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Financial Data Modelling

Financial Data Modelling. Dr Nikolay Nikolaev Department of Computing G oldsmiths College University of London 2018. Lecture 8 (FDM 2018). Option Pricing via Filtering.

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Financial Data Modelling

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  1. Financial Data Modelling Dr Nikolay Nikolaev Department of Computing Goldsmiths College University of London 2018

  2. Lecture 8 (FDM 2018) Option Pricing via Filtering Option pricing is an investment problem whose accurate solutions help to work efficiently with various financial derivatives and hedge against risks. It is a difficult problem because of the typical non-stationarity and stochastic behaviour of the real stock prices. Having an option allows the holder to buy or sell an asset at a specified date (time to maturity) in the future for a certain (strike) price determined in advance. A recognized standard for fair pricing of options is provided by the Black-Scholes formulae based on the assumption that prices follow a geometric Brownian motion. These formulae help us to calculate the value of an option dependent on the current stock price , the strike price , the risk-free interest rate , the time to maturity , and the stock return variance (volatility) . The fair prices of European call and put options are given by the following Black-Scholes equations: / where denotes the cumulative distribution function of the Gaussian distribution.

  3. Lecture 8 (FDM 2018) Strictly speaking, the Black-Scholes formulae are valid upon several conditions: no arbitrage opportunities, continuous trading, no dividends, constant volatility and risk-free interest rate. These conditions, however, lead to uncertainties in the standard Black-Scholes model and researchers have made various efforts to mitigate their effects. Here we present two approavhes to calibration of such option pricing models: • the first approach infers more realistic time-dependent volatility and interest rate using an Extended Kalman Filter (required because of the nonlinear character of the above equations) to estimate these parameters and so to facilitate prediction; • the second approach uses again the stock price and time to maturity as inputs but feeds them directly to a neural network model (without encoding the concrete Black-Scholes equations) to generate forecasts of the call and put prices.

  4. Lecture 8 (FDM 2018) Sequential Estimation of Option Prices The key idea behind this approach to estimation of option prices using the Black-Scholes equatins is to treat the volatility and the interest rate as unobservable (latent) variables is a state-space model. The volatility, the interest rate and the prices are considered Gaussian (with unknown noises) and this allows us to apply the Kalman filter. The Extended Kalman Filter version necessary for the non-linear Black-Scholes equations is passed two inputs: the stock price divided by the strike price, and the time to maturity. The outputs are the corresponding call and put prices normalized also by the strike price, that is we have: The state-space formulation of the Black-Scholes model is as follows: process equation , (0,) observation equation,(0,) where. / and and are two-dimensional matrices.

  5. Lecture 8 (FDM 2018) The application of the filtering recursions requires to perform linearization of the model with the following output derivatives with respect to the parameters: /. / The EKF equations for sequential estimation of the state-space Black-Scholes model become: prediction step: is [] vector + is [] matrix updating step: is [] matrix + is [] vector is [] matrix where the noise matrices should be initialized in advance.

  6. Lecture 8 (FDM 2018) The advantage of this approach to option pricing via filtering is that it infers the volatility which is an unobservable component in the Black-Scholes formulae, which makes possible accurate forecasting of call and put prices. Another alternative to learning the unknown price variance is to use a GARHCH model to replace the state-transition equation in the above state-space Black-Scholes formulation. Using GARCH will improve the quality of approximation of the volatility dynamics. Furthermore, the calculation of analytical derivatives may be avoided with development of Unscented Kalman Filtering. Option Pricing with Neural Networks An alternative approach to option pricing is to map the stock price and the time to maturity variables directly to the call and put prices without using the Black-Scholes equations but with a black-box neural network model. We can develop a two-output neural network architecture having inputs that generates the nonlinear mappings: This option pricing multilayer network can be trained efficiently using he EKF algorithm after linearization with the output derivatives of the error with respect to the weights.

  7. Lecture 8 (FDM 2018) Connectionist Option Pricing Model

  8. Lecture 8 (FDM 2018) Exercise: Option price filtering in Matlab First we need to load: S- Stock prices, C- Call option prices, P- Put Option prices, X- Strike price, and calculate the time to maturity: cc = 1:1:T; tm = (224*ones(size(cc))-cc)/260; Next, we build the inputs and targets (the interest rate and volatility are treated as latent variables and are inferred by the filter), as well as initialize the matrices: Xt= X(1,1); St = S(1,1:T); C = C(1,1:T); P = P(1,1:T); inputs = [St' tm']'; targets = [C' P']'; % Initialization of state and covariance matrices: mf = ones(2,T); mf(1,1) = 0.01; mf(2,1) = 0.15; mfp = mf; Pf = ones(2,2,T); Pf(:,:,1)= diag([0.1 0.1]); Pfp= eye(2); Ax = eye(2); yhat= ones(2,T);

  9. Lecture 8 (FDM 2018) Exercise: Option price filtering in Matlab (continuation…) Then we perform Extended Kalman filtering of the nonlinear model as follows: for t = 2:T % Predicting the latent state: mfp(:,t) = mf(:,t-1); Pfp = Ax*Pf(:,:,t-1)*Ax'+Q; % Forecasting option prices: r = mfp(1,t); s2 = mfp(2,t); % Latent state values St = inputs(1,t); tm = inputs(2,t); % Inputs: stock price and time to maturity d1 = (log(St/Xt)+(r+0.5*(s2*s2))*tm)/(s2*sqrt(tm)); d2 = d1-s2*sqrt(tm); yhat(1,t) = St*normcdf(d1)-Xt*exp(-r*tm)*normcdf(d2); % Predicting the Call price yhat(2,t) = -St*normcdf(-d1)+Xt*exp(-r*tm)*normcdf(-d2); % Predicting the Put price % Computing derivatives of the Black-Scholes model: dCds = St*sqrt(tm)*exp(-d1^2)/sqrt(2*pi); dPds = dCds; dCdr = Xt*tm*exp(-r*tm)*normcdf(d2); dPdr = -Xt*tm*exp(-r*tm)*normcdf(-d2); Jt = [dCdrdPdr; dCdsdPds]'; % Jacobian matrix % Kalman filtering: Kt = Pfp*Jt'*inv(Jt*Pfp*Jt'+R); % Kalman gain mf(:,t) = mfp(:,t)+Kt*(targets(:,t)-yhat(:,t)); Pf(:,:,t) = Pfp-Kt*Jt*Pfp; end

  10. Lecture 8 (FDM 2018) References: M.Niranjan (1996). Sequential tracking in pricing financial options using model based and neural network approaches, In M.C.Mozer, M.I.Jordan and T.Petsche (Eds.), Advances in neural information processing systems, MIT Press, Cambridge, MA, pp.960–966. J.F.G.de Freitas,M.Niranjan and A.H.Gee(2000). Hierarchical Bayesian Models for Regularization in Sequential Learning, Neural Computation, vol.12 (4), pp.933–953.

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