Question

1. Consider a model with only one time period. Assume that there exist a stock and a cash bond in the model. The initial price of the stock is $40. The investor believes that with probability 1/5 the stock price will drop to $20 and with probability 4/5 the stock price will rise to $80 at the end of the time period. The cash bond has an initial price of $100 and it will with certainty deliver $110 at the end of the period. Does this model admit any arbitrage opportunities? Explain your answer. Find a price of a European Put option with a maturity at the end of the time period and a strike price of $60 using risk neutral probabilities and replication principle. Comment on the equivalence of the two approaches.

Answer 1

If $40 be invested in the cash bond at the start of the period, it will return 40 x 1.1 = $44 at the end of the period.

If $40 be invested in the stock at the start of the period, it is expected to return either $20 or $80 at the end of the period.

Since 20 < 44 < 80, the model admits no arbitrage opportunities.

The model would have admitted arbitrage opportunities had the return on the cash bond investment been less than $20 or greater than $80.

For example if the return on cash bond investment had been less than $20, the investor could have shorted $40 nominal of the bond and used the proceeds to buy the stock to lock in a risk less profit at the end of the period.

On the other hand if the return on cash bond investment had been greater than $80, the investor could have shorted the stock at $40 and used the proceeds to buy $40 nominal of the bond to lock in a risk less profit at the end of the period.

Valuing the put using risk-neutral probabilities

Risk-free growth factor = f = 110/100 = 1.1

u = 80/40 = 2, d = 20/40 = 0.5

Risk-neutral probability of up movement = q = (1.1 - 0.5)/(2 - 0.5) = 0.4

Risk-neutral probability of down movement = 1 - q = 1 - 0.4 = 0.6

Strike price of the put = K = $60

Payoff of the put at maturity = Max(K - S1 , 0), where S1 is the stock price at maturity.

= 0, if stock price moves up or 60 - 20 = 40 if stock price moves down.

Therefore price of the put is

{q x0 + (1-q) x 40} / f = {0.4 x 0 + 0.6 x 40} / 1.1 = $21.82

Valuing the put using replicating portfolio strategy

We establish a replicating portfolio by taking a position in $\phi$ shares and $\psi$ cash.

This portfolio will be worth 80$\phi$ + 1.1$\psi$, if stock price moves up or  20$\phi$ + 1.1$\psi$, if the stock price moves down. For the portfolio to be replicating we must have

80$\phi$ + 1.1$\psi$ = 0

and

20$\phi$ + 1.1$\psi$ = 40

subtracting we have $\phi$ = -0.66666667 and $\psi$ = 48.48484848

Therefore the price of the put

= The current value of the portfolio = 40$\phi$ + $\psi$ = 40x(-0.6666666) + 48.48484848

= $21.82

The two valuation approaches give identical answer as the first approach can be derived from the second approach.