Question

1. Consider the one period binomial model and assume 0 < So< 00, S1(H) -- uSo and Si (Τ)-dSo for some 0 〈 1 + r 〈 d 〈 u. P is an arbitrage oportunity. rove or disprove There

Answer 1

There exists an arbitrage opportunity. Please see the proof below:

Let's consider this position at t= 0

1. Borrow an amount B at risk free rate
2. Use this amount to buy the stock
   1. Since current stock price is S₀, we can buy N = B / S₀ number of stock

Thus my initial investment at t=0 is zero.

At t= 1,

Liability on the borrowed amount = Borrowed amount + interest on borrowed amount = B + r x B = B x (1 + r)

Situation 1: Up State when stock price is S₁(H) = uS₀

Value of N number of stock we have = N x S₁(H) = (B / S₀) x uS₀ = uB

Value of the portfolio = Value of N number of stock - Liability on the borrowed amount

= uB - Bx(1+r) = B x [u - (1 + r)]

Since, 0 < 1+r < d < u

Hence, u - (1 + r) > 0

Hence, value of the portfolio = B x [u - (1 + r)] > 0

Since, u - (1 + r) is a positive number and the value of the portfolio in up state is positive, we make money at t = 1 without investing anything at t=0.

Situation 2: Down State when stock price is S₁(T) = dS₀

Value of N number of stock we have = N x S₁(T) = (B / S₀) x dS₀ = dB

Value of the portfolio = Value of N number of stock - Liability on the borrowed amount

= dB - Bx(1+r) = B x [d - (1 + r)]

Since, 0 < 1+r < d < u

Hence, d - (1 + r) > 0

Hence, value of the portfolio = B x [d - (1 + r)] > 0

Since, d - (1 + r) is a positive number and the value of the portfolio in down state is positive, we make money at t = 1 without investing anything at t=0.

In both the possible scenarios, we are making money on this portfolio without any initial investment. Thus there is an arbitrage opprotunity.