dstefan
06-16-2008, 01:04 PM
Chapter 4, Problem 3
Assume we have a one period binomial model for the evolution of the price of an underlying asset between time t and time t + \delta t:
If S(t) is the price of the asset at time t, then the price S(t+\delta t) of the asset at time t + \delta t will be either S(t) u, with (risk-neutral) probability p, or S(t) d, with probability 1-p. Assume that u > 1 and d < 1.
Show that
E_{RN}[S(t+\delta t)] = (p u ~+~ (1-p) d ) ~S(t)
E_{RN}[S^2(t+\delta t)] = (p u^2 ~+~ (1-p) d^2) ~S^2(t)
Solution:
We can regard S(t+\delta t) as a random variable over the probability space S = \{ U, D \} of the possible moves of the price of the asset from time t to time t + \delta t endowed with the risk-neutral probability function P : S \to [0, 1] with P(U) = p and P(D) = 1-p, and given by S(t+\delta t)(U) ~=~ S(t) u; S(t+\delta t)(D) ~=~ S(t) d.
Then,
E_{RN}[S(t+\delta t)] = P(U) \cdot S(t+\delta t)(U) ~+~ P(D) \cdot S(t+\delta t)(D) = (p u + (1-p) d ) ~S(t)
E_{RN}[S^2(t+\delta t)] = P(U) \cdot (S(t+\delta t)(U))^2 ~+~ P(D) \cdot (S(t+\delta t)(D))^2 = (p u^2 + (1-p) d^2) ~S^2(t)
Assume we have a one period binomial model for the evolution of the price of an underlying asset between time t and time t + \delta t:
If S(t) is the price of the asset at time t, then the price S(t+\delta t) of the asset at time t + \delta t will be either S(t) u, with (risk-neutral) probability p, or S(t) d, with probability 1-p. Assume that u > 1 and d < 1.
Show that
E_{RN}[S(t+\delta t)] = (p u ~+~ (1-p) d ) ~S(t)
E_{RN}[S^2(t+\delta t)] = (p u^2 ~+~ (1-p) d^2) ~S^2(t)
Solution:
We can regard S(t+\delta t) as a random variable over the probability space S = \{ U, D \} of the possible moves of the price of the asset from time t to time t + \delta t endowed with the risk-neutral probability function P : S \to [0, 1] with P(U) = p and P(D) = 1-p, and given by S(t+\delta t)(U) ~=~ S(t) u; S(t+\delta t)(D) ~=~ S(t) d.
Then,
E_{RN}[S(t+\delta t)] = P(U) \cdot S(t+\delta t)(U) ~+~ P(D) \cdot S(t+\delta t)(D) = (p u + (1-p) d ) ~S(t)
E_{RN}[S^2(t+\delta t)] = P(U) \cdot (S(t+\delta t)(U))^2 ~+~ P(D) \cdot (S(t+\delta t)(D))^2 = (p u^2 + (1-p) d^2) ~S^2(t)