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An Efficient and Easily Parallelizable Algorithm for Pricing Weather Derivatives

Yusaku YamamotoDept. of Computational Science & Engineering

Nagoya University

LSSC’05 ， SozopolJune 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.

Excessive heatin summer Department stores: increase of air conditioning costs

Electronics companies: larger sales of air conditioners

Little snowIn winter Hotels in ski areas: decrease of revenues

Local governments: decrease of costs for removing snow

Variation of revenues due to weather conditions = weather risks

Financial products to reduce weather risks and stabilize the revenues

Weather derivatives

=

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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 = k ・ max(W – S, 0)– put ：　 Pput = k ・ max(S – W, 0)

Ti

daysN

averagetemperature

period of observation

maximum temperature

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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 > T increases and• The difference Tn > T increases.

• Application– Compensation for air-conditioning costs due to excessive heat, etc.

T: 　 reference temperature

Ti

daysN

T

We focus on the pricing of CDD derivatives

n

n

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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 = E[Pcall] + 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 – β)Θn + βTn–1 + εn

　 　 Θn： temperature of the n-th day in an average year

　　　 εn ～ N [μ ， σ2], 　 i. i. d.

– Widely used for pricing as a simple yet effective model.

• Determination of the parameters– Parameters , and are determined from the past data by least squares fitting.– It is also possible to adjust , and to incorporate the long-term weather predict

ion (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 temperat

ure model.– Compute the expectation value as the average of payoff values ove

r the sample paths.

　　 E[Pcall] = (1/L)∑i=1L Pcall

(i)

• Advantages– Implementation is easy.– Various weather indices can be treated in a unified manner.– Embarrassingly parallel.

Tn

daysN

T

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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

Cn = ∑ i=1n max(0, Ti – T )

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Computing the joint pdf by a recursion formula

• Transition pdf from pn–1 (Tn–1, Cn–1 ) to pn (Tn, Cn)

（　　　　　　　　　　　　　　　　 ）

The transition pdf for can be computed similarly.

When , we have from Cn = Cn–1 + (Tn – T ),

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Computing the joint pdf by a recursion formula

• The recursion formula for pn (Tn, Cn)

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

Convolution of a function with the Gaussian distribution

When ,

• 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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• 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 va

rious financial derivatives (Broadie & Yamamoto, 2003).

• Effect of using the FGT– The computational work at each time step can be reduced from O(M

3) to O(M2).

Acceleration by the fast Gauss transform

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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/τ) with the computational timeτ.

• Comparison with the Monte Carlo method– For the MC method, E = O(1/√τ).

Asymptotically, our method converges faster than the MC method.

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4. Parallelization

• The recursion formulas at each time step

)

When Tn < T,

When Tn ≧T,

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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Tn

Cn

O T

Computation by the recursion formula

Data transfer between the processors

Tn –1

Cn–1

O T

Data mapping at step n–1

PU0PU1PU2PU0PU1PU2

Tn –1

Cn–1

O T

Data transfer

: Data to be transferred

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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– = – 0.56, = – 0.01, = 1.83, k = 20

• Computing environments– Alpha workstation with g77 compiler

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Numerical results (cont’d)

4.94

4.96

4.98

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5.02

5.04

5.06

5.08

5.1

0.1 1 10 100 1000

MCFGTMC lower boundMC upper bound

Time (sec) Time (sec)

Price

13.2

13.25

13.3

13.35

13.4

13.45

13.5

1 10 100 1000 10000

MCFGTMC lower boundMC upper bound

Price

N=10, K=20 N=20, K=40

The MC method needs 100 to 1000 seconds to get an accuracy of 10–2.Our method is more than 10 times faster than the MC method.

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Effect of parallelization

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

N=20N=10

• 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

Number of nodes

Speedup

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6. Conclusion

• Conclusion– We developed a fast and an easily parallelizable algorithm for pricin

g 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