Pricing methods.

LexiFi’s pricing methodology is characterized by combining the full flexibility of LexiFi’s generic contract description with maximal operation speed brought by native code execution.

The first section introduces LexiFi’s pricing methodology and implementation, followed by section 2. Market data manipulation, then, section 3. Quantitative pricing and risk-related tools and the last section 4. covers Implementation techniques.

The list of supported pricing models can be found here.

1. Pricing methodology and implementation

Model coverage and automated selection

All models and quantitative tools are developed “in-house” at LexiFi:

• Detailed documentation
• Optional direct access to LexiFi’s quantitative expertise

Automated model selection and initialization by symbolic analysis of contract description:

• Kind and number of underlyings
• Presence of early-exercise and/or path-dependency features
• Set of all relevant contractual dates

Automatic suggestion of an admissible pricing model with plausible parameters or of the list of all applicable models.

All models are available with both Monte Carlo and PDE implementations (when applicable).

Supported asset classes include equities, interest rates, exchange rates, credit, commodities, inflation.

All equity models support discrete and continuous dividends with term structure and quanto adjustments.

Provision for task-oriented pricing model choice organization thanks to the definition of “Pricing Profiles”.

Calibration

Models can be pre-calibrated, or calibrated on the fly. An automatic cache allows sharing on-the-fly calibration results across multiple pricing jobs.

Dedicated calibration inspectors are available for all models:

• Inspect and verify resulting parameters and surfaces
• Compare model numerical prices with market prices
• Tracing and charting

Monte Carlo

Optional early stopping when a given precision is reached.

Multi-dimensional simulation.

Longstaff Schwartz regression when early exercise pricing is needed:

• Automated selection of state variables
• First pass to infer optimal strategy
• Symbolic rewriting of the payoff to incorporate the strategy
• Second pass, possibly with more trajectories

Pseudo random number generators: Sobol (low-discrepancy) and Mersenne Twister.

Principal component analysis and Brownian bridge.

Variance Reduction through Contract Variates.

LexiFi’s proprietary Adjusters method is available for dramatic precision enhancements, based on an automated decomposition of contracts into:

• A statically replicated part to be priced without model assumptions
• A residual part to be priced numerically

Optional Richardson-Romberg method reduces time discretization bias.

Optional Importance sampling method: automatic multi dimensional modification of the means and standard deviations of the generated random variables reduces the Monte-Carlo price estimator variance (beta).

Extra valuation analytics include:

• Standard deviation and error
• Early redemption and barrier crossing probabilities
• average life-time
• Cash flow details (zero-coupon, forward amount, occurrence probability, conditional expectation)

Note that all previously mentioned probabilities, averages or expectations are evaluated under appropriate forward measures to make them comparable and independent of the implemented choice of numeraire for used model.

Graphical pricing debugger:

• Full transparency on the Monte Carlo simulation, at the level of individual trajectories (underlying realizations, cash flows, values derived from the payoff)
• Inspection of the (possibly joined) distribution of any pricing relevant quantities
• Data may be exported to external CSV files, for further inspection

Partial Differential Equations

Multi dimension (1, 2 or 3).

Automatic detection of path-dependent state variables.

Path dependent variable sampling and various interpolation methods.

Alternating Direction Implicit method: ADI method.

Graphical pricing debugger:

• Full transparency on the PDE simulation
• Showing vectors of prices at each intermediate step

Model-free and nearly-model-free pricing

Model-free pricing:

• Automatic detection by symbolic payoff analysis of any European (or sum of European) payoff(s) depending on only one underlying asset realization on a single date
• Gives a price depending only on observed market prices

Nearly-model-free pricing:

• Multi-dimensional pricing model for even hybrid complex structures (including path dependency and early exercise features)
• Gives a price depending only on observed market prices and a model on asset correlations

Full access to quantities computed by previous static replication models:

• Replication strategies
• Underlying asset densities
• Simple barrier hitting probabilities
• Greeks

2. Market data manipulation

Flexible market data input

• Well documented format and nomenclature for easy import of external market data

• Flexible description of market data, to accommodate various kinds of external data:

  • Support of all usual market data quote kinds
  • Forwards can be quoted through explicit quotes or implicitly with discrete and/or continuous implied dividends
  • Volatilities can be quoted as option price or implied B&S volatilities
  • Implied and historical correlation can easily be expressed
  • etc.
    
    

• Ability to ``tag'' market data in order to differentiate sources etc.

• Smart sources union, with priority rules

Flexible market data transformation

• Possibility to define proxies for equities/indices that do not have enough data

• Various ways to normalize the market data:

  • Yield curve as zero-Coupon
  • Options quotes in relative or absolute strikes
  • CDS quotes as spread or upfront
    
    

• Numerous ways to filter market data depending on maturity or quote kind (FRA, Future, …)

Curve building

• Zero-Coupon / Deposit / Forward - Future / Swap / Basis Swap
• Single-curve or Multi-curve pricing
• FX-based
• Risky curve
• Charting and inspection tools

3. Quantitative pricing and risk-related tools

Automatically price a contract with all suitable models to gain a quick model price dispersion overview.

Proprietary Model Uncertainty tool to provide insights into price precision with respect to observable market data and model choice.

Value at Risk (VaR) and CVaR:

• Historical or Monte-Carlo scenario generation
• Optimized model implementations for simultaneous market-data scenario calculations, when applicable
• Efficient computation using fast first-selection step

XVA/CVA:

• With/without collateral
• Applicable to all asset classes

Greeks:

• Malliavin method
• Finite Differences

Contract Variations:
Get a matrix of prices for parametric variations on a family of contracts.

Contract Solver:
Backsolve ad hoc contract parameters or market data to match a given target price.

Flexible scenario framework:

• Compositional definition of simple or more complex scenarios
• Spot and future scenarios
• Automatic market trajectory interpolation for future scenarios

Value Change Analysis:

• Drill-down into the price differences between two market conditions and/or two dates
• Contractual events, impact of time, impact of changes on spots, volatilities, etc.

4. Implementation techniques

Compilation-based approach

• The best of both worlds:

  • Full flexibility of LexiFi’s generic contract symbolic description
  • Maximal speed by native code execution of the payoff during pricing
    
    

• Automatic support of past life-cycle events, through symbolic rewriting of contract descriptions

• Automatic symbolic detection of closed forms (e.g. for continuous barriers)

LexiFi’s pricing code compiler targets either:

• A custom bytecode language (evaluated by a highly optimized bytecode interpreter), Or
• C source code (to be compiled to native code)

Miscellaneous

• Numerical implementation using state-of-the-art numerical algorithms, carefully reviewed and continuously enhanced and tested by LexiFi’s team
• Algorithms carefully designed for highest speed while enforcing numerical stability, even in parametric corner cases
• Support of multicore architectures for parallel pricing
• Calculations organized to benefit from SIMD CPU instructions, using highly-optimized external numerical libraries