Math
Calculating Option Value

Option Pricing

Option pricing is a difficult aspect of derivative trading. Due to the number of factors influencing the price of an asset and the difficulty of predicting the final price of an asset, the price of an option is very hard to determine. The price for an option must be acceptable for both parties, as one party will always make a smaller profit or even a loss upon exercising. Therefore the price must be a middle course where both parties can agree on and believe that the final profit will be acceptable to them. There are a number of different methods for calculating the price for an option. Where each method has certain advantages and disadvantages over the other. The different methods will be discussed in the following sections to give an understanding of the possibilities of each method. The selection of a suitable method must be a well-considered choice, as each method has his own strengths and weaknesses. Whereas one method can be faster, but less accurate, the different aspects must be contemplated to find the best possible method.

 

Black-Scholes

Black-Scholes is a formula designed to valuate an option, as a function of certain variables. These variables consist of the price of the underlying asset, the strike price, the volatility, remaining time until expiration and risk-free interest rate. By using this formula it is possible to accurately calculate the value of an option, and determine whether an option is over or under valued. Due to this accurate calculation the possibility of arbitrage trading is eliminated. The Black-Scholes method is therefore crucial to the effectiveness of option trading.

 

Employment of Black-Scholes

There are a number of assumption involved in using the Black-Scholes method.
The first assumption is that the option can only be exercised upon the maturity date. This method is based on European options where the exercise date is set on maturity, in contrary to American style options which can be exercised at any moment until the maturity date.
Secondly, the method doesn’t incorporate the transaction costs into its calculations. Everyone trading options pays some form of transaction costs. Especially for individual investors these transaction costs can cause a level of inaccuracy in the calculation.
The third assumption is that the market is effective. Meaning the direction of price movements cannot be predicted. A trader’s prediction may be correct in many cases, but there is no guarantee every prediction will hold up.
Another assumption concerns the volatility. The Black-Scholes method assumes the returns are normally distributed, meaning the volatility is constant over time. Furthermore the risk-free interest is also assumed to remain constant over time.
These assumptions may be invalid in certain markets or for certain underlying assets. This can result in the calculation being inaccurate to some degree. Therefore the value calculated through the Black-Scholes formula, is best employed as a comparison model, rather than a indicator.
Furthermore the Black-Scholes tends to undervalue an options price and thus this slight undervaluing must taken into consideration when planning to buy options. The advantage may be somewhat smaller than the Black-Scholes formula initially portrays.

 

Binomial Model

The Binomial model can be used to calculate the price for an option. The Binomial model is commonly used to valuate American options, which can be exercised upon any moment before the maturity date, because this method can take into consideration the possibility of pre-mature execution in its calculation. It has an advantage over the Black-Scholes method because the mathematical formula is relatively easy compared to Black-Scholes. Furthermore the calculations are more accurate because market developments can be inserted in the ongoing binomial model and thus the calculation will be more in sync with the actual market develoments. The higher accuracy of the Binomial model however comes at a price. This method is more time-consuming than the Black-Scholes method.

 

Employment of the Binomial Model

In contrary to the Black-Scholes model, the Binomial model is an open-form model. It generates not one clear result but a tree of possible asset prices and calculates the corresponding option value upon each selected node of the option pricing tree. There are three calculations involved in creating a binomial option pricing tree.
First off all the possible prices of the assets are calculated. This involves calculating two new prices, one should the assets price rise and one should the asset price fall. The size of this price fluctuation, both up and downward, is determined by the level of volatility.
After calculating the new assets prices, the value of an option can be calculated. This is done by deducting the previous asset’s price from the new asset’s price.
The next calculating involves reducing the option’s value with a percentage of risk-free interest rate. After this calculation the final value of the option can be determined for both the possible up and downward price movement of the asset. When the option’s value in each direction has been calculated the price of the option can be calculated so the option will generate a similar outcome, regardless of the direction an assets price will move. This calculation can then be repeated while using the newly calculated asset value as basis. This will once again create two new possibilities, both with an up and downward price movement.

 

Assumptions

Similar to the Black-Scholes method, there are also a number of assumptions involved in the use of the Binomial model. The following assumptions are active regarding the binomial model.
The most important assumption is that the price can have only two possible outcomes on the following date. It will either move up with a given percentage or move down with a given percentage. However it is impossible to predict with a hundred percent certainty which direction the next price movement will go.
The Binomial model assumes there is a perfect market active. Meaning the market information and prices are accessible to all participants commissions are not calculated into the formula.
Furthermore the Binomial model assumes that the risk-free interest rate remains constant over the entire lifespan of the option.

 

Trinomial Model

The Trinomial model is in many ways similar to the Binomial Model. It is an open-form model, which generates not one answer but rather a number of possible evolutions of the option’s price over the lifespan of the option. These possibilities are then placed into pricing tree, similar to the Binomial model. The difference however is that the Trinomial model takes three possible price movements into consideration, that is the price can rise, the price can fall or the price remains constant. The possibility of the price remaining constant is the factor where the Trinomial model distinguishes itself from the Binomial model. Where the Binomial model gives two possibilities at each node of the pricing tree, the Trinomial model generates three possibilities at each node.
The price of the option will be calculated in a similar way to the Binomial model. By using a formula, the different price possibilities will be generated and a corresponding option’s price will be calculated.