# Calculate Value of a Put Option

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```					Option Pricing
Approaches

Valuation of options

FIN 819
Today’s plan
   Review of what we have learned about
options
   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

FIN 819
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

FIN 819
The basic idea behind the
binomial tree approach
   Suppose we want to value a call option on
IBM 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 present
value of the call option.
   Let the current price of IBM is S and there
are two states when the call option matures:
up and down. If the state is up, the stock
price for IBM is Su; if the state is down, the
price of IBM is Sd.

FIN 819
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

Su=uS and Sd=dS

FIN 819
The risk free security
   The price now and at maturity

Rf
Here Rf=1+rf

1
Rf
now
maturity

FIN 819
The call option payoff
Cu=Max(uS-K,0)

C(K,T)

Cd=Max(dS-K,0)

Now
maturity

FIN 819
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.

FIN 819
Now form a replicating portfolio
(continue)
   Since we have three securities for
investment: the stock of IBM, the risk-
free security, and the call option, how
can we form this portfolio to figure out
the price of the call option on IBM?

FIN 819
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?

FIN 819
How can we get a replicating
portfolio?
   Look at the payoffs for the option and the
portfolio Cu=max(uS-K,0) Portfolio  ΔuS+BRf

Option

B+ΔS
C(K,T)

ΔdS+BRf
Cd=max(dS-K,0)
Now   maturity
now                 Maturity

FIN 819
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
FIN 819
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
Cu  Cd            uCd  dCu
            and B 
(u  d ) S         (u  d ) R f

FIN 819
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.

FIN 819
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

FIN 819
Example of valuing a call
   Suppose that a call 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
0.4 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)

FIN 819
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.

FIN 819
How can we get a replicating
portfolio of a pot option?
   Look at the payoffs for the put option
and the portfolio   Portfolio

Put option                                        ΔuS+BRf
Cu=max(K-uS,0)

B+ΔS
P(K,T)
ΔdS+BRf
Cd=max(K-dS,0)

Now
maturity
now                Maturity

FIN 819
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
FIN 819
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
C  Cd            uC  dCu
 u         and B  d
(u  d ) S        (u  d ) R f

FIN 819
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.

FIN 819
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)

FIN 819
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)

FIN 819
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?

FIN 819
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.

FIN 819
Risk-neutral probability
   The price of call option is
Cu  Cd uCd  dCu
C  S  B         
ud     (u  d ) R f
1 Rf d       u  Rf    
          Cu         Cd 
Rf  u d       ud      

   let p=(Rf-d)/(u-d) < 1. Then
1
C     pCu  (1  p)Cd 
Rf

FIN 819
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.

FIN 819
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.

FIN 819
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?

FIN 819
Solution
    First draw the stock price for each period
and option payoff at the expiration 27.67
p
Stock price            82.67      Option
p
67.43                                  1-p           0
C(K,T)=?
55                                    p
55                                       1-p
1-p              0
44.88
36.62
Three            Six
Now         month
Three   Sixth                                      month
Now
month   month

FIN 819
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
0.473*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

FIN 819
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:

u  e h
d  1/ u
   Where h the interval as a fraction of year. For
example, h=1/4=0.25 if the interval is three month.

FIN 819
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?

FIN 819
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.

FIN 819
A three period binomial tree
u3S

u2dS

S                                    ud2S

d3S

There are three periods. We have four possible values for the stock price

FIN 819
The Black-Scholes formula for a
call option
   The Black-Scholes formula for a European call is
C ( S , K , t , r , )  SN (d1 )  Ke rt N (d1   t )
   Where
ln( S / K )  rt 1
d1                      t
 t         2

N (d )  cumulative normal distributi
on function
t  time to exp iration
r  continuous compounded risk free rate
ly
  stock return volatility per period
S  today' s stock price
K  strike price of the option
FIN 819
The Black-Scholes formula for a
put option
   The Black-Scholes formula for a European put is
P ( S , K , t , r , )  Ke rt N ( t  d1 )  SN ( d1 )
   Where
ln( S / K )  rt 1
d1                      t
 t         2

N (d )  cumulative normal distributi
on function
t  time to exp iration
r  continuous compounded risk free rate
ly
  stock return volatility per period
S  today' s stock price
K  strike price of the option
FIN 819
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

FIN 819
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?

FIN 819
Solution
   The parameter values are
  0.2, t  2, r  0.08
K  70 , S  50
   Then
ln( S / K )  rt 1
d1                    t  0.4825
 t         2
N (d1 )  1  N (d1 )  1  0.685  0.315
N (d1   t )  N (0.765)  0.22
C  \$2.63; P  \$12.27

FIN 819

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