A Survey of Parallel T ree-based Methods on Option Pricing

1. A Survey of Parallel Tree-based Methods on Option Pricing Presenter: LI,Xinying

2. Outline • Introduction • Black-Scholes Model • Binomial Options Pricing Model • Trinomial Options Pricing Model • Improved Binomial Option Pricing • CPU-GPU Hybrid Parallel Binomial • Summary

3. Introduction Stock Currency Bond Underlying Asset!

4. Introduction Option’s price is based on the corresponding underlying asset’s price. A suitable price of option +

5. 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

6. Introduction Central Processing Unit (CPU) Graphics Processing Unit (GPU) CPU: efficient in serial computing CPU: efficient in serial computing

7. Introduction Option pricing: • High demand on calculating speed • Heavy computation volume • The calculation procedure could be parallelized Efficient Algorithm

8. Introduction 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.

9. Outline • Introduction • Black-Scholes Model • Binomial Options Pricing Model • Trinomial Options Pricing Model • Improved Binomial Option Pricing • CPU-GPU Hybrid Parallel Binomial • Summary

10. 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, Vcallis the price for an option call,Vputis the price for an option put, CND(d)is the Cumulative Normal Distribution function,Sis the current option price,Xis the strike price,Tis the time to expiration

11. Outline • Introduction • Black-Scholes Model • BinomialOptionsPricingModel • Trinomial Options Pricing Model • Improved Binomial Option Pricing • CPU-GPU Hybrid Parallel Binomial • Summary

12. 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: • Price tree generation, • Calculation of option value at each final node, • Sequential calculation of the option value at each preceding node.

13. Binomial Options Pricing Model (BOPM) 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]

14. Binomial Options Pricing Model (BOPM) Use of the Model

15. Outline • Introduction • Black-Scholes Model • Binomial Options Pricing Model • Trinomial Options Pricing Model • Improved Binomial Option Pricing • CPU-GPU Hybrid Parallel Binomial • Summary

16. 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.

17. 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, ,(the structure is recombining), m=1 And the corresponding probabilities are:

18. Trinomial Options Pricing Model Use of the Model

19. Outline • Introduction • Black-Scholes Model • Binomial Options Pricing Model • Trinomial Options Pricing Model • Improved Binomial Option Pricing • CPU-GPU Hybrid Parallel Binomial • Summary

20. 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.

21. Improved Binomial Option Pricing

22. Improved Binomial Option Pricing

23. Outline • Introduction • Black-Scholes Model • Binomial Options Pricing Model • Trinomial Options Pricing Model • Improved Binomial Option Pricing • CPU-GPU Hybrid Parallel Binomial • Summary

24. 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 Principle of Hybrid CPU 2 Both share equal workload with each other. 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.

25. CPU-GPU Hybrid Parallel Binomial Speedup plots of the CPU parallel implementation and the hybrid implementation

26. Outline • Introduction • Black-Scholes Model • Binomial Options Pricing Model • Trinomial Options Pricing Model • Improved Binomial Option Pricing • CPU-GPU Hybrid Parallel Binomial • Summary

27. 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.

28. Thank you for your attention!