Computational Intelligence in the Development of Derivative s Pricing,Arbitrage and Hedging

Transcription

1 Computational Intelligence in the Development of Derivative s Pricing,Arbitrage and Hedging Wo-Chiang Lee Department of Finance and Banking,Aletheia University AI-ECON Research Group August 20,2004

2 Outline The Basic concept of Option Pricing Model The Traditional Option Pricing Model NNs in the Derivative Pricing GP in the Derivative Pricing Fuzzy in the Derivative Pricing Concluding Remark

3 Computational Intelligence for Financial Engineering and Financial Applications

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5 The Application field of Financial Derivatives

6 S :Stock price P: E :exercise price σ: volatility r f : risk free rate T : time to maturity Option price D:dividend

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8 real data Model-driven driven approaches fit data Data-driven approaches

9 Model-Driven Approach Data-Driven Approach With a certainty model. Adaptive learning Based on some important assumptions. Serve market price as true price. Based on natural selection,needn t rely on some important assumptions. By way of evolution, widely search space,can find optimal solution.

10 Black-Scholes Option Pricing Formula P d d 1 2 = = = SN(d ln(s/e) d 1 1 σ ) σ T Ee rt + (r+ σ T N(d 2)T Where S underlying asset price P : option price E exercise price T time to maturity volatility r:risk-free rate 2 / 2 ) The Black-Scholes model is the standard approach used for pricing financial options.

11 Fischer Black Born: 1938 Died: earned bachelor's degree in physics earned PhD. from Harvard in applied math joined faculty of University of Chicago Graduate School of Business Published "The Pricing of Options and Corporate Liabilities" 19XX -- Left the University of Chicago to teach at MIT left MIT to work for Goldman Sachs & Co.

12 Myron Scholes Born: Published "The Pricing of Options and Corporate Liabilities" Currently works in the derivatives trading group at Salomon Brothers.

13 The Problems of Black-Scholes Option Pricing Model Black-Scholes model was derived under strict assumptions that do not hold in the real world and model prices exhibit systematic biases from observed option prices Assumes normal distribution of prices. Assumes constant volatility. Assumes constant risk-free rate. Although being theoretically strong, option prices valued by the model often differ from the prices observed in the financial markets. How to solve the problems? Computational Intelligence.

14 The Volatility Models for Option Pricing

15 Model 1: Estimating Volatility from Historical Data 1. Take observations S 0, S 1,..., S n at intervals of τ years 2. Define the continuously compounded return as: u i = 3. Calculate the standard deviation, s, of the u i s 4.The historical volatility estimate is: σˆ = s ln τ S S i i 1

16 Model 2:Implied Volatility The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price The is a one-to-one correspondence between prices and implied volatilities Traders and brokers often quote implied volatilities rather than dollar prices 1.Calculate the initial value of volatility σ σ I = ( i+1) = σ ( i) S 2 ln + γτ X τ 2.Recurative calculate the volatility. f ( σ ( i) ) f ( σ ) P

17 Model 3:GARCH (Generalized ARCH) A high order GARCH(p,q) model p t p t q t q t p j j t j q i i t i t h t = = = + + = = β σ βσ α ε αε ϖ β σ αε ϖ σ Ex: A AR(3)-GARCH(1,1) model = + + = t t t t t t t t r r r r ε σ σ ε

18 Model 4: GJR-GARCH σ 2 = α + p i= 1 β σ i q r 2 2 t i + γ jε t j + j = 1 k = 1 φ D k t k ε 2 t k D t k = 0, if 1, if ε ε t k t k 0 < 0 good bad news news 1.For a leverage effect, we would see k > 0. If k,the news impact is asymmetric. 2.Good news has a impact of bad news has a impact of + k EX:GJR-GARCH(1,1) σ = α + βσ t 1 + γ ε t 1 + φ Dt 1ε t 1

19 Donaldson,R.G. and M. Kamstra(1997) provide the artificial neutral networks to describe the nonlinear relationship between variables-ann-garch model. ( ) = = = = Ψ = s h h t h r k k t k t k q j j t j p i i t i t z D r λ ξ ε φ ε β σ α σ < = k t k t k t if if D ε ε ( ) 1 1 1,,,0,0 exp 1, = = + + = Ψ v d m w w d t w h d h t t Z z λ λ λ ( ) ( ) ( ) 2 ε ε ε E E Z d t d t = [ ] 1 1, ~ 2 1,, + uniform w d λ h ANNs model Sigmoid function Model 5: ANN-GARCH Model

20 Model 6:Neural network to predict volatility

21 Model 7:GP-Volatility Model Chen,S-H. and C.-H.Yeh.(1997). Using Genetic Programming to Model Volatility in Financial Time Series: The Case of Nikkei 225 and S&P 500,", in Proceedings of the 4th JAFEE International Conference on Investments and Derivatives (JIC'97), Aoyoma Gakuin University, Tokyo, Japan, July 29-31, pp

22 Artificial Neural Networks in Derivative s Pricing,Arbitrage and Hedging

23 ANNs for Financial Applications

24 Neural Network Topology Layers of neurons interconnected Non-linear activation functions Weights define strength of information flow

25 Hutchinson. J. A., Lo and T. Poggio (1994), A Nonparametric Approach to Pricing and Hedge Derivative Structure via Learning Networks, Journal of Finance, vol. 49, In this paper,they first propose a nonparametric method(data-driven) for estimating the pricing formula of a derivative asset using learning networks.

26 Advantages: They don t rely on restrictive parametric assumption,such as lognormality or sample path contiunity. They are robust to the specification errors that plague parametric models. They are adaptive and respond to structural changes in the data-generating process in ways that parametric models can not. They are flexible enough to encompass a wide range of derivative securities and fundamental asset price. dynamics, yet they are relatively simple to implement.

27 Drawback: The approach would be inappropriate for thinly traded derivatives. It also be inappropriate for newly created derivatives that Have no similar counterparts among existing securities.

28 Learning Networks:Radial Basis Functions(Backpropagation): P ANN RBF = f ( S / E,1, T t) Based on desired output p and the net s actual output p ann-rbf, adjust weights to all layers w to minimize J(w)=½ p - p ann-rbf 2 Backpropagation: Generalization of iterative LMS approach gradient descent on J(w) Notes Start with random weights Normalize inputs to same scale Too many hidden units can cause overfitting

29 Performance Measure Tracking Error Let V(t) is the dollar value of replicating portfolio at time t,we sell one call option and undertake the usual dynamic trading strategy in stock s and bonds to hedge this call during its life.then V ( t ) = V V ( t ) = e e ( t ) + V V ( T E [ V ( T ( t ) + V τ ) ) ] ( t ) FRBF (0) V (0) = S (0) RBF (0), RBF (0) = S FRBF ( t ) V S ( t ) = S ( t ) RBF ( t ), RBF ( t ) = S V C (0) = FBS (0), bsopm V (0) = ( V (0) + V (0)) B B S ( t )( V ( t ) Tracking error present value ξ rt S rτ B B B C C RBF V RBF ( t τ ))

30 ANNs-Call Warrant Pricing Model 1 Traditional ANNs Model P ANN = f ( S, E, T, σ )

31 MODEL MAE (Training) Comparison of the BS Model and ANNs Model σ σ σ σ σ σ SSE (Training) RMSE (Training) MAE (Test) SSE (Test) RMSE (Test) Lee,Wo-Chiang(2001) Use Nonparametric Network to Pricing Reset Warrant Proceeding on 2001 Taiwan Finance Association Annual Meeting. pp.1-14.

32 2.Genetic Adaptive Neural Networks(GANN) Pricing Model P gann 1 T, S, ( S where = f ( vol, σ 1, E, E ) / E, L) vol.is trading volume.. σ1 is historical volatility. E: is exercise price. T is the time to maturity. S is underlying asset price. (S-E)/E is moneyness. L is liquidity. P gann = f ( vol, 1, E, T, σσ 2 S, ( S Where E) / E, vol.is trading volume σ2 is implied volatility E: is exercise price. 2, L, P, R) T is the time to maturity. S is underlying asset price. (S-E)/E is moneyness. L is liquidity. P is premium ratio. R is leverage ratio. ref:lee,wo-chiang(2002), Applied Genetic Adaptive Neural Network Approach in the Evaluation of Upper-and-Out Call Warrant", International Conference of Artificial and Computational Intelligence(ACI 2002), September 25-

33 Genetic Adaptive Neural Networks Algorithms Topology Optimization number of hidden layers, number of hidden nodes, interconnection pattern Weights Optimization Control Parameter Optimization learning rate, momentum rate, tolerance level, Input Factors Optimization

34 Flowchart of GANN System chromesome population Reproduction,crossover,mutation fitness RMSE 1:choice 0:no choice Combination of chromesome No Convergence? ANNs RMSE Yes Final variables

35 Comparison of ANNs,GANN1 and GANN2 X1=vol X2=σ1 X3=E X4=T X5=S X6=(S-E)/E X7=σ2 X8=L X9=P X10=R ALGORITHMS Final input variables ANN(BS variables) GANN1 GANN2 X2,X3,X4,X5 X2,X3,X4,X6 X3,X5,X6,X7,X9 MEAN(in-sample) VARIANCE E-06 MAX MIN MEAN(out-sample) VARIANCE MAX MIN

36 Genetic Programming in Derivative s Pricing,Arbitrage and Hedging

37 An Adaptive Evolutionary Approach to Option Pricing via Genetic Programming N. K. Chidambaran, Chi-Wen Jevons Lee, and Joaquin R. Trigueros 1998 Conference on Computational Intelligence for Financial Engineering Propose a new methodology of Genetic Programming for better approximating the elusive relationship between the option price and its contract terms and properties of the underlying stock price. I.e. to develop an adaptive evolutionary model of option pricing, is also data driven and non-parametric This method requires minimal assumptions and can easily adapt to changing and uncertain economic environments strongly encouraging and suggest that the Genetic Programming approach works well in practice.

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43 Limit the complexity of the problem by setting a maximum depth size of 17 for the trees used to represent formulas. The search space is, however, still very large, A 17 deep tree is a popular number used to limit the size of tree sizes ractically, we chose the maximum depth size possible without running into excessive computer run times. Note that the Black-Scholes formula is represented by a tree of depth size 12. A depth size of 17, therefore, is large enough to accommodate complicated option pricing formulas and works in practice.

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50 it is a non-parametric data driven approach and requires minimal assumptions. We thus avoid the problems associated with making specific assumptions regarding the stock price process.the Geneti Programming method uses options price data and extracts the implied pricing equation directly. Second, the Genetic Programming method requires less data than other numerical techniques such as Neural Networks (Hutchinson Lo, and Poggio (1994)). The flexibility in adding terms to the parameter set used to develop the functional approximation can also be used to examine whether factors beyond those used in this study, for example, trading volume, skewness and kurtosis of returns, and inflation, are relevant to option pricing. The self-learning and self-improving feature also makes the method robust to changes in the economic

51 Chen,Shu-Heng,Lee,Wo-Chiang and Yeh-Chia-Shen(1999), Hedging Derivative Securities with Genetic Programming, International Journal of Intelligent Systems in Accounting Finance & Management, Vol.4,No.8, pp

52 C = SN r τ f ( d1) Ee N ( d 2 ) * - * S/E CDF exp CDF LOG + * / * sqrt -r * + LOG * / * sqrt S/E r S/E r-0.5 2

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55 Comparison of BS,Linear,ANNs and GP Model RMSE( In sample and out sample)

56 Fuzzy Logic in Derivative s Pricing,Arbitrage and Hedging

57 Pricing European options based on the fuzzy pattern Of Black-Scholes formula-hsien-chung Wu Computers&OperationsResearch31(2004) In the real world, some parameters in the Black Scholes formula cannot always be expected in a precise sense. For instance, the risk-free interest rate r may occur imprecisely. Therefore, the fuzzy sets theory proposed by Zadeh(1965) may be a useful tool for modeling this kind of imprecise problem.

58 Fuzzy Black-Scholes random variables risk-free interest rate : constant or stochastic interest rate. fuzzy interest rate stock price S: is a stochastic process: fuzzy stock price. Volatility: an imprecise data: fuzzy volatility under the considerations of fuzzy interest rate, fuzzy volatility and fuzzy stock price, the option price will turn into a fuzzy number.

59 Fuzzy pattern of Black Scholes formula We can solve the put call price Pt by put-call parity

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61 Fuzzy pattern of Black Scholes formula Since the strike price K and time T are real numbers, they are displayed as the crisp numbers

62 Fuzzy pattern of Black Scholes formula

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65 THE END Thank you for your attention!.