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## 4.4.2 Basic Option Valuation

Like most of modern financial theory, the systematic valuation of options is a surprisingly new field. In the 1960s, traded options only existed on a few of the bigger stocks. Prices were largely set by negotiations, and rules of thumb prevailed. Meanwhile, the question of option pricing was heating up among economic academics.

INSERT BLACK-SCHOLES HISTORY FROM CAPITAL IDEAS HERE.

Thus, Black and Scholes argued the appropriate price for an option depends on a relatively small number of variables:

• the current price of the underlying asset
• the strike price of the option
• the time remaining until the option's expiration
• the risk-free rate of interest
• the underlying asset's volatility.

Absent from the list are predictions of future market prices or volatility, which are captured by the asset's volatility compared to the market and the market-set risk-free interest rate.

Most options are written with stock as the underlying asset, and for call options the formula for the Black-Scholes Option Pricing Model is

 Co = N(d1)Vs - N(d2)E/ert Co = Value of call option N(d) = cumulative normal probability density function Vs = Current value of stock E = Exercise price e = base, natural log (2.71828) r = short-term annual interest rate (continuously compounded) t = length of time to option expiration (in years) d1 = [ln(Vs/E) + t(r + (s2/2)] / (s2t)1/2 d2 = [ln(Vs/E) + t(r + (s2/2)] / (s2t)1/2 ln = natural log s = sigma [standard deviation of annual rate of return on stock (continuously compounded)]

The formula seems daunting, and the theory behind it is complicated. However, we will see that use of the formula boils down to plugging in variables and solving for Co. All of the variables are observable in the Wall Street Journal with the exception of the variance, which will be the variance of the percentage change in daily stock prices over some previous period. (A note of caution: The variance is usually determined using the recent past volatility of the stock. We should not place total confidence in this method as past volatility is not always the best predictor of future volatility).

There are websites where investors can input these variables to price options: WallStreetCom

The proper risk-free rate will be the rate corresponding to the treasury bill maturing closest to the maturity of the option being valued. Use the attached table to find the values for N(d1) and N(d2). For a d1 or d2 that is positive or above the mean of 0, determining N(d1) or N(d2) will require a minor adjustment [1-N(d1 or 2)]. For a negative d1 or d2 , no adjustment is necessary, and N(d1) or N(d2) will simply be the number from the table.

Let's quickly examine the formula in a little more depth. N(d1) represents the ratio of shares of stock to options needed to maintain a fully hedged, or riskless, position. If we can consider the option holder as a levered investor and he or she borrows an amount equal to the exercise price at a rate of r, the second term will then represent this loan, the present value of the exercise price multiplied by the adjustment factor, N(d2). Ultimately, the price of an option will be driven by the short-term interest rate, time to expiration, and volatility of the underlying stock. An increase in any of these will translate into a higher price, but an increase in the risk-free rate will usually have the least impact.

Let's see the formula in action and observe its sophisticated simplicity.

Example

Suppose you are contemplating the purchase of a call option on the stock of Commander Cody Enterprises. The option has a strike price of \$18 and expires in three months. Commander Cody's stock is currently trading at \$23, and you determine that the expected standard deviation of its continuously compounded return is .5. The current rate on a t-bill with similar maturity is 6%. Would you pay \$5.40 for this option?

Answer: Using the Black-Scholes Option Pricing Model, we can approximate the appropriate price for this option. We begin by solving for d1 using the formula:

 = ln(Vs/E) + (r + (s2 /2)) * t square root of (s2*t) = ln(23/18) + (.06 + (.52 /2)) * (91/366) square root of (.52 * (91/366)) = .2451 + .046 .2493 = 1.1677 We then look on the probability table for our z of 1.1677. The table tells us that the N(d1 ) is 0.8790.

Next, we solve for d2 using the formula from above:

 = d1 - square root of (s2 * t) = 1.1677 - square root of (.52 * (91/366)) = 1.1677 - .2493 = .9184 We then look on the probability table for our z of .9184. The table tells us that the N(d2 ) is .8212.

We are almost finished. We now solve for the price of the call option by using the formula from above:

 = S[N(d1)] - E * e-rf*t [N(d2)] = 23(.8790) - 18 * (2.7183-.06*(91/366)) (.8212) = 20.217 - 14.563 = \$5.65

Thus, as we have put the price of the option at \$5.65, you would purchase these underpriced options at \$5.40.

What if the strike price was \$20 instead of \$18? What if the option expired in one year instead of three months? Use the attached spreadsheet to explore these scenarios. This spreadsheet can also be used to check your answers on future problems.

Through use of the Black-Scholes Option Pricing Model, we can also value puts. First, you must determine the value of a call with the same terms as the put to be valued. Once the price of the call is found, finding the price of the put is easy:

 P0 = C0 - S0 + E * e-rt C0 Previously valued call S0 Stock Price E Exercise or strike price e Exponential function = approximately 2.71828 r Continuously compounded annual risk free rate t Time to expiration on an annual basis

Using the Black-Scholes spreadsheet, you can consider the effects of changing variables in the formula.

Student paper

Should unvested stock options be treated as marital property? If so, how should they be valued when dealing with equitable distribution? See Lynn Karlet, Valuing Unvested Stock Options as a Marital Asset in an Equitable Distribution Dissolution: Insights from Wendt v. Wendt.

 4.4.1 Option Fundamentals 4.4.3 Option Applications