Spot

The spot price of an asset $S(t)$ (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 $S(0)$ 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 $C$ and a parent currency $p(C)$ which, currently, will always be the base currency. If $f(C)$ is the foreign currency, and $d(C)$ is the domestic currency, then let $X_{C/A}(t)$ be the spot price of currency $C$ in currency $A$ at time $t$. The asset factor models evolve the FX rate $S(t)=X_{f(C)/d(C)}(t)$, 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 $t$ for delivery at $T$ is the usual no-arbitrage formula: $$F(t,T)=S(t)\frac{Q(t,T)}{D(t,T)}$$

Here $D(t,T)$ 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 $Q(t,T)$ is the discount rate from the dividend rate (for equities) or convenience yield (for commodities).

FX Rates

Forward FX rates for currency $C$ in currency $A$ is given by:

$$F_{C/A}(t,T)=X_{C/A}\frac{D_C(t,T)}{D_A(t,T)}$$

where $D_C(t;T)$ is the discount rate from the interest rate price factor specified by the Interest Rate property on the FX rate price factor for currency $C$. This can be extended to handle the case where a given equity/commodity price $S(t)$ with asset currency $C$ is required in another currency $A$ as follows

$$F_{S/A}(t,T)=S(t)X_{C/A}\frac{Q(t,T)}{D_A(t,T)}$$

Dividend rate interpolation

A initial dividend rate curve $q(t)$ 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):

$$S(0)=F(0,t)D(0,t)+\sum_{0 \lt s_i \le t} H_i D(0,t_i),$$

where $H_i$ is the projected/known dividend paid at time $t_i$ with ex-dividend date $s_i$ (with $s_i \le t_i$). With $Q(0,t)=e^{-q(t)t}$, the implied dividend rate is:

$$q(t)=\frac{1}{t} \log \Biggl( \frac{S(0)}{S(0)-\sum_{0 \lt s_i \le t} H_i D(0,t_i)} \Biggl)$$ .

Since the curve $q(t)t$ is constant on each interval $[s_i,s_{i+1}]$, and is a piecewise linear function of $1/t$, interpolation should also be linear in $1/t$ with flat extrapolation. If $t_1$ and $t_2$ are points on the curve with $t_1 \le t \le t_2$, then

$$q(t)=q(t_1)+\Biggl(\frac{1/t_1 - 1/t}{1/t_1 - 1/t_2}\Biggl)(q(t_2)-q(t_1))$$

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GBMAssetPriceModel

The spot price of an equity or FX rate can be modelled as Geometric Brownian Motion (GBM). The model is specified as follows:

$$dS = \mu S dt + \sigma S dZ$$

Its final form is:

$$S = exp \Big( (\mu-\frac{1}{2}\sigma^2)t + \sigma dW(t) \Big )$$

Where:

• $S$ is the spot price of the asset
• $dZ$ is the standard Brownian motion
• $\mu$ is the constant drift of the asset
• $\sigma$ is the constant volatility of the asset
• $dW(t)$ 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 $\sigma(t)$ and $\mu(t)$ to be functions of time $t$. This can be specified as follows:

$$\frac{dS(t)}{S(t)} = (r(t)-q(t)-v(t)\sigma(t)\rho) dt + \sigma(t) dW(t)$$

Note that no risk premium curve is captured. Its final form is:

$$S(t+\delta) = F(t,t+\delta)exp \Big(\rho(C(t+\delta)-C(t)) -\frac{1}{2}(V(t+\delta)) - V(t)) + \sqrt{V(t+\delta) - V(t)}Z \Big)$$

Where:

• $\sigma(t)$ is the volatility of the asset at time $t$
• $v(t)$ is the Quanto FX Volatility of the asset at time $t$. $\rho$ is then the Quanto FX Correlation
• $V(t) = \int_{0}^{t} \sigma(s)^2 ds$
• $C(t) = \int_{0}^{t} v(s)\sigma(s) ds$
• $r$ is the interest rate in the asset currency
• $q$ is the yield on the asset (If S is a foreign exchange rate, q is the foreign interest rate)
• $F(t,t+\delta)$ is the forward asset price at time t
• $S$ is the spot price of the asset
• $Z$ is a sample from the standard normal distribution
• $\delta$ is the increment in timestep between samples

In the case that the $S(t)$ represents an FX rate, this can be further simplified to:

$$S(t)=S(0)\beta(t)exp\Big(\frac{1}{2}\bar\sigma(t)^2t+\int_0^t\sigma(s)dW(s)\Big)$$

Here $C(t)=\bar\sigma(t)^2t, \beta(t)=exp\Big(\int_0^t(r(s)-q(s))ds\Big), \rho=-1 and v(t)=\sigma(t)$