A Survey of Parallel Tree-based Methods on Option PricingPRESENTER: LI,XINYING

OutlineIntroductionBlack-Scholes ModelBinomial Options Pricing ModelTrinomial Options Pricing ModelImproved Binomial Option PricingCPU-GPU Hybrid Parallel BinomialSummary

Introduction

Stock Bond Currency

Underlying Asset!

Introduction

Option’s price is based on the corresponding underlying asset’s price.

+A suitable price of option

Introduction

According to the Options’ right:

Call Option & Put Option

Option Styles:European OptionAmerican OptionBermudan OptionAsian OptionBarrier OptionBinary OptionExotic OptionVanilla Option

Classification of options

Introduction

CPU: efficient in serial computing

Central Processing Unit (CPU) Graphics Processing Unit (GPU)

GPU: efficient in parallel

computing

CPU: efficient in serial computing

IntroductionOption pricing:

High demand on calculating speed

Heavy computation volume

The calculation procedure could be parallelized

Input: price of the underlying

asset

GPU: parallel computing

Output: option price

Efficient Algorithm

Introduction

StorageAccuracyEfficiency

Properties for evaluating the option pricing method

Therefore, a series of tree-based algorithms have been proposed to optimize the previous ones from different aspects.

OutlineIntroductionBlack-Scholes ModelBinomial Options Pricing ModelTrinomial Options Pricing ModelImproved Binomial Option PricingCPU-GPU Hybrid Parallel BinomialSummary

Black-Scholes Model It was raised by Fischer Black and Myron Scholes in 1973.

From the model, one can deduce the Black-Scholes formula, which gives a theoretical estimate of the price of European-style options.

d1= d2=

Where, Vcall is the price for an option call, Vput is the price for an option put, CND(d) is the Cumulative Normal Distribution function, S is the current option price, X is the strike price, T is the time to expiration

OutlineIntroductionBlack-Scholes ModelBinomial Options Pricing ModelTrinomial Options Pricing ModelImproved Binomial Option PricingCPU-GPU Hybrid Parallel BinomialSummary

Binomial Options Pricing Model (BOPM)

The Binomial Model was first proposed by Cox, Ross and Rubinstein in 1979.

Essentially, the model uses a “discrete-time” model of the varying price over time of the underlying financial instrument.

Option valuation using this method is, as described, a three-step process:

1. Price tree generation,

2. Calculation of option value at each final node,

3. Sequential calculation of the option value at each preceding node.

Binomial Options Pricing Model (BOPM)

1. Sup = S or Sdown = S u = d = =

2. Max [(), 0], for a call option. Max [(), 0], for a put option. Where K is the strike price and is the spot price of the underlying asset at the period

3. Binomial Value = [p]

Binomial Options Pricing Model (BOPM)

Use of the Model

Handling a variety of conditions & Over a

period of time rather than a single point

Slower than the Black-Scholes formula but

more accurate, especially for long-dated

options

Less practical for options with several

sources of uncertainty and complicated

features

OutlineIntroductionBlack-Scholes ModelBinomial Options Pricing ModelTrinomial Options Pricing ModelImproved Binomial Option PricingCPU-GPU Hybrid Parallel BinomialSummary

Trinomial Options Pricing ModelThe Trinomial Tree was developed by Phelim Boyle in 1986.

It is an extension of the Binomial options pricing model, and is conceptually similar.

Under the Trinomial method, at each node, the price has three possible paths: an up, down and stable or middle path.

Trinomial Options Pricing ModelThe price of the underlying asset can be found by multiplying the value at the current node by the appropriate factor u, d or m where,

,(the structure is recombining), m=1

And the corresponding probabilities are:

Trinomial Options Pricing Model

More accurate than the BOPM when fewer time

steps are modelled.

For vanilla options, the binomial model is

preferred due to its simple implementation.

For exotic options, the trinomial model is more

stable and accurate, regardless of step-size.

Use of the Model

OutlineIntroductionBlack-Scholes ModelBinomial Options Pricing ModelTrinomial Options Pricing ModelImproved Binomial Option PricingCPU-GPU Hybrid Parallel BinomialSummary

Improved Binomial Option Pricing

It is proposed by Mohammad Zubair and Ravi Mukkamala in 2008.

This algorithm exploits the underlying memory hierarchy using cache blocking techniques.

Assume cache of the processor running Vanilla algorithm can hold up to m elements of the array. Considering the nested loop which includes the outer and inner loop, we partition the computation into a certain number of blocks. And therefore, we can fetch m elements of the array into cache.

Improved Binomial Option Pricing

Improved Binomial Option Pricing

OutlineIntroductionBlack-Scholes ModelBinomial Options Pricing ModelTrinomial Options Pricing ModelImproved Binomial Option PricingCPU-GPU Hybrid Parallel BinomialSummary

CPU-GPU Hybrid Parallel Binomial

It is proposed by Nan Zhang et al. in 2012.

The hardware devices includes two CPU cores and a GPU.

CPU 1: communication & synchronization

CPU 2

GPU Both share equal workload with each other.

To see the performance of the hybrid algorithm we did two groups of tests where L, the maximum number of levels in a block, was set to 20 and 50, respectively.

Principle of Hybrid

CPU-GPU Hybrid Parallel Binomial

Speedup plots of the CPU parallel implementation and the hybrid implementation

OutlineIntroductionBlack-Scholes ModelBinomial Options Pricing ModelTrinomial Options Pricing ModelImproved Binomial Option PricingCPU-GPU Hybrid Parallel BinomialSummary

Summary In order to improve the calculation efficiency, GPU computation became a promising tool for option pricing.

We mainly focus on the parallel tree-based algorithms on option pricing.

The Black-Scholes Model is the theory basis of all the other algorithms.

All the other tree-based algorithms including the trinomial lattice are based on the method of binomial lattice.

In the future, we will further improve the parallel algorithm on GPU to achieve better accuracy and efficiency on option pricing.

Thank you for your attention!