Option Pricing Models
Also see: Volatilities
A Pricing Model is, generically, any formula/code/function that will generate a MTM (mark to market) on a trade for any trade type.
However, for now, we’ll focus on just pricing models for options.
The most famous one is the ‘Black-Scholes’ model.
As a note, there are different variations on the Black Scholes model for stocks (equities) and a slightly different formula for options on commodities, that does not need to take into account stock dividends, for example.
The Black Sholes model is appropriate for certain types of options, but not others. Specifically, it assumes that you can exercise the option (or not) only once, on the expiration date. This style of option is called ‘European’.
Another variation of an option is ‘American’ style, which allows for an option to be exercised at any time. There are different pricing models that assume American style options. These options, by the way, are always worth the same or more than European style, since you have can exercise the option on the final day, if you choose, thus ‘recreating’ a European option synthetically and since you also have the extra days (all of the days from now to the exercise date) it can only be worth more.
There is also ‘Bermudan’ style options, though these are rarely traded. These have 2 or more exercise dates i.e., so more than European style, which has just one exercise date), and less than every day (which would be American.
The Black Sholes model additionally assumes that it is non-averaging. i.e., the strike price is being compared to a single day’s closing price.
Some options might be averaging, e.g., average of the closing prices for given calendar month, i.e., over the business days in the month. This style is called ‘Asian’ style. Asian options will be worth less than their European equivalent because the volatility of an average will always be less than the volatility for a single value.
Consider, for example, that if you role one 6-sided die, you can easily expect to get a value of 1, 2, 3, 4, 5, or 6. However, if you were to take the average of 100 rolls, you would be surprised if the average was more than 4 or less than 3 (you would expect an average of 3.5).
The point being that you would use another pricing model, i.e., not Black Sholes, for Asian style options.
The Black Sholes model also assumes a single commodity. You can have an option on a spread, such as the difference in price between crude oil and unleaded gas (a ‘crack spread’ option). Spread options would typically require a different pricing model.
‘Swaptions’, which are options on a swap, would require yet another option pricing model.
The simplest option pricing model, Black Sholes, requires the following inputs:
4.1) Option Strike Price. This is easy to get as it is just an attribute of the trade.
4.2) Time in expiration. In years from the current date until the expiration date of the option.
4.3) Market price. This is the projected price of the underlying commodity.
4.4) Interest rate. Use the appropriate interest rate for discounting.
4.5) And lastly, a ‘volatility’. Volatility is a measure of the variation in price changes.
In addition, some spread option models require ‘correlations’, which show how two different commodities move together, e.g., if one goes up, does the other go down or up, but not as much?
5.1) Make sure you can easily see the ‘pricing model inputs’ for an option, i.e., whatever is being used to value an option for a particular valuation.
This isn’t as easy as it sounds or as trivial. For example, if volatilities are defined as different values for different strike prices, as is typical, and you have volatilities defined for crude oil options, for example, at $50, $51, $52, etc., then if you have an option with a strike price of $51.50, it might have a volatility input that is an interpolated average of the volatilities from the $51 and $52 strikes. So need a way to make that clear to users and auditors and anyone else who needs to validate the accuracy of pricing models.
5.2) Would also need to make it easy to swap in other pricing models. Two variations on this:
5.2.1) In one variation, the alternate pricing model uses the same model inputs. So you are just replacing the formula (i.e., the math formula)
5.2.1) In the other variation, the pricing model inputs are different. This involves a different kind of work and can be trickier.