Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-ormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last time, {W t } t [0, ] : Fix a > 0 and large positive integer, define X 0 = 0 and W t = t X j, t = 0,, 2,...,, where {X j } j are i.i.d. random variables that take ± with probability 0.5. he value of X t for general t is defined using linear interpolation. Obviously, the path of W t is continuous. Moreover, we have seen that {W t } is a martingale for time points t = k, 0 k. In fact, the increment W t W s is independent of W u for all t s u +. As, the first-order variation of the path {W t } t [0, ] tends to, but the quadratic variation tends to. It is interesting to know the limiting distribution of W as. he impact of this is that, when we choose a finer partition of the interval [0, ] with a larger, we would like to know what we will obtain in the end. A good tool to study limiting probability distribution is the moment-generating function. Recall that, for a random variable Z, its moment-generating function is defined as φu = Ee uz. 2 Since moment-generating function is the signature of a probability distribution, to study the limiting distribution of W, it suffices to know its limiting moment-generating function. hus, we proceed to define: uw [ ] X j s are i.i.d. φ u = Ee = E exp u X j = E exp u X [ ] = 2 eu + 2 e u. 3 o see the limit lim φ u, it suffices to know lim log φ u. Let x =, then log lim log φ 2 exu + 2 e xu u = lim x 0 + x 2 = 2 u2, 4 where the last equality is a result of L Hôspital s rule. As a result, the limiting distribution of W moment-generating function has φu = e 2 u2. 5 Probability distribution can be uniquely determined by its moment-generating function.
Recall that the moment-generating function of a normal random variable with mean µ and variance σ 2 has moment-generating function e uµ+ 2 u2 σ 2, 5 implies that, the limiting distribution of W, as, is normal with mean 0 and variance. In other words, the limiting distribution has density fx = 2π e x2 2. Remark. he above result is a special case of central limit law. In fact, the result still holds if we replace {X j } by any other i.i.d. random variable with zero mean and unit variance e.g. standard normals, which is quit useful for simulations. 2 Log-ormal Distribution as the Limit of the Binomial Model We play the scaling game for binomial model in this section. Fix a time horizon > 0, choose a large positive integer to partition [0, ] with equal length, so that we have a -period binomial model. We consider zero rate for simplicity: r = 0. Let us fix a positive number σ > 0, we then take the up factor to be u = + σ and the down factor to be d = + σ p = + r d = σ u d 2σ. he risk-neutral probabilities are then = 2 = q. At time, which is the -th step of the -period binomial tree, the stock price is H S = S 0 u H d = S 0 + σ σ, 6 where H / is the number of heads/tails observed in the coin tosses. Suppose we have a simple random walk starting from zero, whose increment is if a head shows up and if a tail shows up, and let us denote the value of the simple random walk at the -th step by M, then On the other hand, hus, we have M = H. 7 = H +. 8 H = 2 + M, = 2 M. 9 Combining 6 with 9 and taking log we obtain that log S = log S 0 + 2 + M log + σ Moreover, using the fact that we obtain that log + x = x 2 x2 + Ox 3, log S = log S 0 + 2 + M σ σ2 2 + O 3 2 + 2 M log σ. 0 + 2 M σ σ2 2 + O 3 2 = log S 0 2 σ2 + O 2 + σ M + On. 2
Recall that in last section, the scaled random walk W = M = X j W, as where W is a normal with zero mean and variance. In conclusion, as, the stock price at time has distribution S = S 0 exp σw 2 σ2. 2 hat is, the log price is normally distributed. 3 Brownian Motion We studied the limiting marginal distribution of scaled random walk and -period binomial model. If we look at the whole path, we obtain Brownian motion. Brownian motion has a random continuous path W t for all t 0 that satisfies W 0 = 0. For all 0 = t 0 < t < t 2 <... t k the increments W t = W t W t 0, W t 2 W t,..., W t k W t k 3 are independent and each of these increments is normally distributed with EW t i+ W t i = 0, 4 VarW t i+ W t i = t i+ t i. 5 In other words, increments of Brownian motion are independent and stationary 2. he joint distribution of the values of Brownian motion at different times can be easily obtained. When the marginal distributions are known, two jointly normal variables are uniquely determined by their covariance. In the case of Brownian motion, W t and W s are zero mean normal variables with variance t and s, respectively. heir covariance is CovW t, W s = EW tw s = min{t, s}. 6 Alternatively, one can also easily obtain the joint moment-generating function of W t, W s using the independence of increments. Without loss of generality, assume t > s, then for u, v R, φu, v = Ee uw t+vw s = Ee uw s+w t W s+vw s = Ee u+vw s+uw t W s independence = Ee u+vw s Ee uw t W s = e 2 u+v2s e 2 u2 t s 7 Let us denote by F t the information available at time t for the Brownian motion. hat is, F t contains all the history of W s for any 0 s t. hen Brownian motion {W t} t 0 is a martingale with respect to the filtration F = {F t } t 0. his is a simple consequence of the fact that increments of Brownian motion have zero mean and are independent of the past. 4 Quadratic Variation of Brownian Motion and Volatility Estimation In Lecture 7 we have shown that, as, the scaled random walk {W k } 0 k has infinite first-order variation and finite quadratic variation. Brownian motion has the same property. For a given time-horizon > 0, we consider an arbitrage partition Π = {t 0, t,..., t } such that 0 = t 0 < t <... < t =. 2 he history does not matter. What matters is the time lapse of the increment. 3
he diameter of the partition Π is defined as the maximum time lapse: Π := max 0 i {t i+ t i }. 8 hen the quadratic variation of Brownian motion {W t} t 0 at time is defined as 3 W := lim W t j+ W t j 2 9 he above limit has a surprisingly simple result. One may get some flavor by taking the expectation of both sides and interchanging 4 the order of limit and expectation: E W = lim EW t j+ W t j 2 = o completely show that W = one just need to show that lim t j+ t j =. 20 Var W = 0. 2 his is not a problem because, using independence of increments, Var EW t j+ W t j 2 = Var [ W t j+ W t j 2] = 2 t j+ t j 2 2 Π t j+ t j = 2 Π 0 +, 22 as Π 0 +. he above result is the fundamental of stochastic calculus. If we use dw t to express the infinitesimal increment of Brownian motion, then we can informally write j=0 In other words, = W = 0 dw t 2. 23 dw t 2 = dt. 24 his is a striking result, as we know that dt 2 = 0. Moreover, one can also show that dw tdt = 0. he fact that the quadratic variation of Brownian motion is simply the time past can be used to estimate the volatility of stock price. Let us consider the classical geometric Brownian motion model for stock price St = S0 exp { σw t + α 2 σ2 t }. 25 where α and σ > 0 are constants. We observe the price process {S t } t 0 and we want to estimate the volatility σ. Fix a time-horizon > 0 say, a day, consider a partition 0 = t 0 < t <... < t =. ote that for any 0 k, log St k+ St k = σw t k+ W t k + α 2 σ2 t k+ t k. 26 3 ote that the limit here is more complicated than that in random walk case: not only, but also Π 0 +. 4 he interchange is legal because, the value of the summation is increasing as we choose more points and finer partitions. In probability this is called the monotone converge theorem. 4
If the partition is fine relatively high-frequency data, then k=0 log St 2 k+ St k = σ 2 W t k+ W t k 2 + terms proportional to t k+ t k 2 or W t k+ W t k t k+ t k k=0 σ 2 dw t 2 = σ 2. 0 Hence we have the estimation ˆσ 2 = j=0 27 log St 2 k+. 28 St k 5