Question

0. Consider the following one-period binomial model for stock price. At t = 0 the stock price is $80 and at t = 1 (t is in years) it could be $70 with probability p > 0 and $y with probability 1 − p. The interest rate is assumed to be 8%.

   1. (1) Determine the range of values for y that precludes arbitrage in this model.

   2. (2)   Assume that y = $83. Construct an arbitrage strategy for this model.1

Answer 1

Part (1)

Up factor = u = y/80

Down factor = d = 70/80 = 0.875

Risk neutral probability of up state, P = (1 + r - d) / (u - d) = (1 + 8% - 0.875) / (u - 0.875) = 0.205 / (u - 0.875)

0 < P < 1

hence, 0 < 0.205 / (u - 0.875) < 1

hence, 0.205 < u - 0.875

Hence, u > 0.205 + 0.875 = 1.08

y / 80 > 1.08

Hence, y > 80 x 1.08 = $ 86.40

Hence, the range of values of y will be: y > 86.40

Part (2)

y = $ 83

Arbitrage strategy:

Arbitrage element	Cash flows at t = 0	Cash flows at t = 1
Short sell the stock	+ 80
Lend the money at risk free rate	- 80	+ 80 x (1 + 8%) = + 86.40
		- 70 in down state - 83 in up state
Net cash flows	0	+ $ 16.40 in down state + $ 3.40 in up state

We are thus making risk-less money at the end of period 1, in either case without any investment at any time. This is the arbitrage opportunity.