U of Warwick2/24/06 Implementing Derivative Valuation Models Cross-Currency Option Pricing using Webservices Arun Verma Quantitative Research Bloomberg

2/24/06 Implementing Derivative Valuation Models U of Warwick

Cross-Currency Option Pricing using Webservices

Arun Verma

Quantitative ResearchBloomberg LP


Outline

• Finance problem overview

• Pricing Model

• Algorithm & Numerical Methods

• Deployment using Web services

• Advanced Pricing Models


The Problem

Given two liquid pairs (MXN-USD, PLN-USD) and associated option quotes, how to form the implied volatility surface for an illiquid “cross-pair” (MXN-PLN) ?

Notes:

1) Heston model is a popular FX options model – fits the volatility surface well for a given currency pair.

2) Can we leverage good properties of Heston model for a cross-FX model needs to model 3 pairs of spot exchange rates?

3) Requirements:

• Need to keep joint dynamics tractable

• The model should be easily identifiable from observed data in the market

• The model should be robust – the parameters should be relatively robust to small changes in market data


Cross-currency pricing in Black-Scholes

In the Black-Scholes model, a constant lognormal volatility is assumed.

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Notation : Primary and Anchor Currencies

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For any cross-FX pair (B-C) of interest, we can pick any one of the many anchor currencies, e.g. A1 or A2. A liquid anchor (USD, EUR or JPY) is useful.

B-A & C-A are denoted as primary pairs (with known volatility surfaces)

Setting I : One anchor currencySetting II : Multiple anchor currencies


Implied Correlation – a Flawed Concept

A natural method is to model the “implied correlation” as a function of delta and maturity.

1) It is a flawed concept since there is no logic for linking delta of a cross pair to the same delta of the primary pairs.

2) A return of 20% on cross can be achieved by (0%,20%), (20%,0%) or (10%,10%) in the primary pairs so linking strike (delta) levels doesn’t make much sense.

3) Moreover, the implied correlation value could be outside its natural bounds of -1 & +1. There may be no arbitrage-violation since this concept has no theoretical underpinnings.

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Critique of the Joint-Heston model

Pros:• Tractability : The primary rates and cross rates all follow Heston dynamics.• Robustness : The model is robustly identified given the market data – no

knowledge of cross is required.

Cons:• The identical clock for two primaries (and indeed the cross) imposes the

constraint that all three vol surfaces will have similar term structure.

• The surfaces will also exhibit similar convexity since they share the vol-vol parameter – Fortunately in the market most pairs have same convexity – the butterfly spreads are typically prices the same across pairs.

• Note that the surfaces can have different skews (slopes) & levels as each exchange rate has their own correlation parameter defining correlation between the common clock and exchange rate increments.


Model Parameters

• The model has 8 free parameters

• is not identifiable from two primary rates volatility surfaces alone.

• It can be estimated from historical data or can be used as a “lever” to get a desired level of output vol.

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Calibration

• The pricing is done using the Fourier Transform (computed via FFT with appropriate discretization, Carr and Madan 98) which is based on computing the characteristic function which is available in closed form (Heston 93).

• The discrete transform derived by Carr and Madan can also be computed using Fractional Fast Fourier Transform which allows for accurate answer for a large range of strikes.

• The calibration problem which is solved numerically the nonlinear least squares problem minimizing the observed and model-implied vol surfaces using a trust-region optimization scheme in the Levenberg-Marquardt framework.

• The optimization functional is formed using sum of squares of price errors weighted by inverse of Black-Scholes Vega for all the options on the two primary FX rates.



Sanity Checks

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Numerical Results

GBP-EUR GBP-USD USD-EUR (cross)

Individual Heston

0.20 % 0.15 % 0.24 %

Joint Heston 0.28 % 0.19 % 0.49 %

RMSE implied vols averaged over 30 days of daily data.

Used correlation parameter to match the short term ATM implied vol.


Cross-FX pricing on the BloombergInputs


Robustness of the Model

Variation of parameters over calendar days –Dashed (separate). Solid (Joint-Heston)


Correlation parameter stability.


Implementation

• The implementation is done using MATLAB technical computing environment which is a powerful package for mathematical modeling.

• The model built in MATLAB is exposed as a COM or .NET assembly using Matlab tools which compile the native .m files into C/C++ and then build the binaries/libraries.

• The Com and .NET assemblies are easily important into a production environment which supports .NET webservices. A MATLAB analytics are included inside a webservice wrapper.

• We have an initiative “Smart Client for Quants” at Bloomberg under which we are publishing matlab models directly to production servers.



Production /Deployment Issues

• The application will consume data provided by Bloomberg Server API. Currently the Server API is provisioned for a defined set of users (generally a firm).

• It is not possible to service requests from an arbitrary group of (potentially all) Bloomberg users, while maintaining the identity (and associated preferences and permissions) of the user.

– A number of Server API rules that are applied by various back end components to limit the scope of data returned:

• A user making a request must be in the group of users associated with the particular Server API installation that initiates the request.

• Data from certain exchanges cannot be delivered through Server API.

– The impact on the servers of supporting large numbers of concurrent requests (>800) on behalf of a potentially large number of users needs to be tested.


Smart Client for Quants

• The “Straight From the Lab” initiative allows Bloomberg quantitative analysts to rapidly deploy applications built around complex calculations to selected customer desktops using Bloomberg Smart Client technology.

• In these deployments the calculations run on a server machine and are exposed as Web Services. These services will acquire data from the Bloomberg Server API.


A smart client application


Advanced Models

Independent stochastic clocks

Adding a Global Economy Factor

Multiple Anchor versions

Copula Models


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Conclusions• The simple joint-Heston model with one anchor currency is

robust and its accuracy is acceptable.

• Can use either a correlation or desired ATM vol level to generate the cross-vol.

• The extensions can be used to exploit information content of additional anchors – these models are costly in terms of computation and are unidentifiable in the single-anchor setting.

• Further extensions using Copulas can be investigated – However, Copulas are hard to generalize for pricing exotic options etc.

• The Web service architecture offers a good platform for sporadic traffic from client requests through out a trading day.

Documents

U of Warwick2/24/06 Implementing Derivative Valuation Models Cross-Currency Option Pricing using Webservices Arun Verma Quantitative Research Bloomberg