Optimal Portfolios for Financial Markets with Wishart Volatility

Abstract

Abstract. We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain explicitly the optimal portfolio strategy and the value function in some parameter settings. In particular when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is indeed to identify when the solution of the HJB equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the influence of the investors' risk aversion on the hedging demand.

Portfolio Optimization in Fractional and Rough Heston Models

“…For similar transformations see e.g. [37,23,1]. Inserting the derivatives and rearranging the terms leads to 0 = cg t + g γr + 1 2…”

“…This is typical in settings where the Brownian motions of stock and volatility process are uncorrelated and where the appreciation rate and the volatility are in a certain relation (see e.g. [1]). It corresponds to the Merton ratio which would exactly be λ in our model.…”

“…It has already be shown to be successful in a number of stochastic volatility models (see e.g. [1,21]). We obtain:…”

“…the extensive textbook [33] and the references therein for a comprehensive overview. In the context of continuous-time portfolio optimization, which is concerned with finding a trading strategy that maximizes expected utility from terminal wealth, the Heston model has among others been dealt with in [37,4,23,27,1] using stochastic control methods and in [21] using martingale methods.…”

Optimal investment and pricing in the presence of defaults

“…Davis & Lleo, 2015, for an overview in the jump-diffusion setting); and, as our results are valid for general factor processes, this technique is applicable to optimal investment and risk-sensitive control problems with matrix-valued factor process (cf. Buraschi, Porchia, & Trojani, 2010;Bäuerle & Li, 2013;Robertson & Xing, 2017), as well as jumps in asset prices.…”