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ECON 4051: Financial Asset Pricing
Lecture 5: No Arbitrage and Asset Pricing (Part II)
Applications to Futures Markets
In the last lecture, we presented a very important and fundamental result
on asset pricing:
Given a price-payoff couple ( ), there are no any arbitrage opportunities in the security markets if and only if there exists at least a state price
vector 0 such that
= > = 1 2 · · ·
In this lecture, we will show you some of its applications in futures markets.
1
Futures, Options, and Derivatives
First, let’s look at some examples: consider one risky security in a two
period and two state economy:
= 300
=0
%
•
&
1000 with
1
2
probability
with
1
2
probability
1
=1
Moreover, suppose there is a financial market for the asset.
Can you imagine what kind of agents could be better off from using this
asset and its market? In other words, what kind of agents could buy such
security at time 0?
Type 1: The agents who have at least 300 today’s consumption
goods.
So, if you have 300 units of consumption goods at time 0 and wish to get
one unit such asset, you could buy it from the financial market. And you are
happy and are better off from this asset and its market.
1
Suppose you have nothing today and will have 400 consumption goods
tomorrow and nobody would like to lend anything to you today. Also, suppose you believe you would be better off if you could buy one unit of such
asset.
What should you do?
Of course, nothing you can do on such current markets with only such asset.
Thus, in order to satisfy this kind agents’ demands who have nothing
today, but will have something tomorrow, we need design new financial asset
and open new financial market for it. This leads us to introduce a new
financial asset:
Definition 1 A forward contract (written) on a security is a contract
to purchase or sell a unit of the security at a specific price and at a
specific time in the future as well as at a specific place.
— The security which the forward contract is written on is called the underlying security.
— The specific price for exchanging the security at the specific time is
called the forward price of the security.
— The specific time for exchanging the security is called the maturity
time.
Note 1: The forward contract is an obligation. That is, at the maturity
date, the seller has to sell the security and the buyer has to buy it at the
specific place and price and at the specific place.
Note 2: At the time of writing the contract, the buyer pays nothing and
the seller gets nothing. And this is why we call it forward contract. That
is, there is no trading happened at the initial time of writing the forward
contract and the two parts just signed the contract. That is all!
Thus, the price of the forward at time 0 is 0 (Can you imagine why?
Since nothing will happen at time 0 and anybody can buy any units
of such forwards!) and the payoff at time 1 is − That is,
0
•
%
&
=0
1000 −
1−
=1
2
In other words,
µ buying one¶forward is equivalent to buying the following asset
1000 −
but having no cost. Is this an arbitrage? No.
with payoffs
1−
Can you imagine why? (Since will satisfy 1 1000!)
The remaining question for forward contract is how to determine the
forward price
Suppose there is one more security: the riskless asset with time 0’s price
1
1+
1
%
1
•
1+
&
1
=0
=1
Thus, there are three assets in the security markets: The underlying asset ,
the riskless asset, and the forward written on the underlying.
The market structure for the three assets is
¡
¢
The underlying the riskless asset, the forward
=
¶
µ
1000 1 1000 −
=
1
1
1−
and the price vector is
⎛
⎝
1
1+
0
⎞
⎠
Can you imagine how to determine ? We try to use the no arbitrage
principle to determine next.
Consider the following two portfolios:
Portfolio 1: Buy one forward contract only.
In this case,
⎞ ⎛ ⎞
⎛
1
0
buy 0 unit of the underlying asset
1 = ⎝ 2 ⎠ = ⎝ 0 ⎠ buy 0 unit of the riskless asset
3
1
buy one unit of the forward contract
3
With this portfolio 1 the cost at time 0 is
⎛
⎞0
⎞0 ⎛ ⎞
⎛
0
1
1
⎝ 1+
⎠ 1 = ⎝ 1+
⎠ ⎝ 0 ⎠=0
1
0
0
and payoff at time 1 is
1 =
µ
1000 1 1000 −
1
1
1−
¶
⎛
⎞
µ
¶
0
1000
−
⎝ 0 ⎠=
= The forward = −
1−
1
Portfolio 2: Buy one unit the underlying security and sell units of
riskless security short.
In this case,
⎞ ⎛
⎛
⎞
1
1
buy 1 unit of the underlying asset
2
⎠
⎝
⎝
⎠
2
−
sell units of the riskless asset short
=
=
3
0
buy 0 unit of the forward contract
With this portfolio 2 its cost at time 0 is
⎛
⎛
⎞
⎞0
⎞0 ⎛
1
1
1
⎝ 1+
⎠ 2 = ⎝ 1+
⎠ ⎝ − ⎠ = −
1 +
0
0
0
and payoff at time 1 is
2 =
µ
1000 1 1000 −
1
1
1−
¶
⎛
⎞
µ
¶
1
⎝ − ⎠ = 1000 −
= The forward = −
1−
0
Thus, the two portfolios will have the same payoffs at time 1. If there are no
arbitrage opportunities in the markets, the costs for the two portfolios must
be the same.
That is,
=⇒ = (1 + )
0=−
1 +
The forward price is the future value of units of time 0’s consumption
good.
4
This is an amazing story! We don’t need to know all agents’
preferences, endowments, and equilibrium allocation, but still we
can price the forward contract! What we need here is just the
common sense: No free lunch (no arbitrage).
Can you imagine what kind of agents could be better off from using this
forward on the asset rather than the asset itself and its market? In other
words, what kind of agents could buy such forward on at time 0?
Type 2: The agents who have nothing at time 0, but will have
something at time 1.
Please remember that the forward contract is an obligation. That is, no
matter whatever would happen at maturity date, you have to buy or sell at
the specific price and the specific time .
Consider the following example
= 300
1000 if state 1
%
•
&
1
=0
if state 2
=1
Suppose the forward price is 350 and you want to buy one unit the forward.
At time 1, if state 2 happens, you will lose 350 − 1 = 349 units of time 1’s
consumption good. That is too bad. That is, the forward contract has a
disadvantage: it is an obligation rather than an option.
Now, suppose you have something today and also will have something
tomorrow, but don’t want to take too much risk at state 2 tomorrow.
What should you do?
Of course, nothing you can do on such current markets with only such two
securities: The assets and the forward on it.
Thus, in order to satisfy this kind agents’ demands who have something
today and also will have something tomorrow, but don’t want to take too
much risk at state 2 tomorrow, we need design new financial asset and open
new financial market for it.
For example, you just want to buy the right, but not the obligation, to
buy the security at time 1 at specific price .
5
That is, after observing which state realized at time 1, you could excise
the right to buy the asset at the specified price if you wish; Also, you could
give up the right to buy the asset at the specified price if you wish. In
other words, to buy or not to buy the asset at the specified price is up to
you. It is your right, but not your obligation.
For the asset
%
&
•
1000 if state 1
1
=0
if state 2
=1
and the specified price = 350
At time 1, if state 1 happens, you will buy the security and pay 350. The
payoff is 1000 − 350 = 650. If state 2 happens, you will give up the right to
buy the security and the payoff is 0. Of course, you are happy, but the seller
is not happy. As a compensation for the seller, you have to pay some units
of consumption goods to the seller at time 0. Of course, must be smaller
than (the price of the security).
That is, the price of the option at time 0 is and the payoffs at time 1
%
&
1000 − 350 = 650
0
=1
The key question is: How to decide the price of this “right” to buy the
security at the specific price 350?
We first define option:
Definition 2 A European call option written on a security gives the buyer
the right, but not the obligation, to buy the underlying security at a specific
price at a specific date and at a specific place.
Note: (1) The specific price is called the exercise price or the strike
price. (2) The specific time is called the maturity date.
6
Definition 3 A European put option written on a security gives the buyer
the right, but not the obligation, to sell the security at a specific price at a
specific time and at a specific place.
Though a European call (or put) gives the right, not the obligation to buy
(or sell) the security, the holder can exercise the option only at the maturity
date if he wants. The holder cannot exercise it earlier. If you want to buy a
right, not an obligation, and also to exercise the right at any time before the
maturity date, what you should buy?
American option.
Definition 4 An American call (put) option written on a security gives the
buyer the right, but not the obligation, to buy (sell) the security at a specific
price at a specific place, and at any time before the maturity date .
Define
– the underlying security’s payoff at time 1;
– the price of the underlying security at time 0;
¯ – the European call’s payoff at time 1 (maturity date);
– the price of the European call at time 0 (initial date);
– the price of the American call at time 0 (initial date);
¯ – the European put’s payoff at time 1 (maturity date);
– the price of the European put at time 0 (initial date);
– the price of the American put at time 0 (initial date);
— the exercise price at time 1, but specified at time 0.
Thus, we have
¯ = max{0 − } = ( − )+
which is a two piece linear function. See the following picture
7
and
¯ = max{0 − } = ( − )+
which is also a two piece linear function. See the following picture
8
For call option, − – the intrinsic value.
Can you draw the picture of the payoffs of a forward contract at time 1?
It is a straight-line.
For buying one forward, its time 1 payoff is
−
9
For selling one forward, its time 1 payoff is
−
10
That is, the payoffs of a forward contract are a linear function. And this is
why we can easily price it by using the no arbitrage principle. How about the
call options? Are a call option’s payoffs a straight line? Not at all. And this
is why we cannot easily price it by using no arbitrage principle. But we can
still price it by using no arbitrage principle. Though ( − )+ and ( − )+
are not linear, they are two piece linear! The process is very complicated
and it will use very advanced mathematical technology. And this is the very
11
famous Black-Scholes formula
= (1 ) − − (2 )
= − (−2 ) − (−1 )
where
1 =
ln () + ( + 2 2)
√
2 =
ln () + ( − 2 2)
√
is the cumulative probability distribution function for a standard normal
distribution, is the riskfree interest rate, and is the maturate date.
What are and 2 ? Here we assume the payoff of the underlying asset at
maturate date is log normally distributed with mean and variance 2
This is the Black-Scholes option price formula. Merton and Scholes got
the Nobel prize in Economics for this formula in 1997. (Very unfortunately,
Black passed away in 1995 just two years before 1997.)
This is a benchmark for pricing options. Right now, there are many
young economists, engineers, mathematicians using this formula to calculate
the prices of many new options every day.
The question is:
Is Black-Scholes option price formula true?
Of course, it is not. Why? Since there are many assumptions under the
formula: For example,
Assumption 1 The interest rate is deterministic.
Is that true? No since it should be stochastic.
Assumption 2 There are no transaction costs.
Is that true? No since there are lots of transaction costs. For example,
tax.
Assumption 3 The payoff at maturate time of the underlying asset is log
normally distributed with constant mean and constant variance 2
Is that true? No. Why is log normal? Even log normal, both and 2
should be stochastic.
12
Can we generalize the Black-Scholes option price formula to cover the
above generalizations? No since there would be no more option price formula
if there are no the above assumptions.
So, we have to balance the two possibilities:
Either we have the Black-Scholes option price formula under the above
assumptions, but it can not fit the data; or we don’t make any assumptions,
but also have no any option price formula!
How about American option? Are the payoffs of an American call option
linear? No, not all! See the following picture
And this is why we still don’t know how to price an American option exactly.
That is, this question is still open!
Definition 5 Say that the call option is
— in the money if − 0;
— at the money if − = 0; and
— out of the money if − 0
Please note that any option has the following two properties:
13
1. If you buy an option, then there must be another person who sells the
option. That is, the net supply (or demand) of the option must be zero.
2. An option payoff depends on the underlying security’s payoff.
That is, an option is a special security having the above two properties.
Actually, it’s value is derived from the underlying assets. Therefore, we call
it a derivative security. This leads us to define
Definition 6 A security is said to be a derivative if
1. its net supply is zero; and
2. its payoffs are derived from some other securities.
The main progress made in the last 30 years in finance is to price different
kinds of derivatives. So, try to design a new financial asset satisfying some
people’s demand, then you will become rich immediately!
If you are interested in this topic, please see the famous book
Options, Futures, and Other Derivatives
written by John Hull (9th edition).
Maybe some of you want to work in Wall Street or Bay Street after
graduating from Carleton, you may need to buy and read carefully the above
book which has been recognized as the bible of this field:
The following is the cover page of the famous book:
14
Can you imagine what kind of agents could be better off from using this
financial derivatives including options and its markets? In other words, what
kind of agents could buy such derivatives at time 0?
Type 3: The agents who have something today, but don’t want to
take too much risk in the future.
So far, we have introduced three types of agents with different demands.
To satisfy their different demands, we have introduced and designed the three
different kinds of securities and the corresponding markets.
Of course, people have more and more demands as time going, therefore
we can imagine that more and more financial products would be designed
and introduced.
An intestine question is the following:
Can we design a new financial product trying to satisfy the following type
agent’s demand?
Type 4: The agent has nothing today and also has nothing in the
future plus does not like to take any risk in the future.
Of course, no such financial product since there is no free lunch in any
financial markets.
2
No Arbitrage and Properties of Options
In this lecture, we are not going to talk about the Black-Scholes formula in
detail, but try to give you some basic idea about pricing European options.
Given the underlying security
(price)
1
2
..
.
%
• −→
& Ω−1
Ω
=0
=1
16
the payoffs of the European call option are
( ) (price)
(1 − )+
(2 − )+
..
.
%
• −→
& (Ω−1 − )+
(Ω − )+
=0
=1
and the payoffs of European put option are
( ) (price)
( − 1 )+
+
% ( − 2 )
..
• −→
.
& ( − Ω−1 )+
( − Ω )+
=0
=1
where ( ), ( ) represent the prices of European call, put option, respectively.
Let ( ), ( ) represent the prices of American call, put option,
respectively.
The key questions are
( ) ( ) =?
and
( ) ( ) =?
With the only very basic assumption: No arbitrage in all the security
markets, we have
Lemma 1 ( ), ( ) must be nonnegative.
Proof. Since ( − )+ ≥ 0 and ( − )+ ≥ 0 ( ) ≥ 0 and ( ) ≥ 0
directly follow from the principle of no arbitrage.
Lemma 2 ( ) is non increasing in ( ) is non decreasing in
17
Proof. Since ∀1 ≥ 2
− 1 ≤ − 2 =⇒
( − 1 )+ ≤ ( − 2 )+ =⇒
( 1 ) ≤ ( 2 )
Similarly, we can prove
( 1 ) ≥ ( 2 )
That is, the high the strike price the low (high) the payoff of the call (put)
option at time 1, therefore, the low (high) price of call (put) option at time
0.
Lemma 3 Both ( ) and ( ) are convex in strike price .
Proof. Omitted!
For given underlying securities
1 2 · · ·
with time 0’s prices
and the strike prices
1 2 · · ·
1 2 · · ·
we can write options on the underlying securities. Denote their prices
(1 1 ) (2 2 ) · · · ( )
For given portfolio
⎛
⎜
⎜
=⎜
⎝
1
2
..
.
⎞
⎟
⎟
⎟ 0
⎠
’s payoff is Thus, (or ) is a new underlying security. We can also
write an option on with strike price > , where
⎛
⎞
1
⎜ 2 ⎟
⎜
⎟
= ⎜ .. ⎟
⎝ . ⎠
18
The question is: What is the relationship between ( > > ) and (1 1 ),
(2 2 ) · · · ( )? We can prove
P
Lemma 4 ( > > ) ≤
=1 ( )
That is, the price of European call option on a portfolio is less than a
portfolio of options on the assets in that portfolio. That is, the option on
a portfolio is cheaper than the portfolio of the options on the assets in the
portfolio.
Since
− ≤ =⇒ ( − )+ ≤ =⇒ ( ) ≤
That is, the payoff of the call option is smaller than the one of the asset at
time 1, therefore, the price of the call option is smaller than the one of the
asset at time 0.
Thus, we have:
Lemma 5 ( ) ≤
Next, suppose there is a riskless asset wish payoffs
1
1+
1
% 1
• −→ ...
& 1
1
=0
=1
Consider the following portfolio : Buy one unit the underlying security, sell
units of the riskless assets short. Its payoff at time 1 is − and its cost
is − 1+
Since the call option’s payoff
( − )+ ≥ − =⇒ ( ) ≥ −
1 +
Therefore, we have
Lemma 6 If there is a riskless asset with interest rate , then
( ) ≥ −
19
1 +
Combining Lemmas 5 and 6, we have
−
≤ ( ) ≤
1 +
Next, we want to know the relationship between the call’s price ( )
and the put’s price written on the same underlying asset and with the same
strike price
To derive the relationship, consider the following two portfolios:
1 : Buy one unit European call option and buy units of riskless assets.
Its payoffs at time 1 are
+
( − ) + =
½
− + = if ≥
0 + = if ≤
and its cost today is
( ) +
1 +
2 : Buy one unit European put option and one unit the underlying security.
Its payoffs at time 1 are
+
( − ) + =
½
0 + = if ≥
− + = if ≤
and its cost at time 0 is
( ) +
Since the two portfolios’ payoffs at time 1 are the same, therefore, the costs
today must be the same as well
( ) +
= ( ) +
1 +
This leads us to have
Lemma 7 (Put-call parity) If there is a riskless asset with interest rate
0, then
( ) +
= ( ) +
1 +
20
3
Pricing Options in a Complete Market
Now, we turn to the option pricing. From the last lecture, if the security
markets are complete and there is no arbitrage, then there exists a unique
state price vector { }Ω
=1 0 such that
P
+
( ) = > ( − )+ = Ω
=1 ( − )
1
=
( − )+
1 +
How to find { }Ω
=1 — the state prices? We look at some examples:
Example 1 (The Lucas tree economy) There are two assets: One risky with
payoffs
with probability
%
(price of the asset) •
&
with probability 1 −
=0
=1
where 1 and one riskless asset (bond) with payoffs
1
1+
%
(price of the bond) •
&
1 if state 1
1 if state 2
=0
=1
where 1 +
Question 1: Are the two markets complete?
From
= ( ) =
and
¯
¯ 1
|| = ¯¯
1
µ
1
1
¶
¯
¯
¯ = − = ( − ) 0
¯
the two markets are complete since the rank of matrix equals the number of
states of nature = 2.
21
Question 2: How to find the two state prices , ?
Based on the Fundamental Theorem of Asset Pricing, the two state prices
, must satisfy
= +
1
1 +
= +
That is,
1 = +
1
=
−
1 +
and
1 =
=
=
=
¶
1
+ =
− +
1 +
− ( − ) +
=⇒
1 +
−1
− (1 + ) 1
1+
=
0
−
−
1 +
(1 + ) − 1
0
−
1 +
µ
Thus, we have
∙
¸
1
−
( ) = + =
( − ) +
−
1 +
We have another method to find the price ( )
As we said earlier, the two security markets are complete. That is, the
European call option is redundant. The next question is:
Question 3: Could we find a portfolio whose payoffs are exactly the same
as the call option?
Let
=
µ
1
2
¶
buy 1 units of the risky asset
buy 2 units of the bond
22
Its payoffs are
¶µ
1
2
¶
=
( − ) = max {0 − } =
½
if state 1
=
if state 2
=
µ
1
1
µ
1 + 2
1 + 2
¶
The call option payoffs are
+
Let
µ
That is,
1 + 2
1 + 2
¶
=
µ
¶
µ
¶
1 + 2 =
1 + 2 =
and
−
( − )
−
= − 1 = −
( − )
−
( − ) − ( − )
= −
=
−
−
( − ) − ( − ) −
=
=
−
−
1 ( − ) = − =⇒ 1 =
2
Its cost is
1 +
2
1 +
1 −
−
+
( − )
1 + −
1 −
−
+
=
−
1 + −
¸
∙
−
1
( − ) +
=
−
1 +
=
Accordingly, the call option’s price is
∙
¸
2
1
−
( − ) +
=
( ) = 1 +
1 +
−
1 +
23
Again, we have found another method to determining the price of European
call option. That is, try to find a portfolio to replicate the option’s payoffs
at maturate date, the price of the option must be equal to the cost of the
portfolio, otherwise there would be some arbitrage opportunities in the market! Amazing! Actually, Merton’s main contribution is this idea for getting
Nobel price.
If the underlying security markets are complete, then all the options are
redundant. However, if the underlying security markets are not complete,
then some options may not be redundant. Thus, the underlying securities
plus the options together may make the markets complete.
Example 2 (The Lucas tree economy)
%
&
•
=0
state 1
state 2
=1
There is only one security in the economy, its payoffs are
%
•
=0
&
1
with 1 2
2
=1
Thus, the market is incomplete!
Consider one European call option written on the security with strike
price 2 , its payoffs at time 1 are
½
1 − 2 if state 1
+
( − 2 ) =
0
if state 2
Is the option redundant?
No since
¯
¯ − 2
|| = ¯¯ 1 1
2
0
¯
¯
¯ = −2 (1 − 2 ) 0
¯
24
Thus, the original security plus the option written on the security constitute
complete markets.
Next, we consider a special security .
Definition 7 Say that a security
⎛
is a state-index security if
⎜
⎜
=⎜
⎝
1
2
..
.
Ω
⎞
⎟
⎟
⎟
⎠
6= 0 for any 6= 0
Without loss of generality, assume
1 2 · · · Ω
Can you imagine how to make market complete? Ω-states, only one stateindex security . We need Ω−1 new securities to complete market. We write
Ω − 1 options on the state-index security as follows:
— Call option 1 with strike price 1 , its payoffs are
⎛
⎞
0
⎜ 2 − 1 ⎟
⎜
⎟
+
1
= ( − 1 ) = ⎜
⎟
..
⎝
⎠
.
Ω − 1
— Call option 2 with strike price 2 , its payoffs are
⎛
⎞
0
⎜
⎟
0
⎜
⎟
⎜
⎟
+
2
−
2 ⎟
= ( − 2 ) = ⎜ 3
⎜
⎟
..
⎝
⎠
.
Ω − 2
···
25
— Call option Ω − 1 with strike price Ω−1 , its payoffs are
⎛
⎞
0
⎜
⎟
0
⎜
⎟
⎜
⎟
..
Ω−1 = ( − Ω−1 )+ = ⎜
⎟
.
⎜
⎟
⎝
⎠
0
Ω − Ω−1
Together, there are Ω securities: one underlying state - index security
plus Ω − 1 European call options written on it. Since
¯
¯
¯
¯ 1
0
···
0
¯
¯
¯
¯ 2 2 − 1 · · ·
¯
¯
0
¯
¯
|| = ¯ 1 · · · Ω−1 ¯ = ¯ ..
¯
..
..
...
¯
¯ .
.
.
¯
¯
¯ Ω Ω − 1 · · · Ω − Ω−1 ¯
= 1 (2 − 1 ) × · · · × (Ω − Ω−1 ) 6= 0
Therefore, the markets are complete.
That is, option has a new function to make market complete.
How to price American options? This is a very challenging question.
Unfortunately, there is no clear answer yet up to now!
Practice Problems
3. There are three securities in an economy as follows:
⎛ ⎞
⎛ ⎞
⎛ ⎞
0
1
1
⎝
⎠
⎝
⎠
⎝
1
0
1 ⎠
=
=
and =
1
1
0
with prices
= = = 1
(a) Prove that there are no any arbitrage opportunities in the economy. Find the state price vector(s).
(b) Consider a European call option written on security with strike
price 12.
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i. Use the state price vector you derived in part (a) to calculate
the price ( 12) of the call option.
⎞
⎛
ii. Find a portfolio = ⎝ ⎠ replicating the European call
option. Find the portfolio’s cost
iii. Is ( 12) equal to ?
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