Quantitative Research

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: Calibration and Implicit Function Theorem

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 in Finance: Root Finding and Least Square Calibration

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.

Bill Pricing

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

Binormal With Correlation by Strike Approach to CMS Spread Pricing

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

Bond Futures: Description and Pricing

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.

Bond Pricing

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.

Deliverable Interest Rate Swap Futures: Pricing in Gaussian HJM 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.

Equity Variance Swap Greeks

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

Equity Variance Swaps with Dividends

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

Forex Options Digital

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.

Forex Options Vanilla and Smile

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.

Hull-White One Factor Model

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.

Inflation Instruments: Zero-Coupon Swaps and Bonds

The most common inflation instruments are described.

Interest Rate Futures and Their Options: Some Pricing Approaches

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.

Libor Market Model with Displaced Diffusion: Implementation

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

Local Volatility

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

Mixed Log-Normal Volatility Model

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

Multi-Curves Framework with Stochastic Spread: A Coherent Approach to STIR Futures and Their Options

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.

Multi-Curves: Variations on a Theme

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.