Physical commodities that encumber difficulties in storage or a lack of speculators that make the arbitrage relationship between spot and forward prices at different tenors weak are (unlike equities or FX) not suited to having the forward price treated as a deterministic function of the spot and carry. Instead, the underlying forward prices themselves must be simulated.
Forward prices are stored as curves (the forward price at time for delivery at time is denoted ) with the initial forward price curve entered in the market data file.
Most derivatives contracts are not written directly on daily energy prices but usually on averages of future prices. Reference prices allow this by being a deterministic function of a Forward price curve, a sampling period delimited by start and end dates ( and ) and a set of sampling dates . It is assumed that sampling periods are contiguous. Reference prices are simulated indirectly via their forward prices and are denoted .
Reference prices are represented by a fixing curve i.e. a mapping of reference dates to the underlying forward price dates. In general, it is possible to contruct reference prices that sample the forward price curve more than once by simply averaging all prices within a reference price window - this is not implemented. Currently, each reference price simply looks up a single forward price using the fixing curve mapping.
The risk-neutral expectation of a sampled price at time when the sampling date is before the forward price date is simply:
After the sampling date, we need to take the sample value into account and the expectation of a fixing price with samples in the past at time but with samples still in the future is:
A reference price with samples is a weighted sum:
where the weight of the fixing is given by the number of samples within the fixing period and is the normalization term .
The set of dates used to compute the reference price is defined by the deal by using a forward price sample which can then specify business days according to a given calendar.
The realized average price is prorated according to the number of sample dates within the realized period (exactly as with unrealized samples). The currency depends on whether FX Averaging is selected or if the deal is a compo. If the deal is a compo, the realized average is in the payoff currency, otherwise, it's in the currency of the forward price.
When an energy deal is specified in a currency other than the price factor currency, each price sample must be converted to the native (deal) currency
Currently, Average FX can only be set to No, meaning that each price sample is converted to deal currency at the prevailing spot FX rate. A fixing price with samples is given in deal currency as:
Where is the forward price in price factor currency and is the price of one unit of price factor currency in deal currency. Here, the realized average of any historical price samples must be in deal currency.
Note that forward FX rates are calculated under the usual risk-neutral measure.
where are the price factor and deal currency discount factors respectively.
The volatility of a reference price with strike price can be estimated using the moment matching technique mentioned earlier (assuming that future reference prices are log-normally distributed).
For valuation date , define to be the least index for which (here ). Then , where
If denotes ( is the sampling date in the reference period and is the price date of fixing in which falls), then the correlation between and (for ) is assumed to be
and is the standard deviation of .
The standard deviation of at time (with ) is then given by
with and being the first and second moments given by:
- are the sample dates in between and
- is the moneyness
- is the forward price volatility at time for delivery at date with expiry and moneyness (and for ).
Spreads on top of reference prices and volatilities are not currently implemented.
Pricing energy composite (compo) deals requires the forward price in payoff currency with the compo-adjusted volatility used at each sampling date for both the reference price and reference price volatility (during moment matching) respectively. The deal currency and the payoff currency must be the same with the Realized Average expressed in payoff currency.
A European option on an energy forward contract can be priced using the Black model with a volatility derived using the moment matching approach described earlier. Consider a European option with a reference price , where is the usual start of the sampling period and is the end of the period and also the expiry date of the option. The value of the option with strike and settlement date is
where is the standard deviation of and is the Black formula.
The time value of a fixed energy cashflow paid at indexed to volume of energy at a fixed price is
The time value of an energy cashflow paid at indexed to volume of energy at a price determined by the reference price is
where is the Price Multiplier and is the Fixed Basis.