In LexiFi Apropos all pricing models can be implemented through different methods, generating a wide spectrum of solutions.

**1) Equities**

All equity models support discrete and continuous dividends, with a term structure in the latter case, and quanto adjustments.

**Static replication**- Available for sums of European payoff-like single-underlying products
- Automatic decomposition
- Quasi closed-form solution

**Black Scholes**- Support of both constant and deterministic volatility term-structure
- Monte Carlo multiple-asset or PDE up to three assets model implementation
- Smart adjuster to improve the price (https://www.lexifi.com/white_papers/adjusters.pdf)

**Local volatility**- Calibration to European call and put quotes using an implied volatility surface fitting
- Calibration based on Andreason-Huge volatility interpolation
- Advanced time discretisation scheme: Runge Kutta, Euler or automatic switch
- Monte Carlo multiple-asset or PDE up to three assets model implementation

**Heston (stochastic volatility)**- Calibration to European call and put quotes using semi-closed formula and numerical integration
- Advanced time discretisation scheme (quadratic exponential)
- Monte Carlo multiple-asset or PDE up to three assets model implementation

**Shifted Black**- Linear local volatility model
- Calibration to European call and put quotes using semi-closed formula
- Fit the skew of the volatility surface
- Monte Carlo large step multiple-asset or PDE up to three assets model implementation

**Heston - Local Stochastic Volatility**- Calibration to European call and put quotes using Heston calibration, Local Volatility calibration followed by either Particle Method or Fokker-Planck equation to compute probability density
- Possibility to give more importance to Local or Stochastic volatility part using mixing weight factor
- Advanced time discretisation scheme for the variance (quadratic exponential)
- Monte Carlo multiple-asset or PDE up to three assets model implementation

**Merton Jump Model**- Jump diffusion extension of the Black-Scholes model
- Calibration to European call and put quotes using semi-closed formula
- Ability to fit short-term smile
- Term structure possible for each parameter
- Large step simulation using efficient Poisson distribution simulation
- Monte Carlo multiple-asset or PDE up to three assets model implementation

**Bates Jump Model**- Jump diffusion extension of the Heston model
- Calibration to European call and put quotes using semi-closed formula and numerical integration
- Ability to fit short and long-term smile
- Advanced time discretisation scheme for the variance (quadratic exponential)
- Monte Carlo multiple-asset implementation

**2) Interest rates**

All interest-rate models are delivered with calibration routines for cap/floor and swaption quotes

**Static replication on Libor or CMS**- Only available for vanilla contracts
- Automatic decomposition
- Convexity Adjustment
- Shifted-SABR interpolation formula
- Quasi-Closed form solution

**Hull-White 1 factor**- Monte Carlo large step or PDE model implementation
- Exact large step simulation using the forward-neutral probability
- Support of constant, exponential and step volatility term structure

**Hull-White 2 factors (G2++)**- Monte Carlo large step or PDE model implementation
- Various calibration modes on caps and swaptions
- Exact large step simulation using the forward-neutral probability
- Support of constant and step volatility term structures

**Cheyette (quasi-Gaussian model)**- Monte Carlo using QE scheme or PDE implementation
- Linear local volatility and linear local volatility with CIR stochastic volatility parameterisations
- Time-dependent parameters
- Captures most shapes of volatility smiles
- Optimised calibration: calibrating first a proxy swap rate market model (SMM) on implied volatility, then bootstrapping the Cheyette model parameters to fit SMM parameters

**Lognormal forward-LIBOR model (LFM)**- Monte Carlo model implementation
- Large step simulation (using Runge-Kutta discretisation)
- Functional volatility and correlation structures
- Dimension reduction using principal component analysis

**Lognormal forward-LIBOR + stochastic volatility model (LFM+SV)**- Large step simulation (using Runge-Kutta discretisation)
- Functional volatility and correlation structures
- Calibration on Cap and Swaption skew and smile
- Dimension reduction using principal component analysis

**Shifted-Lognormal forward-LIBOR model (SLFM)**- Monte Carlo model implementation
- Large step simulation (using Runge-Kutta discretisation)
- Calibration on Cap and Swaption skew and smile
- Dimension reduction using principal component analysis

**Shifted-Lognormal forward-LIBOR model + stochastic volatility (SLFM+SV)**- Large step simulation (using Runge-Kutta discretisation)
- Functional volatility and correlation structures
- Calibration on Cap and Swaption skew and smile
- Dimension reduction using principal component analysis

**Cap CMS Closed form (Hagan)**- Use the method of Hagan for pricing CMS derivatives
- SABR interpolation of swaption volatilities

**3) Inflation**

**Static replication on Index and YoY**- Only available for vanilla contracts
- Automatic decomposition
- Quasi-Closed form solution

**Jarrow-Yildirim**- Calibration on either YoY Cap / Floor or ZC Cap / Floor
- Monte Carlo large steps model implementation

**Heston local volatility on Inflation Index**- https://www.lexifi.com/white_papers/heston_slv_infla.pdf
- Calibration on YoY Cap / Floor and ZC Cap / Floor
- Monte Carlo small steps model implementation

**4) Foreign exchange**

**Static replication**- Only available for vanilla contracts
- Automatic decomposition
- Quasi-Closed form solution

**FX Option Closed Form****Garman-Kohlhagen**- Handle all Black Scholes features, including smart adjusters

**Hull-White 1 factor + Garman-Kohlhagen**- Interest rates are modelled with a Hull-White 1-factor model
- Monte Carlo large steps or PDE model implementation

**Heston - Local Stochastic Volatility**- Calibration to ATM volatility, Risk Reversal and Butterfly quotes using Heston calibration, Local Volatility calibration followed by either Particle Method or Fokker-Planck equation to compute probability density
- Possibility to give more importance to Local or Stochastic volatility part using mixing weight factor
- Advanced time discretisation scheme for the variance (quadratic exponential)
- Monte Carlo multiple-asset or PDE model implementation

**5) Commodities**

**Static replication**- Only available for vanilla contracts
- Automatic decomposition
- Quasi-Closed form solution

**Schwartz 1 factor**- Calibration on Futures, Calls on spot and Calls on Futures
- Handle seasonality
- Monte Carlo large step or PDE model implementation

**Schwartz 2 factors**- Calibration on Futures, Calls on Spot and Calls on Future
- Monte Carlo large step or PDE model implementation

**Gabillon**- Calibration on Futures, Calls on Spot and Calls on Future
- Monte Carlo large step or PDE model implementation

**Clewlow-Strickland**- HJM-like extension of Schwartz 1 factor
- Take Future curve as input (and fit it exactly)
- Calibration on Calls on Spot and Calls on Future
- Monte Carlo large step or PDE model implementation

**Clewlow-Strickland Stochastic Volatility**- HJM-like extension of Schwartz 1 factor
- Take Future curve as input (and fit it exactly)
- Calibration on Spot and Future smile
- Monte Carlo small step

**Forward curve building**- Smooth forward curve building
- Two seasonality effect (i.e. first two harmonics of seasonality effect)

**6) Credit**

**Deterministic intensity**- Deterministic
- Calibration on CDS spreads
- Single risk

**Intensity with copula**- Multiple correlated risk factors
- Calibration on CDS spreads and CDO tranches
- Monte Carlo model implementation

**CDS , CDS Tranches pricer and CDS Swaption pricer**

**7) Hybrids**

**Generic hybrid: equity / interest rate / exchange rate / inflation / credit / commodity**- Equities are modelled with a Black-Scholes model (with a term structure of volatility)
- Interest rates are modelled with a Hull-White 1-factor model
- Exchange rates are modelled with a Garman-Kohlhagen model
- Inflation indices are modelled with a Jarrow-Yildirim model
- Commodities are modelled with a Clewlow-Strickland model
- Credit are modelled using a intensity with copula model
- Monte Carlo large steps model implementation

**Heston - Local Stochastic Volatility**- Handle hybrid FX / Equities / Inflation
- Each underlying can have its own mixing weight factor (handle pure LV for asset and pure Heston for FX for instance)

**Equity Local Volatility / Garman-Kohlhagen**- Hybrid equity / FX
- Monte Carlo or PDE implementation

**Heston - Local Stochastic Volatility / Hull-White**- Handle hybrid FX / Equities / Interest Rate
- Equities follow a Heston - Local Stochastic Volatility model
- FX follow a Heston - Local Stochastic Volatility model
- The short rate follows a Hull-White model with a volatility term-structure
- Calibration with a Particle method (exact calibration)
- Particular Heston / Hull-White and Local Volatility / Hull-White cases are supported

**Deterministic**- Hybrid all asset classes, using forwards for all underlying

**1) General**

- Best-of-two-worlds generic compilation based approach (arbitrary payoff): use full flexibility of LexiFi’s generic MLFi driven contract description while ensuring maximal speed by native code execution
- All models available with Monte Carlo or PDE implementation (when applicable)
- Handle all underlying asset classes (when applicable)
- LexiFi’s unique symbolic contract analysis allows for automatic model suggestion or applicable list of models provision with consistent calibration and simulation parameters
- Automatically detect early-exercise or path-dependency feature for automatic Monte Carlo or PDE numerical model implementation application
- Synchronisation with life-cycle management by design, using LexiFi’s compilation techniques in conjunction with contract life-cycle determination; ensure that pricing simplifies, as contract matures
- Keep precise audit trail of price calculations (including used market data) if needed
- Automatic explanation of differences on contract prices calculated under different market conditions and/or dates
- Use LexiFi’s symbolic-analysis capacity to automatically apply rapid static replication model to any European (or sum of European) payoff(s) depending on only one underlying asset realisation at same date in order to obtain a model-free price depending only on observed prices
- Full access to replication strategies, underlying asset densities and Greeks computed by previous static replication model
- Use LexiFi’s proprietary “adjusters” method for dramatic precision enhancements through automated decomposition of contracts into a statically replicated part and a residual part to be priced numerically

**2) Monte Carlo**

- Generic compilation based approach (arbitrary payoff)
- Multi dimension
- Low-discrepency number generator: Sobol
- Low-discrepancy sequence: Sobol
- Principal component analysis: PCA
- Control variates
- Automatic Longstaff Schwartz regression when early exercise pricing is needed
- Greek computation with automatic method selection (Malliavin or Finite Difference)
- Continuous-barrier closed form
- Decompose a contract price into individual cashflow-adjusted prices

**3) Partial Differential Equation: PDE**

- Generic compilation based approach (arbitrary payoff)
- Multi dimension (up to 3)
- Automatic path dependant variable detection
- Path dependant variable sampling and various interpolation methods
- Alternating Direction Implicit method: ADI method (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2754662)
- Change of variable

**4) Curve building**

- Zero-Coupon / Deposit/ Forward - Future/Swap / Basis Swap
- Single curve or Multi-curve pricing
- FX-based
- Risky curve

**5) Flexible market data input**

- Various forms of market data item description: e.g. equity forward, discrete dividend or implied dividend, numerous choices for volatility-defining items
- Ability to “tag” market data in order to differenciate sources

**6) Flexible market data transformation**

- Various ways to normalise 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, …)

**7) Model selection**

- Automatic heuristic to select adequate pricing model based on payoff
- Adequate pricing method
- Automated selection of calibration and simulation parameters

**8) Detailed pricing results**

- Transparent methodology
- Transparent tools
- Calibration pages with extensive calibration result verification, tracing and charting
- Save calibration results for subsequent pricing or recalibrate on the fly
- Easy-to-inspect, auditable and storable results

**9) Risk**

- Value at Risk (VaR) and CVaR computation with optimised model implementations for simultaneous market-data scenario calculations, when applicable
- Flexible risk-scenario definition and generation
- Greeks computation
- Credit and Debt Valuation Adjustment with/without collateral
- All of them are applicable for each asset class