Covariances can arise at longer horizons with dependencies that occur over time, if, for example, arrival rates of price motion depend on the price levels themselves. In this case one may have, for example, that

$$\begin{aligned} k_{X_{1},X_{2}}(\omega,t,x_{1},x_{2})&=k_{1}\left(\omega,S_{1}(t\_),S_{2}(t\_),t,x_{1}\right) \mathbf{1}_{x_{2}=0}\\ & \quad +k_{2}(\omega,S_{1}(t\_),S_{2}(t\_),t,x_{2})\mathbf{1}_{x_{1}=0}. \end{aligned} $$

A further special case models forces at work that try to keep the ratio within bounds by directly creating just a dependence on the ratio of the prices. In this case,

$$k_{X_{1},X_{2}}(\omega,t,x_{1},x_{2})=k_{1}\left(\omega,\frac{S_{1}(t\_)}{ S_{2}(t\_)},t,x_{1}\right) \mathbf{1}_{x_{2}=0}+k_{2}\left(\omega,\frac{ S_{2}(t\_)}{S_{1}(t\_)},t,x_{2}\right) \mathbf{1}_{x_{1}=0}. $$

By way of a specific example to be simulated, take *k*_{1},*k*_{2} in the bilateral gamma class with parameters depending on the ratio of the levels.

Let *X*_{1}(0)=*X*_{2}(0)=0. Further,i suppose that for | log(*S*_{1}/*S*_{2})|<*α* the arrival rates are not dependent on the ratio. For price relatives within a bound the two processes, conditional on the maintenance of the bound, are just independent bilateral gamma processes. Dependencies may occur when price relatives violate the bound. The next subsection models the creation of dependence via an exponential tilting of the arrival rate functions.

### 3.1 Creating dependence by exponential tilting

The base Lévy measure or arrival rate function is that of a bilateral gamma process introduced, for example, in Madan, Schoutens, and Wang (2017) with

$$k(x)=c_{p}\frac{e^{-\frac{x}{b_{p}}}}{x}\mathbf{1}_{x>0}+c_{n}\frac{e^{- \frac{|x|}{b_{n}}}}{|x|}\mathbf{1}_{x<0}. $$

There will be some constant base drift in the stock over time given the base arrival rate function. With a view to creating dependence, consider changing this drift by exponential tilting that alters the relative rates of positive and negative jumps. The use of such a procedure in well established for premium calculations in Insurance and in the construction of risk neutral distributions for option pricing. We cite in this regard Naik and Lee (1990), Gerber and Shiu (1994), Carr and Wu (2004) and Elliott, Chan and Shiu (2005) among others. Here we employ the procedure to introduce dependence.

On applying an exponential tilt, the arrival rate function shifts to

$$c_{p}\frac{e^{-\left(\frac{1}{b_{p}}-\theta \right)x}}{x}\mathbf{1}_{x>0}+c_{n}\frac{ e^{-\left(\frac{1}{b_{n}}+\theta \right)|x|}}{|x|}\mathbf{1}_{x<0}. $$

One then has another bilateral gamma process with parameters

$$\begin{array}{@{}rcl@{}} \frac{1}{b_{p}^{\prime }} &=&\frac{1}{b_{p}}-\theta \\ \frac{1}{b_{n}^{\prime }} &=&\frac{1}{b_{n}}+\theta, \end{array} $$

or that

$$\begin{array}{@{}rcl@{}} b_{p}^{\prime} &=&\frac{b_{p}}{1-\theta b_{p}} \\ b_{n}^{\prime} &=&\frac{b_{n}}{1+\theta b_{n}}. \end{array} $$

For positivity of parameters it is necessary that either

$$0<\theta <\frac{1}{b_{p}} $$

or \(0<-\theta <\frac {1}{b_{n}}\).

Therefore, for *θ*>0 let

$$\theta =\frac{1}{b_{p}}\eta \text{ for }0<\eta <1 $$

and for *θ*<0 let

$$-\theta =\frac{1}{b_{n}}\eta \text{ for }0<\eta <1. $$

For | log(*S*_{1}/*S*_{2})|<*α* we suppose no tilt and *θ*=0. Mean reversion is organized by taking *θ*<0 for *S*_{1}/*S*_{2}> exp(*α*) and *θ*>0 for *S*_{1}/*S*_{2}< exp(−*α*). More specifically, for *S*_{1}>*S*_{2} let

$$-\theta =\frac{1}{b_{n}}\left(1-\exp \left(-a_{n}\max \left(\frac{S_{1}}{ S_{2}}-e^{\alpha },0\right) \right) \right) $$

while for *S*_{1}<*S*_{2} let

$$\theta =\frac{1}{b_{p}}\left(1-\exp \left(-a_{p}\max \left(e^{-\alpha } - \frac{S_{1}}{S_{2}},0\right) \right) \right). $$

Mean reverting drifts are introduced when the ratio departs from initial levels severely in either direction. Otherwise we have independence. The result is a fourteen parameter model with dependence and parameters

$$\begin{array}{@{}rcl@{}} &&a_{p},b_{p},c_{p},a_{n},b_{n},c_{n},\alpha \\ &&a_{p}^{\prime },b_{p}^{\prime },c_{p}^{\prime },a_{n}^{\prime}, b_{n}^{\prime },c_{n}^{\prime },\alpha^{\prime }, \end{array} $$

where the primed parameters are for *k*_{2}, the Lévy measure for the second stock while the nonprimed parameters are for *k*_{1}, the Lévy measure for the first stock. In the primed case, we have for *S*_{2}>*S*_{1}

$$-\theta^{\prime }=\frac{1}{b_{n}^{\prime }}\left(1-\exp \left(-a_{n}^{\prime }\max \left(\frac{S_{2}}{S_{1}}-e^{\alpha \prime },0\right) \right) \right) $$

and for *S*_{2}<*S*_{1},

$$\theta^{\prime }=\frac{1}{b_{p}^{\prime }}\left(1-\exp \left(-a_{p}^{\prime }\max \left(e^{-\alpha^{\prime }}-\frac{S_{2}}{S_{1}},0\right) \right) \right). $$

When we tilt the Lévy measure to

the effect on the drift is as follows. The original drift is

$$E[\Delta S]=\left(\frac{1}{1-b_{p}}\right)^{c_{p}}\left(\frac{1}{1+b_{n}} \right)^{c_{n}}-1 $$

and it goes to

$$\left(\frac{1}{1-b_{p}^{\prime }}\right)^{c_{p}}\left(\frac{1}{ 1+b_{n}^{\prime }}\right)^{c_{n}}-1 $$

which is

$$\begin{array}{@{}rcl@{}} &&\left(\frac{1}{1-\frac{b_{p}}{1-\theta b_{p}}}\right)^{c_{p}}\left(\frac{1}{1+\frac{b_{n}}{1+\theta b_{n}}}\right)^{c_{n}}-1 \\ &=&\left(\frac{1-\theta b_{p}}{1-(\theta +1)b_{p}}\right)^{c_{p}}\left(\frac{1+\theta b_{n}}{1+(\theta +1)b_{n}}\right)^{c_{n}}-1. \end{array} $$

### 3.2 Simulated data

We simulated 10,000 paths of 252 days for two stock prices with bilateral gamma parameters as those for INTC and IBM on 20170131

$$\begin{array}{lllll} & b_{p} & c_{p} & b_{n} & c_{n} \\ {INTC} & 0.0070 & 1.6988 & 0.0058 & 1.6988 \\ {IBM} & 0.0058 & 2.0106 & 0.0057 & 1.6552 \end{array} $$

and the stocks starting at 100 and the value of *α*=0.05, and the parameters \(a_{p},a_{n},a_{p}^{\prime },a_{n}^{\prime }\) all set equal to 2. Figure 1 presents the conditional drifts for the change in the two stock prices as a function of the ratio of each stock price to the other price.

Observe that for levels below alpha for the log price ratio in absolute value, the process is a pair of independent bilateral gamma processes. When the ratio leaves this region we have mean reversion with a negative drift for a high own price and a positive drift for a low own price.

Figure 2 presents graphs of the two contemporaneous returns and the associated stock prices. We may observe the expected absence of covariance. However, there is dependence as seen by a plot of the two stock prices against each other across a subsample of days and paths presented.

We now train a Gaussian process regression to learn the dependence of the change in the first stock price as a function of the ratio of the two prices, own to other on a subsample of days and paths. Figure 3 presents the result.

We see that a Gaussian process regression is capable of learning the existing dependence.

### 3.3 A two-dimensional analysis of pure discount bond data

We anticipate that bond prices of different maturities must be dependent but may also have no covariation if they are pure jump processes moving at their own times that are not synchronized across the maturity spectrum. The dependence comes from a possible dependence of drift via the dependence of parameters of motion on the prices themselves.

To investigate this further, daily time series of pure discount bonds were constructed from data on yields to maturity each day for a variety of maturities. The data comes at specific maturities that vary each day. By interpolation, pure discount bond prices were derived for the fixed maturities of 1, 3, 6, 9, and 12 months and 2, 5, 10, 15, and 20 years. The data set went from January 3, 2007, to August 29, 2017, for 2782 days. Beginning on December 19, 2007, and employing on a rolling basis of 252 days of past returns on each of the 10 time series, bilateral gamma parameters of motion were estimated for the logarithm of the pure discount bond prices (see the next section for further estimation details). One obtains as a result, 2530 sets of bilateral gamma parameters for each of the ten pure discount bonds.

For an analysis of dependence, consider first just the dependence between the pure discount bond price for maturities of one and five years. From the bilateral gamma parameters we inferred for each day the expected exponential variation in the price as

$$\left(\frac{1}{1-b_{p}}\right)^{c_{p}}\left(\frac{1}{1+b_{n}}\right)^{c_{n}}. $$

For the analysis of the possible nonlinear dependence of drifts on the two continuously compounded rates for the one and five year maturities we estimated a support vector machine regression using a Gaussian kernel. Figure 4 and present the estimated drift response surfaces.

There is a sufficient nonlinearity in the drift response that creates potential dependence in the absence of any covariation as the processes remain independent bilateral gamma processes unless rates move to extreme regions. We observe that when rates are high the price drifts are positive indicating a drop in rates with the opposite occurring when rates are low.