Option Pricing Approaches

1
Option Pricing Approaches

• Valuation of options

2
Todays 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

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.

6
The stock price now and at maturity
Su
uS
S
S
Sd
dS
Now
maturity
Now
maturity
If we define u
Su/S and d Sd/S. Then we have SuuS and
SddS
7
The risk free security

• The price now and at maturity

Rf
Here Rf1rf
1
Rf
now
maturity
8
The call option payoff

CuMax(uS-K,0)
C(K,T)
CdMax(dS-K,0)
Now
maturity
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.

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?

11
Now form a replicating portfolio (continue)

• Suppose we buy ? 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?

12
How can we get a replicating portfolio?

• Look at the payoffs for the option and the
  portfolio

?uSBRf
Portfolio
Cumax(uS-K,0)
Option
B?S
C(K,T)
?dSBRf
Cdmax(dS-K,0)
Now
maturity
now
Maturity
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,
• ?uSBRf Cu
• ?dSBRf Cd

14
Form a replicating portfolio

• Use the following two equations to solve for ?
  and B to get the replicating portfolio
• ?uSBRf Cu
• ?dSBRf Cd
• The solution is

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?S.
• 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.

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 ? shares of stock
  and borrow B dollars from the bank.
• The value of the option is
• ?SB ( the number of shares stock price
  borrowed money), where B is negative

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 41.25. The risk-free
  rate is 4.
• Can you calculate the value of this call option?
  (the value is 8.32) (u1.33,d0.75, Cd0, Cu
  18.33, ?0.57, B-23.1)

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.

19
How can we get a replicating portfolio of a put
option?

• Look at the payoffs for the put option and the
  portfolio

Portfolio
?uSBRf
Put option
Cumax(K-uS,0)
B?S
P(K,T)
?dSBRf
Cdmax(K-dS,0)
Now
maturity
now
Maturity
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,
• ?uSBRf Cu
• ?dSBRf Cd

21
Form a replicating portfolio

• Use the following two equations to solve for ?
  and B to get replicating portfolio
• ?uSBRf Cu
• ?dSBRf Cd
• The solution is

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.

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 41.25. The
  risk-free rate is 4.
• Can you calculate the value of this put option?
  (the value is 7.24) (u1.33,d0.75, Cu0,
  Cd13.75 ?-0.43, B30.8)

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)

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 dont we rub two rocks to
  generate fire now?
• It is clumsy, not efficient
• What have you learned from this example?

26
What can we learn?

• Using the idea in the last slide, to value a call
  option, we dont 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.

27
Risk-neutral probability

• The price of call option is
• let p(Rf-d)/(u-d) lt 1. Then

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.

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
• ( u1.33,d0.75, Rf1.02, p0.47,
  C10.4718.33,
• PV(C1) C1/Rf 8.37.
• Put.
• ( p0.47, C10.5313.75, PV(C1)C1/Rf7.14)

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?

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
0
1-p
C(K,T)?
55
p
1-p
55
1-p
0
44.88
36.62
Three month
Six month
Now
Three month
Sixth month
Now
32
Solution

• Risk-neutral probability is
• p(Rf-d)/(u-d)
• (1.01-0.816)/(1.226-0.816)0.473
• The probability for the payoff of 27.67 is
• 0.4730.473, the probability for other two
  states are 20.473527, and 0.5270.527.
• The expected payoff from the option is
• 0.4730.47327.67
• The present value of this payoff is 6.07
• So the value of the call option is 6.07

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 s, we can calculate u and d as
  following
• Where h the interval as a fraction of year. For
  example, h1/40.25 if the interval is three
  month.

34
Example for u and d

• Using the two-period binomial tree problem in the
  previous example. If s is 40.69,
• Please calculate u and d?
• Please calculate the risk-neutral probability p?
• Please calculate the value of the call option?
• (u1.17,d0.85, p 0.5)

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 n1 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.

36
A three period binomial tree
u3S
u2dS
ud2S
S
d3S
There are three periods. We have four possible
values for the stock price
37
The Black-Scholes formula for a call option

• The Black-Scholes formula for a European call is
• Where

38
The Black-Scholes formula for a put option

• The Black-Scholes formula for a European put is
• Where

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

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?

41
Solution

• The parameter values are
• Then