# valuing options

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.

Intrinsic Value:

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:

(Vs-hVc)(e^(rt))=(Vsu-hVcu)

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)

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.

Also bear in mind that there is much more to the option pricing than this. The most important topics to cover are:

1. Multi-period BOPM

2. Accounting for Dividends

3. Black-Scholes-Merton Option Pricing Model

• Any standard deviation (volatility) would give a chance of less than \$65. A stock with a zero standard deviation of zero isn't a real stock, but a mental exercise. Of no use to the real investor. In my opinion. Apr 11, 2015 at 20:26
• It is not a zero volatility stock. Probable Up and Down movements are calculated using standard deviation of course. Simply in this question the values are already given i.e. lower is \$65 then we find the upper artificially. In reality one would use stock's historical volatility as you said. So to get an upward movement we calculate: u = e^[(v)*(sqrt(t/n)] , where (v) - standard deviation, (t) - time to expiry, (n) - number of sub-periods, and to find downward movement: d = 1/u ; then Vs times u and Vs times d to find \$ values of upper and lower probable limits.
– Andy
Apr 11, 2015 at 20:45
• A bell curve never reaches zero. A standard deviation of \$1, gives a small, but non-zero chance of a \$5 move. If you assume a zero chance of this move, how can the STDev be any number greater than zero? Clearly, I'm being dense here. Apr 11, 2015 at 20:48
• Because there is always some probability of an extreme outcome and bell-shaped curve never reaches zero - we never assume a zero chance of any move. SD is measured from historic data (how true or accurate it is and how perfect is the method to evaluate it is another question) so if in the past the stock had a 10% volatility over a single period of, say 30 days, this is generally accepted as a good indicator of the fact that it is highly unlikely that it would move more than that in the next 30 days. Obviously such moves happen and these are the times when people make or lose fortunes.
– Andy
Apr 11, 2015 at 21:12
• "There is no possibility that the stock will be worth less than \$65 per share in one year" - from this statement, how do you offer the number to use for volatility? Jun 10, 2015 at 21:18

Your strike is \$60 but you are given the information that spot cannot drop below \$65.

The value of a call option in Black Scholes is given by: S * N(d1) -e^{-rT} * X * N(d2)

You can look up the formula on wikipedia where it includes dividends. In essence, a call option is the difference between the Spot rate and the Present value of your strike X, price with some probability adjustments.

The probability of the underlying expiring above strike is 100% (by design of the question). N(d2) measures that probability. Likewise, N(d1) will also be 1, since you do know that there is no way the price will drop anywhere near your strike. This is only possible if Implied Vol is very low. Either way, given this setup, N(d1) = N(d2) = 1.

That way, you end up with S -e^{-rT}*X which is your solution. The Black Scholes model uses continuous interest rates but that is immaterial here.