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Authors: Haitz Borde, Damiano Brigo, Xiaoshan Huang, Andrea Pallavicini, Haitz Borde
Authors: Haitz Borde, Damiano Brigo, Xiaoshan Huang, Andrea Pallavicini, Haitz Borde
Deep learning is a powerful tool whose applications in quantitative finance are growing every day. Yet, artificial neural networks behave as black boxes and this hinders validation and accountability processes. Being able to interpret the inner functioning and the input-output relationship of these networks has become key for the acceptance of such tools. In this paper we focus on the calibration process of a stochastic volatility model, a subject recently tackled by deep learning algorithms. We analyze the Heston model in particular, as this model's properties are well known, resulting in an ideal benchmark case. We investigate the capability of local strategies and global strategies coming from cooperative game theory to explain the trained neural networks, and we find that global strategies such as Shapley values can be effectively used in practice. Our analysis also highlights that Shapley values may help choose the network architecture, as we find that fully-connected neural networks perform better than convolutional neural networks in predicting and interpreting the Heston model prices to parameters relationship.
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Authors: Takuji Arai, Takuji Arai
Authors: Takuji Arai, Takuji Arai
For the Barndorff-Nielsen and Shephard model, we present approximate expressions of call option prices based on the decomposition formula developed by Arai (2021). Besides, some numerical experiments are also implemented to make sure how effective our approximations are.
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Authors: Ruodu Wang, Yuyu Chen, Liyuan Lin, Ruodu Wang
Authors: Ruodu Wang, Yuyu Chen, Liyuan Lin, Ruodu Wang
We study the aggregation of two risks when the marginal distributions are known and the dependence structure is unknown, under the additional constraint that one risk is no larger than the other. Risk aggregation problems with the order constraint are closely related to the recently introduced notion of the directional lower (DL) coupling. The largest aggregate risk in concave order (thus, the smallest aggregate risk in convex order) is attained by the DL coupling. These results are further generalized to calculate the best-case and worst-case values of tail risk measures. In particular, we obtain analytical formulas for bounds on Value-at-Risk. Our numerical results suggest that the new bounds on risk measures with the extra order constraint can greatly improve those with full dependence uncertainty.
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Black-Scholes equation as one of the most celebrated mathematical models has an explicit analytical solution known as the Black-Scholes formula. Later variations of the equation, such as fractional or nonlinear Black-Scholes equations, do not have a closed form expression for the corresponding formula. In that case, one will need asymptotic expansions, including homotopy perturbation method, to give an approximate analytical solution. However, the solution is non-smooth at a special point. We modify the method by {first} performing variable transformations that push the point to infinity. As a test bed, we apply the method to the solvable Black-Scholes equation, where excellent agreement with the exact solution is obtained. We also extend our study to multi-asset basket and quanto options by reducing the cases to single-asset ones. Additionally we provide a novel analytical solution of the single-asset quanto option that is simple and different from the existing expression.
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Authors: Jan Rosenzweig, Jan Rosenzweig
Authors: Jan Rosenzweig, Jan Rosenzweig
Portfolio optimization methods suffer from a catalogue of known problems, mainly due to the facts that pair correlations of asset returns are unstable, and that extremal risk measures such as maximum drawdown are difficult to predict due to the non-Gaussianity of portfolio returns. \\ In order to look at optimal portfolios for arbitrary risk penalty functions, we construct portfolio shapes where the penalty is proportional to a moment of the returns of arbitrary order $p>2$. \\ The resulting component weight in the portfolio scales sub-linearly with its return, with the power-law $w \propto \mu^{1/(p-1)}$. This leads to significantly improved diversification when compared to Kelly portfolios, due to the dilution of the winner-takes-all effect.\\ In the limit of penalty order $p\rightarrow\infty$, we recover the simple trading heuristic whereby assets are allocated a fixed positive weight when their return exceeds the hurdle rate, and zero otherwise. Infinite order power-law portfolios thus fall into the class of perfectly diversified portfolios.
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The so-called Benford's laws are of frequent use in order to observe anomalies and regularities in data sets, in particular, in election results and financial statements. Yet, basic financial market indices have not been much studied, if studied at all, within such a perspective. This paper presents features in the distributions of S\&P500 daily closing values and the corresponding daily log returns over a long time interval, [03/01/1950 - 22/08/2014], amounting to 16265 data points. We address the frequencies of the first, second, and first two significant digits counts and explore the conformance to Benford's laws of these distributions at five different (equal size) levels of disaggregation. The log returns are studied for either positive or negative cases. The results for the S&P500 daily closing values are showing a huge lack of non-conformity, whatever the different levels of disaggregation. Some "first digits" and "first two digits" values are even missing. The causes of this non-conformity are discussed, pointing to the danger in taking Benford's laws for granted in huge databases, whence drawing "definite conclusions". The agreements with Benford's laws are much better for the log returns. Such a disparity in agreements finds an explanation in the data set itself: the inherent trend in the index. To further validate this, daily returns have been simulated calibrating the simulations with the observed data averages and tested against Benford's laws. One finds that not only the trend but also the standard deviation of the distributions are relevant parameters in concluding about conformity with Benford's laws.
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Authors: Chao Wang, Giuseppe Storti, Chao Wang
Authors: Chao Wang, Giuseppe Storti, Chao Wang
A novel forecasting combination and weighted quantile based tail risk forecasting framework is proposed, aiming to reduce the impact of modelling uncertainty in financial tail risk forecasting. The proposed approach is based on a two-step estimation procedure. The first step involves the combination of Value-at-Risk (VaR) forecasts at a grid of different quantile levels. A range of parametric and semi-parametric models is selected as the model universe which is incorporated in the forecasting combination procedure. The quantile forecasting combination weights are estimated by optimizing the quantile loss. In the second step, the Expected Shortfall (ES) is computed as a weighted average of combined quantiles. The quantiles weighting structure used to generate the ES forecast is determined by minimizing a strictly consistent joint VaR and ES loss function of the Fissler-Ziegel class. The proposed framework is applied to six stock market indices and its forecasting performance is compared to each individual model in the model universe and a simple average approach. The forecasting results based on a number of evaluations support the proposed framework.
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Authors: Florian Ziel, Michał Narajewski, Florian Ziel
Authors: Florian Ziel, Michał Narajewski, Florian Ziel
Electricity exchanges offer several trading possibilities for market participants: starting with futures products through the spot market consisting of the auction and continuous part, and ending with the balancing market. This variety of choice creates a new question for traders - when to trade to maximize the gain. This problem is not trivial especially for trading larger volumes as the market participants should also consider their own price impact. The following paper raises this issue considering two markets: the hourly EPEX Day-Ahead Auction and the quarter-hourly EPEX Intraday Auction. We consider a realistic setting which includes a forecasting study and a suitable evaluation. For a meaningful optimization many price scenarios are considered that we obtain using bootstrap with models that are well-known and researched in the electricity price forecasting literature. The own market impact is predicted by mimicking the demand or supply shift in the respectful auction curves. A number of trading strategies is considered, e.g. minimization of the trading costs, risk neutral or risk averse agents. Additionally, we provide theoretical results for risk neutral agents. Especially we show when the optimal trading path coincides with the solution that minimizes transaction costs. The application study is conducted using the German market data, but the presented methods can be easily utilized with other two auction-based markets. They could be also generalized to other market types, what is discussed in the paper as well. The empirical results show that market participants could increase their gains significantly compared to simple benchmark strategies.
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In this paper we study pricing of American put options on the Black and Scholes market with a stochastic interest rate and finite-time maturity. We prove that the option value is a $C^1$ function of the initial time, interest rate and stock price. By means of Ito calculus we rigorously derive the option value's early exercise premium formula and the associated hedging portfolio. We prove the existence of an optimal exercise boundary splitting the state space into continuation and stopping region. The boundary has a parametrisation as a jointly continuous function of time and stock price, and it is the unique solution to an integral equation which we compute numerically. Our results hold for a large class of interest rate models including CIR and Vasicek models. We show a numerical study of the option price and the optimal exercise boundary for Vasicek model.