Dietmar P.J. Leisen

Binomial models for option valuation - examining and improving convergence

Binomial models, which describe the asset price dynamics of the continuous-time model in the limit, serve for approximate valuation of options, especially where formulas cannot be derived analytically due to properties of the considered option type. To evaluate results, one inevitably must understand the convergence properties. In the literature we find various contributions proving convergence of option prices. We examine convergence behaviour and convergence speed. Unfortunately, even in the case of European call options, distorted results occur when calculating prices along the iteration of tree refinements. These convergence patterns are examined and order of convergence one is proven for the Cox-Ross-Rubinstein model as well as for two alternative tree parameter selections from the literature. Furthermore, we define new binomial models, where the calculated option prices converge smoothly to the Black-Scholes solution, and we achieve order of convergence two with much smaller initial error. Notably, only the formulas to determine the up- and down-factors change. Finally, following a recent approach from the literature, all tree approaches are compared with respect to speed and accuracy, calculating the relative root-mean-squared error of approximate option values for a sample of randomly selected parameters across a set of refinements. Here, on average, the same degree of accuracy is achieved 1400 times faster with the new binomial models. We also give some insights into the peculiarities in the valuation of the American put option. Inspecting the numerical results, the approximation of American-type options with the new models exhibits order of convergence one, but with a smaller initial error than with previously existing binomial models, giving the same accuracy on average ten-times faster than previous binomial methods.

Stock Evolution Under Stochastic Volatility: A Discrete Approach

Stochastic volatility appears to be a fact of life in real-world derivatives markets, but it presents huge difficulties for valuation models. Adding a second stochastic variable in addition to the asset price significantly complicates matters. And things become only worse if one wants to model the volatility process realistically, as having a mean-reverting drift of the stock price process as a priced factor. Amercican exercise throws further complications into the situation. A number of useful closed-form and numerical approximation models have been developed over time, but only for particular special cases. In this article, Leisen presents a procedure for constructing a general three-dimensional valuation lattice that can handle a broad range of stochastic volatility models, including those in the literature.

Aggregation of preferences for skewed asset returns

This paper characterizes the equilibrium demand and risk premiums in the presence of skewness risk. We extend the classical mean-variance two-fund separation theorem to a three-fund separation theorem. The additional fund is the skewness portfolio, i.e. a portfolio that gives the optimal hedge of the squared market return; it contributes to the skewness risk premium through co-variation with the squared market return and supports a stochastic discount factor that is quadratic in the market return. When the skewness portfolio does not replicate the squared market return, a tracking error appears; this tracking error contributes to risk premiums through kurtosis and pentosis risk if and only if preferences for skewness are heterogeneous. In addition to the common powers of market returns, this tracking error shows up in stochastic discount factors as priced factors that are products of the tracking error and market returns.

Dynamic Risk Taking with Bonus Schemes

This paper studies dynamic risk taking by a risk-averse manager who receives a bonus; the company may default on its contractual obligations (debt and fixed compensation). We show that risk taking is time independent, and is summarized by the so-called risk aversion of derived utility. We highlight the importance of dynamic aspects and provide a foundation for common qualitative discussions that are based on characteristics of bonus functions. The paper cautions that deferral of fixed compensation may increase risk taking. Finally, we motivate a new bonus scheme that incentivizes the manager to implement the socially optimal risk level.

A Perturbation Approach to Continuous-Time Portfolio Selection

This paper studies portfolio selection in continuous-time models with stochastic investment opportunities. We consider asset allocation problems where preferences are specified as power utility derived from terminal wealth as well as consumption-savings problems with recursive utility Epstein-Zin preferences. The paper approximates the associated dynamic programming problem by perturbing the coefficients of the stochastic dynamics. We represent the Hamilton-Jacobi-Bellman equation as a series of partial differential equations that can be solved iteratively in closed-form through computer algebra software, at any desired accuracy.

Investing for the Long Run

This paper studies long term investing by an investor that maximizes either expected utility from terminal wealth or from consumption. We introduce the concepts of a generalized stochastic discount factor (SDF) and of the minimum price to attain target payouts. The paper finds that the dynamics of the SDF needs to be captured and not the entire market dynamics, which simplifies significantly practical implementations of optimal portfolio strategies. We pay particular attention to the case where the SDF is equal to the inverse of the growth-optimal portfolio in the given market. Then, optimal wealth evolution is closely linked to the growth optimal portfolio. In particular, our concepts allow us to reconcile utility optimization with the practitioner approach of growth investing. We illustrate empirically that our new framework leads to improved lifetime consumption-portfolio choice and asset allocation strategies.

The Shape of Small Sample Biases in Pricing Kernel Estimations

Numerous empirical studies find pricing kernels that are not-monotonically decreasing; the findings are at odds with the pricing kernel being marginal utility of a risk-averse, so-called representative agent. We study in detail the common procedure which estimates the pricing kernel as the ratio of two separate density estimations. In the first step, we analyse theoretically the functional dependence for the ratio of a density to its estimated density; this cautions the reader regarding potential computational issues coupled with statistical techniques. In the second step, we study this quantitatively; we show that small sample biases shape the estimated pricing kernel, and that estimated pricing kernels typically violate the commonly believed monotonicity at the centre even when the true pricing kernel fulfils these. This contributes to an alternative, statistical explanation for the puzzling shape in pricing kernel estimations.

Incentive Contracting For Venture Capital Fund Managers

It is well‐known in the VC literature that VCs should provide so‐called value‐adding activities that nurture their venture companies. This paper presents a model of the VC market that takes account of the so‐called carried interest on the choice of the risk level and actions. We determine the optimal actions of the fund managers and what would be the desired actions of the fund’s investors. Conflicts of interest between the parties will be documented.

A Stochastic Volatility Lattice

Abstract Stochastic volatility models model asset dynamics by a bivariate diffusion process. For practical calculation of prices of financial derivatives lattice models are necessary. In this paper we present a procedure to construct discrete process approximations converging to such models