An Efficient and Easily Parallelizable Algorithm for Pricing Weather Derivatives

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An Efficient and Easily Parallelizable Algorithm
for Pricing Weather Derivatives
Yusaku Yamamoto Dept. of Computational Science
Engineering Nagoya University
LSSC05,Sozopol June 6-10, 2005
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Outline of the talk

• 1. Introduction
• 2. Problem formulation
• 3. A new pricing algorithm based on the fast
  Gauss transform
• 4. Parallelization
• 5. Numerical results
• 6. Conclusion

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1. Introduction

• Weather conditions greatly affect business
  activities.

Electronics companies larger sales of air
conditioners
Excessive heat in summer
Department stores increase of air conditioning
costs
Local governments decrease of costs for removing
snow
Little snow In winter
Hotels in ski areas decrease of revenues
Variation of revenues due to weather conditions
weather risks
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What is a weather derivative?

• Definition
• A financial contract which allows the holder to
  receive a certain amount of money if
  predetermined weather conditions are met.
• Example of a temperature derivative
• The holder will receive 10,000 for one day in
  July for which the maximum temperature is under
  20C
• A manufacturer of air conditioners can use
  this derivative to compensate for a possible loss
  of revenues due to cool summer.
• Market of weather derivatives
• 11 billion in Europe and the USA (in year 2002)

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Specification of a temperature derivative

• Parameters defining a temperature derivative
• Period of observation N days from a specified
  date
• Point of observation
• Temperature index W (C)
• Strike value S (C)
• Tick value k (/C)
• Kind put or call
• Derivative price Q ()
• Temperature indices
• Average temperature
• Maximum temperature
• Cooling Degree Days (CDD)
• Payoff
• call Pcall kmax(W S, 0)
• put Pput kmax(S W, 0)

maximum temperature
Ti
average temperature
N
days
period of observation
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CDD derivatives

• Definition
• A temperature derivative on the CDD(Cooling
  Degree Days) defined by
• Properties
• The payoff increases as
• the number of days for which Tn gt T increases and
• The difference Tn gt T increases.
• Application
• Compensation for air-conditioning costs due to
  excessive heat, etc.

Ti
T
T reference temperature
days
N
We focus on the pricing of CDD derivatives
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2. Problem formulation

• Principles for pricing
• Construct a stochastic model describing the daily
  air temperature
• T1, T2, , TN during the period of
  observation.
• Compute the expectation value of the payoff
  Pcall under the stochastic model.
• Add premium e to the result to get the derivative
  price
• Q EPcall e.
• Stochastic models for daily air temperature
• Historical model (Zeng, 2000)
• Determines Tn by random sampling from the past
  data
• Dischel model (Dischel, 1999)
• Tn is assuemd to follow a non-stationary
  autoregressive model of order 1.
• GARCH models (Cao Wei, 1999)
• Long-term memory models

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The Dischel model

• The model
• Tn follows a non-stationary autoregressive model
  of order 1.
• Tn (1 ß)Tn ßTn1 en
• Tn temperature of the n-th day in an
  average year
• en N µ,s2, i. i. d.
• Widely used for pricing as a simple yet effective
  model.
• Determination of the parameters
• Parameters b, m and s are determined from the
  past data by least squares fitting.
• It is also possible to adjust b, m and s to
  incorporate the long-term weather prediction (Egi
  et al., 2003).

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Pricing by the Monte Carlo method

• Basic algorithm
• Generate a large number of sample paths according
  to the temperature model.
• Compute the expectation value as the average of
  payoff values over the sample paths.
• EPcall (1/L)?i1L Pcall (i)
• Advantages
• Implementation is easy.
• Various weather indices can be treated in a
  unified manner.
• Embarrassingly parallel.

Tn
T
N
days
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Problems with the MC method

• Slow convergence
• Requires 108 sample paths to compute the price to
  3-digit accuracy.
• The computation takes about 10 min on an average
  PC.
• Circumstances where the MC method is too slow
• Real-time pricing
• Pricing of a portfolio consisting of hundreds of
  derivatives
• Objectives of our research
• To develop a pricing algorithm for the CDD
  derivatives that is
• orders of magnitude faster than the MC method,
  and
• easily parallelizable

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3. A new pricing algorithm based on the fast
Gauss transform

• The basic idea (Yamamoto Egi, 2004)
• Define the partial CDD Cn as follows
• Compute the joint probability distribution
  function pn (Tn, Cn) of (Tn, Cn) by a recursion
  formula.
• Compute the expected payoff from pN (TN, CN).
• The joint pdf for the 1st day

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Computing the joint pdf by a recursion formula

• Transition pdf from pn1 (Tn1, Cn1 ) to pn
  (Tn, Cn)

When , we have from Cn Cn1 (Tn
T ),
The transition pdf for can be
computed similarly.
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Computing the joint pdf by a recursion formula

• The recursion formula for pn (Tn, Cn)

Convolution of a function with the Gaussian
distribution
Similarly, the recursion formula for
can be expressed as a convolution of a function
with the Gaussian distribution.

• Computational work (assuming M grid points for
  both Tn and Cn directions)
• O(M2) for each convolution
• O(M3) for computing pn(Tn, Cn) for all values of
  Tn and Cn

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Computing the expected payoff

• The expected payoff can be computed from pN (TN,
  CN) by simple integrations

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Acceleration by the fast Gauss transform

• Computational work of the present method
• O(M3) work to compute M convolutions at each time
  step.
• The fast Gauss transform(Greengard Strain,
  1991)
• Expand the Gaussian in the convolution
• with Hermite functions.
• Reduces the computational work from O(M2) to
  O(M).
• Proved useful to construct fast and accurate
  pricing algorithms for various financial
  derivatives (Broadie Yamamoto, 2003).
• Effect of using the FGT
• The computational work at each time step can be
  reduced from O(M3) to O(M2).

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Convergence of the proposed method

• Computational work
• O(M2) for each step (thanks to the use of the
  FGT)
• Accuracy
• The pricing error can be shown to decrease as
  O(1/M2).
• The error decreases as E O(1/t) with the
  computational timet.
• Comparison with the Monte Carlo method
• For the MC method, E O(1/vt).
• Asymptotically, our method converges faster
  than the MC method.

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4. Parallelization

• The recursion formulas at each time step

The computation of pn (Tn, Cn) for different
values of Cn can be done independently.
Easily parallelizable by partitioning the array
of pn (Tn, Cn) in the Cn direction and
allocating each partial array to one processor.
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Data transfer between the processors
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5. Numerical results

• Target problem
• CDD call derivatives under the Dischel model
• Numerical methods
• Monte Carlo method
• Our method
• Parameters
• Period of observation N days from July 7th (N
  10 or 20)
• Place of observation Tokyo
• Index CDD (T 24 C)
• Strike value K 20 or 40 C
• b 0.56, m 0.01, s 1.83, Qk
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• Computing environments
• Alpha workstation with g77 compiler

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Numerical results (contd)
N10, K20
N20, K40
Price
Price
Time (sec)
Time (sec)
The MC method needs 100 to 1000 seconds to get an
accuracy of 102. Our method is more than 10
times faster than the MC method.
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Effect of parallelization

• Platform
• Cluster of Alpha workstations
• 8 Alpha 21164A processor
• 128MB memory / node
• 100-BaseT network
• Results
• 6 times speedup using 8 nodes.
• Comparable with the MC method, which can achieve
  almost perfect speedup

Speedup
Number of nodes
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6. Conclusion

• Conclusion
• We developed a fast and an easily parallelizable
  algorithm for pricing CDD weather derivatives
  based on the fast Gauss transform.
• Our algorithm is more than 10 times faster than
  the MC method when computing the price of a CDD
  derivative with 5 to 20 monitoring dates.
• Also, it can achieve 6 times speedup on a WS
  cluster with 8 nodes.
• Future work
• Application to other types of temperature
  derivatives
• Application to the pricing of a portfolio
  consisting of a large number of derivatives