45 research outputs found

### Option Pricing with Transaction Costs Using a Markov Chain Approximation

An e cient algorithm is developed to price European options in the pres-
ence of proportional transaction costs, using the optimal portfolio frame-
work of Davis (1997). A fair option price is determined by requiring that
an in nitesimal diversion of funds into the purchase or sale of options
has a neutral e ect on achievable utility. This results in a general option
pricing formula, in which option prices are computed from the solution of
the investor's basic portfolio selection problem, without the need to solve
a more complex optimisation problem involving the insertion of the op-
tion payo into the terminal value function. Option prices are computed
numerically using a Markov chain approximation to the continuous time
singular stochastic optimal control problem, for the case of exponential
utility. Comparisons with approximately replicating strategies are made.
The method results in a uniquely speci ed option price for every initial
holding of stock, and the price lies within bounds which are tight even as
transaction costs become large. A general de nition of an option hedg-
ing strategy for a utility maximising investor is developed. This involves
calculating the perturbation to the optimal portfolio strategy when an
option trade is executed

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### Finite Horizon Portfolio Selection

We study the problem of maximising expected utility of terminal wealth
over a nite horizon, with one risky and one riskless asset available, and
with trades in the risky asset subject to proportional transaction costs.
In a discrete time setting, using a utility function with hyperbolic risk
aversion, we prove that the optimal trading strategy is characterised by
a function of time (t), which represents the ratio of wealth held in the
risky asset to that held in the riskless asset. There is a time varying no
transaction region with boundaries b(t) < s(t), such that the portfo-
lio is only rebalanced when (t) is outside this region. The results are
consistent with similar studies of the in nite horizon problem with in-
termediate consumption, where the no transaction region has a similar,
but time independent, characterisation. We solve the problem numerically
and compute the boundaries of the no transaction region for typical model
parameters. We show how the results can be used to implement option
pricing models with transaction costs based on utility maximisation over
a nite horizo

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### Migration, credit markets, moral hazard, interlinkage.

A fast numerical algorithm is developed to price European options with
proportional transaction costs using the utility maximization framework
of Davis (1997). This approach allows option prices to be computed by
solving the investor's basic portfolio selection problem, without the inser-
tion of the option payo into the terminal value function. The properties
of the value function can then be used to drastically reduce the number of
operations needed to locate the boundaries of the no transaction region,
which leads to very e cient option valuation. The optimization problem
is solved numerically for the case of exponential utility, and comparisons
with approximately replicating strategies reveal tight bounds for option
prices even as transaction costs become large. The computational tech-
nique involves a discrete time Markov chain approximation to a continuous
time singular stochastic optimal control problem. A general de nition of
an option hedging strategy in this framework is developed. This involves
calculating the perturbation to the optimal portfolio strategy when an
option trade is execute

### Long-Range Dependence in Daily Interest Rate

We employ a number of parametric and non-parametric techniques to
establish the existence of long-range dependence in daily interbank o er
rates for four countries. We test for long memory using classical R=S
analysis, variance-time plots and Lo's (1991) modi ed R=S statistic. In
addition we estimate the fractional di erencing parameter using Whittle's
(1951) maximum likelihood estimator and we shu e the data to destroy
long and short memory in turn, and we repeat our non-parametric tests.
From our non-parametric tests we And strong evidence of the presence of
long memory in all four series independently of the chosen statistic. We
nd evidence that supports the assertion of Willinger et al (1999) that
Lo's statistic is biased towards non-rejection of the null hypothesis of no
long-range dependence. The parametric estimation concurs with these
results. Our results suggest that conventional tests for capital market
integration and other similar hypotheses involving nominal interest rates
should be treated with cautio

### Interpolating actions for supersymmetric quantum field theory

A new perturbative scheme has recently been applied to a supersymmetric quantum-field-theory model in which no conventional means for doing analytic calculations existed. We develop an alternative technique and find that it allows a very easy demonstration of the supersymmetric results: ground-state energy density E=0, and fermion-boson mass ratio R=1. Moreover, unlike other techniques, our method can be applied to models with spontaneous supersymmetry breaking, for which we illustrate the broken-supersymmetry results E0 and R1. © 1989 The American Physical Society

### Performance of utility based strategies for hedging basis risk

The performance of optimal strategies for hedging a claim on a non-traded asset is analyzed. The claim is valued and hedged in a utility maximization framework, using exponential utility. A traded asset, correlated with that underlying the claim, is used for hedging, with the correlation $\rho$ typically close to 1. Using a distortion method (Zariphopoulou 2001, Finance and Stochastics 5, 61-82) we derive a nonlinear expectation representation for the claim's ask price and a formula for the optimal hedging strategy. We generate a perturbation expansion for the price and hedging strategy in powers of $\epsilon^{2}=1-\rho^{2}$. The terms in the price expansion are proportional to the central moments of the claim payoff under the minimal martingale measure. The resulting fast computation capability is used to carry out a simulation based test of the optimal hedging program, computing the terminal hedging error over many asset price paths. These errors are compared with those from a naive strategy which uses the traded asset as a proxy for the non-traded one. The distribution of the hedging error acts as a suitable metric to analyze hedging performance. We find that the optimal policy improves hedging performance, in that the hedging error distribution is more sharply peaked around a non-negative profit. The frequency of profits over losses is increased, and this is measured by the median of the distribution, which is always increased by the optimal strategies. An empirical example illustrates the application ofthe method to the hedging of a stock basket using index futures

### A new method for the solution of the Schrodinger equation

We present a new method for the solution of the Schrodinger equation
applicable to problems of non-perturbative nature. The method works by
identifying three different scales in the problem, which then are treated
independently: An asymptotic scale, which depends uniquely on the form of the
potential at large distances; an intermediate scale, still characterized by an
exponential decay of the wave function and, finally, a short distance scale, in
which the wave function is sizable. The key feature of our method is the
introduction of an arbitrary parameter in the last two scales, which is then
used to optimize a perturbative expansion in a suitable parameter. We apply the
method to the quantum anharmonic oscillator and find excellent results.Comment: 4 pages, 4 figures, RevTex

### Optimal exercise of an executive stock option by an insider

We consider an optimal stopping problem arising in connection with the exercise of an executive stock option by an agent with inside information. The agent is assumed to have noisy information on the terminal value of the stock, does not trade the stock or outside securities, and maximises the expected discounted payoff over all stopping times with regard to an enlarged filtration which includes the inside information. This leads to a stopping problem governed by a time-inhomogeneous diffusion and a call-type reward. We establish conditions under which the option value exhibits time decay, and derive the smooth fit condition for the solution to the free boundary problem governing the maximum expected reward, and derive the early exercise decomposition of the value function. The resulting integral equation for the unknown exercise boundary is solved numerically and this shows that the insider may exercise the option before maturity, in situations when an agent without the privileged information may not. Hence we show that early exercise may arise due to the agent having inside information on the future stock price