This thesis considers the problem of optimizing a continuous time portfolio of an investor with a constant risk aversion coe cient, which maximizes the expected utility of his nal wealth. We study optimal investment strategies given investor access not only to bond and stock markets but also to the derivatives market. The problem is solved in a closed form through the stochastic control approach. In the rst analysis, the equivalence between Merton's classic problem and a buy-and-hold strategy that includes a possible investment in the options market compared to the other problem is presented. Then we introduce the solution of the problem in dynamic way when the derivatives are written on Stock that introduces stochastic volatility and it takes into consideration the fact that the price of the Stock can undergo great variations in short temporal instants (jump risks in the stock market). Finally a solution is proposed in the case in which the incomplete markets are completed by derivatives.
In questa tesi si considera il problema di ottimizzazione di un portafoglio a tempo continuo di un investitore con coe ciente di avversione al rischio costante, che massimizza l'utilit a attesa della sua ricchezza nale. Studiamo strategie d'investimento ottimali dando l'accesso agli investitori non solo ai mercati obbligazionari e azionari, ma anche al mercato dei derivati. Il problema viene risolto in forma chiusa attraverso l'approccio di controllo stocastico. In prima analisi viene presentata l'equivalenza tra il problema classico di Merton e una strategia buy-and-hold che include rispetto all'altro problema anche un possibile investimento sul mercato delle opzioni. Viene poi presentata la soluzione del problema in modo dinamico quando i derivati sono scritti su stock che presentano volatilit a stocastica e tengono in considerazione il fatto che il prezzo del titolo possa subire grandi variazioni in brevi istanti temporali (il rischio di salto del titolo). In ne viene proposta una soluzione nel caso in cui mercati incompleti vengano completati da strumenti derivati.
The Merton problem with derivatives
MILANESI, PAOLA
2019/2020
Abstract
This thesis considers the problem of optimizing a continuous time portfolio of an investor with a constant risk aversion coe cient, which maximizes the expected utility of his nal wealth. We study optimal investment strategies given investor access not only to bond and stock markets but also to the derivatives market. The problem is solved in a closed form through the stochastic control approach. In the rst analysis, the equivalence between Merton's classic problem and a buy-and-hold strategy that includes a possible investment in the options market compared to the other problem is presented. Then we introduce the solution of the problem in dynamic way when the derivatives are written on Stock that introduces stochastic volatility and it takes into consideration the fact that the price of the Stock can undergo great variations in short temporal instants (jump risks in the stock market). Finally a solution is proposed in the case in which the incomplete markets are completed by derivatives.File | Dimensione | Formato | |
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https://hdl.handle.net/10589/153308