In this paper we propose to investigate the nature of the excess rate of return of an equity asset taking into account the partial information available on the market. The purpose of the proposed research is to model the information that moves the price on the market with a stochastic process, and by the latter obtain the market sentiment towards the stock concerned. The estimate of the coefficient of risk aversion has become increasingly important for the asset allocation and in the literature it has been presented different methods for estimating the one based on economic considerations. Our approach instead considers the price of the stock and its movements as the sole source of information to derive how much the market is willing to pay for a unit of the stock. Moreover, our model provides an estimation that evolves over time and that in each time is updated with respect to the new information in the market. Our work is based on the model presented by Brody D.C. and Hughston L.P. in their article "Lévy information and the aggregation of risk aversion" [12]. After a first theoretical explanation of the model used, it will be extended to the multifactor case and subsequently to the case where the risk aversion is considered as an Ornstein-Uhlenbeck process. In the multifactor case, the representation will become multi-dimensional and it will be given some explanatory simulations of the theoretical characteristics demonstrated. Moreover it is treated ii the subject of the rate of convergence of our estimator and the intuitive value that our model provides on the market sentiment towards the risk of a given asset. After the first observations, we noticed that it is more responsive to the market to consider a risk aversion that is not deterministic but stochastic, in this specific case an Ornstein-Uhlenbeck process. In the case of stochastic risk aversion the estimation becomes more complex and we have used methods borrowed from the filtering theory, introduced by Wiener [31] in 1948. In particular, the estimation is based on Kalman-Bucy filter, which allows to obtain the best approximation, in the sense of mean square error, of a process given another process that contains the information carried by the price soiled by a white noise. To obtain a daily estimation of the risk aversion, we propose a calibration method applicable every day. It is also proposed a study of the effectiveness of the calibration method based on indexes of sensitivity and stability. To verify the validity of our theoretical results, we have applied the filter to the values of real prices obtained from Bloomberg platform. In order to apply our model we apply our calibration method previously proposed, based on the filtration generated by the price and the minimization of an objective function presented. The application to real data allows to verify the implementation of the model and its accuracy in providing the requested estimation of risk aversion. Finally, we propose a strategy of asset allocation on a single stock which allows us to check if the process obtained brings the real information contained in the price. The innovation we are going to display through this work is the definition and the utilization on real data of an original model that provides a daily estimation of the risk aversion of the market towards the single stock. Consequently it gives an estimation of the excess rate of return of an asset that takes into account all the previous information. In addition, it is proposed a model that takes into account the partial information regarding the stock that move the price without really bring useful information to the valuation of the asset. It is presented new approach to the phenomenological study of the price that can answer to the new challenges of the market. Furthermore some evidences are presented showing that the proposed model corresponds to the reality and describes observed phenomena about the changing opinions in the market and the flow of information about the single stock.
In questo elaborato ci proponiamo di indagare la natura dell’excess rate of return di un titolo equity tenendo conto delle informazioni parziali disponibili sul mercato. L’obbiettivo del modello proposto è quello di modellizzare l’informazione che muove il prezzo sul mercato con un processo stocastico, e da quest’ultimo ricavarne il sentimento del mercato verso lo stock considerato. La stima del coefficiente di risk aversion è divenuta sempre più fondamentale per l’asset allocation, e nella letteratura sono stati presentati diversi metodi per la sua stima, basati su considerazioni economiche. Il nostro approccio invece considera il prezzo dello stock e i suoi movimenti come unica fonte di informazione per ricavare quanto il mercato è disposto a pagare per un’unità di rischio dell’asset. Inoltre il nostro modello fornisce una stima che evolve nel tempo e che a ogni tempo si aggiorna rispetto alle nuove informazioni del mercato. Il nostro elaborato si basa sul modello presentato da Brody D.C. e Hughston L.P. nel loro articolo "Lévy information and the aggregation of risk aversion" [12]. Dopo una prima spiegazione teorica del modello utilizzato, esso verrà esteso al caso di fattori di rischio multipli e successivamente al caso in cui la risk aversion è considerata come un processo di Ornstein-Uhlenbeck. Nel caso multifactor, la rappresentazione diventerà multi dimensionale e verranno fornite alcune simulazioni esplicative delle caratteristiche teoriche dimostrate. Viene trattato l’argomento della velocità di convergenza del nostro estimatore e del valore intuitivo che il nostro modello fornisce sul sentimento del mercato rispetto al rischio di un dato asset. Dopo le prime osservazioni, abbiamo notato che è più corrispondente al mercato considerare un termine di risk aversion che non è deterministico ma stocastico, nel caso specifico un processo Ornstein-Uhlenbeck. Nel caso di risk aversion stocastica la stima diventa più complessa e si sono utilizzati metodi presi a prestito dalla teoria del filtraggio, introdotta da Wiener [31] nel 1948. In particolare, la stima è basata sul filtro di Kalman-Bucy, che permette di ottenere la miglior approssimazione, nel senso di errore quadratico medio, di un processo dato un altro processo che contiene l’informazione portata dal prezzo sporcata da un rumore bianco. Per ottenere la stima della risk aversion giornaliera, abbiamo proposto un metodo di calibrazione applicabile ogni giorno sui dati passati. Inoltre viene proposto uno studio della efficacia del metodo di calibrazione basato su indici di sensitivity e stability. Per verificare la veridicità dei nostri risultati teorici, abbiamo applicato il filtro a valori di prezzi reali, ottenuti dalla piattaforma Bloomberg. Per farlo applichiamo la calibrazione dei parametri precedentemente proposta, basata solamente sulla filtrazione generata dai prezzi e sulla minimizzazione di una funzione obbiettivo presentata. L’applicazione a dati reali permette di verificare l’attuazione del modello e la sua precisione nel fornire la stima richiesta di risk aversion a partire dai dati. Infine, proponiamo una strategia di asset allocation su un singolo stock che ci permette di verificare se il processo ottenuto di risk aversion porta la vera informazione contenuta nel prezzo o no. Il fattore di novità che intendiamo esporre con questo lavoro è la presentazione e l’utilizzo su dati reali di un modello originale che permette di stimare il termine di risk aversion del mercato rispetto al singolo stock ad ogni tempo e di dare una stima dell’excess rate of return di un asset che tenga conto di tutte le informazioni precedenti. Inoltre, si è proposto un modello che tiene conto delle informazioni parziali riguardanti lo stock che ne muovono il prezzo senza veramente portare un’informazione utile alla valutazione dell’asset. Si presenta un nuovo approccio allo studio fenomenologico del prezzo che può rispondere alle nuove sfide del mercato. Inoltre, sono presentate alcune evidenze che dimostrano che il modello proposto corrisponde alla realtà e descrive fenomeni osservati riguardanti le mutevoli opinioni del mercato e il flusso di informazioni relative al singolo titolo.
Information process : properties and applications in a stochastic risk aversion case
BONANNI, DANIELE;STOPPA, STEFANO
2013/2014
Abstract
In this paper we propose to investigate the nature of the excess rate of return of an equity asset taking into account the partial information available on the market. The purpose of the proposed research is to model the information that moves the price on the market with a stochastic process, and by the latter obtain the market sentiment towards the stock concerned. The estimate of the coefficient of risk aversion has become increasingly important for the asset allocation and in the literature it has been presented different methods for estimating the one based on economic considerations. Our approach instead considers the price of the stock and its movements as the sole source of information to derive how much the market is willing to pay for a unit of the stock. Moreover, our model provides an estimation that evolves over time and that in each time is updated with respect to the new information in the market. Our work is based on the model presented by Brody D.C. and Hughston L.P. in their article "Lévy information and the aggregation of risk aversion" [12]. After a first theoretical explanation of the model used, it will be extended to the multifactor case and subsequently to the case where the risk aversion is considered as an Ornstein-Uhlenbeck process. In the multifactor case, the representation will become multi-dimensional and it will be given some explanatory simulations of the theoretical characteristics demonstrated. Moreover it is treated ii the subject of the rate of convergence of our estimator and the intuitive value that our model provides on the market sentiment towards the risk of a given asset. After the first observations, we noticed that it is more responsive to the market to consider a risk aversion that is not deterministic but stochastic, in this specific case an Ornstein-Uhlenbeck process. In the case of stochastic risk aversion the estimation becomes more complex and we have used methods borrowed from the filtering theory, introduced by Wiener [31] in 1948. In particular, the estimation is based on Kalman-Bucy filter, which allows to obtain the best approximation, in the sense of mean square error, of a process given another process that contains the information carried by the price soiled by a white noise. To obtain a daily estimation of the risk aversion, we propose a calibration method applicable every day. It is also proposed a study of the effectiveness of the calibration method based on indexes of sensitivity and stability. To verify the validity of our theoretical results, we have applied the filter to the values of real prices obtained from Bloomberg platform. In order to apply our model we apply our calibration method previously proposed, based on the filtration generated by the price and the minimization of an objective function presented. The application to real data allows to verify the implementation of the model and its accuracy in providing the requested estimation of risk aversion. Finally, we propose a strategy of asset allocation on a single stock which allows us to check if the process obtained brings the real information contained in the price. The innovation we are going to display through this work is the definition and the utilization on real data of an original model that provides a daily estimation of the risk aversion of the market towards the single stock. Consequently it gives an estimation of the excess rate of return of an asset that takes into account all the previous information. In addition, it is proposed a model that takes into account the partial information regarding the stock that move the price without really bring useful information to the valuation of the asset. It is presented new approach to the phenomenological study of the price that can answer to the new challenges of the market. Furthermore some evidences are presented showing that the proposed model corresponds to the reality and describes observed phenomena about the changing opinions in the market and the flow of information about the single stock.| File | Dimensione | Formato | |
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https://hdl.handle.net/10589/102724