This paper develops a continuous-time filtering framework for estimating a hazard rate subject to an unobservable change-point. This framework naturally arises in both financial and insurance applications, where the default intensity of a firm or the mortality rate of an individual may experience a sudden jump at an unobservable time, representing, for instance, a shift in the firm's risk profile or a deterioration in an individual's health status. By employing a progressive enlargement of filtration, we integrate noisy observations of the hazard rate with default-related information. We characterise the filter, i.e. the conditional probability of the change-point given the information flow, as the unique strong solution to a stochastic differential equation driven by the innovation process enriched with the discontinuous component. A sensitivity analysis and a comparison of the filter's behaviour under various information structures are provided. Our framework further allows for the derivation of an explicit formula for the survival probability conditional on partial information. This result applies to the pricing of credit-sensitive financial instruments such as defaultable bonds, credit default swaps, and life insurance contracts. Finally, a numerical analysis illustrates how partial information leads to delayed adjustments in the estimation of the hazard rate and consequently to mispricing of credit-sensitive instruments when compared to a full-information setting.
This paper proposes an innovative Transformer model, Single-directional representative from Transformer (SERT), for US large capital stock pricing. It also innovatively applies the pre-trained Transformer models under the stock pricing and factor investment context. They are compared with standard Transformer models and encoder-only Transformer models in three periods covering the entire COVID-19 pandemic to examine the model adaptivity and suitability during the extreme market fluctuations. Namely, pre-COVID-19 period (mild up-trend), COVID-19 period (sharp up-trend with deep down shock) and 1-year post-COVID-19 (high fluctuation sideways movement). The best proposed SERT model achieves the highest out-of-sample R2, 11.2% and 10.91% respectively, when extreme market fluctuation takes place followed by pre-trained Transformer models (10.38% and 9.15%). Their Trend-following-based strategy wise performance also proves their excellent capability for hedging downside risks during market shocks. The proposed SERT model achieves a Sortino ratio 47% higher than the buy-and-hold benchmark in the equal-weighted portfolio and 28% higher in the value-weighted portfolio when the pandemic period is attended. It proves that Transformer models have a great capability to capture patterns of temporal sparsity data in the asset pricing factor model, especially with considerable volatilities. We also find the softmax signal filter as the common configuration of Transformer models in alternative contexts, which only eliminates differences between models, but does not improve strategy-wise performance, while increasing attention heads improve the model performance insignificantly and applying the 'layer norm first' method do not boost the model performance in our case.
This study investigates the pretrained RNN attention models with the mainstream attention mechanisms such as additive attention, Luong's three attentions, global self-attention (Self-att) and sliding window sparse attention (Sparse-att) for the empirical asset pricing research on top 420 large-cap US stocks. This is the first paper on the large-scale state-of-the-art (SOTA) attention mechanisms applied in the asset pricing context. They overcome the limitations of the traditional machine learning (ML) based asset pricing, such as mis-capturing the temporal dependency and short memory. Moreover, the enforced causal masks in the attention mechanisms address the future data leaking issue ignored by the more advanced attention-based models, such as the classic Transformer. The proposed attention models also consider the temporal sparsity characteristic of asset pricing data and mitigate potential overfitting issues by deploying the simplified model structures. This provides some insights for future empirical economic research. All models are examined in three periods, which cover pre-COVID-19 (mild uptrend), COVID-19 (steep uptrend with a large drawdown) and one year post-COVID-19 (sideways movement with high fluctuations), for testing the stability of these models under extreme market conditions. The study finds that in value-weighted portfolio back testing, Model Self-att and Model Sparse-att exhibit great capabilities in deriving the absolute returns and hedging downside risks, while they achieve an annualized Sortino ratio of 2.0 and 1.80 respectively in the period with COVID-19. And Model Sparse-att performs more stably than Model Self-att from the perspective of absolute portfolio returns with respect to the size of stocks' market capitalization.
In this paper, we consider the discrete-time setting, and the market model described by (S,F,T)$. Herein F is the ``public" flow of information which is available to all agents overtime, S is the discounted price process of d-tradable assets, and T is an arbitrary random time whose occurrence might not be observable via F. Thus, we consider the larger flow G which incorporates F and makes T an observable random time. This framework covers the credit risk theory setting, the life insurance setting and the setting of employee stock option valuation. For the stopped model (S^T,G) and for various vulnerable claims, based on this model, we address the super-hedging pricing valuation problem and its intrinsic Immediate-Profit arbitrage (IP hereafter for short). Our first main contribution lies in singling out the impact of change of prior and/or information on conditional essential supremum, which is a vital tool in super-hedging pricing. The second main contribution consists of describing as explicit as possible how the set of super-hedging prices expands under the stochasticity of T and its risks, and we address the IP arbitrage for (S^T,G) as well. The third main contribution resides in elaborating as explicit as possible pricing formulas for vulnerable claims, and singling out the various informational risks in the prices' dynamics.
We present a functional generative approach to extract risk-neutral densities from market prices of options. Specifically, we model the log-returns on the time-to-maturity continuum as a stochastic curve driven by standard normal. We then use neural nets to represent the term structures of the location, the scale, and the higher-order moments, and impose stringent conditions on the learning process to ensure the neural net-based curve representation is free of static arbitrage. This specification is structurally clear in that it separates the modeling of randomness from the modeling of the term structures of the parameters. It is data adaptive in that we use neural nets to represent the shape of the stochastic curve. It is also generative in that the functional form of the stochastic curve, although parameterized by neural nets, is an explicit and deterministic function of the standard normal. This explicitness allows for the efficient generation of samples to price options across strikes and maturities, without compromising data adaptability. We have validated the effectiveness of this approach by benchmarking it against a comprehensive set of baseline models. Experiments show that the extracted risk-neutral densities accommodate a diverse range of shapes. Its accuracy significantly outperforms the extensive set of baseline models--including three parametric models and nine stochastic process models--in terms of accuracy and stability. The success of this approach is attributed to its capacity to offer flexible term structures for risk-neutral skewness and kurtosis.
This paper considers the pricing of long-term options on assets such as housing, where either government intervention or the economic nature of the asset is assumed to limit large falls in prices. The observed asset price is modelled by a geometric Brownian motion (the 'notional price') reflected at a lower barrier. The resulting observed price has standard dynamics but with localised intervention at the barrier, which allows arbitrage with interim losses; this is funded by the government's unlimited powers of intervention, and its exploitation is subject to credit constraints. Despite the lack of an equivalent martingale measure for the observed price, options on this price can be expressed as compound options on the arbitrage-free notional price, to which standard risk-neutral arguments can be applied. Because option deltas tend to zero when the observed price approaches the barrier, hedging with the observed price gives the same results as hedging with the notional price, and so exactly replicates option payoffs. Hedging schemes are not unique, with the cheapest scheme for any derivative being the one which best exploits the interventions at the barrier. The price of a put is clear: direct replication has a lower initial cost than synthetic replication, and the replication portfolio always has positive value. The price of a call is ambiguous: synthetic replication has a lower initial cost than direct replication, but the replication portfolio may give interim losses, and so the preferred replication strategy (and hence price) of a call may depend on what margin payments need to be made on these losses.
We study the impacts of regime changes and related rule implementations on IPOs initial return for China entrepreneurial boards (ChiNext and STAR). We propose that an initial return contains the issuer fair value and an investors overreaction and examine their magnitudes and determinants. Our findings reveal an evolution of IPO pricing in response to the progression of regulation changes along four dimensions: 1) governing regulation regime, 2) listing day trading restrictions, 3) listing rules for issuers, and 4) participation requirements for investors. We find that the most efficient regulation regime in Chinese IPO pricing has four characteristics: 1) registration system, 2) no hard return caps nor trading curbs that restrict the initial return; 3) more specific listing rules for issuers, and 4) more stringent participation requirements for investors. In all contexts, we show that the registration regime governing the STAR IPOs offers the most efficient pricing.
Les trois derniers siècles du Moyen Âge voient le déploiement, au sein du duché de Bretagne, de multiples « gens de savoir ». Personnages ayant bénéficié d’un enseignement supérieur ou d’une formation sur le terrain, ils déploient, parfois en les cumulant, leurs compétences dans divers domaines : l’Église, l’État ducal, l’administration judiciaire, le notariat, etc. Le présent ouvrage en est l’analyse, réalisée à partir d’un corpus de données prosopographiques constitué de 5 599 individus (Bretons ou étrangers venus en Bretagne), mis au jour grâce aux sources publiées, mais surtout au dépouillement de 5 238 cartons et registres d’archives. Les parcours universitaires et professionnels, les stratégies, les réussites et les échecs, les réseaux, les biens des gens de savoir ont ainsi été mis en exergue. Majoritairement étudiants à Paris, à Angers, et juristes de formation, ils reviennent de plus en plus dans le duché afin d’y faire carrière, en particulier avec le développement de l’État et des administrations ducale et judiciaire, au cours du xve siècle. Très polyvalents, il est fréquent qu’ils exercent deux ou trois fonctions différentes (évêque et chancelier, secrétaire et sénéchal, chanoine et médecin), auxquelles ils accèdent par toutes les techniques et ruses imaginables. Hommes de savoir, hommes de pouvoir, les plus efficaces d’entre eux veillent aussi à se garantir des revenus et un patrimoine mobilier et immobilier parfois très importants.
Transition probability density functions (TPDFs) are fundamental to computational finance, including option pricing and hedging. Advancing recent work in deep learning, we develop novel neural TPDF generators through solving backward Kolmogorov equations in parametric space for cumulative probability functions. The generators are ultra-fast, very accurate and can be trained for any asset model described by stochastic differential equations. These are "single solve", so they do not require retraining when parameters of the stochastic model are changed (e.g. recalibration of volatility). Once trained, the neural TDPF generators can be transferred to less powerful computers where they can be used for e.g. option pricing at speeds as fast as if the TPDF were known in a closed form. We illustrate the computational efficiency of the proposed neural approximations of TPDFs by inserting them into numerical option pricing methods. We demonstrate a wide range of applications including the Black-Scholes-Merton model, the standard Heston model, the SABR model, and jump-diffusion models. These numerical experiments confirm the ultra-fast speed and high accuracy of the developed neural TPDF generators.
The VSTOXX index tracks the expected 30-day volatility of the EURO STOXX 50 equity index. Futures on the VSTOXX index can, therefore, be used to hedge against economic uncertainty. We investigate the effect of trader inventory on the price of VSTOXX futures through a combination of stochastic processes and machine learning methods. We formulate a simple and efficient pricing methodology for VSTOXX futures, which assumes a Heston-type stochastic process for the underlying EURO STOXX 50 market. Under these dynamics, approximate analytical formulas for the implied volatility smile and the VSTOXX index have recently been derived. We use the EURO STOXX 50 option implied volatilities and the VSTOXX index value to estimate the parameters of this Heston model. Following the calibration, we calculate theoretical VSTOXX future prices and compare them to the actual market prices. While theoretical and market prices are usually in line, we also observe time periods, during which the market price does not agree with our Heston model. We collect a variety of market features that could potentially explain the price deviations and calibrate two machine learning models to the price difference: a regularized linear model and a random forest. We find that both models indicate a strong influence of accumulated trader positions on the VSTOXX futures price.
Len Patrick Dominic M. Garces, Gerald H. L. Cheang
We price European and American exchange options where the underlying asset prices are modelled using a Merton (1976) jump-diffusion with a common Heston (1993) stochastic volatility process. Pricing is performed under an equivalent martingale measure obtained by setting the second asset yield process as the numeraire asset, as suggested by Bjerskund and Stensland (1993). Such a choice for the numeraire reduces the exchange option pricing problem, a two-dimensional problem, to pricing a call option written on the ratio of the yield processes of the two assets, a one-dimensional problem. The joint transition density function of the asset yield ratio process and the instantaneous variance process is then determined from the corresponding Kolmogorov backward equation via integral transforms. We then determine integral representations for the European exchange option price and the early exercise premium and state a linked system of integral equations that characterizes the American exchange option price and the associated early exercise boundary. Properties of the early exercise boundary near maturity are also discussed.
This paper explores when the financial market lost the price formation function in prewar Japan in the sense of Fama's (1970) semi-strong form market efficiency using a new dataset. We particularly focus on the relationship between the prewar Japanese financial market and several government policy interventions to explore whether the semi-strong form market efficiency evolves over time. To capture the long-run impact of government policy interventions against the markets, we measure the time-varying joint degree of market efficiency and the time-varying impulse responses based on Ito et al.'s (2014; 2017) generalized least squares-based time-varying vector autoregressive model. The empirical results reveal that (1) the joint degree of market efficiency in the prewar Japanese financial market fluctuated over time because of external events such as policy changes and wars, (2) the semi-strong form EMH is almost supported in the prewar Japanese financial market, (3) Lo's (2004) adaptive market hypothesis is supported in the prewar Japanese financial market even if we consider that the public information affects the financial markets, and (4) the prewar Japanese financial markets lost the price formation function in 1932 and that was a turning point in the market.
Fabio Baschetti, Giacomo Bormetti, Silvia Romagnoli
et al.
The goal of this paper is to investigate the method outlined by one of us (PR) in Cherubini et al. (2009) to compute option prices. We name it the SINC approach. While the COS method by Fang and Osterlee (2009) leverages the Fourier-cosine expansion of truncated densities, the SINC approach builds on the Shannon Sampling Theorem revisited for functions with bounded support. We provide several results which were missing in the early derivation: i) a rigorous proof of the convergence of the SINC formula to the correct option price when the support grows and the number of Fourier frequencies increases; ii) ready to implement formulas for put, Cash-or-Nothing, and Asset-or-Nothing options; iii) a systematic comparison with the COS formula for several log-price models; iv) a numerical challenge against alternative Fast Fourier specifications, such as Carr and Madan (1999) and Lewis (2000); v) an extensive pricing exercise under the rough Heston model of Jaisson and Rosenbaum (2015); vi) formulas to evaluate numerically the moments of a truncated density. The advantages of the SINC approach are numerous. When compared to benchmark methodologies, SINC provides the most accurate and fast pricing computation. The method naturally lends itself to price all options in a smile concurrently by means of Fast Fourier techniques, boosting fast calibration. Pricing requires to resort only to odd moments in the Fourier space. A previous version of this manuscript circulated with the title `Rough Heston: The SINC way'.
The effects of weather on agriculture in recent years have become a major global concern. Hence, the need for an effective weather risk management tool (i.e., weather derivatives) that can hedge crop yields against weather uncertainties. However, most smallholder farmers and agricultural stakeholders are unwilling to pay for the price of weather derivatives (WD) because of the presence of basis risks (product-design and geographical) in the pricing models. To eliminate product-design basis risks, a machine learning ensemble technique was used to determine the relationship between maize yield and weather variables. The results revealed that the most significant weather variable that affected the yield of maize was average temperature. A mean-reverting model with a time-varying speed of mean reversion, seasonal mean, and local volatility that depended on the local average temperature was then proposed. The model was extended to a multi-dimensional model for different but correlated locations. Based on these average temperature models, pricing models for futures, options on futures, and basket futures for cumulative average temperature and growing degree-days are presented. Pricing futures on baskets reduces geographical basis risk, as buyers have the opportunity to select the most appropriate weather stations with their desired weight preference. With these pricing models, farmers and agricultural stakeholders can hedge their crops against the perils of extreme weather.
The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high-order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Lévy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for "near-the-money" option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results we obtain high-order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P500 options data shows it to be consistent with the infinite variation jump component of our models.
Damiano Brigo, Camilla Pisani, Francesco Rapisarda
The Multi Variate Mixture Dynamics model is a tractable, dynamical, arbitrage-free multivariate model characterized by transparency on the dependence structure, since closed form formulae for terminal correlations, average correlations and copula function are available. It also allows for complete decorrelation between assets and instantaneous variances. Each single asset is modelled according to a lognormal mixture dynamics model, and this univariate version is widely used in the industry due to its flexibility and accuracy. The same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allows for consistency of single asset and index/portfolio smile. In this paper, we generalize the MVMD model by introducing shifted dynamics and we propose a definition of implied correlation under this model. We investigate whether the model is able to consistently reproduce the implied volatility of FX cross rates once the single components are calibrated to univariate shifted lognormal mixture dynamics models. We consider in particular the case of the Chinese renminbi FX rate, showing that the shifted MVMD model correctly recovers the CNY/EUR smile given the EUR/USD smile and the USD/CNY smile, thus highlighting that the model can also work as an arbitrage free volatility smile extrapolation tool for cross currencies that may not be liquid or fully observable. We compare the performance of the shifted MVMD model in terms of implied correlation with those of the shifted Simply Correlated Mixture Dynamics model where the dynamics of the single assets are connected naively by introducing correlation among their Brownian motions. Finally, we introduce a model with uncertain volatilities and correlation. The Markovian projection of this model is a generalization of the shifted MVMD model.