Dr. Maddah ENMG 65 Financial Eng g II 10/16/06 Chapter 11 Models of Asset Dynamics () Random Walk A random process, z, is an additive process defined over times t 0, t 1,, t k, t k+1,, such that z( t ) = z( t ) + ε t, k + 1 k + 1 k k t = t + t, k where ε k are iid standard normal r.vs. and z(t 0 ) = 0. For j < k, k 1 z( t ) z( t ) = ε ( t ) t. k j i i= j Therefore, z(t k ) z(t j ) is a normal r.v. with mean 0 and variance k 1 k 1 t var[ ε ( t )] = t = t t. i k j i= j i= j The two random variables z(t k ) z(t j ) and z(t m ) z(t l ), with j < k < l < m, are independent. Brownian motion The limit of a random walk as t 0 is a Brownian motion (or a Wiener process). A Brownian motion is represented symbolically as dz = ε ( t) dt, where ε(t) are iid standard normal r.v. 1
A more rigorous approach is to define a Brownian motion as a stochastic process z(t) that satisfies the following: 1. For any s < t, z(t) z(s) is a normal r.v. with mean 0 and variance t s.. For any z(t ) z(t 1 ) 0 t 1 < t < t 3 < t 4, z(t ) z(t 1 ) and z(t 4 ) z(t 3 ) are independent. 3. Pr{z(0) = 0} = 1. A Brownian motion is not differentiable in time since z( s) z( t) E[( z( s) z( t)) ] 1 E = =, as s t. s t ( s t) s t However, the term dz/dt is utilized and is termed white noise.
Generalized Wiener process A generalized Wiener process is a stochastic process x(t) such that dx( t) = adt + bdz, where z is a Brownian motion, and a and b are constants. Integration gives Ito Process x( t) = x(0) + at + bz( t). This is even more general. It s defined by dx( t) = a( x( t), t) dt + b( x( t), t) dz. No closed-form expression for x(t) exists in general. 3
Lemma (Ito s lemma) Let x(t) be an Ito process. Define the process y(t)=f(x(t), t). Then, F( x, t) F( x, t) 1 F( x, t) F( x, t) dy( t) = a( x, t) + + ( b( x, t)) dt b( x, t) dz. + x t x x Proof. The proof follows by noting that y(t) is obtained from Taylor s series expansion of F(x(t), t) up to first order on t and second order on x. (This is the case since dx is function of dz, which is of the order t ). f x t t f x t t f x t t y( t) = x( t) + t + ( x( t)) ( ( ), ) ( ( ), ) 1 ( ( ), ) x( t) t x ( t) f x t t f x t t f x t t = ( a t + b z) + t + ( a t + b z) ( ( ), ) ( ( ), ) 1 ( ( ), ) x( t) t x ( t) f ( x( t), t) f ( x( t), t) 1 = ( a t + z) + t + x( t) t f x t t x ( t) The result then follows by taking the limit as t 0 lim z = lim ( ε ( t)) t = t 1. that ( ) t 0 t 0 ( ( ), ) [ a ( t ) + ab t z + b ( z ) ] and noting Remark. For a two-variable function f(x, t), the second-order Taylor s series expansion is F( x, t) F( x, t) F( x + x, t + t) = F( x, t) + x + t x t 1 F( x, t) F( x, t) 1 F( x, t) 3 + ( x) + x t + ( t) + O ( x, t), x x t t 3 lim ( x, t) (0,0) O ( t, x) / t + x = 0. Therefore, where ( ) F( x, t) = F( x + x, t + t) F( x, t) F( x, t) F( x, t) 1 F( x, t) F( x, t) 1 F( x, t) x+ t+ x + x t+ t x t x x t t ( ) ( ) 1 An intuitive explanation of this is that (ε(t)) t is a chi-square r.v. with mean t and variance t. 4
Geometric Brownian motion This is the preferred stock price model. It is a continuous version of the multiplicative model defined by the following differential equation on the stock price, S(t), d ln( S( t)) = ν dt + σ dz, where v and σ 0 are constants, and z is a Brownian motion. S(t) is a special case of a generalized Wiener process. Integrating the differential equation gives ln( S( t)) = ln( S(0)) + ν t + σ z( t). Therefore, ln(s(t)) is a normal r.v. with mean ln S(0) + vt and variance σ t. It follows that the stock price S(t) is a lognormal r.v. with parameters S(0) + vt and σ t. Then, E S t e e S e S e E[ln( S ( t )] + var[ln( S ( t)]/ ln( S (0)) + vt+ σ t / ( v+ σ / ) t µ t [ ( )] = = = (0) = (0), stdev S t e e S e e E[ln( S ( t)] + var[ln( S ( t )]/ var[ln( S ( t )] 1/ µ t σ t 1/ [ ( )] = ( 1) = (0) ( 1), where µ v σ / +. Note that the parameter µ can be seen as the expected rate of return for the stock (assuming continuous compounding). The parameter σ is known as the stock price volatility. 5
Geometric Brownian motion in standard Ito form Ito s lemma allows us to determine a differential equation that S(t) satisfies. Note that S(t) = F(x(t), t) = e x(t), where x(t) = ln(s(t)) and dx( t) = νdt + σdz. Note that F( x, t) x( t) F( x, t) x( t) F( x, t) = e, = e, and = 0. x x t Then, Ito s lemma implies that F ( x, t) F( x, t) 1 F( x, t) F( x, t) ds( t) = v + + σ dt σdz + x t x x ( 0 ( σ / ) ) ( ( σ / ) ) ( σ ) = + + + σ x( t) x( t) x( t) ve e dt e dz = + + σ ln( S ( t)) ln( S ( t)) ln( S ( t )) ve e dt e dz = vs( t) + ( / ) S( t) dt + σ S( t) dz. This is the geometric Brownian motion in standard Ito form, ds( t) ds( t) = µ S( t) dt + σ S( t) dz = µ dt + σ dz. S( t) This form provide further justification for seeing µ and σ as the expected ROR of and volatility of the stock. This follows since in a small time t the relative (%) change in the stock price is a normal r.v. with mean µ t and variance σ t. Similar to the multiplicative (discrete time), the change in the relative percent return is independent of price level. 6
Simulating Geometric Brownian motion There are two ways. Both require a long series of small time periods defined by t 0, t 1,, where t k+1 t k = t. The first method utilizes the standard Ito form and proceed to generating a sample path from the equation S( t ) S( t ) = µ S( t ) t + σ S( t ) z = µ S( t ) t + σ S( t ) ε ( t ) t k+ 1 k k k k k k ( µ σε ) S( t ) = S( t ) 1 + t + ( t ) t k+ 1 k k where S(t 0 ) = S 0. The second method works on the differential equation of the ln(s(t)) ln( S( t )) ln( S( t )) = v t + σ z = v t + σε ( t ) t k + 1 k k S t = S t e v t+ σε ( tk ) t ( k + 1) ( k ), where S(t 0 ) = S 0. To simulate the standard normal random variables ε(t k ) one may use the inverse transform method where a random variate ˆ( ε t k ) is generated as 1 ˆ( tk ) ( u( tk )) ε = Φ, where u(t k ) is a random number between 0 and 1 and Φ 1 is the inverse of the standard normal cdf. In Excel, A standard normal random variate is generated by the formula =NORMSINV(RAND())., 7
Binomial Lattice Revisited The discrete-value binomial lattice model is similar to the continuous-time multiplicative model. The parameters u, d, and p can be chosen so that the binomial lattice matches the multiplicative model. One fact that supports this matching is that the logarithmic of the stock price under the binomial lattice model follow a binomial distribution. This binomial distribution is a good approximation to the normal distribution for a large number of time intervals. Hence, for a large number of time intervals both models lead to a lognormal stock price distribution. The matching of the two models is done by equating the mean and variance of the stock price logarithm after one period, S 1, assuming that the initial price is S(0) = 1. Let the parameters of the multiplicative model be v and σ, and let t be the time interval. Then, matching leads to E[ln S ] = p ln u + (1 p) ln d = v t, 1 var[ln S1] = p(ln u) + (1 p)(ln d) [ p ln u + (1 p)ln d] = t. Assuming ln d = ln u, and solving for ln u and p gives with σ some approximation 1 1 v σ t σ t p = + t, u = e, d = e. σ Remark. With these parameters, the binomial lattice model approximate the geometric Brownian motion for small t. 8