Lecture 10
Foundations of Finance
Lecture 10: Options: Valuation.
I.
II.
III.
IV.
Reading.
Preliminaries.
No Arbitrage Pricing Bounds.
Black Scholes Model.
0
Lecture 9-10
IV.
Foundations of Finance
Forward and Futures Contracts.
A.
Forward Contracts.
1.
Definition.
a.
A forward contract on an asset is an agreement between the buyer
and seller to exchange cash for the asset at a predetermined price
(the forward price) at a predetermined date (the settlement date).
b.
The asset underlying a futures contract is often referred to the
“underlying” and its current price is referred to as the “spot” price.
c.
The buyer of the forward contract agrees today to buy the asset on
the settlement date at the forward price.
Position opened at time 0
Long 1 Forward Contract on the Stock with
delivery at T at a forward price FT(0)
Time 0
Time T
0
35
S(T)<FT(0)
S(T)$FT(0)
S(T)-FT(0)
S(T)-FT(0)
S(T)-FT(0)
Lecture 9-10
Foundations of Finance
d.
e.
f.
The seller agrees today to sell the asset at that price on that date.
No money changes hands until the settlement date. In fact, the
forward price is set so that neither party needs to be paid any
money today to enter into the agreement.
Can see that the sum of the buyer’s and the seller’s payoff is zero.
Position opened at time 0
Time 0
Time T
S(T)<FT(0)
S(T)$FT(0)
Long 1 Forward Contract on the Stock with
delivery at T at a forward price FT(0)
0
S(T)-FT(0)
S(T)-FT(0)
S(T)-FT(0)
Short 1 Forward Contract on the Stock with
delivery at T at a forward price FT(0)
0
FT(0)-S(T)
FT(0)-S(T)
FT(0)-S(T)
Sum
0
0
0
0
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Lecture 9-10
Foundations of Finance
2.
Payoff Diagram.
a.
Let T be the settlement date, S(t) be the price of the asset at time t
and FT(0) be the forward price agreed to at time 0 for delivery at
time T.
b.
At time T, can see that the buyer of the forward contract gets
S(T)-FT(0); the seller gets FT(0)-S(T).
c.
Can see that the payoff to both the buyer and seller at time T could
be negative.
d.
Can see that the sum of the payoffs to the buyer and the seller
equal zero.
e.
The cash flows to the buyer and the seller at time 0 and at time T
are given above.
f.
Can see that the forward price at the settlement date must equal the
spot price: FT(T)=S(T).
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Lecture 9-10
Foundations of Finance
3.
Market Characteristics: Currency Markets.
a.
The most organized forward markets are currency markets.
b.
Example: WSJ. On Tuesday, 4/5/05, the British pound was quoted
as $1.8806/£ for immediate delivery. For delivery 6-months
forward, the quote was $1.8661/£. When a buyer and seller
entered into this contract on 4/05/05, the buyer of the pounds
agrees to pay the seller $1.8661 in 6 months (10/05) for each £ and
the £'s are delivered at that time. This may differ from the spot
exchange rate that prevails in 10/05. No money changes hands on
4/05/05.
c.
The forward market in FX is a telephone dealer market that exists
as an adjunct to the spot market.
d.
It is only open to banks and other institutional players.
e.
All transactions are customized as to size, currency and delivery
terms.
f.
The buyer and seller bear each others credit risk. So if one
defaults the other has to bear the loss.
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Foundations of Finance
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Lecture 9-10
B.
Foundations of Finance
Futures Contracts vs Forward Contracts.
1.
Futures contracts are sold on metals, agricultural commodities, oil,
livestock and financial securities. Financial futures exist on stocks, bonds,
T-bills, notes, Federal funds, Libor loans and municipal bonds.
2.
Daily Resettlement.
a.
Forward and futures contracts are essentially the same except for
the daily resettlement feature of futures contracts.
b.
With a forward contract, no money changes hands until the
settlement date.
c.
With a futures contract:
(1)
on the day that the futures contract is bought or sold: no
money is paid or received.
(2)
each day after the contract is bought or sold:
(a)
a positive change in the futures price from the
previous day has to be paid by the seller to the
buyer of the futures contract.
(b)
a negative change has to be paid by the buyer to the
seller.
(3)
the futures price on the settlement day is set equal to the
spot price on that day.
3.
Why futures and forward contacts are essentially the same?
a.
Suppose that for a given settlement date, the forward and futures
prices are the same.
b.
Consider an investor who buys either contract on this day and
holds it til the settlement day.
c.
Her total cash flow is the same under both contracts.
(1)
It is the spot price on the settlement day less the
futures/forward price on the day the position is opened.
(2)
This explains why futures and forward contracts are
essentially the same.
d.
However, the timing is different depending on the contract.
(1)
With the forward contact, the cash flow occurs on the
settlement date: there are no intermediate cash flows.
(2)
With the futures contract, cash flows occur over the period
that the position is open due to daily resettlement
(a)
When the position is left open til the settlement
date, the sum of these cash flows equals the spot
price on the settlement day less the futures/forward
price on the day the position is opened.
(3)
This timing difference explains why futures and forward
prices are typically slightly different.
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Foundations of Finance
4.
Examples:
a.
Jun 06 futures contract on the S&P 500: Bloomberg on 4/6/05.
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Foundations of Finance
b.
Apr 06 futures contract on gold: Bloomberg on 4/06/05.
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Foundations of Finance
5.
6.
Other differences: Future vs Forward Contract.
a.
Standardization.
(1)
Forward contracts are tailored to the needs of the parties
involved.
(2)
Futures contracts traded on the CME (the "Merc") have
standardized unit sizes and standardized maturity dates.
b.
Exchange traded.
(1)
Forward contracts are usually traded in dealer markets.
(2)
Merc futures contracts are traded only on the Merc. There
is no trading off the Merc (by law). This is different from
the situation with stocks: IBM, an NYSE-listed company
often trades away from the NYSE.
c.
Credit risk is borne by a clearing house.
(1)
In a forward market, you must be concerned with the credit
worthiness of your counterparty.
(2)
In a futures market, the clearing house (an affiliate of the
exchange) interposes itself between the two parties. The
clearing house sells to the buyer and buys from the seller.
(a)
The buyer's contract is with the clearing house; the
seller's contract is with the clearing house.
(b)
The clearing house does not bear any risk other than
the credit risk: it is long one contract and short one
contract.
(3)
How does the clearing house manage the credit risk?
(a)
All exchange members are required to post funds
with the clearing house.
(b)
An exchange member is responsible for its
customers' transactions.
d.
Margin.
(1)
In a forward contract, no money changes hands until
settlement.
(2)
In a futures transaction, the buyer and seller are each
required to post a small amount of the face value of the
contract with their brokers. In turn, their brokers are
required to post margin with the clearing house.
Comment: the standardization, exchange trading facility and reduced
credit risk serve to increase liquidity in the futures market, and make it
accessible to many smaller investors who could not directly participate in
the forward market.
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Lecture 9-10
V.
Foundations of Finance
Features of Derivatives.
A.
Zero Net Supply.
1.
Stocks and bonds are said to be in positive net supply. They are claims on
real assets. On balance the economy is long IBM stock, for example.
2.
Derivatives (forwards, futures and options) are in zero net supply. For
everyone who is long a forward contract, the opposite side is short. When
you add up the long and short positions, they sum to zero.
B.
Zero Sum Game.
1.
As has been noted, if the long side of a derivative contract gains, the short
side loses. One side's profit is the other's loss. The sum of the profits to
both sides is zero.
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Lecture 9-10
VI.
Corrected
Foundations of Finance
Motives for Holding Derivatives.
A.
Speculation.
1.
Example: Using call options to create a leveraged position in the
underlying
a.
XYZ has a share price S(0)=50 and a European call option on
XYZ with strike price X=50 and maturity date T= 6 months has a
value C(0)=5. Bond equivalent yield on a 6-month t-bill is 10%.
b.
You have $5000 and consider 3 strategies:
(1)
Buy 100 shares of XYZ.
(2)
Buy 1000 call options with X=$50.
(3)
Buy 100 call options with X=$50 and invest $4500 in 6month t-bills.
c.
Payoffs and returns on the strategies as a function of S(T):
Payoff
35
40
45
50
55
60
65
70
75
80
(1)
3500
4000
4500
5000
5500
6000
6500
7000
7500
8000
(2)
0
0
0
0
5000
10000
15000
20000
25000
30000
(3)
4725
4725
4725
4725
5225
5725
6225
6725
7225
7725
35
40
45
50
55
60
65
70
75
80
(1)
-30%
-20%
-10%
0%
10%
20%
30%
40%
50%
60%
(2)
-100%
-100%
-100%
-100%
0%
100%
200%
300%
400%
500%
(3)
-5.5%
-5.5%
-5.5%
-5.5%
4.5%
14.5%
24.5%
34.5%
44.5%
54.5%
Return
d.
Strategy (2):
(1)
When the price falls below 50, you lose everything.
(2)
But when price increases only modestly, e.g. an increase
from 50 to 60 (only a 20% increase), your return goes from
0% to 100%!
(3)
This is called leverage and is the speculative feature of
options.
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2.
Currency Example. Today is 4/5/05. The buyer of the £'s for delivery on
10/05 may simply be betting that by the 10/05, the £ will have increased in
value. Suppose the buyer agrees to buy £10000 on 10/05 (at a price of
$18661). If she has no other exposure to £'s, she gains if the spot price of
the pound is high on 10/05; she loses if the spot price is low.
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Lecture 9-10
B.
Foundations of Finance
Hedging.
1.
Example.
a.
Can combine the underlying asset with puts and calls to create a
riskless position.
b.
Exploits put-call parity.
2.
Currency Example (cont). The buyer of the £'s may owe £10000 in
October but is receiving a fixed number of $ (say $19000) at that time. By
entering into the forward contract on 4/5/05, she removes the exchange
rate risk that she would otherwise face.
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48
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Lecture 9-10
C.
Foundations of Finance
Risk Management.
1.
Can use derivatives to manage the risks of a portfolio.
2.
Portfolio Insurance.
a.
Can combine the underlying asset with a long position in a put on
the asset to put a floor on the value of the position equal to the
strike price of the put.
(1)
Known as portfolio insurance.
(2)
Uses a protective put.
(3)
Going long a call with the same strike price and long a
discount bond with a face value equal to the strike price
gives the exact same payoff.
b.
Example: XYZ (cont)
(1)
Strategy (3) is an example of using call options to obtain
portfolio insurance.
c.
Example. Suppose you invested 100 times the value of the S&P
500 index in the market at the end of September 2004 at a price of
$1125. With the S&P 500 at $1181 on 4/11/05, you decide you
want to put a lower bound on your 1-year return so you buy
European puts with an expiration date of 9/05 on 100 times the
value of the S&P 500:
(1)
If concerned about staying above zero, buy puts with a
strike price of $1150 at a price of $27. Then your dollar
return is bounded below by $1150-$27-$1125 = - $2 which
is approximately zero.
(2)
If you don’t want to lose more than 5% of your investment,
buy puts with a strike price of $1100 at a price of $16.
Then your dollar return is bounded below at $1100-$16$1125 = - $41 which is a smaller dollar loss than the
$56.25 needed for a 5% loss on $1125.
50
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Lecture 10
Foundations of Finance
Lecture 10: Options: Valuation.
I.
Reading.
A.
BKM, Chapter 20, Section 20.4.
B.
BKM, Chapter 21, ignore Section 21.3 and skim Section 21.5.
II.
Preliminaries.
A.
Up until now, we have been concerned with the payoffs of put and call options at
maturity. This handout is concerned with:
1.
The value of a call or put option prior to maturity.
2.
Whether an American call option or an American put option should be
exercised prior to maturity.
B.
The results in this handout refer to non-dividend paying underlying assets unless
otherwise stated.
C.
Notation.
1.
S(0) be the value of the underlying at time 0.
2.
dT(0) be the discount factor for a T-year discount bond available at time 0.
3.
CX,T(0) be the time 0 price of a European call option with exercise price X
and expiration date T.
4.
cX,T(0) be the time 0 price of an American call option with exercise price
X and expiration date T.
5.
PX,T(0) be the time 0 price of a European put option with exercise price X
and expiration date T.
6.
pX,T(0) be the time 0 price of an American put option with exercise price X
and expiration date T.
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Lecture 10
III.
Foundations of Finance
No Arbitrage Pricing Bounds.
A.
Call Options.
1.
Floor on the Value of a European Call.
a.
Have already established that a European call option must have a
nonnegative price since its cashflow at expiration is nonnegative:
CX,T(0) $ 0.
b.
Consider the following two investment strategies:
Strategy
Time 0
Time T
S(T)<50
S(T)$50
Buy underlying at 0 and sell at T
-S(0)
S(T)
S(T)
Sell a T period discount bond with face
value of 50 and hold to maturity
50 dT(0)
-50
-50
Net Cash Flow
-S(0) +50 dT(0)
S(T)-50
S(T)-50
Strategy
Time 0
Buy a European call option at 0 with
exercise price of 50 and hold until
expiration at T
Time T
-C50,T(0)
2
S(T)<50
S(T)$50
0
S(T)-50
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c.
Can see that the second strategy always produces a cash flow equal
to or greater than the first strategy:
(1)
If S(T) $ 50, both strategies generate the same cash flow at
T: [S(T)-50].
(2)
If S(T) < 50, the first strategy generates [S(T)-50] while the
second strategy generates 0 which is >[S(T)-50].
d.
For there not to exist an arbitrage opportunity, the second strategy
must cost more than the first one; i.e., C50,T(0) $ S(0)-50 dT(0).
This restriction is a floor on the call’s value.
e.
So more generally, CX,T(0) $ S(0) - X dT(0) and CX,T(0) $ 0 which
implies that
CX,T(0) $ max{S(0) - X dT(0), 0}.
The RHS is known as the intrinsic value of the option while the
difference between the LHS and RHS is known as its time value.
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Lecture 10
Foundations of Finance
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Foundations of Finance
f.
Example. Bloomberg screen for 4/13/05 Dell options. May 05
options expire 5/21/05. Dell is not paying any dividend between
4/13/05 and 5/21/05. The “Fin Rate” is the riskfree rate and is
2.76%. Assuming this is a continuously compounded annual rate,
it implies a discount factor on a 38-day discount bond of
e-0.0276x38/365 = 0.9971. The Dell stock price on 4/13/05 is 36.91.
(1)
For X=35,
max{S(0) - X dT(0), 0} = max{36.91-35 x 0.9971, 0} = max{2.01, 0} = 2.01 < 2.45
(2)
Strategy
which is the ask price of the call on the screen.
Suppose the ask price of the 35 call on 4/13/05 is 1.90
which violates the floor of 2.01. There is an arbitrage
opportunity which involves buying the call (since it is
undervalued) and selling the first strategy described above:
4/13/05
5/21/05
S(5/21/05)<35
S(5/21/05)$35
Buy 1 Dell call option on 4/13/05 with
exercise price of 35 and hold until
expiration on 5/21/05
-1.90
0
S(5/21/05)-35
Sell 1 Dell share on 4/13/05 and close out
on 5/21/05
36.91
-S(5/21/05)
-S(5/21/05)
On 4/13/05, buy a discount bond with
face value of 35 maturing on 5/21/05 and
hold to maturity
-35 x 0.9971
= -34.90
35
35
Net Cash Flow
0.11
35-S(5/21/05)>0
0
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Lecture 10
Foundations of Finance
2.
Early Exercise of an American Call.
a.
Know that an American option must be worth at least as much as a
European option with the same expiry date and exercise price:
cX,T(0) $ CX,T(0).
b.
The value of a European option with same expiry date and exercise
price can be thought of as the value of holding the American
option til the exercise date.
c.
If this value exceeds the value of exercising the American option
now, and this is true for any date prior to T, then it is optimal to
hold the American option til maturity.
d.
In general, the floor on the value of the call associated with not
exercising prior to maturity is always greater than or equal to the
value of exercising the call now (since dT(0) <1):
cX,T(0) $ max{S(0) - X dT(0), 0}$ max{S(0) - X, 0}.
e.
So the holder of an American call option on a non-dividend paying
underlying never wants to exercise early: prior to the expiration
date, the holder always prefers to sell the option rather than
exercise it.
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Lecture 10
B.
Foundations of Finance
Put Options.
1.
Floor on the Value of a European Put.
a.
Consider the following two investment strategies:
Strategy
Time 0
Time T
S(T)<50
S(T)$50
Sell underlying at 0 and close out at T
S(0)
-S(T)
-S(T)
Buy a T period discount bond with face
value of 50 and hold to maturity
-50 dT(0)
50
50
Net Cash Flow
S(0) -50 dT(0)
50-S(T)
50-S(T)
Strategy
Time 0
Buy a European put option at 0 with
exercise price of 50 and hold until
expiration at T
Time T
-P50,T(0)
9
S(T)<50
S(T)$50
[50-S(T)]
0
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Foundations of Finance
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b.
Can see that the second strategy always produces a cash flow equal
to or greater than the first strategy:
(1)
If S(T) < 50, both strategies generate the same cash flow at
T: [50-S(T)].
(2)
If S(T) $ 50, the first strategy generates [50-S(T)] while the
second strategy generates 0 which is $ [50-S(T)].
c.
For there not to exist an arbitrage opportunity, the second strategy
must cost more than the first one; i.e., P50,T(0) $ 50 dT(0) - S(0).
d.
Combining this floor with the nonnegativity restriction, obtain:
P50,T(0) $ max{50 dT(0) - S(0), 0}.
e.
For general exercise price X, obtain the following floor:
PX,T(0) $ max{X dT(0) - S(0), 0}.
The RHS is known as the intrinsic value of the option while the
difference between the LHS and RHS is known as its time value.
f.
Example (cont). Bloomberg screen. Dell options. May 05 options
expire 5/21/05.
(1)
For X=35,
max{X dT(0) - S(0), 0} = max{35 x 0.9971-36.91, 0} = max{-2.01, 0} = 0 < 0.40
which is the ask price of the put.
(2)
For X=40,
max{X dT(0) - S(0), 0} = max{40 x 0.9971- 36.91, 0} = max{2.97, 0} = 2.97 < 3.20
which is the ask price of the put.
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Lecture 10
Foundations of Finance
2.
Early Exercise of an American Put.
a.
Know that an American option must be worth at least as much as a
European option with the same expiry date and exercise price:
pX,T(0) $ PX,T(0).
b.
The value of a European option with same expiry date and exercise
price can be thought of as the value of holding the American
option til the exercise date.
c.
In general, the floor on the value of the put associated with not
exercising prior to maturity is less than the value of exercising the
put now (since dT(0) <1):
pX,T(0) $ max{X - S(0), 0} $ max{X dT(0) - S(0), 0}.
d.
e.
So the holder of an American put option on a non-dividend paying
underlying may want to exercise early.
In fact, it can be shown that for S(0) sufficiently small, the holder
of an American put on a non-dividend paying underlying prefers to
exercise immediately.
13
Lecture 10
C.
Foundations of Finance
Put Call Parity.
1.
Consider the following two investment strategies:
Strategy
Time 0
Buy underlying at 0 and sell at T
-S(0)
Strategy
Time 0
Time T
S(T)<50
S(T)$50
S(T)
S(T)
Time T
S(T)<50
S(T)$50
Buy a European call option at 0 with
exercise price of 50 and hold until
expiration at T
-C50,T(0)
0
S(T)-50
Write a European put option at 0 with
exercise price of 50 and hold until
expiration at T
P50,T(0)
-[50-S(T)]
0
Buy a T period discount bond with face
value of 50 and hold to maturity
-50 dT(0)
50
50
Net Cash Flow
-C50,T(0)+P50,T(0)
-50 dT(0)
S(T)
S(T)
2.
Can see that these two strategies have the same cash flows. The law of
one price says that these strategies must have the same price.
3.
Thus, get a relation between the price of a European call and put with the
same exercise date and price and the price of the underlying and the
present value of the exercise price:
S(0) = C50,T(0) - P50,T(0) + 50dT(0).
4.
For general exercise price X,
S(0) = CX,T(0) - PX,T(0) + X dT(0).
5.
6.
This relation is known as put call parity.
Can also see the relation using payoff diagrams. Sum the payoff diagrams
for the second strategy and you get the payoff at T from holding the
underlying.
7.
Example. Dell price today is 36.91. The price of a discount bond (face
value of 100) maturing in 38 days is 99.71. A European call expiring in
38 days with an exercise price of 35 has a price of 2.40 today. What is the
price today of a European put expiring in 38 days with an exercise price of
35?
P35,38day(0) = C35,38day(0) - S(0) + 35 d38day(0) = 2.40 - 36.91 +35 x 0.9971 = 0.39.
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Lecture 10
IV.
Foundations of Finance
Black Scholes Model.
A.
Assumptions.
1.
Yield curve is flat through time at the same interest rate. So there is no
interest rate uncertainty
2.
Underlying asset return is lognormally distributed with constant volatility
and does not pay dividends.
3.
Continuous trading is possible.
4.
No transaction costs, taxes or other market imperfections.
B.
Notation. Above notation holds. Additionally:
1.
r’ is the continuously compounded annual riskfree rate. So $1 invested
today at the riskless rate is worth $1 er’ t in t years time.
2.
σ is the volatility of the continuously compounded annual return on the
underlying asset.
C.
Formula for European Call Options.
1.
The value of a European call option is given by:
CX,T(0) = S(0) N(d1) - X e -r’ T N(d2)
where N(.) is the cumulative Normal distribution function (see BKM,
Table 20.2);
d1 '
ln[S(0)/X] % {r ) % σ2/2} T
σ T
; and, d2 ' d1 & σ T .
18
Lecture 10
Foundations of Finance
2.
Factors affecting the value of the call.
a.
S(0): CX,T(0) is monotonically increasing in S(0) as expected.
b.
X: CX,T(0) is monotonically decreasing in X as expected.
c.
σ: CX,T(0) is monotonically increasing in σ. Why?
(1)
Option feature of the call truncates the payoff at 0 when the
underlying’s value is less than the strike price.
(2)
When σ increases, the volatility of S(T) increases.
(3)
The call option holder benefits from the greater upside
potential of S(T) but does not bear the greater downside
potential due to the truncation of the option payoff at 0.
(4)
Thus, the value of the call increases relative to S(0).
States are Equally Likely
Stock S(t)
Call C50,T(t)
Put P50,T(t)
Payoff at T - State 1
40
0
10
Payoff at T - State 2
60
10
0
E[Payoff at T]
50
5
5
σ[Payoff at T]
10
States are Equally Likely
Stock S(t)
Call C50,T(t)
Put P50,T(t)
Payoff at T - State 1
20
0
30
Payoff at T - State 2
80
30
0
E[Payoff at T]
50
15
15
σ[Payoff at T]
3.
30
d.
T: CX,T(0) is monotonically increasing in T. Why?
(1)
The exercise price does not have to be paid until time T.
When T increases, the current value of X paid at T
decreases making the option more valuable for given S(0).
(2)
Second, with a longer time to maturity the volatility of S(T)
increases for given σ. So the value of the call today
increases for the same reason that an increase in σ increases
the call’s value today.
(3)
Both effects are acting in the same direction.
e.
r’: CX,T(0) is monotonically increasing in r’. Why?
(1)
The exercise price does not have to be paid until time T.
When r’ increases, the current value of X paid at T
decreases making the option more valuable for given S(0).
Factors not affecting the value of the call.
19
Lecture 10
Foundations of Finance
a.
the expected return on the underlying asset. Why?
(1)
When the expected return on the underlying increases, the
expected return on the call also increases.
(2)
Since the current underlying’s price S(0) remains equal to
the current value of the underlying despite its higher
expected return, C(0) remains the current value of the call
option despite its higher expected return.
States are Equally Likely
Stock S(t)
Payoff at T - State 1
20
25
0
0
Payoff at T - State 2
80
85
30
35
E[Payoff at T]
50
55
15
17.5
σ[Payoff at T]
30
30
Price(0)
40
E[Return]
25%
D.
Call C50,T(t)
10
37.5%
50%
75%
Value of an American call option.
1.
It was shown above that the holder of an American call option on a nondividend paying asset would never exercise early.
2.
Thus, the value of an American call equals the value of the European call
with the same exercise price and date.
3.
So the value of an American call is also given by the Black Scholes call
option formula.
20
Lecture 10
Foundations of Finance
E.
European Put Options.
1.
Once the value of the European call with same exercise price and date has
been determined, put call parity can be used to determine the value of a
European put.
2.
Factors affecting the value of the European put.
a.
S(0): PX,T(0) is monotonically decreasing in S(0) as expected.
b.
X: PX,T(0) is monotonically increasing in X as expected.
c.
σ: PX,T(0) is monotonically increasing in σ. Why?
(1)
Option feature of the put truncates the payoff at 0 when the
underlying’s value is more than the strike price.
(2)
When σ increases, the volatility of S(T) increases.
(3)
The put option holder benefits from the greater downside
potential of S(T) but does not bear the greater upside
potential due to the truncation of the option payoff at 0.
(4)
Thus, the value of the put increases relative to S(0).
d.
T: affect on PX,T(0) is ambiguous. Why?
(1)
The exercise price is not received until time T. When T
increases, the current value of X received at T decreases
making the option less valuable for given S(0).
(2)
But acting in the other direction, a longer time to maturity
increases the volatility of S(T) for a given σ. When T
increases, the value of the put today also increases for the
same reason that an increase in σ (for fixed T) increases the
put’s value today.
(3)
It is not clear which of these effects dominates.
e.
r’: PX,T(0) is monotonically decreasing in r’. Why?
(1)
The exercise price is not received until time T. When r’
increases, the current value of X received at T decreases
making the option less valuable for given S(0).
F.
American Put Options.
1.
No closed form solution exists to the value of an American put option.
2.
A value can be obtained numerically.
G.
Implied Volatility.
1.
All the inputs into the Black-Scholes model are readily observable except
the volatility of the return on the underlying.
2.
The value of σ which together with the other Black-Scholes inputs gives a
Black-Scholes call price equal to the market’s prevailing call price is
known as the implied volatility of the underlying.
21