Model Calibration with Artificial Neural Networks

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1 Introduction This document contains five proposals for MSc internship. The internships will be supervised by members of the Pricing Model Validation team of Rabobank, which main task is to validate value and sensitivities of pricing models for financial that are used to price trades in the bank s trading and banking books. In case of interest please send an with grade list and CV to the following 3 persons: Marko.Iskra@Rabobank.com Sebastiaan.Borst@Rabobank.com Erik.van.Raaij@Rabobank.com Model Calibration with Artificial Neural Networks There are several important aspects that need to be taken into account when considering the applicability of a pricing model. One of the main practical considerations is the speed by which the pricing model can be calibrated. Model calibration is the process by which model parameters are adjusted to best fit known observations. One pricing model might be preferable compared to another pricing model from a theoretical perspective, however if that model takes too long to calibrate it will not be used in practice. Artificial neural networks (ANNs) are a family of machine-learning techniques. In general, ANNs can be seen as an extension of regression. An ANN is simply a network of regression units stacked in a particular configuration. Each regression unit, called a neuron, takes input from the previous layer, combines that input according to a rule, and applies a function on the result. In ANNs independent regression units are stacked together in layers, with layers stacked on top of each other. The use of ANNs have become quite prevalent in many stage-of-the-art solutions for image and speed recognition. ANNs have been used for model calibration as well. In finance, ANNs have been widely used for time series forecasting. However, the use of ANNs to calibrate financial models is very limited. Generally neural networks are used to approximate a function that has a large number of input parameters. Therefore, this method lends itself to solve the calibration problem for pricing models. In fact, ANNs provide a method that will perform the calibration significantly faster regardless of the pricing model, hence removing the calibration speed from a pricing model s practicality. The aim of this research proposal is to implement an ANN which can be used to perform the calibration of a sophisticated pricing model for which the standard way of calibrating is time consuming. The research will consist out of two main phases. The first phase of the research will be largely based on [1]. In the first phase a single-factor Hull White model needs to be calibrated to ATM swaptions using an ANN. In the second phase of the research the focus will be on the calibration of a more sophisticated pricing model using an ANN. Phase 1: 1. Get acquainted with the basic concepts of artificial neural networks in the context of the calibration of pricing models.

2 2. Implementation of a single-factor Hull White model that can be calibrated to ATM swaptions. 3. Implementation of an ANN that can be used to calibrate a single-factor Hull White model to ATM swaptions. 4. Training the neural network. Generally the data that is available which can be used for the calibration of pricing models is not large enough in order to train the neural network. Therefore, this step will also include the generation of a training set via statistical sampling. 5. Compare the calibration results obtained with ANN to the benchmark model. Phase 2: 6. In phase 2 steps (2)-(5) are repeated only now for a more sophisticated pricing model. The choice of the sophisticated model will depend on the remaining available time after phase 1 has been completed. References [1] A. Hernandez, Model calibration with neural networks, Risk, June 2017 [2] A. Hernandez, Model Calibration: Global Optimizer vs. Neural Network, SSRN abstract id= [3] T. Mares, E. Janouchovà and A. Kucerova, Artificial neural networks in calibration of nonlinear mechanical models, Preprint, available at [4] R. J. Frank, N. Davey and S. P. Hunt, Time series prediction and neural networks, Journal of Intelligent and Robotic Systems 31, 2011 [5] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4, 1991

3 Using Cheapest-to-Deliver Collateral for Accurate Derivative Valuation Since the financial crisis of 2007/2008 the number of derivative contracts that have collateral agreements has increased dramatically. The collateral agreements mitigate counterparty credit risk and constrain the contagion effects that a default has on possible defaults of other counterparties. The Credit Support Annex (CSA) specifies all the details of the collateral agreement. The collaterals that are specified in the CSA may include: Cash; Securities and assets; Choices of cash in one or multiple currencies. When a CSA with cash collateral is in place, the party with a negative market-to-market value on the contract posts cash to a separate account. This cash will become property of the counterparty in case of default. However, interest has to be paid on the collateral, which brings along funding costs. These funding costs fundamentally change the way collateralized contracts are priced and can have a large impact on the value of the contract. If a CSA allows for posting collateral in different currencies, then the party posting collateral has a choice of which currency they post. If a CSA allows for free replacement then the already posted collateral can be replaced at any point in time to another currency. As such the party posting collateral has, now and at each future point in time, a choice of which currency to post collateral in. The optimal currency is the currency that has the lowest funding cost. Generally the party would want to post the collateral in the optimal currency, often referred to as the currency that is Cheapest-to-Deliver (CtD). The choice of which currency to post collateral in leads to optionality and the value of this optionality can be of a great order. Therefore, it is important to take this optionality into account. 1. Investigate the influence of collateral on pricing derivatives. 2. Investigate the main assumptions used in the CtD framework that are typically used in practice, amongst others: Deterministic FX basis spread; FX forward rate is assumed to be independent on the collateral currencies. 3. Price linear and exotic products under the assumptions stated in (2). 4. Relax the assumptions under point 2. This will mean that the CtD framework needs to be re-derived. The pricing formulas will become more involved. 5. Implementation of a new CtD framework which can be used to price linear and exotic products. Furthermore, efficiency should be taken into consideration. Compare the obtained results with the current CtD framework. 6. Investigate other CtD features/assumptions, e.g. free-replacement vs no freereplacement, asymmetrical collateralization. 7. If time allows development of an original alternative model. References [1] Vladimir Piterbarg. Funding beyond discounting: collateral agreements and derivatives pricing. Risk, 23(2):97, [2] Masaaki Fujii and Akihiko Takahashi. Choice of collateral currency. Risk, 24(1):120, [3] Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi, et al. A note on construction of multiple swap curves with and without collateral. FSA Research

4 Review, 6( ), [4] Masaaki Fujii, Yasufumi Shimada, and Akihiko Takahashi. A market model of interest rates with dynamic basis spreads in the presence of collateral and multiple currencies. Wilmott, 2011(54):61{73, [5] Vladimir Piterbarg. Cooking with collateral. Risk, 25(8):46, [6] Alexandre Antonov and Vladimir Piterbarg. Collateral choice option valuation. Available at SSRN , [7] Alexandre Antonov and Vladimir Piterbarg. Options for collateral options. Risk, page 66, [8] Vladimir Sankovich and Qinghua Zhu. Collateral option valuation made easy. Risk, 2015.

5 Pricing commodity derivatives with stochastic interest rates Currently there are multiple types of commodity derivatives that are traded over the counter. Examples of those are commodity swaps, European options, Asian options, digital options, spread options and barrier options. Instead of physical commodities, often futures contracts are the exchange traded contracts. As a result of this the futures contract prices are used as underlying processes to price commodity derivatives. When pricing commodity derivatives interest rates are often assumed to be independent from the commodity or even assumed to be deterministic. With this assumption, commodity swaps can be priced straightforward without a model for the underlying futures dynamics. For non-linear commodity products still a model is needed to take into account market implied volatilities. The question is if correlations between interest rates and commodity futures do have a significant impact for the pricing of commodity derivatives. For this research proposal we are interested in having more insight in this. In the literature correlation between commodity and interest rates is for example investigated in [1] for Commodity Put and Call options. Here a stochastic volatility process for commodity futures is used together with a stochastic interest rate process. In the research also the impact on non-european options will be investigated and if there are alternative models for the one presented in [1]. 1. Implement the model that is presented in [1]. 2. Are there any interesting alternative underlying models than the model used in [1]? 3. Can commodity swap still be priced with an analytical formula without assuming underlying dynamics in case the model of [1] is used or in case of an alternative model? 4. What is the impact of stochastic interest rates on Commodity swaps? a. in case of single currency commodity swaps. b. in case the commodity futures are in a different currency than the pay-out currency of the swap. 5. What is the impact of stochastic interest rate in case of Asian Options, digital options, spread options and barrier options? Depending on the available time a subset of products can be done. [1] B. Cheng et al., Pricing of long-dated commodity derivatives: Do stochastic interest rates matter? Journal of Banking and Finance (2017).