Modeling Uncertainty in Financial Markets

Transcription

1 Modeling Uncertainty in Financial Markets Peter Ritchken 1 Modeling Uncertainty in Financial Markets In this module we review the basic stochastic model used to represent uncertainty in the equity markets. Topics Random Walk Models and the Geometric Wiener Process Empirical Evidence Binomial Approximation of the Geometric Wiener Process Simulating Stock Price Processes. Peter Ritchken 2 1

2 Modeling Uncertainty in Financial Markets Prices reflect all known information If markets were not efficient, some information would not be used in price determination. Astute investors could use this information to identify predictable patterns. Their strategy would soon become apparant to other traders and abnormal profits wouls soon be eliminated. Peter Ritchken 3 Price Process is Markov. At date t, for all t > t : Prob {S(t ) < x S(t)} = Prob {S(t ) < x Entire History} The particular path a stock takes to reach its current level does not provide any information on the likelihood of falling in a particular interval in the future. The probability of falling in the specific interval is only determined by the current level of the price, not its history. If the current level is an all time high, or an all time low, makes no difference to the above computation. Peter Ritchken 4 2

3 Additive Random Walk S(0) S(1) = S(0)+V(1) S(2) = S(1) + V(2) = S(0) + V(1) + V(2)... S(n) = S(0) + V(1) + V(2) + V(3) V(n) E {S(n)} = S(0) + E{V(1)} + E{V(2)} E{V(n)} Var{ S(n)} = Var{ V(1) }+ Var{ V(2) }+...+ Var{ V(n)} Peter Ritchken 5 Additive Random Walk Model Assume: E{V(i)} = µ t Var{V(i)} = σ 2 t Then: E{S(n)} = S(0) + nµ t = S(0) + µt Var{S(n)} = nσ 2 t = σ 2 T Peter Ritchken 6 3

4 A Normal Random Walk Model S(0) = 100 µ t = $2. 0 σ 2 t = $ The change in the stock price, V(1), over the small time increment, is a normal random variable with mean 2 and variance 9. Let s = V ( 1) s µ t Then Z = σ t Or s = µ t + σ t Z s = µ t + σ w µ is the drift term, σ is the volatility and w is the Wiener increment. Peter Ritchken 7 A Multiplicative Random Walk S(1) = S(0) R(1) S(2) = S(1)R(2) = S(0)R(1)R(2)... S(n) = S(0)R(1)R(2)...R(n) S(n)/S(0) = R(1)R(2)...R(n) ln(s(n)/s(0)) is the logarithmic return over the time period [t(0),t(n)] n n ln( S( n)/ S(0)) = lnri ( ) = i= r 1 i= 1 i Peter Ritchken 8 4

5 Multiplicative Processes Assume that the distribution of the logarithmic returms are independent of the stock price and statistically independent. Also r i = ln{r(i)} is normal with mean α t and variance σ 2 t. Then after the n time increment, or at date T, we have: or ln{ S( n) / S( 0 )} = αt + σ TZ S( n) e T S( 0 ) = α + σ TZ Peter Ritchken 9 The Geometric Wiener Process. As the time partition becomes finer and finer, the above process converges to a continuous time process called geometric Wiener process. Equivalently, the logarithmic returns follow a Wiener process. An alternative way of writing the dynamics for the process is ds = µ dt+ σdw s It can be shown that the drift term here s linked to the drift term of the logarithmic return by the equation: µ = α + σ 2 / 2. The expected price of the stock can be shown to be: µ E{ S( T)} = S ( 0) e T We shall call µ the expected return and α the expected continuously compounded return. Peter Ritchken 10 5

6 Estimating The parameters of a Geometric Wiener Process. Collect daily closing prices. Compute the daily price relatives, and take their logarithms. Compute the mean and standard deviation of these logarithms. Annualize the standard deviation by multiplying by the square root of 260. (260 represents the number of business days in the year.) Peter Ritchken 11 Empirical Evidence of Stock Return Behavior. The Fat Tail Problem Kurtosis depends on the time increments. Volatility Clustering Conclusions Peter Ritchken 12 6

7 A Simple Binomial Approximation To a Geometric Wiener Process. Construct a lattice of prices over discrete time partitions such that as the partition becomes finer and finer, the underlying process converges to a GWP. We can do this as follows: S(0)=S S(1,1)=uS S(1,0)= ds S(2,2)=uuS S(2,1)=udS S(2,0)=ddS S(3,3)=uuuS S(3,2)=uudS S(3,1)=uddS S(3,0)=dddS Peter Ritchken 13 A Simple Binomial Approximation To a Geometric Wiener Process. u = eσ t d = 1/ u p e µ t d = u d Peter Ritchken 14 7

8 Advantages of the Lattice Rather than work with a continuous process we can work with a simple discrete process. This process is called a lattice or tree. The tree has the property that it can be made indistinguishable from the true continuous time process. It will be easy to price options on a lattice. It will make transparent some of the hedge strategies. Peter Ritchken 15 Simulating a Geometric Wiener Process S(0) 100 mu 0.1 sigma 0.3 Time(yrs) 1 #Periods 52 Delta t alpha Time Z-value Stock Price Peter Ritchken 16 8

9 Simulating a Geometric Wiener Process S( t) = S(0) e α+σ t tz t Peter Ritchken 17 Simulating a Geometric Wiener Process Simulating Paths of the Process: Why is this important? How will we use this methodology? Simulating a terminal distribution for the stock price process. Why is this important? How will we use this methodology? Peter Ritchken 18 9

10 Problems, Examples, Illustrations Peter Ritchken 19 Extensions of Geometric Wiener Processes Could have other representations of uncertainty over time. For example, consider the dynamics of bond prices. Consider the dynamics of exchange rates. Consider the dynamics of stocks that pay continuous dividends. Consider jumps, and stochastic volatility. Peter Ritchken 20 10