Report

A Survey of Parallel Tree-based Methods on Option Pricing PRESENTER: LI,XINYING Outline Introduction Black-Scholes Model Binomial Options Pricing Model Trinomial Options Pricing Model Improved Binomial Option Pricing CPU-GPU Hybrid Parallel Binomial Summary 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 Classification of options According to the Options’ right: Call Option & Put Option Option Styles: European Option American Option Bermudan Option Asian Option Barrier Option Binary Option Exotic Option Vanilla Option Introduction Central Processing Unit (CPU) Graphics Processing Unit (GPU) CPU: CPU: efficient in serial efficient computing in serial computin g GPU: efficient in parallel computing Introduction Option pricing: High demand on calculating speed Heavy computation volume The calculation procedure could be parallelized Input: price of the underlying asset GPU: parallel computing Efficient Algorithm Output: option price Introduction Properties for evaluating the option pricing method Efficiency Storage Accuracy Therefore, a series of tree-based algorithms have been proposed to optimize the previous ones from different aspects. Outline Introduction Black-Scholes Model Binomial Options Pricing Model Trinomial Options Pricing Model Improved Binomial Option Pricing CPU-GPU Hybrid Parallel Binomial Summary 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. = ∙ 1 − ∙ − ∙ (2 ) log XS d1= + + 2 2 log XS d2= +(− 2 ) 2 √ = ∙ − ∙ −2 − ∙ (−1 ) 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 Outline Introduction Black-Scholes Model Binomial Options Pricing Model Trinomial Options Pricing Model Improved Binomial Option Pricing CPU-GPU Hybrid Parallel Binomial Summary 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× u or Sdown = S× d u = 1 d = − = 2. Max [( − ), 0], for a call option. Max [(K − ), 0], for a put option. Where K is the strike price and is the spot price of the underlying asset at the ℎ period 3. Binomial Value = [p× option up + 1 − p × option down]× exp −r × ∆t (−)∆ − 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 BlackScholes formula but more accurate, especially for long-dated options Less practical for options with several sources of uncertainty and complicated features Outline Introduction Black-Scholes Model Binomial Options Pricing Model Trinomial Options Pricing Model Improved Binomial Option Pricing CPU-GPU Hybrid Parallel Binomial Summary Trinomial Options Pricing Model The 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 Model The 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, = 2∆ ,d = − 2∆ (the structure is recombining), m=1 And the corresponding probabilities are: = − ∆/2 = ∆/2 ∆/2 − − ∆/2 − − − 2 ∆/2 − ∆/2 ∆/2 − − ∆/2 = 1 − + 2 Trinomial Options Pricing Model Use of the 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. Outline Introduction Black-Scholes Model Binomial Options Pricing Model Trinomial Options Pricing Model Improved Binomial Option Pricing CPU-GPU Hybrid Parallel Binomial Summary 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 Outline Introduction Black-Scholes Model Binomial Options Pricing Model Trinomial Options Pricing Model Improved Binomial Option Pricing CPU-GPU Hybrid Parallel Binomial Summary 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 Both share equal workload with each other. Principle of Hybrid GPU 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. CPU-GPU Hybrid Parallel Binomial Speedup plots of the CPU parallel implementation and the hybrid implementation Outline Introduction Black-Scholes Model Binomial Options Pricing Model Trinomial Options Pricing Model Improved Binomial Option Pricing CPU-GPU Hybrid Parallel Binomial Summary 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!