111 research outputs found
Optimal Investment with Transaction Costs and Stochastic Volatility
Two major financial market complexities are transaction costs and uncertain
volatility, and we analyze their joint impact on the problem of portfolio
optimization. When volatility is constant, the transaction costs optimal
investment problem has a long history, especially in the use of asymptotic
approximations when the cost is small. Under stochastic volatility, but with no
transaction costs, the Merton problem under general utility functions can also
be analyzed with asymptotic methods. Here, we look at the long-run growth rate
problem when both complexities are present, using separation of time scales
approximations. This leads to perturbation analysis of an eigenvalue problem.
We find the first term in the asymptotic expansion in the time scale parameter,
of the optimal long-term growth rate, and of the optimal strategy, for fixed
small transaction costs.Comment: 27 pages, 4 figure
Evaluating Incentive Options
We provide an analytical and flexible framework to evaluate incentive options. Our model not only considers vesting periods and trading and hedging restrictions on the holders, but also specifically includes provisions of reloading and resetting to capture the fact that firms tend to grant more options after existing options are either exercised or become deep out of the money. By treating the incentive option as a flow of barrier options, we are able to obtain a near-explicit formula for the option value. Our model allows us to discuss many issues related to incentive options such as their issuing cost, exercising strategies, and induced incentives. Especially, we highlight some significant interactions among different features of incentive optionsExecutive Stock Options, Incentives, Resetting and Reloading, Subjective Valuation
Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations
We consider the problem of optimal portfolio selection under forward
investment performance criteria in an incomplete market. The dynamics of the
prices of the traded assets depend on a pair of stochastic factors, namely, a
slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic
volatility). We analyze the associated forward performance SPDE and provide
explicit formulae for the leading order and first order correction terms for
the forward investment process and the optimal feedback portfolios. They both
depend on the investor's initial preferences and the dynamically changing
investment opportunities. The leading order terms resemble their time-monotone
counterparts, but with the appropriate stochastic time changes resulting from
averaging phenomena. The first-order terms compile the reaction of the investor
to both the changes in the market input and his recent performance. Our
analysis is based on an expansion of the underlying ill-posed HJB equation, and
it is justified by means of an appropriate remainder estimate.Comment: 26 page
American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics
American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where scaling parameter value is equal to unity, fast and slow scale approximations are equally accurate
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