1. Assume that the stock price is of the form S(t) = S(0)exp(µt − 12σ2t +σW(t)). Generate a sequence of daily prices over two months. Then given this sample try to recover the volatility coeﬃcient σ.

2. Assume A(0) = 1. Consider the following strategy:

x(t) =0 for t ∈ [0, 12T)

x(t)=1 for t ∈ [12T,T]

Find y(t) so that (x(t),y(t)) is self-ﬁnancing. Is this an admissible strategy? Is this a martingale strategy?

3. Prove that it is impossible to ﬁnd a self-ﬁnancing strategy (x(t),y(t)) such that V(x,y)(t) = tS(t) where S(t) = S(0)exp(µt − 12σ2t + σW(t)), A(t) = ert. 4. Assume that two stock prices satisfy the

equations

dS1 = µ1S1dt + σ1S1dW dS2 = µ2S2dt + σ2S2dW

Discuss the problem of ﬁnding a single martingale probability for both processes.

Namely

a) Find a condition guaranteeing existence of a martingale measure, common

for both stocks.

b) If this condition is violated, try to construct an arbitrage strategy. Hint: try do design a self-ﬁnancing strategy with zero noise term and return beating the risk free rate.

You will need to generalise the notion of self-ﬁnancing strategy to involve two stocks, and to formulate and prove, the characterisation using the discounted values, together with the formula for the money market account position.

5. Assume the S(t) is the stock price in the Black-Scholes model with µ = 10%, σ = 20%; assume r = 5% and let T = 1, K = 100.

Find the price of a digital option with strike price K (the payoﬀ is H = 1{S(T)≥K}) using the general pricing formula: H(0) = e−rTEQ(H), by means of the Monte-Carlo simulation. That is, simulate 1000 prices S(T) in Excel using NORMSINV(RAND()) as the generator for a standard normal random variable X (arrange the values of X in the column A1-A1000). Then use the average of the payoﬀ as an approximation to the expectation.

Find the theoretical price of this option and compare with the approxima-tion.

Extend your approximation to 10,000 samples (optional).

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