CrossCurrency Option Pricing using Webservices Arun Verma Quantitative Research Bloomberg LP

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Cross-Currency Option Pricing using Webservices
Arun VermaQuantitative ResearchBloomberg
LP
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Outline

• Finance problem overview
• Pricing Model
• Algorithm Numerical Methods
• Deployment using Web services
• Advanced Pricing Models

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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
• Heston model is a popular FX options model fits
  the volatility surface well for a given currency
  pair.
• Can we leverage good properties of Heston model
  for a cross-FX model needs to model 3 pairs of
  spot exchange rates?
• 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

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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
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
A1
B
C
A2
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Implied Correlation a Flawed Concept

• A natural method is to model the implied
  correlation as a function of delta and maturity.
• 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.
• 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 doesnt
  make much sense.
• 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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Joint-Heston Dynamics Primary rates dynamics
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Joint-Heston Dynamics Cross rate dynamics
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Joint-Heston Dynamics Change of Measure
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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.

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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.

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Sanity Checks
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Numerical Results

• RMSE implied vols averaged over 30 days of daily
  data.
• Used correlation parameter to match the short
  term ATM implied vol.

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Cross-FX pricing on the Bloomberg
Inputs
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Robustness of the Model
Variation of parameters over calendar days
Dashed (separate). Solid (Joint-Heston)
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Correlation parameter stability

• .

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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.

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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 (gt800) on behalf
  of a potentially large number of users needs to
  be tested.

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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.

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A smart client application
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Advanced Models

• Independent stochastic clocks
• Adding a Global Economy Factor
• Multiple Anchor versions
• Copula Models

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Extension I Independent stochastic clocks
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Extension I Independent stochastic clocks
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Extension II Additional Global Factor
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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.