# Valuation Techniques -- Option Pricing

Overview:

- Define an option and differentiate major types.
- Identify the factors that affect the value of an option and the role of these factors.
- Describe the black Scholes and binomial option pricing models.

One of the most difficult financial instruments to value has been options which is the right to buy or sell an asset. In order to solve this issue, several pricing options models have been developed. This discussion, as mentioned in Wiley CPAexcel defines options and the factors that play a role in establishing the value of an option. It also identifies and describes two of the most common option pricing models such as the Black Scholes and the binomial pricing models.

**Introduction –**

An option is a contract that entitles the owner (holder) to buy (call option) or sell (put option) an asset (e.g. stock) at a stated price within a specified period. Financial options are a form of derivative instrument (contract).

- Under terms of an American-style option, the option can be exercised any time prior to expiration.
- Under terms of a European-style option, the option can be exercised only at the expiration (maturity date).

**B. Valuing options**

- An option may or may not have value.
- Valuing an option, including determining it has no value, is based on six factors:
- Current stock price relative to the exercise price of the option – the difference between the current price and the exercise price affects the value of the option. The impact of the difference depends on whether the option is a call option or a put option.
- Call option (a contract that gives the right to buy) – A current price above the exercise (or strike price) increases the option value, the option is considered “in the money”. The greater the excess, the greater the option value.
- Put option (a contract that gives the right to sell) – A current price below the exercise price increases the option value; once again the option is considered “in the money”. The lower the current price relative to the exercise price, the greater the option value.

- The longer the time to expiration, the greater the option value (because there is a long time for the price of the stock to go up).
- The risk-free rate of return in the market – The higher the risk-free rate, the greater the option value.
- A measure of risk for the optioned security, such as standard deviation – The larger the standard deviation, the greater the option value (because of the price of the stock which is more volatile, goes up higher and down further than its market changes).
- Exercise price
- Dividend payment on the optioned stock – the smaller the dividend payments, the greater the option value (because more earnings are being retained).

- Current stock price relative to the exercise price of the option – the difference between the current price and the exercise price affects the value of the option. The impact of the difference depends on whether the option is a call option or a put option.
- There is a direct relationship between these factors and the fair value of an option.

**II. Black-Scholes Model **

A. The original Black-Scholes model was developed to value options under specific conditions. Therefore, its appropriate for:

- European call options, which permit exercise only at the expiration date
- Options for stocks that pay no dividends
- Options for stocks whose price increases in small increments
- Discounting the exercise price using the risk-free rate, which is assumed to remain constant.

B. As with other models used to estimate the fair value of an option, the Black-Scholes method uses the six factors listed above. The advantage of the Black-Scholes model is the addition of two elements:

- Probability factors for:

a. The likelihood that the price of the stock will pay off within the time till expiration, and

b. The likelihood that the option will be exercised

2. Discounting of the exercise price

C. Many of the limitations (condition constraints) in the original Black-scholes model have been overcome by subsequent modifications, so that today modified Black-Scholes models are widely used in valuing options.

D. The underlying theory of the Black-Scholes method and the related computation can be somewhat complex. Therefore, computer applications should be used in order for it to be used effectively.

**III. Binomial Option Pricing Model (BOPM) **

A. The binomial option pricing model (BOPM) is a generalized numerical method for the valuation of options.

- The BOPM uses a “tree” to estimate value at a number of time points between the valuation date and the expiration of the option.
- Each time point where the tree “branches” off represents a possible price for the underlying stock at that time.
- Valuation is performed iteratively, starting at each of the final nodes (these that may be reached at the time of expiration), and then working backward through the tree towards the first node (valuation date).
- The value computed at each stage is the value of the option at that point in time, including the single value at the valuation date.

B**. **There is a distinct BOPM** **process.

- The BOPM process consists of three basic steps:
- Generate a price tree
- Calculate the option value at each tree end node
- Sequentially calculate the option value at each preceding node (tree branch).

- The process can be illustrated using a simple one-year option:
- Assume an option for a share of stock that expires in one year and has an exercise price of $100. An evaluation estimates that at the end of the year the underlying stock could have a price as high as $120 and as low as $80.
- Probabilities are assigned to each of the possible outcomes to develop an expected value for the option.
- For Example: Assume that a .60 probability is assigned to the $120 high value and a .40 probability (1.00-.60) is assigned to the $80 low value. The entity’s cost of funds is 10%.
- Expected value = [(.60 x $20) + (.40 x $0)]/1.10
- = [$12 + $0]/1.10
- Option value = $12/1.10=$10.91

- While the one-period model is an extreme simplification of the use of the binomial option pricing model, it illustrates the approach. In practice, the entire time period of the option would be divided into multiple sub-periods, with the expected outcome of each period being the input for the prior period.
- The binomial option pricing model (BOPM) can be used for American-style options, which permits exercise any time up until the expiration date, and the time when the underlying stock pays dividends. The original Black-Scholes model does not accommodate these options.

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