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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