1. Fi8000 Option Valuation I
Milind Shrikhande

2. 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}

3. 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}

4. 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:

5. 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

6. 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.
all the possible cash flows in every possible state and time are positive or zero - CFtj ≥ 0 for every t and j.
at least one cash flow is strictly positive - there exists a pair ( t , j ) for which CFtj > 0.

7. 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%.

8. 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.

9. The Construction of an Arbitrage Transaction
Constructing the arbitrage strategy:
Move all the terms to one side of the equation so their sum will be positive;
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.

10. Arbitrage – an Example
In this case we move all terms to the LHS:

11. Arbitrage – an Example Time: → t = 0 t = T Strategy: ↓ State: →
ST < X = 40
ST > X = 40
Short stock
+S=$39
-ST
Write put
+P=$6.5
-(X-ST)
Buy call
-C=(-$7.25)
(ST-X)
Buy bond
-X/(1+rf)=(-$37.736) X
Total CF
S+P-C-X/(1+rf)
= > 0
-ST -(X-ST)+X
= 0

12. 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)

13. 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

14. 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

15. 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.

16. The Value of a Call OptionS
S-PV(X)
PV(X) S

17. 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?

18. 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

19. 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.

20. 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)

21. 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.

22. The Value of a Put Option
PV(X)
PV(X)
PV(X)-S
PV(X) S

23. 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?

24. 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

25. 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.

26. C(X1) ≥ C(X2) and P(X1) ≤ P(X2)
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)

27. 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

28. 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).

29. 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

30. 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…

31. 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=$48.5
Sd=S·(1+d)

32. Binomial Option Pricing Model Example
The call option price
Assume X= $50,
T= 1 year (1 period)
Cu= $5
= Max{55-50,0}
Cu= Max{Su-X,0} C C
Cd= $0
= Max{ ,0}
Cd= Max{Sd-X,0}

33. Binomial Option Pricing Model Example
The bond price
Assume r= 6%
$1.06
(1+r) $1 1
$1.06
(1+r)

34. 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 = $5
N·$ B·$1.06 = $0
Obviously, this portfolio should also have the same price as the call option at t=0:
N·$50 + B·$1 = C
We get N=0.7692, B=( ) and the call option price is C=$

35. A Different Replication
The price of $1 in the “up” state:
The price of $1 in the “down” state: $0 $1 qd qu $1 $0

36. 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 = $50
qu·$ 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·$0 = C
We get qu=0.6531, qd = and the call option price is C=$

37. Binomial Option Pricing Model Example
The put option price
Assume X= $50,
T= 1 year (1 period)
Pu= $0
= Max{50-55,0}
Pu= Max{X-Su,0} P P
Pd= $1.5
= Max{ ,0}
Pd= Max{X-Sd,0}

38. 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 = $50
qu·$ 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·$1.5 = P
We get qu=0.6531, qd = and the put option price is P=$

39. 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?

40. The Stock Price S= $50, u= 10% and d= (-3%) Suu=$60.5 Su=$55
Sud=Sdu=$53.35
Sd=$48.5
Sdd=$47.05

41. The Bond Price r= 6% (for each period) $1.1236 $1.06 $1.1236 $1 $1.06

42. The Call Option Price X= $50 and T= 2 periods Cuu=Max{60.5-50,0}=$11.5
Cud=Max{ ,0}=$3.35 C Cd
Cdd=Max{ ,0}=$0

43. 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 = $50
qu·$ 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.

44. The Call Option Price qu= 0.6531 and qd= 0.2903 Cuu=$11.5 Cu Cud=$3.35Cd
Cdd=$0

45. 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?

46. 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

47. The European Put Option Price
qu= and qd=
Puu=$0 Pu
Pud=$0 P Pd
Pdd=$2.955
Pu = * *0 = $0
Pd = * *2.955 = $0.8578
PEU = * * = $0.2490

48. The American Put Option Price
qu= and qd=
Puu=$0 Pu
Pud=$0
PAm Pd
Pdd=$2.955

49. If Max{ X-Su,0 } > Pu(European) => Exercise
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

50. The American Put Option Price
qu= and qd=
Puu=$0 Pu
Pud=$0 P Pd
Pdd=$2.955
Pu = Max{ * *0 , } = $0
Pd = Max{ * *2.955 , } = = $1.5
PAm = Max{ * *1.5 , } = $ > $ = PEu

51. Determinants of the Values of Call and Put Options
Variable
C – Call Value
P – Put Value
S – stock price
Increase
Decrease
X – exercise price
σ – stock price volatility
T – time to expiration
r – risk-free interest rate
Div – dividend payouts

52. 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