Dr Maddah ENMG 65 Financial Eng g II 0/6/06 Chater Models of Asset Dynamics () Overview Stock rice evolution over time is commonly modeled with one of two rocesses: The binomial lattice and geometric Brownian motion Both of these are stochastic rocesses that have the Markovian roerty Geometric Brownian motion is a secial case of Ito rocess Binomial lattices are discrete-time, discrete-value Ie, o There are limited time instants where the stock rice can change; o There are limited values that the stock rice can take On the other hand geometric Brownian motion is continuous in time and in values With small time intervals for stock rice changes and arameters chosen aroriately, a binomial lattice can reasonably aroximate a geometric Brownian motion Binomial lattices are analytically simler than Geometric Brownian motion rocesses They are of great use in ractical comutational works
Binomial lattice model A binomial lattice model is defined over a basic eriod length t (eg, t = day =/365 years) where the rice can change If S is the rice at the beginning of a eriod, the rice at the end of the eriod is us w and ds w, where u > and d < This model can be extended for any number of eriods A roer choice of the arameters, u, and d is such that the binomial lattice aroximates a geometric Brownian motion Define the mean and variance of the yearly growth rate as v = E[ln(S T /S 0 )] and σ = var[ln(s T /S 0 )], where S 0 is the initial stock rice and S T is the rice at the end of year Then, the binomial lattice arameters are chosen as v σ t = + t, u = e, d = e σ σ t
The additive model This model allows the stock rice to change over a continuum of values but at discrete times The stock rice at time k +, S(k+) is given function of the stock rice at time k, S(k), as S( k + ) = S( k) + ε k, where ε k are indeendent and identically distributed normal random variables with mean 0 and variance σ It can be easily shown that k = + i i= S( k) S(0) ε It follows that S(k) is a normal random variable with E[S(k)] = S(0) and k σ kσ i= var[ S ] = = k The additive model is simle and easy to work with However, it lacks realism because (i) The normal distribution allows negative stock rices; (ii) The robability that the rice changes by a certain amount is the same regardless of the rice level 3
Lognormal random variables A random variable Y is said to be lognormal if X = ln(y) is a normal random variable Alternatively, Y is a lognormal r v if Y = e X, where X is a normal r v If X = ln(y) is normal with mean λ and variance σ, then the density function of Y is (ln y λ ) fy y e y y πσ σ ( ) =, > 0 dlnorm y, 0, dlnorm( y, 0, ) 3 dlnorm y, 0, 08 06 04 0 0 0 3 4 5 The mean and variance of Y are given by E Y e Y e e λ+ σ / λ+ σ σ [ ] =, var[ ] = ( ) The mode of the distribution is m = e λ σ A good book to learn about lognormal and other distributions is Simulation Modeling and Analysis by Law and Kelton 4
The multilicative model Similar to the additive model, this model is continuous in value but discrete in time The stock rice at time k +, S(k+) is given function of the stock rice at time k, S(k), as S( k + ) = S( k) ε k, where ε k are iid lognormal random variables with arameters v and σ The logarithmic of the stock rice in a multilicative model follows an additive model ln S( k + ) = ln S( k) + ln( ε k ) Using the additive model results, we find that ln(s(k)) is a normal random variable with mean and variance E S k S vk S k kσ [ln ( )] = ln (0) +, var[ln ( )] = Therefore, the stock rice, S(k) is a lognormal rv Note that in this model the stock rice cannot be negative In addition, the roblem with stock rice difference over a eriod being indeendent of the rice level is eliminated With the multilicative model ln S(k+)/ S(k) has the same normal distribution This imlies that the robability of a given relative (ercent) change of the rice is the same at all rice levels 5
Real stock distribution To validate the multilicative model, test the goodness of fit of a normal distribution to lns(k+)/s(k) Chooses a eriod length (eg week, day), and then note the stock rice at the beginning of each eriod Then, form the histogram of the logarithmic of the ratio of two consecutive observations, ln S(k+)/S(k) It is observed (according to our text) that most stocks are actually close to the multilicative model Eg, the histogram below is for American Airlines stock, ln S(k+)/S(k), from 98 to 99 with a eriod length of week A normal fit with the same vaeiance is also shown Two tyical differences between fitted and actual distribution are that the actual distribution is skinnier around the mean and has a fatter tail 6
Tyical values and estimation of arameters Tyical annual values for the arameters of the multilicative models are v = E[ln(S T /S 0 )] = % and σ = stdev[ln(s T /S 0 )] = 5% If the observation eriod length is art of the year then these values scale down to v = v and σ = σ Eg, tyical weekly arameters values are v /5 = 03% and σ /5 = 08% Estimate v and σ based on observations from n+ eriods of length each are as follows n k = 0 [ ] vˆ = ln S( k + ) ln S( k) = ln S( n) / S(0), n n ˆ σ n = { ln [ S( k + ) / S( k) ] vˆ } n k = 0 The error on these estimates are characterized by ˆ var[ v ] = / n, ˆ σ n var[ σ ] = σ /( ) With the tyical values of v % and σ 5% It turns out that estimating σ accurately requires much less data than estimating v accurately (Remember Mean Blur?) 7