Dynamic behaviour of optimal portfolio with stochastic volatility

Abstract

In the existing literature, little is known about the dynamic behaviour of the optimal portfolio in terms of market inputs and arbitrary stochastic factor dynamics in an incomplete market with a stochastic volatility. In this paper, to study optimal portfolio behaviour, we compute and analyze the mean and the variance of the optimal portfolio and of their adjustment speed in terms of market inputs in an incomplete market. The incompleteness arises from the additional source of uncertainty of the volatility in Heston’s stochastic volatility model. Conducting sensitivity analysis for the mean and the variance of the optimal portfolio process as well as its adjustment speed to the market parameters, we find several interesting behavioural patterns of investors towards asset price and its volatility shocks. Our results are robust and convergent by the agreement from two simulation methods for different time step increments and the number of Monte Carlo simulation paths.

Keywords:

• Portfolio selection
• portfolio distribution
• dynamic behaviour
• Heston Stochastic volatility model
• incomplete market

JEL CODES:

• G11
• G12
• C63

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Expression of Concern

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Here short-selling is allowed. That is, we do not impose that $\pi (t)\in [0,1].$

2 For the formal definition of a self-financing portfolio process please see, for example, Korn and Korn (2001) page 63.

3 An increasing utility implies that the investor prefers more to less, while a concave utility is associated with a risk-averse investor. The symbols ($\prime$) and ($″$) denote first and second derivative, respectively.

4 It is known that, when γ converges to 1 (or $\gamma \to 1$), the power utility function (5) simplifies to the log utility ln(x).

5 When the stock variance v(t) is constant, (6) becomes the well-know Merton’s portfolio rule.

6 Note (10) is a special case of (11) when $v\to \theta .$

7 For time-dependent excess returns, similar results can be obtained.

Additional information

Funding

The work is sponsored by K. C. Wong Magna Fund in Ningbo University, Ningbo-CASS collaborative grant NTZKT201701, Research Academy of Belt & Road of Ningbo University and Foundation of Zhejiang Educational Committee under grant Y201737633.