Consider a market with one risk-free asset and one risky asset that follows the binomial tree model. Suppose that whenever stock goes up, you can predict that it will go down at the next step. Find a self-financing (but not necessarily predictable) strategy with V(0) = 0, V(1) ≥ 0 and 0 # V(2) ≥ 0.

Question

Consider a market with one risk-free asset and one risky asset that follows the binomial tree model. Suppose that whenever stock goes up, you can predict that it will go down at the next step. Find a self-financing (but not necessarily predictable) strategy with [latex]V (0) = 0, V (1) ≥ 0[/latex] and [latex]0 ≠ V (2) ≥ 0.[/latex]

Accepted Answer

If you happen to know that an increase in stock price will be followed by a fall at the next step, then adopt this strategy:

At time 0 do not invest in either asset.
At time 1 check whether the stock has gone up or down. If down, then again do not invest in either asset. But if stock has gone up, then sell short one share for S(0)u, investing the proceeds risk-free.

Clearly, the time 0 and time 1 value of this strategy is V (0) = V (1) = 0. If stock goes down at the first step, then the value at time 2 will be V (2) = 0. But if stock goes up at the first step, then it will go down at the second step and [latex]V (2) = S(0)u(r − d)[/latex] will be positive, as required, since [latex]u > r > d.[/latex] (The notation is as in Section 3.2.) Clearly, this is not a predictable strategy, which means that no arbitrage has been achieved.

Question 4.4: Consider a market with one risk-free asset and one risky ass...

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