Machine Learning for Quantitative Finance

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1 Machine Learning for Quantitative Finance Fast derivative pricing Sofie Reyners Joint work with Jan De Spiegeleer, Dilip Madan and Wim Schoutens

2 Derivative pricing is time-consuming... Vanilla option pricing European-type Fast Fourier transform American-type Tree methods Exotic option pricing Monte Carlo simulations 1 Machine Learning for Quantitative Finance

3 ... but time is money! time-consuming algorithms continuously moving markets prices are outdated when available, overnight calculations cannot be performed in one night,... 2 Machine Learning for Quantitative Finance

4 Let a machine learn the pricing function machine learning product, market and model parameters model price time-consuming method Expensive pricing function is summarized with machine learning. 3 Machine Learning for Quantitative Finance

5 Let a machine learn the pricing function machine learning product, market and model parameters model price time-consuming method When training is completed, prediction is extremely fast! 3 Machine Learning for Quantitative Finance

6

7 Gaussian process regression (GPR) Consider a training set (X, y) = {(x i, y i ) i = 1,..., n}. Find a relation between inputs and outputs: y i = f(x i ) + ε i where f(x) is a Gaussian process and ε i N (0, σ 2 n) are i.i.d. random variables representing the noise in the data. 4 Machine Learning for Quantitative Finance

8 Gaussian process A Gaussian process f(x) is a, possibly infinite, collection of random variables, any finite subset of it having a joint Gaussian distribution. Mean function: m(x) = E [ f(x) ] Kernel function: k(x, x ) = Cov(f(x), f(x )) = f(x) GP (m(x), k(x, x )) 5 Machine Learning for Quantitative Finance

9 Gaussian process If f(x) GP (0, k(x, x )), then f N (0, K(X, X)) where (X, f) = {(x i, f i ) i = 1,..., n} is a sample from f(x) and k(x 1, x 1 )... k(x 1, x n ) K(X, X) =..... k(x n, x 1 )... k(x n, x n ) 6 Machine Learning for Quantitative Finance

10 GPR: a Bayesian method Don t model the relation as one function, but as a distribution over functions. Procedure: 1 Start from a prior GP prior knowledge: smooth function, periodic function,... prior distribution over functions 2 Include observed data points 3 Compute a posterior GP 7 Machine Learning for Quantitative Finance

11 Posterior distribution Only consider functions that agree with the data. Take new inputs X, with corresponding (unknown) function values f Joint distribution of training outputs and function values: [ ] ( [ ]) y K(X, X) + σ 2 N 0, n I K(X, X ) K(X, X) K(X, X ) f 8 Machine Learning for Quantitative Finance

12 Posterior distribution Condition on the observations: ( ) f X, X, y N µ, Σ with µ = K(X, X) [ K(X, X) + σ 2 ni ] 1 y Σ = K(X, X ) K(X, X) [ K(X, X) + σ 2 ni ] 1 K(X, X ) 9 Machine Learning for Quantitative Finance

13 Kernel function Squared exponential kernel function ( x x k(x, x ) = σf 2 2 ) exp 2l 2 with hyperparameters σ f and l: σ 2 f = signal variance l = length-scale parameter Hyperparameters (including σ n ) are estimated from the training data, usually with MLE. 10 Machine Learning for Quantitative Finance

14 Mean function Often set to zero, but can be modelled using basis functions h(x). g(x) = f(x) + h(x) T β GP (h(x) T β, k(x, x )) where f(x) GP (0, k(x, x )) β should be estimated from the training data Common choice: h(x) = (1, x, x 2 ) 11 Machine Learning for Quantitative Finance

15 Application set-up Construct a training set: product, market and model parameters time-consuming method model price sample n random combinations x i compute n corresponding prices y i Fit a Gaussian process regression (GPR) model. Fast prediction of new model prices. 12 Machine Learning for Quantitative Finance

16 Pricing European call options Training set: Product/market VG Heston K [40%, 160%] σ [0.05, 0.45] κ [1.4, 2.6] T [11M, 1Y ] ν [0.55, 0.95] ρ [ 0.85, 0.55] r [1.5%, 2.5%] θ [ 0.35, 0.05] θ [0.45, 0.75] q [0%, 5%] η [0.01, 0.1] v 0 [0.01, 0.1] sample n values of each parameter calculate n FFT-based model prices 13 Machine Learning for Quantitative Finance

17 Pricing European call options Fit the GPR model K, T, r q, model parameters GPR FFT model price Construct a test set: Similarly as training set Slightly smaller parameter intervals 14 Machine Learning for Quantitative Finance

18 Out-of-sample prediction (a) Variance Gamma (b) Heston model trained on points, tested on points. 15 Machine Learning for Quantitative Finance

19 Performance summary VG Heston Size of training set In-sample prediction MAE AAE e e e e e e-04 Out-of-sample prediction MAE AAE e e e e e e-04 Speed-up with MAE = max { EC F F T (i) EC GP R(i), i = 1,..., n} AAE = 1 n EC F F T (i) EC GP R(i) n i=1 16 Machine Learning for Quantitative Finance

20 Pricing American options Put options with strike K and maturity T Use binomial tree model (daily steps) with volatility σ [0.05, 0.55] K, T, r, q, σ GPR binomial tree model price 17 Machine Learning for Quantitative Finance

21 Out-of-sample performance MAE AAE e-04 Speed-up 70 model trained on points, tested on points. 18 Machine Learning for Quantitative Finance

22 Pricing barrier options Down-and-out barrier put options with barrier level H, strike K and maturity T with H [55%, 99%] Use Monte Carlo simulation, according to Heston s model H, K, T, r, q, GPR MC Heston κ, ρ, θ, η, v 0 model price 19 Machine Learning for Quantitative Finance

23 Out-of-sample performance MAE AAE e-04 Speed-up 5850 model trained and tested on points. 20 Machine Learning for Quantitative Finance

24 Conclusion Time-consuming pricing methods Gaussian process regression Matrix inversion Hyperparameter optimization Apply GPR on existing methods Speed-up of several orders of magnitude Some trade-off with accuracy 21 Machine Learning for Quantitative Finance

25 Thank you! More information: De Spiegeleer, J., Madan, D. B., Reyners, S. and Schoutens, W. (2018), Machine Learning for Quantitative Finance: Fast Derivative Pricing, Hedging and Fitting, Quantitative Finance, forthcoming. 22 Machine Learning for Quantitative Finance

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