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# Ch 8 Pricing European Options The Lognormal Model

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Ch 8 Pricing European Options The Lognormal Model

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 Jamada El Oula 24, 1431

International Financial Markets and the FirmLognormal Option Pricing Model page 8-2

Binomial model: time is discrete, S is from a discrete scale.

?C?Cn+1t+?t

hedge ratio = = ?S?Sn+1t+?t Black-Scholes-Merton: time is continuous, S is lognormal

(i.e. from a continuous scale).

?C?Ct+dtt

hedge ratio = = ?S?St+dtt 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 C = n,j

q C + (1-q) Cn+1,j+1n+1,j

1+r

BSM: numerical solution of a partial differential equation

P. Sercu and R. Uppal Version January 1994 Printout Jamada El Oula 24, 1431

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

CS

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

P. Sercu and R. Uppal Version January 1994 Printout Jamada El Oula 24, 1431

International Financial Markets and the FirmLognormal Option Pricing Model page 8-4 4. The distribution of the percentage changes in the exchange is

lognormalor 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. ]

P. Sercu and R. Uppal Version January 1994 Printout Jamada El Oula 24, 1431

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]:

?(C 1. Compute the expected value of the option at maturity, Et

). T

2. Correct this expected value for risk. That is, compute

????CEQ(C ) from E(C ), by replacing E(S ) with CEQ(S ) = tTtTtTtT

F. t,T

3. Discount the risk-adjusted expected future value at the risk-

free rate to determine the call’s value at time zero, C. That t

is,

?(C)CEQtT

(1) C = t1+rt,T

P. Sercu and R. Uppal Version January 1994 Printout Jamada El Oula 24, 1431

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 may be 38 39 40 41 42 43 44 45 46 47 T

with prob 0 .05 .10 .15 .20 .20 .15 .10 .05 0 t,T

? then C = 0 0 0 0 0 0 1 2 3 4 T

with prob 0 .05 .10 .15 .20 .20 .15 .10 .05 0 t,T

? E(C ) = 0 + 0 + ... + (1 ? 0.15) + (2 ? 0.1) tT

+ (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]

P. Sercu and R. Uppal Version January 1994 Printout Jamada El Oula 24, 1431

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

?

?(5) E(C ) = S ? Prob(S) X ? ?tTTt,TT

S=XT

?

Prob(S) ?t,TT

S=XT

= [Sum A] X ? [Sum B]

partial mean X prob of ending in the money

P. Sercu and R. Uppal Version January 1994 Printout Jamada El Oula 24, 1431

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

??(lnS ) by µ, and sd(lnS ) by ?. Then Denote EtTt,TtTt,T

?

??(7) E(C ) = S f(S; µ,?)dS X tTTTt,Tt,TT?

X

?

?f(S; µ,?)dS Tt,Tt,TT?

X

= [Integral A] X ? [Integral B]

partial mean X prob of ending

in the money

[après maintes péripéties:]

?(8) = E(S ) N(d' ) X N(d' ) tT12where N(d' ) denotes the cumulative standard normal probability: i

P. Sercu and R. Uppal Version January 1994 Printout Jamada El Oula 24, 1431

International Financial Markets and the FirmLognormal Option Pricing Model page 8-9

n(z)

N(d)

z

d0

?(S)EtT21 + ?ln t,T2X

(9) d' = , d' = d' ?. 121t,T ?t,T

P. Sercu and R. Uppal Version January 1994 Printout Jamada El Oula 24, 1431

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

(S) by F: correction means replacing EtTt,T

?(10) CEQ(C ) = F N(d) X N(d) tTt,T12

F1t,T?

?ln + t,T2X

(11a) d = 1?t,T

(11b) d = d ? 21t,T

2.3. Step 3: Discounting the Risk-adjusted Expiration value

of the Call at the Risk-free Rate.

?(C)CEQtT

(12) C = t1 + rt,T

FXt,T

= N(d) N(d) 121 + r1 + rt,Tt,T

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