Suppose the martingale model holds and dividends follow a random walk D t+1 = D
ID: 3248468 • Letter: S
Question
Suppose the martingale model holds and dividends follow a random walk Dt+1 = Dt+wt+1, where wt is white noise with variance 2. Suppose that the information set of investors at time t contains all past dividend realizations. Finally, assume that investors require a constant rate of return k for holding stock, Et [Rt+1] = k, with k > 0.
(a) Let denote the discount factor between period t and t + 1. Express in terms of the variables defined above.
(b) Conditional on information available at period-t, what is the expected value of dividends at time t + j for j = 1, 5, 10?
(c) Find the period-t log price-dividend ratio implied by this model.
(d) Calculate the variance of Pt+1 Pt conditional on information available at period-t.
Explanation / Answer
A basic definition of a discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X1, X2, X3, ... that satisfies for any time n,
E ( | X n | ) < {displaystyle mathbf {E} ( ert X_{n} ert )<infty } mathbf {E} ( ert X_{n} ert )<infty
E ( X n + 1 X 1 , … , X n ) = X n . {displaystyle mathbf {E} (X_{n+1}mid X_{1},ldots ,X_{n})=X_{n}.} mathbf {E} (X_{n+1}mid X_{1},ldots ,X_{n})=X_{n}.
That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation.
Martingale sequences with respect to another sequenceEdit
More generally, a sequence Y1, Y2, Y3 ... is said to be a martingale with respect to another sequence X1, X2, X3 ... if for all n
E ( | Y n | ) < {displaystyle mathbf {E} ( ert Y_{n} ert )<infty } mathbf {E} ( ert Y_{n} ert )<infty
E ( Y n + 1 X 1 , … , X n ) = Y n . {displaystyle mathbf {E} (Y_{n+1}mid X_{1},ldots ,X_{n})=Y_{n}.} mathbf {E} (Y_{n+1}mid X_{1},ldots ,X_{n})=Y_{n}.
Similarly, a continuous-time martingale with respect to the stochastic process Xt is a stochastic process Yt such that for all t
E ( | Y t | ) < {displaystyle mathbf {E} ( ert Y_{t} ert )<infty } mathbf {E} ( ert Y_{t} ert )<infty
E ( Y t { X , s } ) = Y s s t . {displaystyle mathbf {E} (Y_{t}mid {X_{ au }, au leq s})=Y_{s}quad orall sleq t.} mathbf {E} (Y_{t}mid {X_{ au }, au leq s})=Y_{s}quad orall sleq t.
This expresses the property that the conditional expectation of an observation at time t, given all the observations up to time s {displaystyle s} s, is equal to the observation at time s (of course, provided that s t).
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.