Spot
The spot price of an asset (assuming immediate delivery) is expressed in the asset currency. The asset currency is specified by the corresponding property:
- Currency for equity and commodity prices
- Domestic Currency for FX rates
The initial price is given by its Spot property.
FX Rates
There is just one FX rate price factor for each currency pair (including the base currency). FX rate price factors always have a currency and a parent currency which, currently, will always be the base currency. If is the foreign currency, and is the domestic currency, then let be the spot price of currency in currency at time . The asset factor models evolve the FX rate , effectively making the domestic currency of all FX rates the same as the base currency
Forward rates
For equity and commodity prices, the forward price at time for delivery at is the usual no-arbitrage formula:
Here is the discount rate from the asset's interest rate price factor (i.e. its repo rate specified by the Interest Rate price factor) and is the discount rate from the dividend rate (for equities) or convenience yield (for commodities).
FX Rates
Forward FX rates for currency in currency is given by:
where is the discount rate from the interest rate price factor specified by the Interest Rate property on the FX rate price factor for currency . This can be extended to handle the case where a given equity/commodity price with asset currency is required in another currency as follows
Dividend rate interpolation
A initial dividend rate curve can be derived from discrete dividends using the no-arbitrage relationship between spot and forward prices. The spot price is the present-value forward price plus the dividends the forward purchaser does not get (but the spot purchaser does):
where is the projected/known dividend paid at time with ex-dividend date (with ). With , the implied dividend rate is:
.
Since the curve is constant on each interval , and is a piecewise linear function of , interpolation should also be linear in with flat extrapolation. If and are points on the curve with , then
GBMAssetPriceModel
The spot price of an equity or FX rate can be modelled as Geometric Brownian Motion (GBM). The model is specified as follows:
Its final form is:
Where:
- is the spot price of the asset
- is the standard Brownian motion
- is the constant drift of the asset
- is the constant volatility of the asset
- is a standard Wiener Process
GBMAssetPriceTSModelImplied
GBM with constant drift and vol may not be suited to model risk-neutral asset prices. A generalization that allows this would be to modify the volatility and to be functions of time . This can be specified as follows:
Note that no risk premium curve is captured. Its final form is:
Where:
- is the volatility of the asset at time
- is the Quanto FX Volatility of the asset at time . is then the Quanto FX Correlation
- is the interest rate in the asset currency
- is the yield on the asset (If S is a foreign exchange rate, q is the foreign interest rate)
- is the forward asset price at time t
- is the spot price of the asset
- is a sample from the standard normal distribution
- is the increment in timestep between samples
In the case that the represents an FX rate, this can be further simplified to:
Here