage
when borrowing in floating-rate markets. It makes sense for a company to
go to the market where it has a comparative advantage, but this may
imply to borrow floating when it wanted to borrow fixed (or vice versa).
Thus the swap is used to transform a floating rate loan to a fixed rate loan
(or vice versa).
Arturo Leccadito (Unical) Introduzione ai Derivati 20 / 36Example
Suppose
AAACorp wants to borrow floating
BBBCorp wants to borrow fixed
and they have been offered the rates
Fixed Floating
AAACorp 4.00% 6-month LIBOR + 0.30%
BBBCorp 5.20% 6-month LIBOR + 1.0%
Since the difference between the two fixed rates is bigger than the
difference between the two floating rates, BBBCorp has a compara-
tive advantage in the floating-rate market and AAACorp has a com-
parative advantage in the fixed-rate market. A possible swap is:
3.95%
4%
AAACorp BBBCorp
LIBOR+1%
LIBOR
Arturo Leccadito (Unical) Introduzione ai Derivati 21 / 36Example cont’d
At the end of the day
AAACorp ends up paying LIBOR+ 4% – 3.95% = LIBOR +0.05%
which is 0.25% less than the floating rate it was offered
BBBCorp ends up paying LIBOR+ 1% – LIBOR + 3.95% = 4.95%
which is 0.25% less than the fixed rate it was offered
Arturo Leccadito (Unical) Introduzione ai Derivati 22 / 36Ways Derivatives are Used
To hedge risks. For instance
◮ A US company will pay £10 million for imports from Britain in 3
months and decides to hedge using a long position in a forward contract
◮ An investor owns 1000 Microsoft shares currently worth $28 per share.
A two-month put with a strike price of $27.50 costs $1. The investor
decides to hedge by buying 1000 put options. The value of his holding
is always above $27500 ($26500 when the cost of the option is taken
into account)
Arturo Leccadito (Unical) Introduzione ai Derivati 23 / 36Two more hedging examples:
◮ Due to an exceptionally high number of new employees, pension fund A
will receive £10 million in pension contributions one year from now.
The fund will want to invest this amount of money in the FTSE 100
Index, but it is worried the price of the index may become over-inflated
over the next year. They decide to buy at-the-money European Call
options expiring in one year time. The fund this way acquires the right
(but not the obligation) to buy shares of the index in one year time at
the current price. Effectively the found has bought protection against
price increases.
◮ Fund B has £20 million invested in the same index but has to make a
payment in one year time, due to an exceptionally high number of
retiring employees. The fund could be in trouble if the index price falls
in one year. Hence they decide to buy at-the-money European Put
options maturing in one year time. This way the fund acquires
protection against price decreases.
Arturo Leccadito (Unical) Introduzione ai Derivati 24 / 36To speculate (take a view on the future direction of the market)
◮ An investor with $4,000 to invest feels that Amazon.com’s stock price
will increase over the next 2 months. The current stock price is $40
and the price of a 2-month call option with a strike of 45 is $2. He
could buy 100 shares or 2000 call options. Profit and losses from the
strategies are illustrated in the following table for two different price
scenarios.
Price after 2 months
Strategy 30 50
Buy shares -1000 1000
Buy call options -4000 6000
This property is known as Financial Leverage.
Arturo Leccadito (Unical) Introduzione ai Derivati 25 / 36Positions in an Option & the Underlying
Profit Profit
K
K ST ST
a) Long Stock + Short Call b) Short Stock + Long Call
Profit Profit
K
ST K ST
c) Long Stock + Long Put d) Short Stock + Short Put
Arturo Leccadito (Unical) Introduzione ai Derivati 26 / 36Bull Spread Using Calls
A spread trading strategy involves taking a position in two or more options
of the same type (i.e. two or more calls or two or more puts).
A trader who enters a bull spread is hoping that the stock price will
increase.
Profit
K1 K2 ST
Arturo Leccadito (Unical) Introduzione ai Derivati 27 / 36Bull Spread Using Puts
Profit K1
K2 ST
Arturo Leccadito (Unical) Introduzione ai Derivati 28 / 36Bear Spread Using Calls
A trader who enters a bear spread is hoping that the stock price will
decrease.
Profit
K1 K2
ST
Arturo Leccadito (Unical) Introduzione ai Derivati 29 / 36Bear Spread Using Puts
Profit
K2
K1 ST
Arturo Leccadito (Unical) Introduzione ai Derivati 30 / 36Box Spread
A Box Spread is combination of a bull call spread and a bear put
spread.
If all options are European a box spread is worth the present value of
the difference between the strike prices. This is because the payoff is
for K1 < K2 :
max{ST − K1 , 0} − max{ST − K2 , 0}
| {z }
payoff of a bull call spread
+ max{K2 − ST , 0} − max{K1 − ST , 0}
| {z }
payoff of a bear put spread
= max{ST − K1 , 0} − max{K1 − ST , 0}
| {z }
=ST −K1
− (max{ST − K2 , 0} − max{K2 − ST , 0})
| {z }
=ST −K2
= K2 − K1 .
Arturo Leccadito (Unical) Introduzione ai Derivati 31 / 36Butterfly Spread Using Calls
A trader who enters a butterfly spread is hoping that the stock price will
stay close to K2 (typically the option with strike K2 is at-the-money). A
significant price move leads to a small lost.
Profit
K1 K2 K3 ST
Arturo Leccadito (Unical) Introduzione ai Derivati 32 / 36Butterfly Spread Using Puts
Profit
K1 K2 K3 ST
Arturo Leccadito (Unical) Introduzione ai Derivati 33 / 36Combinations
A combination involves investing in both calls and puts on the same stock.
A bottom straddle involves buying a call and a put with the same strike K
and maturity. Payoff: |ST − K |. A top straddle is the reverse position.
Profit
K
ST
Arturo Leccadito (Unical) Introduzione ai Derivati 34 / 36Strip & Strap
Strip: long position in a call and two puts (same strike, same maturity).
The trader hopes that there will be a big stock move (with a decrease
more likely than an increase).
Strap: long position in two calls and a put (same strike, same maturity).
The trader hopes that there will be a big stock move (with an increase
more likely than a decrease). Profit
Profit
K ST K ST
Strip Strap
Arturo Leccadito (Unical) Introduzione ai Derivati 35 / 36A Strangle Combination
Similar to a straddle, but the call strike price, K2 , is bigger than the put
strike price, K1 . The price has to move farther in a strangle than in a
straddle for the buyer to make a profit.
Profit
K1 K2
ST
Arturo Leccadito (Unical) Introduzione ai Derivati 36 / 36You can also read
