Question

4. Consider the one-period binomial model, and let V1 S1- That is, the derivative security pays the stock price at time t-1. Find the time t = 0 no-arbitrage price of the derivative, Vo

Answer 1

There are multiple ways to solve this problem.

Method 1:

The value of the derivative at t=1 is V₁ = S₁

That means i have to create a replicating portfolio today that will result into a value of S₁ at the end of time period 1. The replicating portfolio should be nothing but 1 number of stock today. This portfolio of 1 number of stock will assume the value of S₁ at the end of period 1.

So, V₀ = value of the derivative today = value of the replicating portfolio today = value of 1 number of stock today = S₀

Hence, V₀ = S₀

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Method 2:

The symbols have usual meanings.

Let's revisit the assumptions and the results of 1 period binomial model where a stock with current price of S₀ can move to S_u = S₀ x u in up state and S_d = S₀ x d in down state.

Assume continuous compounding risk free interest rate r.

Risk neutral probability of up state at the end of 1 period: P_u = (e^r - d) / (u - d) and

Risk neutral probability of down state at the end of 1 period: P_d = (u - e^r) / (u - d)

Expected value of the stock at the end of 1 period, S₁ = P_u x S_u + P_d x S_d

$=\frac{(e^{r} - d)}{(u - d)}\times S_{0}\times u + \frac{(d- e^{r})}{(u - d)}\times S_{0}\times d = S_{0}\times e^{r}$

Hence the expected value of the derivative at the end of 1 period = V₁ = S₁ = S₀ x e^r

Hence value of the derivative today = V₀ = PV of V₁ = V₁.e^-r = S₀ x e^r x e^-r = S₀