# Theoretical Pricing Models: Binomial Option Pricing and the Black-Scholes Formula

Although several factors have been considered in what determines an option's worth, it is intuitively obvious that *what actually determines the worth of an option is the probability that the option will be in the money by expiration*, and *by how much*. Everything else can be subsumed under these 2 variables. If a given variable increases the option premium, it is because it increases 1 or both factors. Thus, a longer time until expiration or a greater volatility increases premiums because of the greater chance the option will be in the money by expiration, and by a larger amount; likewise, premiums are low for an option way out of the money, because there is little chance the underlying asset will reach the strike price by expiration.

While prices and time intervals are easy enough to measure, what cannot be known with certainty is the volatility of the underlying asset, and therefore, the probability that an option will be in the money or by how much, before expiration. **Historical volatility** is not necessarily a good indicator of future volatility, although it does provide some measure of volatility.

Various pricing models have been developed trying to more accurately gauge the true worth of options, or to price them better initially, when they are first created.

The **binomial option pricing model** starts by evaluating what a call premium should be if the underlying asset can only be 1 of 2 prices by expiration. A variable that can only be 1 of 2 values is known as a **binomial random variable**. By subdividing the time into smaller time intervals with 2 possible prices that are closer together, a more accurate option premium can be calculated. As the number of time periods increases, the distribution of possible stock prices approaches a **normal distribution** — the familiar bell curve.

## The Black-Scholes Formula

In 1973, United States economists Myron Scholes and Fisher Black developed a mathematical formula for calculating the price of an option based on variables such as the current price of the underlying asset, time until expiration, and by how much the prices varied over time, i.e., its volatility. Robert C Merton expanded on this pricing model.

To simplify, these economists based their mathematical model on certain assumptions:

- A "no arbitrage rule", meaning that prices reflect all information available about the underlying asset today and in the future. So, an arbitrageur could not earn guaranteed profits by hedging against future risk.
- An option contract can be constructed to offset any risk of any portfolio of assets.
- The fluctuation of asset prices is random, but based on a normal distribution, meaning that prices do not vary much over a brief period.

The **Black-Scholes formula** is the most widely used formula to calculate option premiums. Much easier to use than the binomial option pricing model, it, nonetheless, depends on assessing the volatility of the underlying asset, which is denoted by the standard deviation, σ, of the underlying asset prices about the current price.

Black-Scholes Formula: C_{0} = S_{0}N(d_{1}) - Xe^{-rt}N(d_{2})

d_{1} | = | ln(S_{0}/X) + (r + σ^{2}/2)Tσ√T |

d_{2} | = | d_{1} - σ√T |

- C
_{0}= current call premium. - S
_{0 }= current stock price. - N(d) = the probability that a value in a normal distribution will be less than d.
- N(d
_{2}) = the probability that the option will be in the money by expiration. - X = strike price.
- T = time (in years) until expiration.
- r = risk-free interest rate.
- e = 2.71828, the natural log base.
- ln = natural logarithm function.
- σ = standard deviation of the stock's annualized continuously compounded rate of return.

Note that:

- Xe
^{-rt}= X/e^{rt}= the present value of the strike price using a continuously compounded interest rate.

Requirements for validity:

- The stock pays no dividend before expiration.
- No changes in interest rate and variance before expiration.
- No discontinuous jumps in stock price, as would occur in a tender offer, for instance.

Although the Black-Scholes formula calculates the premium for a call, the put premium can be calculated by using the put-call parity formula.

Note from this formula, that the standard deviation, **σ**, which measures volatility, can be calculated if the other variables are known. This is called the **implied volatility**, because it is implied by the other variables. Some traders compare the implied volatility with the observed volatility to judge whether an option is fairly priced.

## Why Volatility Increases Time Value and Option Premiums

Volatility is the unknown change in price of the underlying security over time, and is what gives options any value at all. For instance, consider a hypothetical stock with a price that never changes. An option based on such a stock would never have any time value, because the underlying is always the same price, therefore, no one would want the option — neither put nor call — if it was out of the money. On the other hand, no one would sell an option based on this stock that was in the money, because it would certainly be exercised. Now consider another hypothetical scenario where the stock changed price according to some formula, so that anyone could calculate the stock price with certainty at any time. Again, there would be no time value to any option based on this stock, because its price at any time can be known by anyone. For instance, if this stock were $50 today, and it was known for certain that, before a specific expiration date, it would be $60, then no one would write a call with a $50 strike, unless they were getting $10 per share (although maybe they would charge a little less to get the money sooner) and no one would buy this call unless it were discounted enough to at least equal prevailing interest rates. Thus, it is the unknown fluctuation in the underlying prices that gives options value.

Stock | Prices | ||||
---|---|---|---|---|---|

SSS | 40 | 45 | 50 | 55 | 60 |

VVV | 30 | 40 | 50 | 60 | 70 |

Now consider 2 hypothetical stocks, currently at $50 per share. Stock SSS is relatively stable, and has ranged between $40 and $60 per share over the past year, whereas stock VVV is more volatile, and has ranged from $30 to $70. Further, assume that the chance is 1/5 that either stock will be at some specific price within its historical range, listed in the table, at expiration. Obviously, a call for VVV with a strike of $50 will command a higher premium than the same call for stock SSS for the same expiration date, because there is a 20% chance that the VVV call premium will be worth $20 per share, and a 40% chance that it will be worth at least $10 per share, the most that the call for SSS will be worth. There is a chance that VVV will be at $30 per share, and that SSS will be no less than $40 per share, but this doesn't matter, because if the stock price is ≤ the strike price, then the options will expire worthless, and the chance that they will expire worthless is 50% for both stocks. Thus, if the VVV call has a chance of paying $20 per share, but the most that the SSS call will pay is $10 per share, and the chance that they will expire worthless is the same, then it makes sense that the VVV call will command a higher premium, because it has a greater potential payoff.