Problem Set 6, problem 1 Q&A

Here is a Q&A I just had with a Finance 4366 student about Problem Set 6, problem 1:

Student: Dr. Garven, I am a little confused about problem 1 for PS6. Do you want us to write out a huge binomial tree table with 10 steps and have that over the 6-month period of the call? Or do just want us to use an equation to get the price of the call?  Thanks, Student

Dr. Garven: With the Cox-Ross-Rubinstein (CRR) model, there is no point in writing out the entire binomial tree for a large number of timesteps involving either a European call option or a European put option. Since European options cannot be exercised prior to expiration, CRR provides a simplified framework for determining the complete set of terminal nodes at which the option expires in-the-money (see p. 34 of the Binomial Trees lecture note). Once you can identify all terminal in-the-money nodes, then calculate the payoff at each of these nodes using the appropriate equation (which is Max(0, S-K) for calls and Max(0, K-S) for puts). Once you know all of the payoffs, then multiply each payoff by the risk-neutral probability of the node at which the payoff occurs. Add all of these probability-adjusted payoffs together to determine the certainty-equivalent, or risk-neutral expected value of the option at expiration, and then the arbitrage-free price of the option corresponds to the present value of the certainty-equivalent.

For call options, the solution procedure that I describe here is captured by the CRR call option pricing equation which appears on p. 33 of the Binomial Trees lecture note; the version of the equation that appears on p. 35 is algebraically equivalent and shown in order to reinforce the concept that prices determined under the CRR model converge to Black-Scholes-Merton (BSM) model prices (see p. 36 for the BSM call option pricing equation).

Of course, once you know the CRR price for the call option, you can either use the CRR put pricing equation shown on page 37 of the Binomial Trees lecture note or simply apply the appropriate version of the put-call parity equation; for purposes of this problem set, the latter approach is much simpler and will suffice.

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