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Fi8000
Option Valuation I
Milind Shrikhande
A Call Option
A European call option gives the buyer
of the option a right to purchase the
underlying asset, at the exercise price
on the expiration date.
It is optimal to exercise the call option if the
stock price exceeds the strike price:
CT = Max {ST – X, 0}
A Put Option
A European put option gives the buyer
of the option a right to sell the
underlying asset, at the exercise price
on the expiration date.
It is optimal to exercise the put option if the
stock price is below the strike price:
PT = Max {X - ST , 0}
The Put Call Parity
If two portfolios have the same payoffs in
every possible state and time in the future,
their prices must be equal:
X
C SP
(1 rf )T
Arbitrage – the Law of One Price
If two assets have the same payoffs in every
possible state in the future and their prices
are not equal, there is an opportunity to
make an arbitrage profit.
We say that there exists an arbitrage
opportunity if we identify that:
There is no initial investment
There is no risk of loss
There is a positive probability of profit
Arbitrage – a Technical Definition
Let CFtj be the cash flow of an investment
strategy at time t and state j. If the following
conditions are met this strategy generates
an arbitrage profit.
(i) all the possible cash flows in every
possible state and time are positive or zero -
CFtj ≥ 0 for every t and j.
(ii) at least one cash flow is strictly positive -
there exists a pair ( t , j ) for which CFtj > 0.
Arbitrage – an Example
Is there an arbitrage opportunity if the
following are the market prices of the assets:
The price of one share of stock is $39;
The price of a call option on that stock,
which expires in one year and has an
exercise price of $40, is $7.25;
The price of a put option on that stock,
which expires in one year and has an
exercise price of $40, is $6.50;
The annual risk free rate is 6%.
Arbitrage – an Example
In this case we must check whether the put
call parity holds. Since we can see that this
parity relation is violated, we will show that
there is an arbitrage opportunity.
X $40
C $7.25 $44.986
(1 rf )T
(1 0.06)1
S P $39 $6.50 $45.5
The Construction of
an Arbitrage Transaction
Constructing the arbitrage strategy:
1. Move all the terms to one side of the equation
so their sum will be positive;
2. For each asset, use the sign as an indicator of
the appropriate investment in the asset. If the
sign is negative then the cash flow at time t=0 is
negative (which means that you buy the stock,
bond or option). If the sign is positive reverse
the position.
Arbitrage – an Example
In this case we move all terms to the LHS:
X
S P C $45.5 $44.986 $0.514 0
(1 rf )T
i.e.
X
S P C 0
(1 rf )T
Arbitrage – an Example
Time: → t=0 t=T
Strategy: ↓ State: → ST < X = 40 ST > X = 40
Short stock +S=$39 -ST -ST
Write put +P=$6.5 -(X-ST) 0
Buy call -C=(-$7.25) 0 (ST-X)
Buy bond -X/(1+rf)=(-$37.736) X X
Total CF S+P-C-X/(1+rf) -ST -(X-ST)+X -ST -(X-ST)+X
= 0.514 > 0 =0 =0
Valuation of Options
☺Quantitative Valuation
☺Binomial model (an algorithm)
☺Black-Scholes model (a formula)
☺Arbitrage Restrictions on the Values of Options
☺European Call and Put options
☺American vs. European Call option
☺American vs. European Put option
☺The Put-Call Parity
☺The option price and the exercise price
☺Option Price Convexity (options with the same
expiration date but different exercise prices)
Notation
S = the price of the underlying asset (stock)
C = the price of a call option
P = the price of a put option
X or K = the exercise or strike price
T = the expiration date
t = a time index
The Value of a Call Option
Assumptions:
1. A European Call option on a stock
2. The stock pays no dividends before
expiration
3. The stock is traded
4. A risk free bond is traded
Arbitrage restrictions:
Max{ S-PV(X) , 0 } ≤ CEU < S
The Value of a Call Option
Max{ S-PV(X) , 0 } ≤ CEU ≤ S
☺ 0 ≤ C : the owner has a right but not an
obligation.
☺ S-PV(X) ≤ C : arbitrage proof.
☺ C ≤ S : you will not pay more than $S, the
market price of the stock, for an option to buy
that stock for $X. Buying the stock itself is
always an alternative to buying the call option.
The Value of a Call Option
C
S S-PV(X)
0 PV(X) S
The Value of a Call Option
Example:
☺ The current stock price is $83
☺ The stock will not pay dividends in the next six
months
☺A call option on that stock is traded for $3
☺ The exercise price is $80
☺ The expiration of the option is in 6 months
☺ The 6 months risk free rate is 5%
Is there an opportunity to make an arbitrage profit?
The Value of a Call Option
Example:
☺ Check for arbitrage opportunities
☺ Show the investment strategy and
calculate the arbitrage profits
☺ Show the general proof for this case
The Value of a Call Option
Assumptions:
1. A Call option on a stock
2. The stock pays no dividends before expiration
3. The stock is traded
4. A risk free bond is traded
Arbitrage restriction:
C(European) = C(American)
Hint: compare the payoff from immediate exercise to the lower
bound of the European call option price.
The Value of a Put Option
Assumptions:
1. A European Put option on a stock
2. The stock pays no dividends before
expiration
3. The stock is traded
4. A risk free bond is traded
Arbitrage restrictions:
Max{ PV(X)-S , 0 } ≤ PEU < PV(X)
The Value of a Put Option
Max{ PV(X)-S , 0 } ≤ PEU < PV(X)
☺ 0 ≤ P : the owner has a right but not an
obligation.
☺ PV(X)-S ≤ P : arbitrage proof.
☺ P < PV(X) : the highest profit from the put
option is realized when the stock price is zero.
That profit is $X realized on date T, which is
PV($X) today.
The Value of a Put Option
P
PV(X)
PV(X)
PV(X)-S
0 PV(X) S
The Value of a Put Option
Example:
☺ The current stock price is $75
☺ The stock will not pay dividends in the next six
months
☺A put option on that stock is traded for $1
☺ The exercise price is $80
☺ The expiration of the option is in 6 months
☺ The 6 months risk free rate is 5%
Is there an opportunity to make an arbitrage profit?
The Value of a Put Option
Example:
☺ Check for arbitrage opportunities
☺ Show the investment strategy and
calculate the arbitrage profits
☺ Show the general proof for this case
The Value of a Put Option
Assumptions:
1. A put option on a stock
2. The stock pays no dividends before
expiration
3. The stock is traded
4. A risk free bond is traded
For a put option we always have:
P(European) < P(American)
Note: in this case an example is enough.
Exercise Prices
Let X1 < X2 be two different exercise prices;
C1, C2 be the appropriate call option prices; and
P1, P2 be the appropriate put option prices
(we assume that all the options are European, on
the same stock S that pays no dividends, with the
same expiration date T).
Then,
C(X1) ≥ C(X2) and P(X1) ≤ P(X2)
Example
Show that if there are two call options on the
same stock (that pays no dividends), and
both have the same expiration date but
different exercise prices as follows, there is
an opportunity to make an arbitrage profit.
X1= $40 and X2= $50
C1= $3 and C2= $4
Option Price Convexity
Let X1 < X2 < X3 be three different exercise prices,
and C1, C2, C3 be the appropriate call option
prices, and P1, P2, P3 be the appropriate put option
prices (all the options are European, on the same
stock S that pays no dividends, with the same
expiration date T).
If X 2 X 1 (1 ) X 3
Then C2 C1 (1 )C3
and P2 P (1 ) P3
1
Example
Show that if there are three call options on
the same stock (that pays no dividends),
and all three have the same expiration date
but different exercise prices as follows, there
is an opportunity to make an arbitrage profit.
X1= $40, X2= $50 and X3= $60
C1= $4.6, C2= $4 and C3= $3
Binomial Option Pricing Model
Assumptions:
☺ A single period
☺ Two dates: time t=0 and time t=1
☺ The future (time 1) stock price has only two
possible values
☺ The price can go up or down
☺ The perfect market assumptions
☺ No transactions costs, borrowing and lending at the
risk free interest rate, no taxes…
Binomial Option Pricing Model
Example
The stock price Assume S= $50,
u= 10% and d= (-3%)
Su=$55
Su=S·(1+u)
S=$50
S
Sd=S·(1+d) Sd=$48.5
Binomial Option Pricing Model
Example
The call option price Assume X= $50,
T= 1 year (1 period)
Cu= $5
Cu= Max{Su-X,0}
= Max{55-50,0}
C
C
Cd= Max{Sd-X,0} Cd= $0
= Max{48.5-50,0}
Binomial Option Pricing Model
Example
The bond price Assume r= 6%
$1.06
(1+r)
$1
1
(1+r) $1.06
Replicating Portfolio
At time t=0, we can create a portfolio of N shares
of the stock and an investment of B dollars in the
risk-free bond. The payoff of the portfolio will
replicate the t=1 payoffs of the call option:
N·$55 + B·$1.06 = $5
N·$48.5 + B·$1.06 = $0
Obviously, this portfolio should also have the same
price as the call option at t=0:
N·$50 + B·$1 = C
We get N=0.7692, B=(-35.1959) and the call option price is C=$3.2656.
A Different Replication
The price of $1 in the The price of $1 in the
“up” state: “down” state:
$0
$1
qd
qu
$0 $1
Replicating Portfolios Using the
State Prices
We can replicate the t=1 payoffs of the stock and
the bond using the state prices:
qu·$55 + qd·$48.5 = $50
qu·$1.06 + qd·$1.06 = $1
Obviously, once we solve for the two state prices
we can price any other asset such as the call
option:
qu·$5 + qd·$0 = C
We get qu=0.6531, qd =0.2903 and the call option price is C=$3.2656.
Binomial Option Pricing Model
Example
The put option price Assume X= $50,
T= 1 year (1 period)
Pu= $0
Pu= Max{X-Su,0}
= Max{50-55,0}
P
P
Pd= Max{X-Sd,0} Pd= $1.5
= Max{50-48.5,0}
Replicating Portfolios Using the
State Prices
We can replicate the t=1 payoffs of the stock and
the bond using the state prices:
qu·$55 + qd·$48.5 = $50
qu·$1.06 + qd·$1.06 = $1
But the assets are exactly the same and so are the
state prices. The put option price is:
qu·$0 + qd·$1.5 = P
We get qu=0.6531, qd =0.2903 and the put option price is P=$0.4354.
Two Period Example
☺ Assume that the current stock price is $50,
and it can either go up 10% or down 3% in
each period.
☺ The one period risk-free interest rate is
6%.
☺ What is the price of a European call option
on that stock, with an exercise price of $50
and expiration in two periods?
The Stock Price
S= $50, u= 10% and d= (-3%)
Suu=$60.5
Su=$55
S=$50 Sud=Sdu=$53.35
Sd=$48.5
Sdd=$47.05
The Bond Price
r= 6% (for each period)
$1.1236
$1.06
$1 $1.1236
$1.06
$1.1236
The Call Option Price
X= $50 and T= 2 periods
Cuu=Max{60.5-50,0}=$11.5
Cu
C Cud=Max{53.35-50,0}=$3.35
Cd
Cdd=Max{47.05-50,0}=$0
State Prices in the Two Period Tree
We can replicate the t=1 payoffs of the stock and the bond
using the state prices:
qu·$55 + qd·$48.5 = $50
qu·$1.06 + qd·$1.06 = $1
Note that if u, d and r are the same, our solution for the
state prices will not change (regardless of the price levels
of the stock and the bond):
qu·S·(1+u) + qd·S ·(1+d) = S
qu ·(1+r)t + qd ·(1+r)t = (1+r)(t-1)
Therefore, we can use the same state-prices in the two
period tree.
The Call Option Price
qu= 0.6531 and qd= 0.2903
Cuu=$11.5
Cu
C Cud=$3.35
Cd
Cdd=$0
Two Period Example
☺ What is the price of a European put option on
that stock, with an exercise price of $50 and
expiration in two periods?
☺ What is the price of an American call option on
that stock, with an exercise price of $50 and
expiration in two periods?
☺ What is the price of an American put option on
that stock, with an exercise price of $50 and
expiration in two periods?
Two Period Example
☺ European put option - use the tree or the
put-call parity
☺ What is the price of an American call
option - if there are no dividends …
☺ American put option – use the tree
The European Put Option Price
qu= 0.6531 and qd= 0.2903
Puu=$0
Pu
P Pud=$0
Pd
Pdd=$2.955
Pu = 0.6531*0 + 0.2903*0 = $0
Pd = 0.6531*0 + 0.2903*2.955 = $0.8578
PEU = 0.6531*0 + 0.2903*0.8578 = $0.2490
The American Put Option Price
qu= 0.6531 and qd= 0.2903
Puu=$0
Pu
PAm Pud=$0
Pd
Pdd=$2.955
American Put Option
Note that at time t=1 the option buyer will
decide whether to exercise the option or
keep it till expiration.
If the payoff from immediate exercise is
higher than the option value the optimal
strategy is to exercise:
If Max{ X-Su,0 } > Pu(European) => Exercise
The American Put Option Price
qu= 0.6531 and qd= 0.2903
Puu=$0
Pu
P Pud=$0
Pd
Pdd=$2.955
Pu = Max{ 0.6531*0 + 0.2903*0 , 50-55 } = $0
Pd = Max{ 0.6531*0 + 0.2903*2.955 , 50-48.5 } = 50-48.5 = $1.5
PAm = Max{ 0.6531*0 + 0.2903*1.5 , 50-50 } = $0.4354 > $0.2490 = PEu
Determinants of the Values
of Call and Put Options
Variable C – Call Value P – Put Value
S – stock price Increase Decrease
X – exercise price Decrease Increase
σ – stock price volatility Increase Increase
T – time to expiration Increase Increase
r – risk-free interest rate Increase Decrease
Div – dividend payouts Decrease Increase
Practice Problems
BKM Ch. 21:
End of chapter - 1, 2
Example 21.1 and concept check Q #4
(pages 755-6)
Practice set: 17-24, 25-35
