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Johannes Muhle-Karbe

Publications and source records attributed to Johannes Muhle-Karbe.

50 records · Page 3Linked to original sources
arXiv2013

Portfolio Selection with Small Transaction Costs and Binding Portfolio Constraints

An investor with constant relative risk aversion and an infinite planning horizon trades a risky and a safe asset with constant investment opportunities, in the presence of small transaction costs and a binding exogenous portfolio constraint. We explicitly derive the optimal trading policy, its welfare, and implied trading volume. As an application, we study the problem of selecting a prime broker among alternatives with different lending rates and margin requirements. Moreover, we discuss how changing regulatory constraints affect the deposit rates offered for illiquid loans.

q-fin.PM↗
arXiv2012

Option Pricing and Hedging with Small Transaction Costs

An investor with constant absolute risk aversion trades a risky asset with general Itô-dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading-order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.

q-fin.PR↗
arXiv2012

Long Horizons, High Risk Aversion, and Endogeneous Spreads

For an investor with constant absolute risk aversion and a long horizon, who trades in a market with constant investment opportunities and small proportional transaction costs, we obtain explicitly the optimal investment policy, its implied welfare, liquidity premium, and trading volume. We identify these quantities as the limits of their isoelastic counterparts for high levels of risk aversion. The results are robust with respect to finite horizons, and extend to multiple uncorrelated risky assets. In this setting, we study a Stackelberg equilibrium, led by a risk-neutral, monopolistic market maker who sets the spread as to maximize profits. The resulting endogenous spread depends on investment opportunities only, and is of the order of a few percentage points for realistic parameter values.

q-fin.PM↗
arXiv2011

Option Pricing in Multivariate Stochastic Volatility Models of OU Type

We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets while still being highly analytically tractable. First we derive the characteristic function and give conditions that ensure its analyticity and absolute integrability in some open complex strip around zero. Therefore we can use Fourier methods to compute the prices of multi-asset options efficiently. To show the applicability of our results, we propose a concrete specification, the OU-Wishart model, where the dynamics of each individual asset coincide with the popular Gamma-OU BNS model. This model can be well calibrated to market prices, which we illustrate with an example using options on the exchange rates of some major currencies. Finally, we show that covariance swaps can also be priced in closed form.

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arXiv2011

Utility Maximization, Risk Aversion, and Stochastic Dominance

Consider an investor trading dynamically to maximize expected utility from terminal wealth. Our aim is to study the dependence between her risk aversion and the distribution of the optimal terminal payoff. Economic intuition suggests that high risk aversion leads to a rather concentrated distribution, whereas lower risk aversion results in a higher average payoff at the expense of a more widespread distribution. Dybvig and Wang [J. Econ. Theory, 2011, to appear] find that this idea can indeed be turned into a rigorous mathematical statement in one-period models. More specifically, they show that lower risk aversion leads to a payoff which is larger in terms of second order stochastic dominance. In the present study, we extend their results to (weakly) complete continuous-time models. We also complement an ad-hoc counterexample of Dybvig and Wang, by showing that these results are "fragile", in the sense that they fail in essentially any model, if the latter is perturbed on a set of arbitrarily small probability. On the other hand, we establish that they hold for power investors in models with (conditionally) independent increments.

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arXiv2011

Asymptotics and Duality for the Davis and Norman Problem

We revisit the problem of maximizing expected logarithmic utility from consumption over an infinite horizon in the Black-Scholes model with proportional transaction costs, as studied in the seminal paper of Davis and Norman [Math. Operation Research, 15, 1990]. Similarly to Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20, 2010], we tackle this problem by determining a shadow price, that is, a frictionless price process with values in the bid-ask spread which leads to the same optimization problem. However, we use a different parametrization, which facilitates computation and verification. Moreover, for small transaction costs, we determine fractional Taylor expansions of arbitrary order for the boundaries of the no-trade region and the value function. This extends work of Janecek and Shreve [Finance Stoch., 8, 2004], who determined the leading terms of these power series.

q-fin.PM↗
arXiv2011

Small-Time Asymptotics of Option Prices and First Absolute Moments

We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price process $S$ follows a general martingale. This is equivalent to studying the first centered absolute moment of $S$. We show that if $S$ has a continuous part, the leading term is of order $\sqrt{T}$ in time $T$ and depends only on the initial value of the volatility. Furthermore, the term is linear in $T$ if and only if $S$ is of finite variation. The leading terms for pure-jump processes with infinite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation of $S$ so that calculations are necessary only for the class of Lévy processes.

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arXiv2011

On the Performance of Delta Hedging Strategies in Exponential Lévy Models

We consider the performance of non-optimal hedging strategies in exponential Lévy models. Given that both the payoff of the contingent claim and the hedging strategy admit suitable integral representations, we use the Laplace transform approach of Hubalek et al. (2006) to derive semi-explicit formulas for the resulting mean squared hedging error in terms of the cumulant generating function of the underlying Lévy process. In two numerical examples, we apply these results to compare the efficiency of the Black-Scholes hedge and the model delta to the mean-variance optimal hedge in a normal inverse Gaussian and a diffusion-extended CGMY Lévy model.

q-fin.CP↗
arXiv2010

A characterization of the martingale property of exponentially affine processes

We consider local martingales which are standard or stochastic exponentials M of one component X of a multivariate affine process in the sense of Duffie, Filipovic and Schachermayer (2003). By completing their characterization of conservative affine processes, we provide deterministic necessary and sufficient conditions in terms of the parameters of X for M to be a true martingale.

math.PR↗
arXiv2010

Asymptotic and Exact Pricing of Options on Variance

We consider the pricing of derivatives written on the discretely sampled realized variance of an underlying security. In the literature, the realized variance is usually approximated by its continuous-time limit, the quadratic variation of the underlying log-price. Here, we characterize the small-time limits of options on both objects. We find that the difference between them strongly depends on whether or not the stock price process has jumps. Subsequently, we propose two new methods to evaluate the price of options on the discretely sampled realized variance. One of the methods is approximative; it is based on correcting prices of options on quadratic variation by our asymptotic results. The other method is exact; it uses a novel randomization approach and applies Fourier-Laplace techniques. We compare the methods and illustrate our results by some numerical examples.

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arXiv2010

Existence of Shadow Prices in Finite Probability Spaces

A shadow price is a process lying within the bid/ask prices of a market with proportional transaction costs, such that maximizing expected utility from consumption in the frictionless market with this price process leads to the same maximal utility as in the original market with transaction costs. For finite probability spaces, this note provides an elementary proof for the existence of such a shadow price.

q-fin.PM↗
arXiv2010

The dual optimizer for the growth-optimal portfolio under transaction costs

We consider the maximization of the long-term growth rate in the Black-Scholes model under proportional transaction costs as in Taksar, Klass and Assaf [Math. Oper. Res. 13, 1988]. Similarly as in Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20, 2010] for optimal consumption over an infinite horizon, we tackle this problem by determining a shadow price, which is the solution of the dual problem. It can be calculated explicitly up to determining the root of a deterministic function. This in turn allows to explicitly compute fractional Taylor expansions, both for the no-trade region of the optimal strategy and for the optimal growth rate.

q-fin.PM↗
arXiv2009

Utility maximization in models with conditionally independent increments

We consider the problem of maximizing expected utility from terminal wealth in models with stochastic factors. Using martingale methods and a conditioning argument, we determine the optimal strategy for power utility under the assumption that the increments of the asset price are independent conditionally on the factor process.

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