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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.

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 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

    SSSS+dS 3. Trading is continuous.

    [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

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

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