Question for Abraham Weishaus...

[Burnt Orange]

... or anyone who can help. This question is in reference to Exercise 3.14 of the ASM Study Manual for SOA Exam MFE. Given the strike price, stock price on a nondividend paying stock and the risk free rate, we are asked to calculate the change in a European call premium if the volatility increases.

I understand that I first must calculate u and d, then p*, to calculate the Call premium. This is where I'm a little confused. On the solution, you only take into account the up moment (a positive payoff), and not the down movement. In this exercise, dS > K (also a positive payoff). Shouldn't this too be taken into account when calculating the premium? Is there an implicit assumption that no matter what the down movement on the binomial tree is, the payoff is 0? In otherwords, why does the solution only consider the up movement when calculating premiums or this an error?

Thank you for your time and any insight you can give me.

[Burnt Orange]

I should clarify:

K = strike price
S = current stock price
u = up movement of the stock price after one period on the binomial tree
d = down movement of the stock price after one period on the binomial tree

Thanks

[Captain Nemo]

While I don't have the ASM study manual, yes, you must consider the possibility that the down movement provides a payoff when pricing a European call option, or conversely, the possibility that the up movement provides a payoff when pricing a European put option.

This leads to three possible situations when trying to determine whether early exercise is optimal for the 1-period binomial model of an American option.

Note that if both up and down movements provide a payoff, delta for the option is either e^(-delta*(T-t)) for a call option or -e^(-delta*(T-t)) for a put option at that node - which makes constructing the replicating portfolio rather easy.

[Abraham Weishaus]

Yes, the solution is wrong. Since the option pays off either way, its value is simply the definite payoff you'll get (25e^{0.01}-20) discounted. Since the payoff is the same with either volatility, the answer to the question is 0.