Corporate Finance, Module 21: “Option Valuation”

Homework Assignment

(The attached PDF file has better formatting.)

Binomial Tree Pricing Method

A firm’s current share price is $80; one year from now, its share price will either fall to $76 or rise to $92. The risk-free rate is 5%, and one-year European call and put options on the stock have an exercise price of $85.

A.    What is the upward movement if the stock price rises to $92? (Express this as a factor of $92 / $80.) Call this upward movement by the symbol U.
B.    What is the value of the call option at its maturity if the stock price rises to $92? (The call option gives the investor the right to buy the stock for the strike price of $85. How much would the investor pay to buy a stock for $85 if its price is $92?) Call this price Call+, or the value of the call option at maturity if the stock price rises.
C.    What is the value of the put option if the stock price rises to $92? (The put option gives the investor the right to sell the stock for the strike price of $85. How much would the investor pay to sell a stock for $85 if its price is $92?) Call this price Put+, or the value of the put option at maturity if the stock price rises.
D.    What is the downward movement if the stock price falls to $76? Call this downward movement by the symbol D.
E.    What is the value of the call option if the stock price falls to $76? (Call–)
F.    What is the value of the put option if the stock price falls to $76? (Put–)
G.    The actual probability of a rise in the stock price is not relevant for options pricing. We discuss it here to differentiate it from the risk-neutral probability. Suppose the expected return on the stock is 12% per annum. The stock has only two possible values at the end of the year, $92 or $76. If the probability of rising to $92 is P, it must be that P × U + (1 – P) × D = 1.120. We solved for U and D earlier; now solve for P.
H.    What is the risk-neutral probability of a rise in the stock price? If all investors are risk-neutral, the expected return from the stock is 5% per annum, not 12% per annum. To determine the risk-neutral probability, solve P × U + (1 – P) × D = 1.050. We use this value of P in the remaining parts of this homework assignment.
I.    We used U and D as factors; we can also express them as percentage returns. If U = $92 / $80 – 1 and D = $76 / $80 – 1, then P × U + (1 – P) × D = 5%. (There is nothing to write for this part; it is informative.)
J.    What is the expected value of the call option at its maturity in a risk-neutral world? (We solved for the values of the call option at its maturity if the stock price moves up or down. Using the value of P, solve for the expected value of the call option at its maturity in the risk-neutral world: P × Call+ + (1 – P) × Call–)
K.    What is the present value of the call option? (In a risk-neutral world, all discounting is done at the risk-free interest rate. Discount the value obtained above at a 5% rate.)
L.    What is the expected value of the put option at expiration in a risk-neutral world?
M.    What is the present value of the put option?
N.    Verify that the put call parity relation holds. Using the present values of the call option and put option, show that call + PV(exercise price) = put + stock price.

Part N : Put-Call parity relation
call + PV(exercise price) = Put + stock price

call + PV(exercise price) =$3.333 (from part K) + 85/1.05 = 84.29
put + stock price = 4.29 (from part M) + 80 = 84.29

h) I got p = 10.  I think this may be where I got off track.  I used this formula 0.15p + .05 +.05p = 1.050 because the stock could increase by 15% and could drop by 5%

J) use p=10 to get $70

K) PV($70) = 66.67 not the 3.33 that you posted.

The formula of Part H is the same as part G , except the right hand side is 1.05 instead of 1.12.

The formula is :
P x (1+ 15%) + (1-P)x(1- 5%)=1.05
==> P x 1.15 + (1-P) x 0.95 = 1.05
==> 1.15P + 0.95 - 0.95P=1.05
==> P=0.5

P (The probability of a rise in the stoc price)have to be <=1.

Or you can use the formula P x 15% + (1-p) x(-5%)=5%, you will get the
same answer P=0.5

[NEAS: Yes]

Just a note: I personally don't like the formula

expected return = p*ru + (1-p)*rl

where ru is the return for the up case, and rl is the return (lose) for the down case.  Because you have to REMEMBER that rl is negative.

instead, I like (1+expected return) = p*(1+ru)+(1-p)(1+rl) better.

I'm pretty sure from the text that C and E are 0.  It makes sense that the options are worthless in these situations.  They are not intended to be negative, right?  Anyone have any thoughts?

[NEAS: Correct]

The price of an option is non-negative, since you don't need to call if you wouldn't make any money.  You are never forced to call, hence the name option.

when you ask for Call+, do you mean the payoff amount or is it the price of the call option, since you describe the answer as either in section B of the list

I interpreted B as the price that the option is worth, which would be the price of the stock minus the strike price ($7)

[NEAS: Correct. At maturity, the final stock price is known. If it is more than the strike price, the value of the call option is the stock price minus the strike price.]