Question

In Questions 1 and 2, we have a stock that is currently priced at $50 a share, and in one period, it will be worth either $45 or $55. Let’s call this stock, CBA. Suppose that the probability that CBA price will increase is 99% and the probability that it will decrease is 1%.

Now, suppose, we have another stock that is currently priced at $50 a share, and in one period, it will be worth either $45 or $55. Let’s call this stock, ZYX. Suppose that the probability that ZYX price will increase is 1% and the probability that it will decrease is 99%. Suppose that just like in Questions 1 and 2, we also have European call and put options on ZYX with one period to expiration and an exercise price of $50.

What is the relation between the binomial prices of European call on CBA and the European call on ZYX? What is the relation between the binomial prices of European put on CBA and the European put on ZYX? EXPLAIN.

Answer 1

Let the continuously compounded risk free rate be r and assume that both the stocks are non-dividend paying

u= 1.1

d=0.9

So, the risk neutral probability q = (exp(rt) -d)/(u-d)

Now, since the risk neutral probability is the same for stock CBA and stock XYZ , the actual probability does not matter and the one period price of the European call options on both the stocks will be the same.

This happens because it is impossible to find two stocks which have same prices today and same expected prices after one period but different true probabilities of going up and going down. For example, in this case, existence of CBA and XYZ stocks as given is impossible because either the spot price of CBA should go up or spot price of XYZ should go down as per the given information.