A novel Monte Carlo approach to hybrid local volatility models

We present in a Monte Carlo simulation framework, a novel approach for the evaluation of hybrid local volatility [Risk, 1994, 7, 18–20], [Int. J. Theor. Appl. Finance, 1998, 1, 61–110] models. In particular, we consider the stochastic local volatility model—see e.g. Lipton et al. [Quant. Finance, 2014, 14, 1899–1922], Piterbarg [Risk, 2007, April, 84–89], Tataru and Fisher [Quantitative Development Group, Bloomberg Version 1, 2010], Lipton [Risk, 2002, 15, 61–66]—and the local volatility model incorporating stochastic interest rates—see e.g. Atlan [ArXiV preprint math/0604316, 2006], Piterbarg [Risk, 2006, 19, 66–71], Deelstra and Rayée [Appl. Math. Finance, 2012, 1–23], Ren et al. [Risk, 2007, 20, 138–143]. For both model classes a particular (conditional) expectation needs to be evaluated which cannot be extracted from the market and is expensive to compute. We establish accurate and ‘cheap to evaluate’ approximations for the expectations by means of the stochastic collocation method [SIAM J. Numer. Anal., 2007, 45, 1005–1034], [SIAM J. Sci. Comput., 2005, 27, 1118–1139], [Math. Models Methods Appl. Sci., 2012, 22, 1–33], [SIAM J. Numer. Anal., 2008, 46, 2309–2345], [J. Biomech. Eng., 2011, 133, 031001], which was recently applied in the financial context [Available at SSRN 2529691, 2014], [J. Comput. Finance, 2016, 20, 1–19], combined with standard regression techniques. Monte Carlo pricing experiments confirm that our method is highly accurate and fast.     

One-state variable binomial models for European-/American-style geometric Asian options

Abstract

This paper is concerned with geometric Asian options whose pay-offs depend on the geometric average of the underlying asset prices. Following the Cox et al (1979 J. Financial Economics 7 229-63) arbitrage arguments, we develop one-state variable binomial models for the options on the basis of the idea of Cheuk and Vorst (1997 J. Int. Money Finance 16 173-87). The models are more efficient and faster than those lattice methods (for the options) proposed by Hull and White (1993 J. Derivatives 1 21-31), Ritchken et al (1993 Manage. Sci. 39 1202-13), Barraquand and Pudet (1996 Math. Finance 6 17-51) and Cho and Lee (1997 J. Financial Eng. 6 179-91). We also establish the equivalence of the models and certain difference schemes.     

Book Reviews

Organized Uncertainty: Designing a World of Risk Management. Michael Power. Oxford University Press, 2007. xviii and 248pp. ISBN 978–0–9–925394–4. £24.99.

Intellectual Capital Reporting: Lessons from Hong Kong and Australia. J. Guthrie, R. Petty and F. Ricceri. The Institute of Chartered Accountants of Scotland, 2007, vii and 118pp. ISBN 978–1–904574–27–9. £15

The Routledge Companion to Fair Value and Financial Reporting. P. Walton (ed.). Routledge, 2007. xviii and 404 pp. ISBN 978–0–415–42356–4. £95.

UK Reporting of Intellectual Capital. Jeffrey Unerman, James Guthrie and Ludmila Striukova. ICAEW Centre for Business Performance, 2007. 68 pp. ISBN 978 1 84152 507 5. £20.     

Sequential Quasi-Credibility for Scale Dispersion Models

The sequential approach to credibility, developed by Landsman and Makov [(1999a) On stochastic approximation and credibility. Scand. Actuarial J. 1, 15-31; (1999b) Sequential credibility evaluation for symmetric location claim distributions. Insurance: Math. Econ. 24, 291-300] is extended to the scale dispersion family, which contains distributions often used in actuarial science: log-normal, Weibull, Half normal, Stable, Pareto, to mention only a few. For members of this family a sequential quasi-credibility formula is devised, which can also be used for heavy tailed claims. The results are illustrated by a study of log-normal claims.     

A Note on s. or n.s. cPp

Abstract

1. In a s. or n.s. cPp (stationary or non-stationary compound Poisson process) the probability for occurrence of m events, while the parameter (one-or more-dimensional) passes from zero to τ ₀ as measured on an absolute scale (the τ-scale), is defined as a mean of Poisson probabilities with intensities, which are distributed with distribution functions defining another random process, called the primary process with respect to the s. or n.s cPp. The stationarity (in the weak sence) and the non-stationarity of the primary process imply the same properties of the s. or n.s. cPp.     

Algebraic structure of vector fields in financial diffusion models and its applications

High-order discretization schemes of SDEs using free Lie algebra-valued random variables are introduced by Kusuoka [Adv. Math. Econ., 2004, 5, 69–83], [Adv. Math. Econ., 2013, 17, 71–120], Lyons–Victoir [Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 2004, 460, 169–198], Ninomiya–Victoir [Appl. Math. Finance, 2008, 15, 107–121] and Ninomiya–Ninomiya [Finance Stochast., 2009, 13, 415–443]. These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical methods. The authors have found a special Lie algebraic structure on the vector fields in the major financial diffusion models. Using this structure, we can solve the flows associated with vector fields analytically and efficiently. Numerical examples show that our method reduces the computation time drastically.     

CALL FOR PAPERS

Organized Uncertainty: Designing a World of Risk Management. Michael Power. Oxford University Press, 2007. xviii and 248pp. ISBN 978–0–9–925394–4. £24.99.

Intellectual Capital Reporting: Lessons from Hong Kong and Australia. J. Guthrie, R. Petty and F. Ricceri. The Institute of Chartered Accountants of Scotland, 2007, vii and 118pp. ISBN 978–1–904574–27–9. £15

The Routledge Companion to Fair Value and Financial Reporting. P. Walton (ed.). Routledge, 2007. xviii and 404 pp. ISBN 978–0–415–42356–4. £95.

UK Reporting of Intellectual Capital. Jeffrey Unerman, James Guthrie and Ludmila Striukova. ICAEW Centre for Business Performance, 2007. 68 pp. ISBN 978 1 84152 507 5. £20.     

Time-varying long-run mean of commodity prices and the modeling of futures term structures

The exploration of the mean-reversion of commodity prices is important for inventory management, inflation forecasting and contingent claim pricing. Bessembinder et al. [J. Finance, 1995, 50, 361–375] document the mean-reversion of commodity spot prices using futures term structure data; however, mean-reversion to a constant level is rejected in nearly all studies using historical spot price time series. This indicates that the spot prices revert to a stochastic long-run mean. Recognizing this, I propose a reduced-form model with the stochastic long-run mean as a separate factor. This model fits the futures dynamics better than do classical models such as the Gibson–Schwartz [J. Finance, 1990, 45, 959–976] model and the Casassus–Collin-Dufresne [J. Finance, 2005, 60, 2283–2331] model with a constant interest rate. An application for option pricing is also presented in this paper.     

An improved convolution algorithm for discretely sampled Asian options

We suggest an improved FFT pricing algorithm for discretely sampled Asian options with general independently distributed returns in the underlying. Our work complements the studies of Carverhill and Clewlow [Risk, 1990, 3(4), 25–29], Benhamou [J. Comput. Finance, 2002, 6(1), 49–68], and Fusai and Meucci [J. Bank. Finance, 2008, 32(10), 2076–2088], and, if we restrict our attention only to log-normally distributed returns, also Ve?e? [Risk, 2002, 15(6), 113–116]. While the existing convolution algorithms compute the density of the underlying state variable by moving forward on a suitably defined state space grid, our new algorithm uses backward price convolution, which resembles classical lattice pricing algorithms. For the first time in the literature we provide an analytical upper bound for the pricing error caused by the truncation of the state space grid and by the curtailment of the integration range. We highlight the benefits of the new scheme and benchmark its performance against existing finite difference, Monte Carlo, and forward density convolution algorithms.     

Nils Ekholm

Abstract

The problem of χ² tests of a linear hypothesis H₀ for ‘matched samples’ in attribute data has been discussed earlier by the author (Bennett, 1967, 1968). This note presents corresponding results for the hypothesis that the multinomial probabilities p satisfy (c ?1) functional restrictions: F ₁(p) = 0, ... , F _C?1(p) = 0. An explicit relationship between the usual ‘goodness-of-fit’ χ² and the modified minimum χ² (=χ*²) of Jeffreys (1938) and Neyman (1949) is demonstrated for this situation. An example of the test for the 2 × 2 × 2 contingency table is given and compared with the solution of Bartlett (1935).     

Truncated inverse binomial sampling

Abstract

1.Introductions Inverse Binomial Sampling.

From an infinite population of a's and b's, in proportions p and q = 1-p respectively, individuals are drawn at random until M a's are found. Thus the sample size, n, is a random variable. Its well-known distribution is      

Numerical evaluation of the compound Poisson distribution: Recursion or Fast Fourier Transform?

1. The problem

The finite vector p=(p ₁,p ₂, ...,p_s ) defines a probability distribution on the integers 1,2, ...,s.     

Remark on a theorem by Gurland

Abstract

Let X ₁,X ₂,...,X _n be a random sample of size from a distribution with probability density function p(x|θ), where the unknown parameter θ belongs to a non-degenerate interval I. The unknown true value of θ will be denoted by θ₀.     

Real-time market microstructure analysis: online transaction cost analysis

Motivated by the practical challenge in monitoring the performance of a large number of algorithmic trading orders, this paper provides a methodology that leads to automatic discovery of causes that lie behind poor trading performance. It also gives theoretical foundations to a generic framework for real-time trading analysis. The common acronym for investigating the causes of bad and good performance of trading is transaction cost analysis Rosenthal [Performance Metrics for Algorithmic Traders, 2009]). Automated algorithms take care of most of the traded flows on electronic markets (more than 70% in the US, 45% in Europe and 35% in Japan in 2012). Academic literature provides different ways to formalize these algorithms and show how optimal they can be from a mean-variance (like in Almgren and Chriss [J. Risk, 2000, 3(2), 5–39]), a stochastic control (e.g. Guéant et al. [Math. Financ. Econ., 2013, 7(4), 477–507]), an impulse control (see Bouchard et al. [SIAM J. Financ. Math., 2011, 2(1), 404–438]) or a statistical learning (as used in Laruelle et al. [Math. Financ. Econ., 2013, 7(3), 359–403]) viewpoint. This paper is agnostic about the way the algorithm has been built and provides a theoretical formalism to identify in real-time the market conditions that influenced its efficiency or inefficiency. For a given set of characteristics describing the market context, selected by a practitioner, we first show how a set of additional derived explanatory factors, called anomaly detectors, can be created for each market order (following for instance Cristianini and Shawe-Taylor [An Introduction to Support Vector Machines and Other Kernel-based Learning Methods, 2000]). We then will present an online methodology to quantify how this extended set of factors, at any given time, predicts (i.e. have influence, in the sense of predictive power or information defined in Basseville and Nikiforov [Detection of Abrupt Changes: Theory and Application, 1993], Shannon [Bell Syst. Tech. J., 1948, 27, 379–423] and Alkoot and Kittler [Pattern Recogn. Lett., 1999, 20(11), 1361–1369]) which of the orders are underperforming while calculating the predictive power of this explanatory factor set. Armed with this information, which we call influence analysis, we intend to empower the order monitoring user to take appropriate action on any affected orders by re-calibrating the trading algorithms working the order through new parameters, pausing their execution or taking over more direct trading control. Also we intend that use of this method can be taken advantage of to automatically adjust their trading action in the post trade analysis of algorithms.     

Smart Monte Carlo: various tricks using Malliavin calculus

Abstract

Current Monte Carlo pricing engines may face a computational challenge for the Greeks, not only because of their time consumption but also their poor convergence when using a finite difference estimate with a brute force perturbation. The same story may apply to conditional expectation. In this short paper, following Fournié et al (Fournié E, Lasry J M, Lebuchoux J, Lions P L and Touzi N 1999 Finance Stochastics 3 391-412), we explain how to tackle this issue using Malliavin calculus to smoothen the payoff to estimate. We discuss the relationship with the likelihood ratio method of Broadie and Glasserman (Broadie M and Glasserman P 1996 Manag. Sci. 42 269-85). We show by numerical results the efficiency of this method and discuss when it is appropriate or not to use it. We see how to apply this method to the Heston model.     

One Approach to Dual Randomness in Life Insurance

Abstract

1. The Unnatural Hypothesis of a Constant Rate of Interest

There are loan contracts which assume a constant interest during several years and thereafter payment of the amount borrowed, but nowadays clauses are as a rule admitted giving the debtor right of conversion or repayment after a certain period, generally ten years. Low interest loans can be considered as perpetuities from a practical point of view, as long as no possibility is meant to exist that the market rate will fall under their nominal rate. Such a loan—as e.g. Consols—with the nominal rate i ₀ ought to be valued at a discount if the market rate is higher, say i > i ₀, the value being equal to the fraction i ₀ : i. But constant rates are no rule in practice.     

Comments on Harald Bohman's paper

Abstract

We indicate by S _t a non-decreasing function having the differential dS_t υ defined for all t?0 (S₀=0); P _t a non-decreasing continuous function with an existing second differential for all t ?0.     

On the asymptotic normality of some unbiased estimates

Abstract

Let X _m(n) =(X _j , n, ..., X _{j m,n} ) be a subset of observations of a sample X_n = (X_1n X _2n ... , X _nn ). Here the X_jn 'S in Xn are not necessarily independent or identically distributed, and m(n) mayor may not tend to infinity as n tends to infinity. Suppose the joint density function h_n =h_n (x _m (n); θ) of the X _jn 's in Xm(n) is completely specified except the values of the parameters in the parameter vector θ = (θ1 θ2, ... , θ _k ), where θ belongs to a non-degenerate open subset H of the k-dimensional Euclidean space Rk and k?m(n).     

Eksamen i Statistik ved Københavns Universitet

Abstract

Extract

d1. Bestern karakteristikkerne for den partielle differentialigning Gør rede for, at der ved begyndelsebetingelsen z=2√x for 0<x<+∞,y=0 fastlægges netop en løsning til (*) i et passende område ω i xy-planen, og bestem denne løsning (herunder et brugbart område ω).