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This chart shows evolution of an initially ATM 1-year equity-like option in Q- and in P-measure. Under Q-measure, volatility is 15%; it is used both to evolve the underlying and price the option on the underlying. Under P-measure, the underlying is evolved with historical vol of 10%, while log of implied volatility is assumed to follow an AR(1) process with initial value and long-term mean equaling 15%, the Q-measure volatility. The log vol-of-vol under P is set to 150%, mean-reversion speed 20% and equity/vol driver correlation is -90% under P. Drifts are 1% and 3% for Q and P.

This chart contrasts a GBM model for an equity-like asset with a log-GARCH(1,1) model for the same asset's return. Both models are driven with same white noise. GARCH long-term volatility is set equal to that of GBM, 15%. The GARCH parameters are p = 0.3 and q = 0.05. We expect the lines for GBM and GARCH to cross, such that GARCH volatility spikes are offset by volatility dips so as to keep the expected volatility equal to that of the GBM. While exposure will not be much dissimilar, GARCH model generates "historical stochastic volatility".

These charts compare two dynammic credit spread models, where credit is driven by migration between IG and HY dynamics. Migration probabilies are set to 0.95. In the first model, both IG and HY spreads follow log-AR(1) processes, with AR coefficients of 0.99 and 0.95 for IG and HY and with residuals correlation of 0.9. In the second model, the spreads follow a joint log-VAR(1) model with residual correlation of 0. The VAR(1) matrix is entries are all set to [[0.98, 0.01],[0.3, .96]], such that the eigenvalues match the AR coefficients of the first model. VAR(1) is a much richer model, allowing finer control over the residual auto- and regular correlation. Toggle "CDS OFF" to observe only AR(1) vs VAR(1) current path, drawn by the same white noise. CDS spread is assumed to have an N(0,3bp) idiosyncratic basis to its current driver.

Click on the series' key in the chart legend to toggle the series on/off on the chart.

In modern world, many derivatives have become the first-class tradeables and they are not driven entirely by the underlyings. As such, market risk analysis of portfolios containing derivatives requires joint modelling of underlyings and derivatives under P-measure. Such models are essentially dynamic and require simulation of the time series and not just distributions of returns. We can construct such models for a wide range of markets.

Derivatives' payoffs have become less exotic, but increased liquidity in markets for different underlyings allows structuring more hybrids to hedge or bet on correlation risk between market factors. Hybrid models that are able to price such products are only conditionally risk-neutral and they necessarily depend on parameters that can only be observed historically. Model risk becomes essential and we would be happy to help tackling such risk.

Pricing marginal XVA is a corporate finance problem. It is solved in practice by treating XVA as a hybrid derivative. The model is exposed to a huge number of unobserved parameters, and jump diffusions are needed to capture realistic wrong-way risk. Cases of low-leveraged counterparties are special, as marginal XVA affects prior leverage. AAD optimizes certain parts of calculation, but bump-and-recalc is still necessary for scenario analysis. Our experience of delivering XVA solutions is at your disposal.

Quantitative analysis function in large organizations cannot be separated into quant and IT anymore. This is because calculations are implemented as large distributed workflows and intermediate calculation results need to be manipulated non-linearly to obtain final results. The architecture and data model have to go far beyond the in-memory calculation graphs, especially if global BAU function is taken in consideration. We can help you delivering such architecture.