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Option
Pricing Models
Also see: Volatilities
Overview
5) Considerations for ETRM/CTRM (Energy/Commodities Trading and
Risk Management) system design
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) Considerations for
CTRM system design
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.