This page contains various working papers that have been written by the OpenGamma quantitative development team. They outline the some of the modelling and pricing methodologies used in the OpenGamma analytics library. This is not a comprehensive list of every instrument or pricing technique that is available; however, this page will be added to regularly.

Adjoint Algorithmic Differentiation is an efficient way to obtain financial instrument price derivatives with respect to the data inputs. Often the differentiation does not cover the full pricing process when a model calibration is performed. Thanks to the implicit function theorem, the differentiation of the solver embedded in the calibration is not required to differentiate the full pricing process. An efficient approach to the full differentiation process is described.

Algorithmic Differentiation is an efficient way to compute derivatives of a value with respect to the data inputs. In finance the model calibration to market data can be an important part of the valuation process. In presence of calibration, when obtained through exact equation solving or optimisation, very efficient implementation can be done using the implicit function theorem with the standard AD approach. Previous results discussing the exact case are here extended to the case of calibration obtained by a least-square approach.

The details of the implementation of pricing for (Treasury) bills is provided.

The pricing of CMS spreads using a binormal with correlation by strike approach is described.

The descriptions of standard bond futures in major currencies are provided. The standard pricing approach based on cheapest-to-deliver is described. A method taking into account the delivery option, based on a one-factor HJM model, is also described.

The details of the implementation of pricing for fixed coupon bonds and floating rate notes are provided. The different day count and yield conventions used by sovereign bonds are described.

Description and pricing method for Brazilian swaps is provided.

Over the last several years rapid changes in global contracts and local conventions for credit default swaps (CDS) contracts have been made with the implementation of CDS Big Bang and Small Bang protocols in order to increase transparency and efficiency of the market. In this note we present conventions and market standards for single-name CDS.

The purpose of this document is to present analytical formulas for the convexity adjustment within the year-on-year inflation swap and the zero-coupon inflation swap with payment delay. This adjustment is also used in the calculation of the forward for inflation-linked options. The framework used in this note is the forward price index market model.

CME will soon be proposing a new product: Deliverable Interest Rate Swap Futures. This note analyses the pricing of such futures in the Gaussian multi-factor HJM model and multi-curves framework. We also provide numerical examples of prices and hedging with those futures.

We present what Greeks can (and should) be calculated for Equity Variance Swaps with and without discrete dividends.

A discussion paper on how to price equity variance swaps in the presence of known (cash and proportional) dividends.

Some pricing methods for forex digital options are described. The price in the Garman-Kohlhagen model is first described. The call-spread approximation approach is described.

The pricing of vanilla Forex options using the Garman-Kohlhagen formula is described. The smile using the standard market quotes at-the-money, risk-reversal and strangle is also described.

This paper is a followup to The Pricing and Risk Management of Credit Default Swaps, with a Focus on the ISDA Model. Here we show how to price indices (portfolios of CDSs) from the calibrated credit curves of the constituent names, and how to adjust those curves to match the market price of a index (basis adjustment). We then show how to price forward starting single-name CDSs and indices, since these are the underlying instruments for options on single-name CDSs and indices. The pricing of these options is the main focus of this paper. The model we implement for index options was first described by Pedersen, and we give full implementation details and examples. We discuss the common risk factors (the Greeks) that are calculated for these options, given various ways that they may be calculated and show results for some example options. Finally we show some comparisons between our numbers and those displayed on Bloomberg’s CDSO screens.

Details regarding the implementation of the Hull-White one factor model are provided. The details concern the model description and parameters, the vanilla instruments pricing and the Monte Carlo implementation.

The purpose of this document is to present analytical formulas for the inflation caps and floors (year on year and zero-coupon). The framework of the note is the forward price index market model.

The most common inflation instruments are described.

The most common inflation linear instruments and the curve construction are described in this note.

Exchange-traded interest rate futures and their options are described. The future options include those paying an up-front premium and those with future-like daily margining.

Conventions and market standards for the most common interest rate instruments.

The Libor Market Model (LMM) with displaced diffusion is described. The details of the implementation are provided.

Details of computing a local volatility surface from market data, then numerically solving different PDE representations to reproduce the market prices, and compute greeks.

We present two versions of a Mixed Log-Normal Volatility Model. The first is applicable to produce a single, arbitrage free, volatility smile (i.e. it produces a single terminal Probability Density Function (PDF) for some expiry) but not a consistent volatility surface. The second version can produce an arbitrage free implied volatility surface and the corresponding local volatility surface via simple analytic expressions. These surfaces are useful for testing other numerical methods, in particular methods to calculate the fair value of variance swaps as the true value is known from the parameters of the model via another simple expression

This note is dedicated to the impact of collateral on the multi-curve framework. The pricing formula in presence of collateral are described in a generic way encompassing several financial realities. The multiple currency collateral case is also described, including the convexity adjustment required. The pricing of STIR futures in this framework is analysed in detail.

The development of the multi-curve framework has mainly concentrated on swaps and related products. By opposition, this contribution focuses on STIR futures and their options. They are analysed in a stochastic multiplicative spread multi-curve framework which allows a simultaneous modelling of the Ibor rates and of the cash-account required for futures with continuous margining. The framework proposes a coherent pricing of cap/floor, futures and options on futures.

The multi-curves framework is often implemented in a way to recycle one-curve formulas; there are no fundamental reasons behind the choice. Here we present different approaches to the multi- curves framework. They vary by the choice of building blocks instruments (Ibor coupon or futures) and the definition of curve (pseudo-discount factors or direct forward rate). The features of the different approaches are described.

We present details of the 1D PDE solver used in the OpenGamma Platform, showing how it can price European and American options, with and without barrier features.

Detailed mathematical documentation for pricing European options using Fast Fourier Transform methods where the characteristic function of the log of the terminal asset price exists in closed form.

This note reviews interpolation algorithms based on piecewise polynomial functions. We first introduce two basic interpolations: piecewise linear interpolation and cubic spline interpolation. Since the latter suffers from overshooting problems and nonlocal dependencies, we discuss alternative interpolation methods aiming at shape-preserving of given data and avoiding the nonlocality. Finally, two-dimensional extensions of the spline interpolations are considered.

In the paper we detail the reduced form or hazard rate method of pricing credit default swaps, which is a market standard. We then show exactly how the ISDA standard CDS model works, and how it can be independently implemented. Particular attention is paid to the accrual on default formula: We show that the original formula in the standard model is slightly wrong, but more importantly the proposed fix by Markit is also incorrect and gives a larger error than the original formula. We finish by discussing the common risk factors used by CDS traders, and how these numbers can be calculated analytically from the ISDA model.

Technical details for Constant Maturity Swap pricing by replication on cash-settled swaptions. Valid for linear standard CMS and non-linear TEC-like CMS.

This documentation describes the way sensitivities to curves and other parameters are computed. In particular we describe rate deltas or Bucketed PV01.

An implementation of smile extrapolation for high strikes is described. The main smile is described by an implied volatility function, e.g. SABR. The extrapolation described is available for cap/floor and swaption pricing in the OpenGamma library.

Pricing methods for swaps and caps/floors with fixing in arrears or payment delays are proposed. The methods are based on replication.

Implementation details for the pricing of European swaptions in different frameworks are presented.