Hulley, H. & Schweizer, M. 2010, 'M6 - On minimal market models and minimal Martingale measures' in Chiarella, C. & Novikov, A. (eds), Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, Springer, Germany, pp. 35-51.
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The well-known absence-of-arbitrage condition NFLVR from the fundamental theorem of asset pricing splits into two conditions, called NA and NUPBR. We give a literature overview of several equivalent reformulations of NUPBR; these include existence of a growth-optimal portfolio, existence of the numeraire portfolio, and for continuous asset prices the structure condition (SC). As a consequence, the minimal market model of E. Platen is seen to be directly linked to the minimal martingale measure. We then show that reciprocals of stochastic exponentials of continuous local martingales are time changes of a squared Bessel process of dimension 4. This directly gives a very specific probabilistic structure for minimal market models.
Hulley, H. 2010, 'The economic plausibility of strict local Martingales in financial modelling' in Chiarella, C. & Novikov, A. (eds), Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, Springer, Germany, pp. 53-75.
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The context for this article is a continuous financial market consisting of a risk-free savings account and a single non-dividend-paying risky security. We present two concrete models for this market, in which strict local martingales play decisive roles. The first admits an equivalent risk-neutral probability measure under which the discounted price of the risky security is a strict local martingale, while the second model does not even admit an equivalent risk-neutral probability measure, since the putative density process for such a measure is itself a strict local martingale. We highlight a number of apparent anomalies associated with both models that may offend the sensibilities of the classically-educated reader. However, we also demonstrate that these issues are easily resolved if one thinks economically about the models in the right way. In particular, we argue that there is nothing inherently objectionable about either model.
Glover, K. & Hulley, H. 2011, 'The limits of arbitrage and the term structure of stock index futures mispricing'.
Hulley, H. 2011, 'The impact of idiosyncratic risk on mutual fund fees and performance'.
Hulley, H. & Platen, E. 2011, 'A visual criterion for identifying Ito diffusions as martingales or strict local martingales', Seminar on Stochastic Analysis, Random Fields and Applications VI, Springer, Ascona, Switzerland, pp. 147-157.
It is often important, in applications of stochastic calculus to financial modelling, to know whether a given local martingale is a martingale or a strict local martingale. We address this problem in the context of a time-homogenous diffusion process with a finite lower boundary, presented as the solution of a driftless stochastic differential equation. Our main theorem demonstrates that the question of whether or not this process is a martingale may be decided simply by examining the slope of a certain increasing function. Further results establish the connection between our theorem and other results in the literature, while a number of examples are provided to illustrate the use of our criterion.
Hulley, H. 2010, 'Local martingales obtained by discounting'.
Casavecchia, L. & Hulley, H. 2010, 'The effect of idiosyncratic risk-taking on mutual fund performance and fees', Financial Management Association 2010 Meetings, Financial Management Association, New York, USA, pp. 1-50.
We identify for the first time the crucial role played by idiosyncratic risk as a determinant of performance persistence, flow-performance sensitivity and management fees charged to fund shareholders. Using a sample of US equity mutual funds, we show that high idiosyncratic volatility indirectly captures the aggressiveness of fund investment strategies. We document that funds characterized by high idiosyncratic risk exhibit high probabilities of transitioning into the tails of the performance distribution. In particular, these high transition probabilities in performance cause funds characterized by high idiosyncratic risk to jump more frequently from one tail of the performance distribution to the other, making them appear as if they do not significantly underperform as opposed to funds with low levels of idiosyncratic risk. Consistent with the model of Berk and Green (2004), we argue that idiosyncratic risk is a confusing factor and significantly compromises investors ability to clearly quantify managerial skills. Since investors learn about managerial abilities from past returns and chase performance accordingly, we should expect high noise in performance to reduce the precision of investors priors about these abilities. As a result, in the presence of switching costs and search costs, investors may optimally choose to wait to receive a better signal before (re-) allocating their capital. We document in fact that the sensitivity of flows to performance significantly and monotonically plunges for those funds engaging in high idiosyncratic risk, irrespective of their performance rankings.
Thorp, S.J., Hulley, H., McKibbin, R. & Pedersen, A. 2009, 'Means-tested income support, portfolio choice and decumulation in retirement'.
Hulley, H. 2009, 'The economic plausibility of strict local martingales in financial modelling'.
Hulley, H. 2008, 'A Chapman-Kolmogorov algorithm for barrier options with moving barriers'.
Hulley, H. 2008, 'Conditions for Martingales, with Applications in Finance'.
Hulley, H. & Platen, E. 2008, 'Laplace transform identities for diffusions, with applications to rebates and barrier options', Banach Centre Publications: Advances in Mathematics of Finance, Polska Akademia Nauk, Warszawa, Poland, pp. 139-157.
Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also obtain a number of useful expressions for the Laplace transforms of some functions of first-passage times for the diffusion. These results are applied to the special case of squared Bessel processes with killing or reflecting boundaries. In particular, we demonstrate how the above-mentioned integral identity enables us to derive the transition density of a squared Bessel process killed at the origin, without the need to invert a Laplace transform. Finally, as an application, we consider the problem of pricing barrier options on an index described by the minimal market model.
Hulley, H. 2006, 'A survey and reassessment of the constant elasticity of variance model', 5th National Symposium on Financial Mathematics.
Hulley, H. 2006, 'Hedging basis risk using quadratic criteria', Quantitative Methods in Finance 2006 Conference.
Hulley, H. 2005, 'A simulation-based analysis of equity index models', Quantitative Methods in Finance 2005 Conference, -, -.
Hulley, H., Heath, D.P. & Platen, E. 2005, 'A comparative study of performance robustness for equity index models', 4th National Symposium on Financial Mathematics, -, -.
Hulley, H., Heath, D.P. & Platen, E. 2005, 'A comparative study of performance robustness for equity index models', Mathematics in Finance International Conference, -, -.
This paper considers a market containing both continuous and discrete noise. Modest assumptions ensure the existence of a growth optimal portfolio. Non-negative self-financing trading strategies,
when benchmarked by this portfolio, are local martingales under the real-world measure. This justifies the fair pricing approach, which expresses derivative prices in terms of real-world conditional
expectations of benchmarked payoffs. Two models for benchmarked primary security accounts are presented, and fair pricing formulas for some common contingent claims are derived.
Hulley, H. 2004, 'Fair pricing in a benchmark model with jumps', -, -, -.
Glover, K., Hulley, H. & Peskir, G. 2013, 'THREE-DIMENSIONAL BROWNIAN MOTION AND THE GOLDEN RATIO RULE', ANNALS OF APPLIED PROBABILITY, vol. 23, no. 3, pp. 895-922.
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Hulley, H., McKibbin, R., Pedersen, A. & Thorp, S.J. 2013, 'Means-tested public pensions, portfolio choice and decumulation in retirement', The Economic Record, vol. 89, no. 284, pp. 31-51.
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Age Pension means-testing buffers retired households against shocks to wealth and may influence decumulation patterns and portfolio allocations. Simulations from a simple model of optimal consumption and allocation strategies for a means-tested retired household indicate that, relative to benchmark, eligible and near-eligible households should optimally decumulate faster, and choose more risky portfolios, especially early in retirement. Empirical modelling of a Household, Income and Labour Dynamics in Australia panel of pensioner households confirms a riskier portfolio allocation by wealthier retired households. Poorer pensioner households decumulate at around 5 per cent p.a. on average; however, better-off households continue to add around 3 per cent p.a. to wealth, even when facing a steeper implicit tax rate on wealth.
Glover, K. & Hulley, H. 2013, 'Optimal Prediction of the Last-Passage Time of a Transient Diffusion'.
We identify the integrable stopping time $\tau_*$ with minimal $L^1$-distance
to the last-passage time $\gamma_z$ to a given level $z>0$, for an arbitrary
non-negative time-homogeneous transient diffusion $X$. We demonstrate that
$\tau_*$ is in fact the first time that $X$ assumes a value outside a half-open
interval $[0,r_*)$. The upper boundary $r_*>z$ of this interval is
characterised either as the solution for a one-dimensional optimisation
problem, or as part of the solution for a free-boundary problem. A number of
concrete examples illustrate the result.
In the years following the publication of Black and Scholes (J Political Econ, 81(3), 637-654, 1973), numerous alternative models have been proposed for pricing and hedging equity derivatives. Prominent examples include stochastic volatility models, jump-diffusion models, and models based on Lvy processes. These all have their own shortcomings, and evidence suggests that none is up to the task of satisfactorily pricing and hedging extremely long-dated claims. Since they all fall within the ambit of risk-neutral valuation, it is natural to speculate that the deficiencies of these models are (at least in part) attributable to the constraints imposed by the risk-neutral approach itself. To investigate this idea, we present a simple two-parameter model for a diversified equity accumulation index. Although our model does not admit an equivalent risk-neutral probability measure, it nevertheless fulfils a minimal no-arbitrage condition for an economically viable financial market. Furthermore, we demonstrate that contingent claims can be priced and hedged, without the need for an equivalent change of probability measure. Convenient formulae for the prices and hedge ratios of a number of standard European claims are derived, and a series of hedge experiments for extremely long-dated claims on the S&P 500 total return index are conducted. Our model serves also as a convenient medium for illustrating and clarifying several points on asset price bubbles and the economics of arbitrage.
This paper considers the problem of when a local martingale is a martingale or a universally integrable martingale, for the case of time-homogeneous scalar diffusions. Necessary and suffcient
conditions of a geometric nature are obtained for answering this question. These results are widely applicable to problems in stochastic finance. For example, in order to apply risk-neutral pricing, one
must first check that the chosen density process for an equivalent change of probability measure is in fact a martingale. If not, risk-neutral pricing is infeasible. Furthermore, even if the density
process is a martingale, the possibility remains that the discounted price of some security could be a strict local martingale under the equivalent risk-neutral probability measure. In this case,
well-known identities for option prices, such as put-call parity, may fail. Using our results, we examine a number of basic asset price models, and identify those that suffer from the above-mentioned
Hulley, H., Miller, S. & Platen, E. 2005, 'Benchmarking and fair pricing applied to two marker models', The Kyoto Economic Review, vol. 74, no. 1, pp. 85-118.
This paper considers a market containing both continuous and discrete noise. Modest assumptions ensure the existence of a growth optimal portfolio. Non-negative self-financing trading strategies, when benchmarked by this portfolio, are local martingales unde the real-world measure. This justifies the fair pricing approach, which expresses derivative prices in terms of real-world conditional expectations of benchmarked pay-offs. Two models for benchmarked primary security accounts are presentated, and fair pricing formulas for some common contingent claims are derived.
Hulley, H. & Platen, E. 2011, 'A Visual Criterion for Identifying Ito Diffusions as Martingales or Strict Local Martingales'.
Thorp, S.J., Hulley, H., McKibbin, R. & Pedersen, A. 2009, 'Means-tested income support, portfolio choice and decumulation in retirement', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
Research Paper Number: 248 Abstract: We investigate the impact of means tested public income transfers on post-retirement decumulation and portfolio choice using theoretical simulations and panel data on Australian Age Pensioners. Means tested public pension payments in Australia have broad coverage and give insight into the incentive responsiveness of well-o, as well as poorer households. Via numerical solutions to a discrete time, fi?nite horizon dynamic programming problem, we simulate the optimal consumption and portfolio allocation strategies for a retired household subject to assets and income tests. Relative to benchmark, means tested households should optimally decumulate faster early in retirement, and choose more risky portfolios. Panel data tests on inferred wealth for pensioner households show evidence of more rapid spending early in retirement. However they also show that better-o households continue to accumulate, even when facing a steeper implicit tax rate on wealth than applies to poorer households. Wealthier households also hold riskier portfolios. Results from tests for Lorenz dominance of the panel wealth distribution show no decrease in wealth inequality over the ?five years of the study.
Thorp, S.J., Hulley, H., McKibbin, R. & Pedersen, A. 2009, 'Means-tested income support, portfolio choice and decumulation in retirement', Working Paper Series, Centre for Applied Macroeconomic Analysis.
Hulley, H. & McWalter, T. 2008, 'Quadratic hedging of basis risk', Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney.
Research Paper Number: 225 Abstract: This paper examines a simple basis risk model based on correlated geometric Brownian motions. We apply quadratic criteria to minimize basis risk and hedge in an optimal manner. Initially, we derive the Follmer-Schweizer decomposition of a European claim. This allows pricing and hedging under the minimal martingale measure, corresponding to the local risk-minimizing strategy. Furthermore, since the mean-variance tradeoff process is deterministic in our setup, the minimal martingale- and variance-optimal martingale measures coincide. Consequently, the mean-variance optimal strategy is easily constructed. Simple closed-form pricing and hedging formulae for put and call options are derived. Due to market incompleteness, these formulae depend on the drift parameters of the processes. By making a further equilibrium assumption, we derive an approximate hedging formula, which does not require knowledge of these parameters. The hedging strategies are tested using Monte Carlo experiments, and are compared with recent results achieved using a utility maximization approach.
Hulley, H. & Platen, E. 2007, 'Laplace Transform Identities for Diffusions, with Applications to Rebates and Barrier Options'.
Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also
obtain a number of useful expressions for the Laplace transforms of some functions of first-passage times for the diffusion. These results are applied to the special case of squared Bessel processes with
killing or reflecting boundaries. In particular, we demonstrate how the above-mentioned integral identity enables us to derive the transition density of a squared Bessel process killed at the origin,
without the need to invert a Laplace transform. Finally, as an application, we consider the problem of pricing barrier options on an index described by the minimal market model.
ISSN 1441-8010 www.business.uts.edu.au/qfrc/research/research_papers/rp155.pdf