So in my finance textbook I have the following question: T-bills currently yield 5.5 percent. Stock in Nina Manufacturing is currently selling for $70/share. There is no possibility that the stock will be worth less than $65 per share in one year. What is the value of a call option with a $60 exercise price? What is the intrinsic value? The answer book says that the value is found like this: C0 = $70 – [$60/1.055] = $13.13. I don't understand how this works. Could anyone explain? Thanks!
Below I will try to explain two most common Binomial Option Pricing Models (BOPM) used.
First of all, BOPM splits time to expiry into N equal sub-periods and assumes that in each period the underlying security price may rise or fall by a known proportion, so the value of an option in any sub-period is a function of its possible values in the following sub period. Therefore the current value of an option is found by working backwards from expiry date through sub-periods to current time.
There is not enough information in the question from your textbook so we may assume that what you are asked to do is to find a value of a call option using just a Single Period BOPM.
Here are two ways of doing this:
First of all let's summarize your information:
Current Share Price (Vs) = $70
Strike or exercise price (X) = $60
Risk-free rate (r) = 5.5% or 0.055
Time to maturity (t) = 12 months
Downward movement in share price for the period (d) = $65 / $70 = 0.928571429
Upward movement in share price for the period (u) = 1/d = 1/0.928571429 = 1.076923077
"u" can be translated to $ multiplying by Vs => 1.076923077 * $70 = $75.38 which is the maximum probable share price in 12 months time. If you need more clarification here - the minimum and maximum future share prices are calculated from stocks past volatility which is a measure of risk. But because your textbook question does not seem to be asking this - you probably don't have to bother too much about it yet.
Just in case someone reading this is unclear - the Value of an option on maturity is the difference between the exercise (strike) price and the value of a share at the time of the option maturity. This is also called an intrinsic value.
Note that American Option can be exercised prior to it's maturity in this case the intrinsic value it simply the diference between strike price and the underlying share price at the time of an exercise.
But the Value of an option at period 0 (also called option price) is a price you would normally pay in order to buy it. So, say, with a strike of $60 and Share Price of $70 the intrinsic value is $10, whereas if Share Price was $50 the intrinsic value would be $0. The option price or the value of a call option in both cases would be fixed.
So we also need to find intrinsic option values when price falls to the lowest probable and rises to the maximum probable (Vcd and Vcu respectively)
(Vcd) = $65-$60 = $5 (remember if Strike was $70 then Vcd would be $0 because nobody would exercise an option that is out of the money)
(Vcu) = $75.38-$60 = $15.38
1. Setting up a hedge ratio:
h = Vs*(u-d)/(Vcu-Vcd)
h = 70*(1.076923077-0.928571429)/(15.38-5) = 1
That means we have to write (sell) 1 option for each share purchased in order to hedge the risks. You can make a simple calculation to check this, but I'm not going to go into too much detail here as the equestion is not about hedging.
Because this position is risk-free in equilibrium it should pay a risk-free rate (5.5%).
Then, the formula to price an option (Vc) using the hedging approach is:
Where (Vc) is the value of the call option, (h) is the hedge ratio, (Vs) - Current Share Price, (Vsu) - highest probable share price, (r) - risk-free rate, (t) - time in years, (Vcu) - value of a call option on maturity at the highest probable share price.
Therefore solving for (Vc):
(70-1*Vc)(e^(0.055*(12/12))) = (75.38-1*15.38) =>
(70-Vc)*1.056540615 = 60 =>
70-Vc = 60/1.056540615 =>
Vc = 70 - (60/1.056540615)
Which is similar to the formula given in your textbook, so I must assume that using 1+r would be simply a very close approximation of the formula above.
Then it is easy to find that Vc = 13.2108911402 ~ $13.21
2. Risk-neutral valuation:
Another way to calculate (Vc) is using a risk-neutral approach.
We first introduce a variable (p) which is a risk-neutral probability of an increase in share price.
p = (e^(r*t)-d)/(u-d)
so in your case:
p = (1.056540615-0.928571429)/(1.076923077-0.928571429) = 0.862607107
Therefore using (p) the (Vc) would be equal:
Vc = [pVcu+(1-p)Vcd]/(e^(rt)) =>
Vc = [(0.862607107*15.38)+(0.137392893*5)]/1.056540615 =>
Vc = 13.2071229185 ~ $13.21
As you can see it is very close to the hedging approach.
I hope this answers your questions.
Also bear in mind that there is much more to the option pricing than this. The most important topics to cover are:
Accounting for Dividends
Black-Scholes-Merton Option Pricing Model