Ch. 8 by hcj

VIEWS: 18 PAGES: 44

									International Financial Markets and the FirmLognormal Option Pricing Model                page 8-1




Ch. 8.                  Pricing European Options: The
                        Lognormal Model



        1.   Assumptions of the Continuous-time Option Pricing Model
        2.   A Discrete-Time Derivation of the Continuous-Time Model
             2.1.   Step 1: Computing the Expected Value of a Call Option
                      2.1.1. A Discrete-Scale Example
                      2.1.1.The Expected Call Value when the Spot Rate is
                             Lognormal
             2.2.   Step 2: Correcting the Call’s Expected Expiration
                    Value for Risk
             2.3.   Step 3: Discounting the Risk-adjusted Expiration value
                    of the Call at the Risk-free Rate.
             2.4.   Standard Notational Convention for the Continuous-
                    time Call Pricing Model.
        3.   How to Use the Continuous-Time Option Valuation Formula
             3.1.   A Numerical Example
             3.2.   How to Use the Formula for Delta-hedging
        4.   Related Option Pricing Models
             4.1.   The Value of European Put Option
             4.2.   The Value of European Options on a Futures Contract
             4.3.   The Value of European Currency Options with
                    Stochastic Interest Rates
        5.   Conclusions
        Appendix A: Derivation of the Expected Expiration Value of the
             Call Option
        Appendix B: Stochastic Calculus and the Black-Scholes
             Differential Equation




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    •   Binomial model: time is discrete, S is from a discrete scale.
                                           ∆Cn+1 ∆Ct+∆t
                             hedge ratio = ∆S     = ∆S
                                              n+1      t+∆t

• Black-Scholes-Merton: time is continuous, S is lognormal
  (i.e. from a continuous scale).
                                           ∂Ct+dt                    ∂Ct
                             hedge ratio = ∂S                      = ∂S
                                              t+dt                      t

• Samuelson-Rubinstein-Brennan:                                 time is discrete, S is
  lognormal.



Links
• The BSM and SRB models yield the same formula for
  European options. The bimonial model converges to this
  formula.

• The binomial and BSM approach can be used for more
  complicated options, like American options
  • binomial:           stepwise,                               using                 Cn,j               =
     q Cn+1,j+1 + (1-q) Cn+1,j
 1+r

    • BSM: numerical solution of a partial differential equation




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1.        Assumptions of the Continuous-time Option Pricing
          Model

1. The process for the exchange rate is continuous.

2. The value of the option is a continuous and twice
  differentiable function of the underlying process S.
     [Thus: (1) over a short time interval the changes in the exchange rate will be
     small, and (2) the effect of a small change in the spot rate on the call price is
     always well-defined. Thus, hedging works.]

      contract value
                                                                                               value forward
                                                                                                      contract




                                                                                               exposure line =
                                                                                             tangency line in S




                          C
                                                                                                     S
                                          S–dS S S+dS

3. Trading is continuous.

   [You can adjuste the hedge all the time the option price is always correct.]




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4. The distribution of the percentage changes in the exchange is
  lognormal—or the continuously compounded change in the
  spot rate is normally distributed.

5. The risk-free rate(s), and the variance of the ("continuously
  compounded") percentage changes in the spot rate are
  constant over the option's life.

   [4 and 5 correspond to the assumption in the binomial model that the process is
   multiplicative and that u, d, and (1+r), (1+r*) are constant over time. ]




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2.       A Discrete-Time Derivation of the Continuous-Time
         Model

Samuelson [1967], Rubinstein [1976], Brennan [1979]:

                                                             ˜
1. Compute the expected value of the option at maturity, E t(C
   T).


2. Correct this expected value for risk. That is, compute
         ˜           ˜                     ˜              ˜
   CEQt(C T) from Et(C T), by replacing Et(S T) with CEQt(S T) =
   Ft,T.

3. Discount the risk-adjusted expected future value at the risk-
   free rate to determine the call’s value at time zero, Ct. That
   is,

                                              CEQt(CT)˜
(1)                                      Ct = 1+r
                                                  t,T




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               2.1. Step 1: Computing the Expected Value
                             of a Call Option

2.1.1.        A Discrete-Scale Example
Consider a call with X = 43.

  ˜
  S T may be 38 39 40 41 42 43 44 45 46 47
  with probt,T 0 .05 .10 .15 .20 .20 .15 .10 .05 0

       ˜
  then C T =             0 0 0 0 0 0 1 2 3                                                     4
  with probt,T           0 .05 .10 .15 .20 .20 .15 .10 .05                                     0

                     ˜
                  Et(C T) = 0 + 0 + ... + (1  0.15) + (2  0.1)
                          + (3  0.05) + (4  0) = 0.5


                        = (43 – 43)  0.20 + (44 – 43)  0.15
                        + (45 – 43)  0.10 + (46 – 43)  0.05


      = [(43  0.20) + (44  0.15) + (45  0.1) + (46  0.05)]
                   – 43  [.20 + 0.15 + 0.1 + 0.05]




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                                
(5)                ˜
                Et(C T) =  ST  Probt,T(ST)                                      –X
                              ST=X
 
 Probt,T(ST)
ST=X

                   =            [Sum A]                  –         X  [Sum B]
                                 partial mean                  X  prob of ending in the money




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2.1.1.        The Expected Call Value when the Spot Rate is
              Lognormal

            ˜                       ˜
Denote Et(lnS T) by µt,T, and sdt(lnS T) by t,T. Then


              ∞
       ˜
(7) Et(C T) = ST f(ST; µt,T,t,T)dST
                                                                                                 – X
                         X
∞
f(S ; µ , )dS
   T   t,T t,T T
X




                    =        [Integral A]                         – X  [Integral B]
                             partial mean                         – X  prob of ending
                                                                             in the money
[après maintes péripéties:]

(8)                      ˜
                    = Et(S T) N(d' )
                                 1                                – X N(d' )
                                                                         2


where N(d'i ) denotes the cumulative standard normal probability:




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                                                     n(z)

                        N(d)




                                                                                                         z
                                                d     0

                                               ˜
                                            Et(ST) 1 2
                                       ln     X + 2 t,T
(9)                         d' =                         , d' = d'1 – t,T .
                             1
                                               t,T         2




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2.2. Step 2: Correcting the Call’s Expected Expiration Value
                           for Risk

In the BSM or binomial logic, and also in the SRB model, risk
correction means replacing Et(ST) by Ft,T:



(10)                            ˜
                           CEQt(C T) = Ft,T N(d1) – X N(d2)
                                               Ft,T 1 
                                            ln X + 2 t,T
(11a)                               d1 =
                                                 t,T

(11b)                                      d2 = d1 – t,T



 2.3. Step 3: Discounting the Risk-adjusted Expiration value
               of the Call at the Risk-free Rate.

                       CEQt(CT) ˜
(12)             Ct = 1 + r
                            t,T


                      Ft,T                               X
                  =+r
                  1 t,T                    N(d1) – 1 + r                       N(d2)
                                                         t,T


                            St                  X
(13)                    = 1 + r * N(d1) – 1 + r                              N(d2)
                               t,T              t,T



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    2.4. Standard Notational Convention for the Continuous-
                   time Call Pricing Model.

The convention in the literature and among practitioners is to
quote all data on an annualized basis.

• The p.a. variance is typically denoted by the (non-subscripted)
  symbol 2. Thus,
                                           2            2
                                        t,T  =  (T-t),

• The riskfree rate is typically a continuously compounded, p.a.
  interest rate, denoted by the (non-subscripted) symbol r (HC)
  and r* or r' (FC). Thus,
                   (1 + rt,T) = er (T-t) and (1 + rt,T ) = er* (T-t)
                                                    *


        Example
        life is 201 days                 T–t = 201/365 = 0.55 years

        volatility 14.14% p.a.                   variance = 0.14142 = 0.02 p.a.
                                                    2
                                                 t,T  = .55  .02 = .011

        r (p.a., cc): 9.7347%                    1 + rt,T = e0.097347  0.55 = 1.055

        r* (p.a., cc): 5.9031%                   1 + rt,T = e0.059031  0.55 = 1.033
                                                       *




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Thus:
                                     1+rt,T
                            Ft,T = St *
                                   1+r                    = St e(r-r*) (T-t) 
                                           t,T


                               Ft,T             St
                            ln X           = ln X + (r–r*) (T–t).



(14)                    Ct = St e–r* (T–t) N(d1) – X e–r (T–t) N(d2)


                                                                      1
                               ln(St/X) + (r–r*)(T–t) + 2 2(T–t)
                    d1 =
                                                    (T–t)

                                     d2 = d1 –  (T–t)




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3.       How to Use the Continuous-Time Option Valuation
         Formula

                              3.1. A Numerical Example

data:

• S = USD/DEM 0.45, or 45 cents, and X = USD/DEM 0.43, or
  43 cents.

     We use exchange rates expressed in cents, and keep in mind that the Black-
     Scholes-Merton formula then yields an option premium that is likewise
     expressed in cents.




• The option expires in 210 days. Thus, (T-t) = 201/365 = 0.55
  years



• The volatility is 14.14%. This is a per annum figure, so the
  effective variance equals 2(T–t) = .55  .02 = .011, and the
  effective standard deviation is  r(T–t) = 0.1414 0.55 =
  0.10488.



• The riskfree simple interest rates are 10% on USD, and 6% on
  DEM. Thus, 1 + rt,T = 1.055, and 1 + rt,T = 1.033.
                                           *



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Thus:
                    1+rt,T                         1.055
           Ft,T = St *
                  1+r t ,T                    = 45 1.033 = 45.958 (cents)

              ln(F/X) = ln(45.958/43) = ln(1.0688) = .066536
                                                    1
                             0.066536 + 2 0.011
                        d1 =      0.10488       = 0.686824

                              d2 = d1 – 0.10488 = 0.583098.

                    N(d1) = 0.753935 ,                      N(d2) = 0.720086
                          St                  45
                        1 + r*t ,T   =       1.033 = 43.5624 (cents)
                            X                                43
       X e –r (T–t) = 1 + r                             =   1.055 = 40.7583 (cents)
                            t,T




            Ct = (43.5624  0.753935) – (40.7583  0.720086)

                                       = 3.48432 (cents)




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            3.2. How to Use the Formula for Delta-hedging

When valuing the call, we have in fact computed the value of a
portfolio containing a certain amount of DEM and USD T-bills:
• The second term in the solution of Ct corresponds to a
  purchase of –N(d2) = –0.7196 USD T-bills, each having a face
  value of 43 cents. So the amount invested in domestic T-bills
  is
                   N(d1)         .7196
             –43  1+r * = –43  1.055 = –29.33 UScents ,
                      t,T


    That is, one takes out a riskfree loan of 29.33.


• The first term corresponds to buying N(d1) = 0.7538 DEM T-
  bills. Since each DEM T-bill costs DEM 1/1.033, the amount
  of DEM required in order to buy these T-bills is
                             N(d1)    .7538
                             1+rt,T
                                 *
                                    = 1.033 = DEM 0.73

                    N(d1)
            ∆spot = 1+r * = number of foreign currency units
                       t,T


                        to be bought spot and invested until T
            N(d1)
   ∆frwd = 1+r                      = number of currency units to be bought
               t,T

                                forward (for delivery at T)

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4.       Related Option Pricing Models

In this section we describe how the currency call pricing model
given in (13) or (14) is related to other option valuation models
in the finance literature.

                   4.1. The Value of European Put Option

We can find the value of a European put by starting from the
Put-Call-Parity Theorem and the substituting the valuation
formula for the call:


                           X                       St
(16)              Pt = 1+r                    –
                                                  1+rt,T
                                                      *
                                                            +   Ct
                           t,T


                              X                     St             St
                        = 1+r                  +
                                                   1+rt,T
                                                       *
                                                              +
                                                                  1+rt,T (N(d1))
                                                                      *
                                                                                          –
                              t,T
     X
1+rt,T (N(d2))
                             X                                         St
                        = 1+r (1–N(d2))                           –
                                                                      1+rt,T (1–N(d1))
                                                                          *
                             t,T


                              X                         St
                        = 1+r                 N(–d2) – 1+r * N(–d1)
                              t,T                                     t,T


                              X                        St
                        = 1+r                 N(e1) – 1+r * N(e2)
                              t,T                                 t,T




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                        = X e–r(T–t) N(e1) – S e–r*(T–t) N(e2)




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 4.2. The Value of European Options on a Futures Contract

When interest rates are known, forward and futures prices are
identical; and even when interest rates are uncertain we can
ignore the difference between both prices as trivial. Thus, we set
ft,T = Ft,T.



• If the futures contract and the option expire at the same time
  (T), then the option on the forward contract has the same
  payoff as the option on the cash. Thus, it must have the same
  present value, too.

    We use (11) and (12), which is our option price expressed in
    terms of Ft,T rather than St:

(17)           Value of a European call on a futures contract expiring at T:

                                ft,T                      X
                        = 1+r                 N(d1) – 1+r                    N(d2)
                                t,T                       t,T

                                          1
               ln(ft,T/X) + 2 t,T2
    where d1 =                                                     and       d2 = d1 – t,T
                  t,T



• If the futures contract expires at T 2>T, use ft,T2 instead of ft,T,
  but do not change T in the variance: use t,T2, or 2(T-t)).



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        4.3. The Value of European Currency Options with
                     Stochastic Interest Rates

Merton uses B(t,T) to denote 1/(1+r t,T) and B*(t,T) to denote
1/(1+r* ,T), and then shows that the model still holds when bond
       t

prices are not fully predictable—as long as the bond's day-by-day
variance is known in advance.

[Popular assumption: the day-by-day variance is proportional to the bond's
duration.]

Then (Merton (1973); Grabbe (1983)):
(19)                    C = S B*(t,T) N(d1) – X B(t,T) N(d2)
                                          1                           1
                           ln F/X ± 2 T,t               ln F/X ± 2 2 (T–t)
                   di =
                                  t,T               =
                                                                  T–t
                    2 = annualized average variance of dlnF



• we should use the average variance of the forward rate (for
  delivery at T) rather than the variance of the spot rate.

• This variance typically higher because domestic and foreign
  interest rates are uncertain and imperfectly correlated.

• Grabbe’s model therefore prices options higher than the
  Garman-Kohlhagen model.


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5.       Warnings

1. The model assumes continuous rebalancing. In practice
   rebalancing occurs far less frequently. Thus, a linear hedge
   will not mimic the call price as perfectly as one would like.

      Hint: use Gamma-hedging:

                                     ∂C      ∂C      1 ∂2C
                                   ˜
                                ∆C – ∂t ∆t + ∂S ∆S + 2 2 (∆S) 2
                                                       ∂S
                                                                    ∂C
      Form a portfolio of FC and FC-options that has the same delta ∂t and the
                   ∂2C
      same gamma         as the option to be hedged/replicated.
                   ∂S2




2. Sudden jumps in the exchange rate, and changes in its
   volatility, are not taken into account by the model. Thus,
   delta-hedging will not protect us against jumps in the
   exchange rate, or changes in volatility.



3. The assumption that the variance of the log exchange rate,
     2
   t,T , is proportional to the horizon T–t may be inappropriate
   in the long run.




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4. Interest rates are not constant. This is important for long-
   term options.

      Hint: use the Merton-Grabbe model with the higher .



5. Exchange rate changes have distributions that are fat-tailed;
   that is, the probability of extreme events is somewhat higher
   than the lognormal model predicts. As a result, options
   should probably be priced higher than what the lognormal
   model predicts, because options thrive on uncertainty.




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                    Appendix A: Integrals A & B
            A.1. The Probability of Exercising (Integral B)

                     ˜
The probability that S T ≥ X can be read off immediately from the
standard normal tables:



                                       ˜
                                    lnST – µt,T                    lnX – µt,T
      ˜
(A.2)(S T ≥ X)  (
                                    t,T
                                                                ≥
                                                                 t,T
                                                                                                  )

                                                                     lnX – µt,T
                       ˜
                 Prob(S T ≥ X) = prob(z ≥
                                       ˜
                                                                  t,T
                                                                                                     )

                                                                   µt,T – lnX
(A3)                                           = prob(z ≤
                                                      ˜                              )
                                                                    t,T
                                                         µt,T – lnX
(A.4)                           N(d'2) , d'2 =
                                                          t,T



        In short, we have evaluated Integral B in (7) as follows:
                        ∞
(A.5)             X     
                                                            ˜
                            f(ST; µt,T, t,T) dST = X Probt (S T ≥ X)
                        X


                                                           = X N(d'2)


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                                                       ˜
                              A.2. The Partial Mean of S T



              ˜
Lemma: Assume S is lognormal with mean  and stdev . Then
                                                       1
                  S f(S; µ, ) = exp{µ + 2 2} f(S; µ+2, ) .



                                           ˜
Proof: The density for a lognormal variate S , we have
                                                           1      lnS – µ 2
                        f(S; µ, ) = k exp{– 2 [                          ]}
                                                                     
                        1
where k =                     . Thus, the integrand is
                        2

                                                                     1       lnS – µ 2
(A.6) S f(S; µ, ) = exp{lnS} k exp{– 2 [                                            ]}
                                                                                
                                                           1     lnS – µ 2
                                  = k exp{lnS – 2 [                      ]}
                                                                    




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Rearrange the argument of the exponential and "complete the
square":
                                                 1   (lnS – µ)2
                                         lnS –   2       2
                            1     –2 lnS 2 + [(lnS)2 – 2µlnS + µ2]
                    = –     2                     2
                                    1   –2 lnS (µ+2) + (lnS)2 + µ2
(A.7)                    = –        2              2

       1   –2 lnS (µ+2) + (lnS)2 + µ2 + [2µ2 + 4] – [2µ2 + 4]
   –   2                             2

           1   (lnS)2 – 2 lnS (µ+2) + (µ2 + 2µ2 + 4) – 2µ2 – 4
   = –     2                             2

                              1     [lnS – (µ + 2)]2 – 2 µ 2 – 4
                        = –   2                   2
                                1         lnS – (µ + 2) 2       1
(A.8)                   = –     2       [                ] + µ + 2 2
                                                 




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Thus
                                            1                       1       lnS – (µ + 2) 2
    S f(S; µ, ) = exp{µ +                  2        } k exp{–
                                                     2
                                                                    2     [                ]}
                                                                                   
                                            1
(A.9)                   = exp{µ + 2 2} f(S; µ+2, )

where f(S; µ+2, ) is the density of a lognormally distributed
variable with a shifted mean:  has been replaced by µ+2.



                                                       ˜
Corollary 1: The expected value of a lognormal variate S , is
                              1
  ˜                                 ˜                  ˜
E(S ) = exp{µ + 2 2} where  = E(lnS ) and 2 = var(lnS ).

Proof:
                                             ∞
(A.10)                           ˜
                               E(S )           
                                                   S f(S; µ, ) dS
                                                0


                                                                     ∞
                                                         1
                                       = exp{µ +         2    2}    
                                                                         f(S; µ+2, ) dS
                                                                      0

                                                          1
                                       = exp{µ + 2 2}

QED




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                                    ˜
Corollary 2: The partial mean of S from S=X to S= is given
                               (µ+2) – lnX
      ˜
by Et(S T) N(d'1), where d'1 =
                                    
                                            .

Proof. We apply the Lemma, and then Corollary 1.
                                      
                                       S f(S; µ, ) dS
                                      
(A.11)
                                      X


                                                 
                                          1
                        = exp{µ +         2   2}  f(S; µ+2, ) dS )
                                                  
                                                    X


                                        
                                     ˜ 
                                 = E(S )  f(S; µ+2, ) dS
                                               X


Thus:

(A.12)                                         ˜
                                   Part A = Et(S T) N(d' )
                                                       1


where
                      (µ+2) – lnX
(A.13)           d' =              ; and d' = d' – . QED.
                  1
                                         2    1




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  A.3. The Link Between the Probabilities and the Expected
                         Spot Rate

Combining our results of Sections A.1 and A.2, we obtain



(A.14)                     ˜         ˜
                        Et(C T) = Et(S T) N(d'1) – X N(d'2)



Using the first Corollary of Section A.2, we can write the d’1 and
                                        ˜
d’2-factors as explicit functions of Et(S T):
                                           µ + 2 – lnX
(A.15)                               d'1 =
                                                
                                           1          1
                                    µ + 2 2 + 2 2 – lnX
                                =
                                                     
                                                       1
                                       ˜
                                  ln(E(S)) + 2 2 – lnX
                                =
                                            

                                             ˜
                                           E(S) 1
                                         ln X + 2 2
                                       =
                                               

                                                  ˜
                                                E(S) 1
                                              ln X – 2 2
(A.16)                        d'2 = d'1 –  =
                                                   


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Appendix B: Stochastic Calculus and
the   Black-Scholes-Merton       Differential
         Equation



                                      B.1. Ito Processes

Consider an additive, continuous normal random walk in
continuous time. As the process is continuous, we can select any
observation frequency we like.



• Annual observations:
                           x t+1 = xt + a +   t,t+1
                           ˜                  ˜
        x = the level of the random walk variable
        a = the annual growth or "drift"
         = the standard deviation of the annual growth, and
         t,t+1 N(0,1).
        ˜
                ˜


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• N-annual observations:
            x t+N = xt + a N +  ( t,t+1 +  t+1,t+2 +... +  t+(N–1),t+N)
            ˜                     ˜         ˜                ˜

               = xt + a N + 
    t,t+1 + t+1,t+2 +... + t+(N–1),t+N
    ˜        ˜               ˜
  N
                  N

                    = xt + a N +  N  t,t+N
                                     ˜

                          t,t+1 + t+1,t+2 +... + t+(N–1),t+N
                          ˜        ˜               ˜
where  t,t+N =
      ˜
                                        N                                           ˜   N(0,1)




• Quarterly observations: can we set N = 1/4 in the above?

                                            ?1                   1
                             x   t+1/4 – xt =
                                              4 a+               4  
                             ˜                                       ˜


    Yes, because this is consistent with the annual observations:
                           a
     • a quarterly mean of 4 implies that the mean of the sum of
                                             a
       four quarterly changes will be 4  4 a = the annual
       expected change.




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                               2
     • a quarterly variance of 4 implies that the variance of the
                                                          2
       sum of four quarterly changes will be equal to 4  4  2,
       the annual variance.




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• Continuous observations: set N = dt

(B1)                dx = a dt +  dt  ,  a standard normal
                                     ˜   ˜

                         = a dt +  dz

    where dz  dt  (the standard Wiener Process) .
                  ˜


   Example
   Think of dt as one day, that is, 1/365 year. For each of the next
   365 days, you intend to make a random drawing , compute dz
   =  1/365 =   0.0523, and cumulate these figures. What
   can you say about the sum after 365 days?

   The sum of 365 such drawings is a standard normal:
                   365
         var(∑i=1 dzt+i/365) = 365  var(dz) = 365  (1/365) = 1. •



• A process with continuously changing drift and variance:

   Example: geometric random walk
   Assume that, at each instant, the mean and standard deviation
   of dS are proportional to the price level S; that is, a t =  St and
   t =  St, where  and  are constants.

                                 dS = [S  dt + [S  dz



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                                     dS
                                    S =  dt +  dz . •




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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-33




                                      B.2. Ito's Lemma

                          ∂f      ∂f
Let f = f(x,y). Then df ≠ ∂x dx + ∂y dy if {dz, dy} are random.



B.2.1. Ito’s Lemma Applied to a Function of One Ito Process
Consider a univariate function f = f(x).
                                                                1
                 f(x1) – f(x0) = fx (x1 – x0) + 2 fxx (x1 – x0)2
                                      1
                                   + 6 fxxx (x1 – x0)3 + ...
                                          1
                    ∆f = fx ∆x + 2 fxx (∆x)2 + 1 fxxx (∆x)3 + ...
                                               6

                                          1
                        df = fx dx + 2 fxx (dx)2 + 1 fxxx (dx)3 + ...
                                                   6


• Deterministic math: if dx is infinitesimally small, then (dx)2,
  (dx)3, etc. are negligible relative to dx itself. So df = fx dx.

• If dx = a dt +  dz, then

            (dx)2 = (a dt +  dz) (a dt +  dz)

                        = a2 (dt)2 + 2 (a dt) ( dz) + ( dz)2

                    = a2 (dt)2 + 2 (a dt) ( dt) + (  dt)2
                                             ˜            ˜

                    = a2 (dt)2 + 2 a  (dt)3/2 + 2 2 dt
                                       ˜                ˜




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    Time being deterministic, any power larger than unity of dt can
    be ignored. Thus,

                                        (dx)2 = 2  2 dt
                                                   ˜

                                 1                                    1
(B.2)        df = fx dx + 2 fxx (dx)2 = fx dx + 2 fxx 2  2 dt
                                                         ˜

    New term 2  2 dt?
                ˜

    • emerges whenever there is randomness and non-linearity.

    • its mean equals 2 dt because E( 2) = 1,  being a standard
                                      ˜         ˜
      normal.

    • its variance equals var( 2) 4 (dt)2, but this is negligible.
                              ˜

Thus:                   (dx)2 = 2  2 dt = E(2  2 dt) = 2 dt
                                   ˜             ˜

and
                                                        1
(B.3)                            df = fx dx + 2 fxx 2 dt



   Example of (B.3) in the geometric case
           dx
   Suppose x =  dt +  dz or dx = x ( dt +  dz).

                                                  1
(B.4)                        df = fx dx + 2 fxx 2 dt
                                                                      1
                                  = fx x ( dt +  dz) + 2 fxx x2 2 dt . •


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   Example: the geometric case implies lognormality of x
                       dx
   Let f(x)=ln(x), and x =  dt +  dz. Then

                                        1             1
                                   fx = x and fxx = – x2 .

                                                     1
   Thus                 dln(x) = fx dx + 2 fxx 2 dt
                                                                      1
                                   = fx x ( dt +  dz)+ 2 fxx x2 2 dt
                                       1                                   1 1
                                                                              
                                   =   x    x ( dt +  dz) –              2 x2       x2 2 dt
                                                                                   

                                                 1
                                   = ( – 2 2) dt +  dz

   We conclude:
                                             1
   • The p.a. mean  =  – 2 2 of the continuously compounded
      return, d lnx, is systematically lower than the mean of the
      p.a. simple return dx/x, which is .

   • Over short intervals, the random component of the
     continuously compounded return is indistinguishable from
     the random component in the simple return;

   • Over any small interval dt, dln(x) is normal. So over any
     finite interval ∆t, ∆ln(x) will be normal, because it is the



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      "sum" of many normal instantaneous changes. Thus, x itself
      is lognormal. •




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B.2.2. Ito’s Lemma Applied to a Function of Two Lognormal
         Processes
Let f = f(x,y) where dx = ax dt + x dzx , dy = ay dt + y dzy .

(B.5)                                   df = fx dx + fy dy
                        1
                  +2        {fxx (dx)2 + 2 fxy dx dy + fyy (dy)2} + ...
As before: (dx)2 = x2 dt and (dy)2 = y2 dt = instantaneous
variances. Analogously, dx dy is the instantaneous covariance:
                            dx dy = (ax dt + x dzx) (ay dt + y dzy)

                                     = (x dzx) (y dzy)

                                     = (x  x dt ) (y  y dt )
                                           ˜            ˜

                                     = x y  x  y dt
                                             ˜ ˜

• E( x  y)  , the correlation coefficient. Thus, E(dx dy) = x
     ˜ ˜
  y  dt, the instantaneous covariance between dx and dy.

• Obviously, the variance of x y  x  y dt around its mean will
                                    ˜ ˜
  again be of order (dt)2, implying that we can set x y  x  y dt
                                                          ˜ ˜
  equal to its mean, x y dt.



(B.6)                                   df = fx dx + fy dy



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                         1
                        +2   {fxx x2     + 2 fxy  x y + fyy y2} dt




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   Example: stock returns in domestic and foreign currency,
   with lognormals
    Consider the price process f = VS, where V is a foreign-
    currency stock price and S is an exchange rate, and where
             dV                                               dS
             V = V dt + V dzV and                           S = S dt + S dzS

          d VS
   What is VS , the return on the stock measured in HC?

   Ito's Lemma:
(B.7)                   df = fv V (v + dzv) + fs S (s + dzs)
                1
            + 2 {fvv V2 v2 + 2 fvs V S  v s + fss S2 s2} dt .

    We have fV = S, fS = V, fSS = 0, fSV = 1, fVV = 0:
                                               1
        dVS = S dV + V dS +                    2    {0 + 2 VS SV V S + 0} dt,

                        d VS         dV  dS
(B.8)                    VS        = V + S + SV V S dt .

                                                        dV
    Special case: return on the foreign riskfree asset ( V = r*dt):

               d VS                   dS
                VS          = r* dt + S (cross-term is too small) .




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   Example: returns on a portfolio
   Consider the price process f = nxX + nyY, where X is the price
   of one asset, Y is the price of another asset, and nx and ny are
   the number of shares of each asset held.
                                             dX      dY
                   df = nx dX + ny dY = nx X X + nYY Y

                              df  nXX dX  nYY dY
                              f  = f  X  + f Y

     (No 2nd-order terms because f is linear in X and Y.)



B.2.3. Ito’s Lemma Applied to a Function of an Ito Process
         and Time
Consider f=f(S,t), where t is calendar time.
                                                              1
(B.9)                      df = ft dt + fS dS +               2   fSS S2 dt



   Example: the Lognormal Case
   If dS = S ( dt +  dz), then the change in f equals
                                                                    1
(B.10)         df = ft dt + fS S ( dt +  dz) +                    2    fSS S2 S2 dt




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      B.3. The Black-Scholes-Merton Differential Equation



B.3.1. Derivation of the Differential Equation
If C = C(S, t) where S is lognormal, then
                                                           1
(B.10)                  dC = Ct dt + CS dS +               2    CSS S2 S2 dt



Hedge this by buying N units of forex worth (initially) NS.
Change in the value of the hedge:

                                   dH = N (dS + S r*dt)
Set N = –CS, and add this to the call.

• Initial value of the portfolio:
                                        p = C + (–CS) S.

• Change of p over an instant dt:
(B.11) dp = dC – CS (dS + S r*dt)
                                                  1
                    = [Ctdt + CSdS + 2 CSS S2 S2 dt] – CS(dS + Sr*dt)
                                                 1
                    = [Ct – CS S r*+             2    CSS S2 S2] dt




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• No arbitrage:
                            dp
(B.12)                      p = r dt               or      dp = p r dt

                                      1
             [Ct – CS S r*+           2    CSS S2 S2] dt = [C – CS S] r dt
                                                               1
(B.13)                  C r = Ct + CS S (r–r*) +               2    CSS S2 S2



This is the fundamental pde for any contingent claim C that pays
out no income stream between t and T (i.e., there are only capital
gains), and depends on S.




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B.3.2. Interpretation as a Risk-Adjusted version of the
        Expected Capital gain on the Option
Ito's Lemma:
                                                               1
                    dC = Ct dt + CS dS +                       2     CSS S2 S2 dt

                                            dS                  1
                             = Ct dt + CS S S +                 2    CSS S2 S2 dt

Thus: expected gain on the option is
                                                                     1
(B.14)              E(dC) = Ct dt + CS S S dt +                     2       CSS S2 S2 dt
                                                                1
(B.15)                  C C = Ct + CS S S +                   2    CSS S2 S2

Black-Scholes:
                                                                 1
(B.13)                  C r = Ct + CS S (r–r*) +                 2   CSS S2 S2



• (r*– r) replaces the true drift of S, S in equation (B.15).

• the domestic risk-free rate replaces the true expected return on
  the contingent claim, C in equation (B.15).



Thus: if we replace the true drift of S, s, by r-r*—that is, if we
replace Et(St+dt) by Ft,t+dt—, we can discount the future option
value at r rather than at c.

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