As with all financial assets the price of an option should equal the expected value of the option. However, unlike other financial assets it is impossible to figure out expected cash flows and discount them using the opportunity cost of capital as discount rate. In particular the latter is impossible, as the risk of an option changes every time the underlying stock price moves.

Black and Scholes solved this problem by introducing a simple option valuation model, which applies the principle of value additivity to create an option equivalent. The option equivalent is combining stocks and borrowing, such that they yield the same payoff as the option. As the value of stocks and borrowing arrangements is easily assessed and they yield the same payoff as the option, the price of the option must equal the combined price on the stock and borrowing arrangement.

Example:

- How to set up an option equivalent

- Consider a 3-month Google call option issued at the money with an exercise price of $400.

- For simplicity, assume that the stock can either fall to $300 or rise to $500.

- Consider the payoff to the option given the two possible outcomes:

- As the payoff to the option equals the payoff to the alternative of buying 0.5 stock and borrowing $150 (i.e. the option equivalent), the price must be identical. Thus, the value of the option is equal to the value of 0.5 stocks minus the present value of the $150 bank loan.

- If the 3-month interest rate is 1%, the value of the call option on the Google stock is:

• Value of call = Value of 0.5 shares - PV(Loan)

= 0.5 • $400-$150/1.01 = 51.5

The option equivalent approach uses a hedge ratio or option delta to construct a replicating portfolio, which can be priced. The option delta is defined as the spread in option value over the spread in stock prices:

Example:

- In the prior example with the 3-month option on the Google stock the option delta is equal to:

- Thus, the options equivalent buys 0.5 shares in Google and borrow $150 to replicate the payoffs from the option on the Google stock.

9.3.1. Binominal method of option pricing

The binominal model of option pricing is a simple way to illustrate the above insights. The model assumes that in each period the stock price can either go up or down. By increasing the number of periods in the model the number of possible stock prices increases.

Example:

- Two-period binominal method for a 6-month Google call-option with a exercise price of $400 issued at the money.

- In the first 3-month period the stock price of Google can either increase to $469.4 or decrease to $340.9. In the second 3-month period the stock price can again either increase or decrease. If the stock price increased in the first period, then the stock price in period two will either be $550.9 or $400. Moreover, if the stock price decreased in the first period it can either increase to $400 or decrease to $290.5.

- To find the value of the Google call-option, start in month 6 and work backwards to the present. Number in parenthesis reflects the value of the option.

- In Month 6 the value of the option is equal to Max[0, Stock price - exercise price]. Thus, when the stock price is equal to $550.9 the option is worth $150.9 (i.e. $550.9 - $400) when exercised. When the stock price is equal to $400 the value of the option is 0, whereas if the stock price falls below the exercise price the option is not exercised and, hence, the value is equal to zero.

- In Month 3 suppose that the stock price is equal to $469.4. In this case, investors would know that the future stock price in Month 6 will be $550.9 or $400 and the corresponding option prices are $150.9 and $0, respectively. To find the option value, simply set up the option equivalent by calculating the option delta, which is equal to the spread of possible option prices over the spread of possible stock prices. In this case the option delta equals 1 as ($150.9-$0)/($550.9-$400) = 1. Given the option delta find the amount of bank loan needed:

- Since the above portfolio has identical cash flows to the option, the price on the option is equal to the sum of market values.

• Value of Google call option in month 3 = $469.4 - $400/1.01 = $73.4

- If the stock price in Month 3 has fallen to $340.9 the option will not be exercised and the value of the option is equal to $0.

- Option value today is given by setting up the option equivalent (again). Thus, first calculate the option equivalent. In this case the option delta equals 0.57 as ($73.4-$0)/( $469.4-$340.9) = 0.57.

- As today's value of the option is the equal to the present value of the option equivalent, the option price =

$400 • 0.57 - $194.7/1.01 =$35.7.

To construct the binominal three, the binominal method of option prices relates the future value of the stock to the standard deviation of stock returns, 0, and the length of period, h, measured in years:

In the prior example the upside and downside change to the Google stock price was +17.35% (469.4/400 -1= 0.1735) and -14.78% (340.9/400 - 1 = -0.1478), respectively. The percentage upside and downside change is determined by the standard deviation on return to the Google stock, which is equal to 32%. Since each period is 3 month (i.e. 0.25 year) the changes must equal:

Multiplying the current stock price, $400, with the upside and downside change yields the stock prices of $469.4 and $340.9 in Month 3. Similarly, the stock prices in Month 6 is the current stock price conditional on whether the stock price increased or decreased in the first period.

9.3.2. Black-Scholes Model of option pricing

The starting point of the Black-Scholes model of option pricing is the insight from the binominal model: If the option's life is subdivided into an infinite number of sub-periods by making the time intervals shorter, the binominal three would include a continuum of possible stock prices at maturity.

The Black-Scholes formula calculates the option value for an infinite number of sub-periods.

Black-Scholes Formula for Option Pricing

where

• N(d1) = Cumulative normal density function of (d1)

• P = Stock Price

• N(d2) = Cumulative normal density function of (d2)

| • PV(EX) = Present Value of Strike or Exercise price = EX • e-rt

The Black-Scholes formula has four important assumptions:

- Price of underlying asset follows a lognormal random walk

- Investors can hedge continuously and without costs

- Risk free rate is known

- Underlying asset does not pay dividend

Example

- Use Black-Scholes' formula to value the 6-month Google call-option

- Current stock price (P) is equal to 400

- Exercise price (EX) is equal to 400

- Standard deviation (a) on the Google stock is 0.32

- Time to maturity (t) is 0.5 (measured in years, hence 6 months = 0.5 years)

- 6-month interest rate is 2 percent

- Find option value in five steps:

• Step 1: Calculate the present value of the exercise price PV(EX) = EX • e-rt = 400 • e-0.04 0.5 = 392.08

• Step 2: Calculate d1:

• Step 3: Calculate d2:

• Step 4: Find N(d1) and N(d2):

- N(X) is the probability that a normally distributed variable is less than X. The function is avail in Excel (the Normdist function) as well as on most financial calculators.