Multi-currency BGM Pricing Model

The Brace-Gatarek-Musiela (BGM) model is a multi-factor log-normal model. This model applies to both currencies.

Multi-currency BGM Pricing Model

The Brace-Gatarek-Musiela (BGM) model is a multi-factor log-normal model. This model applies to both currencies. Its principle is to fix a tenor , for instance 3 months, and to assume that each Libor rate at date , has a log-normal distribution in the “forward-neutral” probability of maturity . The present model uses 4 factors, which we may assume independent.

Let be the FRA from to as observed at date . The diffusion of with respect to t is given, as in the Heath-Jarrow-Morton (HJM) model, by a combination of factors:

where the Brownian motions are independent and the drift is adapted to make the model arbitrage free. The factors are defined by their “relative” size with respect to and by a “volatility triangle”:

The “relative” factors are parameterized in the following way:

The sum of the squares of all the factors is imposed to be identically 1 so that the local volatility of each single Libor future rate is precisely . The parameters , , , and are input in the HJM spreadsheet under the following names:

shift sector 1 sector 2

slope 1 slope 2

The volatility triangle bears this name because it is defined for . It is discretized (see next sect.) and its values on the different cells are calibrated to match the market price of caplets and swaptions, except for diagonal values () which are input by the user (the so-called “s-vols”).

A series of dates , spaced approximately by the tenor , is fixed, for instance the IMM maturity dates. At each date , the only available information is the short rate , the spot Libor rate , the series of future rates , where is the first index such that , and the stub rate , which applies on the period . In practice, the time is discretized and the short rate is that which applies over the time period of a diffusion step.

Discount factors are computed from the rates as follows:

Option values are discounted risk-neutral expectations of their pay-off. In a stochastic interest rate environment, the discounting should be taken as the accumulation of the spot rate . We define the numeraire at date actualized in 0 as:

Again, a discretized version of this formula is used in practice (see sect. II).

The arbitrage theory states that:

The Bayesian rule implies:

The above structure applies to both currencies. In the sequel, superscripts and will specify whether we refer to domestic or foreign rates.

We denote by the exchange rate at date . It follows a diffusion process defined by:

Short rates and are processes described in the previous section.

The volatility can either be a time dependent parameter (deterministic volatility) or itself a process (stochastic volatility). In the latter case, its diffusion equation is:

This process is positive. As indicated, parameters and can be time dependent. The first one is the Mean Reversion, the second one is the Expected Spot Volatility and the third one is the Volatility of Volatility or Vvol. The two Brownian motions and can be correlated, with a time dependent correlation . Parameters and must be positive, but can have either sign.

In terms of volatility surface, the Vvol introduces a positive smile, the correlation induces a skew and the mean reversion makes the smile decrease with maturity. The expected spot volatility drives the term structure.

As mentioned previously, Brownian motions are uncorrelated, as well as . However, we allow and to be correlated with a possibly time dependent correlation . In order to avoid almost useless complexity, we assume that and are not correlated for non-equal indices and .

The Brownian motion can be correlated to interest rate ones, with possibly time dependent correlations . In order to avoid non-positive definite correlation matrices, the correlation of with is set to the product . If we decompose into a component totally correlated to and another independent, then this is the same thing as saying that the independent component is also independent of the interest rate Brownian motions (ref https://finpricing.com/lib/EqCallable.html).

There is one theoretical subtlety about multi-currency models. Risk-neutral probabilities differ in both currencies, because numeraires are different. In the domestic risk-neutral probability, the expectation of the daily discounted value of a unit of domestic currency is equal to the domestic discount factor:

The same applies to a unit of foreign currency, and this yields:

By the Bayesian rule, we get:

In the foreign risk-neutral expectation, one would have:

If domestic rates and FX are correlated, this shows that expectations and cannot coincide. Similarly, the non-discounted expectation of the exchange rate is not equal to the forward exchange rate:

The price of an option is also the risk-neutral expectation of its discounted pay-off. Consequently, if the pay-off is set in domestic currency, the price of the option in domestic currency is:

However, if it is set in foreign currency, then the price of the option in domestic currency is:

and, in foreign currency:

In the forward risk-neutral probability, one would have:

A discretized version of these formulas will be used in the next section to compute option prices and sensitivities.