Modeling via Stochastic Processes in Finance Dimbinirina Ramarimbahoaka Department of Mathematics and Statistics University of Calgary AMAT 621 - Fall 2012 October 15, 2012
Question: What are appropriate mathematical objects that enable us to model asset prices? The S&P500 index for 2000-2010.
Appears to be continuous. Far from being differentiable. Do not have a deterministic dynamics.... It evolves randomly over time. ===> Stochastic Processes
Definition of Stochastic Processes Assume there exists a common probability space (Ω, A, P). A collection of random variables X t0, X t1, can be conveniently used to describe the evolution of an observed asset price over any given set of observation times t 0 < t 1 < Definition We call a family X = {X t, t I } of random variables X t R a stochastic process where the quantity P(X ti1 x i1,, X tij x ij ) determines its probability law. i j {0, 1, }, x ij R and t ij I. It is a sequence of observations from a probability distribution.
Certain Classes of Stochastic Processes Stationary Processes: Their joint distributions are all invariant under time displacements. In particular, the random variable X t have the same distribution for all t I. Example: Interest rates, dividend rates, inflation rates,... Processes with Stationary Independent Increments: The random increments X tj+1 X tj are independent. These increments are assumed to be stationary, that is X t+h X t has the same distribution as X h X 0.
Markov Processes Satisfy the Markov Property : The future evolution of the process depends only on its present state, therefore, not depending on the past history. In discrete time: P(X tn+1 = x X t0,, X tn ) = P(X tn+1 = x X tn ) Important examples: processes with independent increments.
Natural interpretation: The present price of a stock encapsulates all the information contained in the knowledge of past prices. This does not exclude the possibility of using certain statistical properties of the stock price history to determine model parameter.
Discrete Time Markov Processes: Markov Chains Continuous time with finite set of states: Continuous time Markov chains Continuous time Markov Processes
For 0 s < t < and a continuous time Markov chain with state space {x 1,, x n }, the transition probability matrix from a state to another is given by: P(s, t) = (p i,j (s, t)) N i,j=1 with p i,j (s, t) = P(X t = x j X s = x i ). Let p i (t) = P(X t = x i ) and p(t) = (p 1 (t),, p N (t)) T, then we have the relation: p(t) T = p(s) T P(s, t).
Interest rate example. We consider the interest rate process X = {X t, t [0, )} with state space {0.005, 0.006} with probabilities (p 1 (t), p 2 (t)) T = p(t) and a transition probability matrix: P(t) = ( 1+e 10t 2 1 e 10t 2 1 e 10t 2 1+e 10t 2 Consider the case where at time 0 we have p(0) = (0.5, 0.5) T. Then for any time t > 0, ) p(t) T = p(0) T P(0, t) = p(0) T. The above interest rate example form a stationary process with stationary probability vector (0.5, 0.5) T.
Wiener Processes Discovered by Robert Brown when observing the motion of pollen grains, and in mathematics it is named in honor of Norbert Wiener. Most important continuous process with independent increments. Definition A process W with Gaussian stationary independent increments and continuous sample path for which: W 0 = 0, E(W t ) = 0, Var(W t W s ) = t s for all t [0, ) and s [0, t].
The most famous asset price model that uses the Wiener process is the Black-Scholes Model where stock prices follow the Geometric Brownian motion: and the saving account (B t ): ds t = µ t S t dt + σ t S t dw t db t = rb t dt W : Wiener process, µ: drift coefficient (rate of change in the conditional mean of the price process), σ: diffusion coefficient (volatility, average size of fluctuations of the price process).
Assumptions behind the Black-Scholes model are: The price follows the lognormal random walk. That is, the natural logarithm of the stock prices are normally distributed. The risk-free rate and the asset volatility are deterministic. Hedging a portfolio does not involve transaction costs. The underlying asset does not pay any dividend up to the maturity time of the option. There are no arbitrage opportunities. We can buy and sell underlying assets as much as we want. A result of the Black-Scholes model is a pricing formula in closed form called Black-Scholes formula. The formula do not depends on the expected growth rate anymore.
References R.Elliott and E.Kopp Mathematics of Financial Markets. Springer. E.Platen A Benchmark Approach to Quantitative Finance. Springer. Web Wikipedia.