Generative modelling of financial time series with structured noise and MMD-based signature learning

A Heston model experiment

The Heston model is defined by the following stochastic differential equations:

where S t is the asset price, v t is the variance of the asset price, μ is the drift, κ is the mean reversion rate, θ is the long-term variance, σ is the volatility of the variance, W t 1 and W t 2 are Brownian motions where d ⁢ W t 1 ⁢ d ⁢ W t 2 = ρ ⁢ d ⁢ t for some correlation parameter ρ ∈ [ - 1 , 1 ] .

We set the parameters μ = 0.2 , κ = 1 , θ = 0.25 , σ = 0.7 , ρ = - 0.7 and the initial variance v 0 = 0.09 . Using the Quantlib python library https://quantlib-python-docs.readthedocs.io/en/latest/, we generate 6,400 samples of length 250 with a time step of 1/252. The Quantlib implementation uses first order Euler discretisation to simulate trajectories of the Heston model.

The models are trained using a simplified version of Algorithm 1, where the conditioning process is omitted the LSTM cell state and hidden state and only the time augmentation is used instead of both lead-lag and time augmentation.

To simulate noise, we generate independent paths using the same parameters for the Heston process, except with μ = 0 , then calculate

to obtain the noise at time t i . For the i.i.d. standard Gaussian noise, we use the same Quantlib Heston implementation but set κ = σ = 1 ⁢ e - 9 and θ = v 0 which effectively results in the noise having a mean of zero and variance of one. This method of simulating the i.i.d. Gaussian noise allows the use of the same random seed and generate noise sequences that are only different in variance so as to provides a fair comparison with the noise that has changing variance. We use a noise dimension of 2 as the Heston model has two Brownian motions.

The truncation level was set to m = 5 and the static kernel was a rational quadratic kernel with α = 1 and l = 0.1 .

B Moments of returns

We briefly present the formulas used to calculate the empirical estimates of the moments of the returns in Section 5.6. Let ( r 1 , r 2 , … , r n ) be a sample of log returns; we define r ¯ := 1 n ⁢ ∑ i = 1 n r i and m i := 1 n ⁢ ∑ j = 1 n ( r j - r ¯ ) i . We will assume that there are 252 business days in a year.

The first four moments of the log returns are named and calculated as follows:

1. The mean is annualised by business day convention which we calculate it as

   Ann. returns = 252 × r ¯ .

2. Similarly volatility is also annualised by convention and calculated as

   𝐕𝐨𝐥𝐚𝐭𝐢𝐥𝐢𝐭𝐲 = 252 × m 2 .

3. The skew is calculated as

   𝐒𝐤𝐞𝐰 = m 3 m 2 3 2 .

4. The kurtosis is calculated as

   𝐊𝐮𝐫𝐭𝐨𝐬𝐢𝐬 = m 4 m 2 2 - 3 .