Lecture 3 Sergei Fedotov 091 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 091 010 1 / 6
Lecture 3 1 Distribution for lns(t) Solution to Stochastic Differential Equation for Stock Price 3 Examples Sergei Fedotov (University of Manchester) 091 010 / 6
Differential of ln S Example 1. Find the stochastic differential equation (SDE) for by using Itô s Lemma: f = lns ( ) df = f f t + µs S + 1 σ S f dt + σs f S S dw Sergei Fedotov (University of Manchester) 091 010 3 / 6
Differential of ln S Example 1. Find the stochastic differential equation (SDE) for by using Itô s Lemma: f = lns ( ) df = f f t + µs S + 1 σ S f dt + σs f S S dw We obtain df = This is a constant coefficient SDE. dt + σdw Sergei Fedotov (University of Manchester) 091 010 3 / 6
Differential of ln S Example 1. Find the stochastic differential equation (SDE) for by using Itô s Lemma: f = lns ( ) df = f f t + µs S + 1 σ S f dt + σs f S S dw We obtain df = This is a constant coefficient SDE. dt + σdw Integration from 0 to t gives ) f f 0 = (µ σ since W (0) = 0. Sergei Fedotov (University of Manchester) 091 010 3 / 6
Normal distribution for lns(t) We obtain for lns(t) lns(t) lns 0 = where S 0 = S(0) is the initial stock price. Sergei Fedotov (University of Manchester) 091 010 4 / 6
Normal distribution for lns(t) We obtain for lns(t) lns(t) lns 0 = where S 0 = S(0) is the initial stock price. ( lns(t) has a normal distribution with mean lns 0 + variance σ t. µ σ ) t and Sergei Fedotov (University of Manchester) 091 010 4 / 6
Normal distribution for lns(t) We obtain for lns(t) lns(t) lns 0 = where S 0 = S(0) is the initial stock price. ( lns(t) has a normal distribution with mean lns 0 + variance σ t. µ σ ) t and Example. Consider a stock with an initial price of 40, an expected return of 16% and a volatility of 0%. Find the probability distribution of lns in six months. Sergei Fedotov (University of Manchester) 091 010 4 / 6
Normal distribution for lns(t) We obtain for lns(t) lns(t) lns 0 = where S 0 = S(0) is the initial stock price. ( lns(t) has a normal distribution with mean lns 0 + variance σ t. µ σ ) t and Example. Consider a stock with an initial price of 40, an expected return of 16% and a volatility of 0%. Find the probability distribution of lns in six months. We have lns(t) N ( lns 0 + ) ) (µ σ T,σ T Sergei Fedotov (University of Manchester) 091 010 4 / 6
Normal distribution for lns(t) We obtain for lns(t) lns(t) lns 0 = where S 0 = S(0) is the initial stock price. ( lns(t) has a normal distribution with mean lns 0 + variance σ t. µ σ ) t and Example. Consider a stock with an initial price of 40, an expected return of 16% and a volatility of 0%. Find the probability distribution of lns in six months. We have lns(t) N ( lns 0 + Answer: ln S(0.5) N (3.759, 0.00) ) ) (µ σ T,σ T Sergei Fedotov (University of Manchester) 091 010 4 / 6
Probability density function for ln S(t) Recall that if the random variable X has a normal distribution with mean µ and variance σ, then the probability density function is ( ) 1 p(x) = exp (x µ) πσ σ Sergei Fedotov (University of Manchester) 091 010 5 / 6
Probability density function for ln S(t) Recall that if the random variable X has a normal distribution with mean µ and variance σ, then the probability density function is ( ) 1 p(x) = exp (x µ) πσ σ The probability density function of X = lns(t) is ( 1 πσ t exp (x lns 0 (µ σ /)t) ) σ t Sergei Fedotov (University of Manchester) 091 010 5 / 6
Exact expression for stock price S(t) Definition. The model of a stock ds = µsdt + σsdw is known as a geometric Brownian motion. Sergei Fedotov (University of Manchester) 091 010 6 / 6
Exact expression for stock price S(t) Definition. The model of a stock ds = µsdt + σsdw is known as a geometric Brownian motion. The random function S(t) can be found from ln(s(t)/s 0 ) = Sergei Fedotov (University of Manchester) 091 010 6 / 6
Exact expression for stock price S(t) Definition. The model of a stock ds = µsdt + σsdw is known as a geometric Brownian motion. The random function S(t) can be found from ln(s(t)/s 0 ) = Stock price at time t: S(t) = S 0 e ) (µ σ t+σw(t) Sergei Fedotov (University of Manchester) 091 010 6 / 6
Exact expression for stock price S(t) Definition. The model of a stock ds = µsdt + σsdw is known as a geometric Brownian motion. The random function S(t) can be found from ln(s(t)/s 0 ) = Stock price at time t: S(t) = S 0 e ) (µ σ t+σw(t) Or S(t) = S 0 e ) (µ σ t+σ tx where X N (0,1) Sergei Fedotov (University of Manchester) 091 010 6 / 6