Question

Consider a stock priced at $30 with a standard deviation of 0.3. The risk-free rate is 0.05. There are put and call options available at exercise prices of 30 and a time to expiration of six months. The calls are priced at $2.89 and the puts cost $2.15. There are no dividends on the stock and the options are European.   

What is the profit from the transaction from a finance perspective (adjusting for the TVM)?

$7.00 ?

$4.11 ?

$4.04 ?

$3.96 ?

none of the above ?

Assume that the stock begins paying a dividend of 3% and the price of the call option falls to $2.75. What is the price of the put in this situation?

Answer 1

From simple Cox Rubinstein Binomial Formula

European Call with S=30, X=30, Time=6 Months, Sd= 0.30 and Risk Free Rate of 5% is

u= e^{SD(Root of T)} and d= 1/u

u= e^{0.3(root of 0.5)} = 1.2363

d= 0.8089

So the price will either be S_u = 37.089

or S_d = 24.267

Value of C_u = 7.089 and C_d = 0

Probability of Up Move P = (e^rt -d)/(u-d) = (1.02531- 0.8089)/(1.2363-0.8089) = 0.506364 or 50.64%

and Probability of Down Move (1-P) = 0.493636 or 49.36%

Call Price = PV of (7.089*50.64%) = PV of 3.58987 =$3.50122 or $3.50

Simlarly for Put Option, C_u = 0 and C_d= 5.733

Put Price = PV of (5.733*49.36%) = PV of 2.829957 =$2.76

And as the call and put price are lower, we need to buy them,

So, Invest $5.04 and pay $5.167613 at settlement

If the price is 37.089 then you receive 7.089 and if it is 24.267 then you receive 5.7333 and the probabilities are known, so expected payoff is $6.42

And the profit for one put and call is $1.25203

If dividend is added, the r in above equations is substituted with 2% instead of 5% (r-q = 5-3 = 2)

The call and put option prices will go up, but as they fell, you should invest.

Good luck