Correlation
Modelling.

Modelling the dependence between two assets.

The second project of my internship focused on the pricing of two types of derivatives based on two underlying assets: Worst Of (WO) and Best Of (BO) products.

In order to price these products accurately, it is necessary to model the dependence between the two underlying assets. This is where correlation plays a key role.

The objective of the project was therefore to study and model the correlation between two financial assets using historical market data and to determine an appropriate correlation parameter for the pricing of Worst Of and Best Of products.

Worst Of and Best Of.

A Worst Of or Best Of product is a structured product whose payoff at maturity depends on the performance of the least or best performing asset in a basket.

For a basket composed of two assets, the payoff is determined by the minimum or maximum performance of the two underlying assets.

In practice, the payoff can also depend on initial prices, strike levels, barriers, coupons, or other contractual features specified in the term sheet.

Why correlation matters.

Consider a basket composed of two stocks, Kering (KER) and LVMH (MC). Both companies operate in the luxury goods sector and are listed on the Paris stock exchange.

Since they belong to the same industry and can be affected by similar economic factors, their stock prices tend to exhibit a relatively strong positive correlation over time.

In contrast, Kering and Société Générale (GLE) operate in very different sectors and are generally less dependent on the same economic factors.

If two assets are perfectly positively correlated, with a correlation equal to 1, they tend to move in the same direction. Conversely, negative correlation means that their performances tend to move in opposite directions.

Returns and correlation matrix.

Let \(S_t^i\) denote the closing price of asset \(i\) at time \(t\). The logarithmic return of the price is defined as:

For \(N\) assets, the return vector is:

The correlation matrix is denoted by:

During the internship, a 5-days log-return was used in order to reduce the noise of the data.

This measure provides a weekly log return.

Training and test samples.

The work was based on historical market data. Two different samples were created, one for training and one for testing.

The benchmark used during the study was the price of Worst Of and Best Of assets, which are sometimes quoted by the market on Bloomberg.

Another comparison was also made with data provided by ICE. The methodology supplied by ICE remained strictly confidential.

Constant Historical Correlation.

The first model considered was the constant historical correlation model.

The historical correlation model assumes that the dependence structure remains constant over the estimation period.

The historical correlation matrix is therefore:

One important advantage of this approach is reproducibility. If the inputs do not change, the Pricing Team can reproduce the same price.

However, this approach raises several questions, particularly regarding the observation window to use and the weight that should be given to each observation.

Exponentially Weighted Moving Average.

The Exponentially Weighted Moving Average (EWMA) model introduces time variation by assigning a larger weight to recent observations.

The parameter \(\lambda \in [0,1]\) is the decay factor. It is a hyperparameter of the model and was optimized using market data.

For two assets, the correlation is:

The model has the advantage of remaining reproducible while giving more importance to recent observations.

During the study, a value of approximately \(\lambda=0.93\) provided a good approximation to the correlation given by the market when comparing the resulting prices with observed market prices.

Dynamic Conditional Correlation.

The DCC-GARCH model was considered in order to capture the time-varying dependence between the two underlying assets.

Unlike the constant historical correlation approach, the DCC-GARCH model allows the correlation to evolve over time according to information contained in recent market returns.

The individual volatility dynamics were first modelled using a GARCH(1,1) specification.

The resulting standardized residuals were then used to estimate the dynamic correlation parameters.

In the illustrative calibration presented in the report:

The resulting correlation fluctuates over time. For illustration, it can be considered to vary approximately between 0.45 and 0.80, with an average level around 0.65.

The main advantage of DCC-GARCH is therefore that it does not impose a constant correlation throughout the observation period.

Ornstein–Uhlenbeck Model.

The Ornstein–Uhlenbeck (OU) model was considered as an alternative approach for modelling the time-varying dependence between the two underlying assets.

Its main characteristic is its mean-reverting behaviour. When the process moves away from its long-term equilibrium level, a force tends to bring it back towards this level.

The discrete-time representation used for simulation is:

where:

The process tends towards its long-term level \(\theta\), while the stochastic component introduces fluctuations around this level.

From the latent factor to correlation.

The OU process itself cannot directly represent a correlation coefficient because \(X_t\) takes real values whereas correlation must satisfy:

To solve this problem, the latent factor is transformed using the hyperbolic tangent function.

Equivalently:

The long-term level of the correlation is obtained by applying the same transformation to the equilibrium level \(\theta\).

Dynamic correlation in simulation.

The resulting process \(\rho_t\) can be introduced into the correlation matrix used for the Monte Carlo simulation of the two underlying assets.

The evolution of \(\rho_t\) therefore directly affects the dependence between the simulated returns of the two assets.

The OU model can consequently be incorporated into the Monte Carlo framework used to price Worst Of and Best Of products.

Comparison of the models.

EWMA as the main approach.

The comparison of the different models showed a trade-off between reproducibility and the ability to capture changes in market dependence.

The EWMA model was ultimately selected as the main approach for the Pricing Team.

This choice was mainly motivated by its ability to capture changes in correlations while remaining simple and, most importantly, reproducible.

The conclusion was obtained by pricing Worst Of and Best Of products with different correlations and comparing the results with prices observed on the market.

Possible extension.

As a continuation of this work, a Markov Switching model could be considered.

This type of model could capture changes of regime in correlation, especially during market crashes when stocks can become strongly correlated.