Sheet 02 - HW1 PDF

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A. Minca

Financial Engineering with Stochastic Calculus I

Fall 2020

Assignment Sheet 2

1. Consider a discrete-time (t = 0, 1, 2, ...) market consisting of a riskless bond with price Bt = (1 + r)t and a stock index with price St = S0 V1 · ... · Vt where V1 , V2 , ... are independent and identically distributed positive random variables with E[Vt 2] < ∞ and E[1/Vt2] < ∞. Suppose that an investor with initial wealth X0 = 1 employs a self-financing trading strategy which at each time t invests a fixed proportion λ of the current wealth Xt into the stock index. a) Find a formula for Xt in terms of V1 , ..., Vt . b) The average logarithmic return of the strategy at time n is n1 log Xn . Show that lim

n→∞

1 log Xn = c(λ) n

for some deterministic function c(·) of the parameter λ. We are interested in finding an optimal proportion λ∗ which maximizes the long-term average logarithmic return c(λ). c) Show that c(·) is a concave function. d) By computing c′ (0) and c′ (1), find conditions on V which guarantee that there exists an optimizer λ∗ ∈ (0, 1). 2. Implement in a language of your choice a pricing and hedging algorithm for an European option in a Binomial model with N periods. You can choose a call option, a put option, or any other payoff you are interested in. You should include your code with the submission, a print screen which verifies point 3 below, and the graph at point 4. The parameters of the model are the interest rate r, u and d that govern the price change in each period, N the number of periods. You may use a payoff function of your choice. a) Compute the premium of the option using the recursive algorithm in the lecture notes b) Compute the hedging strategy, i.e, the delta for each period and possible value of the stock in that period.

c) Simulate under the real probability measure a path of the stock and verify that the value of the replicating portfolio at time N equates the payoff of the option. d) Fix now a time interval T, say T = 1 and set h = T /N . Let σ = .25. Plot the value of your option as you increase N for √ the parameters r =√.1 ∗ h (where .1 is the annual interest rate), u = 1 + r + σ h and d = 1 + r − σ h. 3. Let X be a random variable on a probability space (Ω, F, P ) and G ⊂ F a sub-σ algebra (i.e. there is more information in F than in G ) a) Show that Y ∗ := E[X|G] satisfies E [(X − Y ∗ )2 ] ≤ E [(X − Y )2 ]

(1)

for all G-measurable random variables Y . Moreover, show that if we have equality in (1) for some Y , then Y = Y ∗ a.s. Hint. Consider E[(X − Y )2 − (X − Y ∗ )2 |G] and arrive at the expectation of the square of a random variable. Remark. This result says that the conditional expectation E[X|G] minimizes the variance of the error Y − X among all G-measurable estimates Y for X . b) Let Y be a G-measurable random variable. Show that  Cov X − E[X|G], Y = 0.

Due on Monday October 5 2020, 5 pm (EOD)....