1. Option Pricing Approaches
Valuation of options
FIN 819-lecture 7

2. Today’s plan Review of what we have learned about options
We first discuss a simple business ethics case.
We discuss two ways of valuing options
Binomial tree (two states)
Simple idea
Risk-neutral valuation
The Black-Scholes formula (infinite number of states)
Understanding the intuition
How to apply this formula

3. Business ethics
Suppose that you have applied to two jobs: A and B. Now you have received the offer letter for job A and have to make a decision now about whether or not to take job A. But you like job B much more and the decision for job B will be made one week later.
What is your decision?

4. What have we learned in the last lecture?
Options
Financial and real options
European and American options
Rights to exercise and obligations to deliver the underlying asset
Position diagrams
Draw position diagrams for a given portfolio
Given position diagrams, figure out the portfolio
No arbitrage argument
Put-call parity
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5. The basic idea behind the binomial tree approach
Suppose we want to value a call option on ABC stock with a strike price of K and maturity T. We let C(K,T) be the value of this call option.
Remember C(K,T) is the price for the call or the value of the call option at time zero.
Let the current price of ABC is S and there are two states when the call option matures: up and down. If the state is up, the stock price for ABC is Su; if the state is down, the price of ABC is Sd.
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6. The stock price now and at maturitySu uS S S Sd dS
Now
maturity
Now
maturity
If we define:
u = Su/S and d = Sd/S. Then we have
Su=uS and Sd=dS
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7. The risk free security The price now and at maturity Rf Here Rf=1+rf 1
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8. The call option payoff Cu=Max(uS-K,0) C(K,T) Cd=Max(dS-K,0) Now
maturity
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9. Now form a replicating portfolio
A portfolio is called the replicating portfolio of an option if the portfolio and the option have exactly the same payoff in each state of future.
By using no arbitrage argument, the cost or price of the replication portfolio is the same as the value of the option.
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10. Now form a replicating portfolio (continue)
Since we have three securities for investment: the stock of ABM, the risk-free security, and the call option, how can we form this portfolio to figure out the price of the call option on ABC?
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11. Now form a replicating portfolio (continue)
Suppose we buy Δ shares of stock and borrow B dollars from the bank to form a portfolio.
What is the payoff for the this portfolio for each state when the option matures?
What is the cost of this portfolio?
How can we make sure that this portfolio is the replicating portfolio of the option?
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12. How can we get a replicating portfolio?
Look at the payoffs for the option and the portfolio
ΔuS+BRf
Portfolio
Cu=max(uS-K,0)
Option
B+ΔS
C(K,T)
ΔdS+BRf
Cd=max(dS-K,0)
Now
maturity
now
Maturity
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13. Form a replicating portfolio
From the payoffs in the previous slide for the call option and the portfolio, to make sure that the portfolio is the replicating portfolio of the option, the option and the portfolio must have exactly the same payoff in each state at the expiration date.
That is,
ΔuS+BRf = Cu
ΔdS+BRf = Cd
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14. Form a replicating portfolio
Use the following two equations to solve for Δ and B to get the replicating portfolio:
ΔuS+BRf = Cu
ΔdS+BRf = Cd
The solution is
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15. To get the value of the call option
By no arbitrage argument, the value or the price of the option is the cost of the replicating portfolio, B+ΔS.
Can you believe that valuing the option is so simple?
Can you summarize the procedure to do it?
This procedure walks you through the way of understanding the concept of no arbitrage argument.
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16. Summary
Using the no arbitrage argument, we can see the cash flows from investing in a call option can be replicated by investing in stocks and risk-free bond. Specifically, we can buy Δ shares of stock and borrow B dollars from the bank.
The value of the option is
Δ*S+B ( the number of shares *stock price –borrowed money), where B is negative
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17. Example of valuing a call
Suppose that a call on ABC has a strike price of $55 and maturity of six-month. The current stock price for ABC is $55. At the expiration state, there is a probability of 0.4 that the stock price is $73.33, and there is a probability of 0.6 that the stock price is $ The risk-free rate is 4%.
Can you calculate the value of this call option? (the value is $8.32) (u=1.33,d=0.75, Cd=0, Cu= $18.33, Δ=0.57, B=-$23.1)
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18. How to value a put using the similar idea
We can use the similar idea to value a European put.
Before you look at my next two slides, can you do it yourself?
Still try to form a replicating portfolio so that the put option and the portfolio have the exactly the same payoff in each state at the expiration date.
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19. How can we get a replicating portfolio of a put option?
Look at the payoffs for the put option and the portfolio
Portfolio
ΔuS+BRf
Put option
Cu=max(K-uS,0)
B+ΔS
P(K,T)
ΔdS+BRf
Cd=max(K-dS,0)
Now
maturity
now
Maturity
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20. Form a replicating portfolio
From the payoffs in the previous slide for the put option and the portfolio, to make sure that the portfolio is the replicating portfolio of the option, the put option and the portfolio must have exactly the same payoff in each state at the expiration date.
That is,
ΔuS+BRf = Cu
ΔdS+BRf = Cd
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21. Form a replicating portfolio
Use the following two equations to solve for Δ and B to get replicating portfolio:
ΔuS+BRf = Cu
ΔdS+BRf = Cd
The solution is
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22. What happens?
You can see that the formula for calculating the value of a put option is exactly the same as the formula for a call option?
Where is the difference?
The difference is the calculation of the payoff or cash flows in each state.
To get this, please try the valuation of put option in the next slide.
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23. Example of valuing a put
Suppose that a European put on IBM has a strike price of $55 and maturity of six-month. The current stock price is $55. At the expiration state, there is a probability of 0.5 that the stock price is $73.33, and there is a probability of 0.5 that the stock price is $ The risk-free rate is 4%.
Can you calculate the value of this put option? (the value is $7.24) (u=1.33,d=0.75, Cu=0, Cd=$13.75 Δ=-0.43, B=$30.8)
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24. Example of valuing a put option (continue)
Recall that the value of call option with the same strike price and maturity is $8.32.
Can you use this call option value and the put-call parity to calculate the value of the put option?
Do you get the same results? ( if not, you have trouble)
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25. Can you learn something more?
Everybody knows how to set fire by using match.
Long, long time ago, our ancestors found that rubbing two rocks will generate heat and thus can yield fire, but why don’t we rub two rocks to generate fire now?
It is clumsy, not efficient
What have you learned from this example?
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26. What can we learn?
Using the idea in the last slide, to value a call option, we don’t need to figure out the replicating portfolio by calculating the number of shares and the amount of money to borrow. Instead we can jump to calculate the value of the call option using the way in the next slide.
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27. Risk-neutral probability
The price of call option is
let p=(Rf-d)/(u-d) < 1. Then
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28. Risk-neutral probability (continue)
Now we can see that the value of the call option is just the expected cash flow discounted by the risk-free rate.
For this reason, p is the risk-neutral probability for payoff Cu, and (1-p) is the risk-neutral probability for payoff Cd.
In this way, we just directly calculate the risk-neutral probability and payoff in each state. Then using the risk-free rate as a discount rate to discount the expected cash flow to get the value of the call option.
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29. Examples for risk-neutral probability
Using the risk neutral probability approach to calculate the values of the call and put options in the previous two examples.
Call
( u=1.33,d=0.75, Rf=1.02, p=0.47, C1=0.47*18.33,
PV(C1) = C1/Rf= $8.37.
Put.
( p=0.47, C1=0.53*13.75, PV(C1)=C1/Rf=$7.14)
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30. Two-period binomial tree
Suppose that we want to value a call option with a strike price of $55 and maturity of six-month. The current stock price is $55. In each three months, there is a probability of 0.3 and 0.7, respectively, that the stock price will go up by 22.6% and fall by 18.4%. The risk-free rate is 4%.
Do you know how to value this call?
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31. Solution
First draw the stock price for each period and option payoff at the expiration
27.67 p
Stock price
Option
82.67 p
67.43
1-p
C(K,T)=? 55 p
1-p 55
1-p
44.88
36.62
Three
month
Six
month
Now
Three
month
Sixth
month
Now
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32. Solution Risk-neutral probability is
p=(Rf-d)/(u-d)
=( )/( )=0.473
The probability for the payoff of is
0.473*0.473, the probability for other two states are 2*0.473*527, and 0.527*0.527.
The expected payoff from the option is
0.473*0.473*27.67=
The present value of this payoff is 6.07
So the value of the call option is $6.07
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33. How to calculate u and d
In the risk-neutral valuation, it is important to know how to decide the values of u and d, which are used in the calculation of the risk-neutral probability.
In practice, if we know the volatility of the stock return of σ, we can calculate u and d as following:
Where h the interval as a fraction of year. For example, h=1/4=0.25 if the interval is three month.
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34. Example for u and d
Using the two-period binomial tree problem in the previous example. If σ is 40.69%,
Please calculate u and d?
Please calculate the risk-neutral probability p?
Please calculate the value of the call option?
(u=1.17,d=0.85, p= 0.5)
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35. The motivation for the Black-Scholes formula
In the real world, there are far more than two possible values for a stock price at the expiration of the options. However, we can get as many possible states as possible if we split the year into smaller periods. If there are n periods, there are n+1 values for a stock price. When n is approaching infinity, the value of a European call option on a non-dividend paying stock converges to the well-known Black-Scholes formula.
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36. A three period binomial tree
u3S
u2dS S
ud2S
d3S
There are three periods. We have four possible values for the stock price
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37. The Black-Scholes formula for a call option
The Black-Scholes formula for a European call is
Where
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38. The Black-Scholes formula for a put option
The Black-Scholes formula for a European put is
Where
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39. The Black-Scholes formula (continue)
One way to understand the Black-Scholes formula is to find the present value of the payoff of the call option if you are sure that you can exercise the option at maturity, that is, S-exp(-rt)K.
Comparing this present value of this payoff to the Black-Scholes formula, we know that N(d1) can be regarded as the probability that the option will be exercised at maturity
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40. An example
Microsoft sells for $50 per share. Its return volatility is 20% annually. What is the value of a call option on Microsoft with a strike price of $70 and maturing two years from now suppose that the risk-free rate is 8%?
What is the value of a put option on Microsoft with a strike price of $70 and maturing in two years?
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41. Solution
The parameter values are
Then
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