# Ch. 8 by hcj

VIEWS: 18 PAGES: 44

• pg 1
```									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

P. Sercu and R. Uppal                  Version January 1994              Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model                page 8-2

•   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.

• 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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International Financial Markets and the FirmLognormal Option Pricing Model                page 8-3

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

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

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International Financial Markets and the FirmLognormal Option Pricing Model                page 8-4

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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International Financial Markets and the FirmLognormal Option Pricing Model                page 8-5

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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International Financial Markets and the FirmLognormal Option Pricing Model                page 8-6

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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International Financial Markets and the FirmLognormal Option Pricing Model                page 8-7

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

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:

P. Sercu and R. Uppal                  Version January 1994              Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model                page 8-9

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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-10

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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-11

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
*

P. Sercu and R. Uppal                  Version January 1994              Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model                page 8-12

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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-13

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.
*

P. Sercu and R. Uppal                  Version January 1994              Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model               page 8-14

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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-15

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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International Financial Markets and the FirmLognormal Option Pricing Model                    page 8-16

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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-17

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

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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-18

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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International Financial Markets and the FirmLognormal Option Pricing Model                 page 8-19

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

[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.

P. Sercu and R. Uppal                  Version January 1994                Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model               page 8-20

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.

P. Sercu and R. Uppal                  Version January 1994              Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model               page 8-21

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.

P. Sercu and R. Uppal                  Version January 1994              Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model               page 8-22

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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-23

˜
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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-24

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


P. Sercu and R. Uppal                     Version January 1994           Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model                page 8-25

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

P. Sercu and R. Uppal                  Version January 1994               Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model               page 8-26

˜
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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-27

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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-28

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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-30

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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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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International Financial Markets and the FirmLognormal Option Pricing Model                page 8-34

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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International Financial Markets and the FirmLognormal Option Pricing Model                page 8-35

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

P. Sercu and R. Uppal                  Version January 1994               Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model               page 8-36

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

P. Sercu and R. Uppal                  Version January 1994              Printout Rabi' Awwal 12, 1431
International Financial Markets and the FirmLognormal Option Pricing Model               page 8-37

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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International Financial Markets and the FirmLognormal Option Pricing Model               page 8-43

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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