Question

You construct a one-period binomial tree to model the price movements of a stock.

You are given:

1. The length of one period is 6 months.
2. The current price of the stock is 100.
3. The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 3%.

Suppose:

• u denotes one plus the rate of gain on the stock if the stock price goes up.
• d denotes one plus the rate of loss on the stock if the stock price goes down.
• r denotes the continuously compounded risk-free interest rate.

Which of the following parameters would give rise to an arbitrage opportunity?

1. u=1.011,d=0.822,r=0.05

2. u=1.021,d=0.981,r=0.06

3. u=1.025,d=1.002,r=0.07

4. u=1.028,d=1.010,r=0.08

5. u=1.029,d=1.022,r=0.09

PlEASE PROVIDE EXPLANATION

Answer 1

No arbitrage opportunity exist when it satisfy the condition

0<q<1 where q is risk neutral probability

q ={exp(r) - d}/ (u-d) where "u" is the up factor , "d" is the down factor , R is the intereest rate .

so the required condition for no arbitrage opportunity is

d<exp(r)<u

(i) u =1.011 , d =0.822 r =0.05

exp(0.05) =1.0512

0.822<1.0512 1.011 arbitrage opportunity exist

(iI) u=1.021 , d=0.981 , r=0.06

exp(0.06) =1.061

0.981<1.0611.021 arbitrage opportunity exist

(iii) u =1.025 d = 1.002 r= 0.07

exp(0.07) =1.0725

1.002<1.07251.025 arbitrage opportunity exist

(iV) u = 1.028 d =1.010 r=0.08

exp(0.08) =1.083287

1.010<1.0832871.028 arbitrage oppotunity exist

(v) u = 1.029 d =1.022 r = 0.09

exp(0.09) =1.0941

1.022<1.09411.029 arbitrage opportunity exist

all above parameters give rise to arbitrage opportunity as they can short sell the share and deposit the amount in bank they earn more than investing in share