# International Journal of Theoretical and Applied Finance

Published by World Scientific Publishing

Online ISSN: 0219-0249

Published by World Scientific Publishing

Online ISSN: 0219-0249

Publications

Conference Paper

We provide an option pricing formula based on an arbitrarily given
stock distribution, where the problem of optimally hedging the payoff on
a European call option is considered through a self-financing trading
strategy. An optimal hedging problem is solved on a trinomial lattice by
assigning suitable probabilities on the lattice, where the underlying
stock price distribution is derived directly from empirical stock price
data which may possess heavy tails. We show that these probabilities are
obtained from a network flow optimization. Numerical experiments
illustrate that our formula generates the implied volatility smile, in
contrast to the Black-Scholes formula

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Article

This paper analyzes the short- and long-term effects of the September 11, 2001 terrorist attacks on a comprehensive sample of stock market indices from 33 industrial and emerging economies. From a finance-theoretic point of view, we employ the international capital asset pricing model (ICAPM) to analyze the incidence of the 9/11 event. Consistent with expectations, we document statistically negative short-term stock market reactions to the 9/11 event for 28 countries. More importantly, we find increases in the level of systematic risk for 10 stock markets which attest to the presence of negative permanent effects emanating for the 9/11 event. However, a great many capital markets (including the US, Canada, Japan, China, Russia, and the largest European economies) did not experience statistically significant increases in systematic risk in the post-9/11 period. The decisiveness of the evidence clearly points in the direction of resilience and flexibility of the world capital markets.

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Article

S&P 500 index data sampled at one-minute intervals over the course of 11.5 years (January 1989- May 2000) is analyzed, and in particular the Hurst parameter over segments of stationarity (the time period over which the Hurst parameter is almost constant) is estimated. An asymptotically unbiased and efficient estimator using the log-scale spectrum is employed. The estimator is asymptotically Gaussian and the variance of the estimate that is obtained from a data segment of $N$ points is of order $\frac{1}{N}$. Wavelet analysis is tailor made for the high frequency data set, since it has low computational complexity due to the pyramidal algorithm for computing the detail coefficients. This estimator is robust to additive non-stationarities, and here it is shown to exhibit some degree of robustness to multiplicative non-stationarities, such as seasonalities and volatility persistence, as well. This analysis shows that the market became more efficient in the period 1997-2000.

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Article

We address the issue of finding a strategy to sustain structural profitability of an investment project, whose production activity depends on the market price of a number of underlying commodities. Depending on the fluctuating prices of these commodities, the activity will either continue until the project's profitability reaches a critical low level at which it is stopped and starts again when it becomes profitable. But, if the structural nonprofitability remains for a while, the investment project will face the risk to be abandoned or be definitely closed. We suggest a general probabilistic set up to model profitability as a function of the market price of a set of commodities, and find the related optimal strategy to sustain it, under the constraint that the project faces the abandonment risk when being nonprofitable under a fixed finite time interval. When the market price dynamics is described by a diffusion process, we show that the optimal strategy is related to viscosity solutions of a system of two variational inequalities with inter-connected obstacles.

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Article

This paper uses a conditional law of large numbers and a conditional central limit theorem to provide simplified asymptotic valuation formulas for credit derivatives on baskets, including synthetic and cash-flow CDOs. In particular, approximate pricing procedures are provided for synthetic and cash-flow CDOs. In the process, this paper also clarifies the relation between the "top-down" and "bottom-up" approaches for pricing credit derivatives.

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Article

We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener process, Lebesgue measure, and a Poisson random measure, then all compensators of totally inaccessible stopping times are absolutely continuous with respect to the minimal filtration generated by X. However Cinlar and Jacod have shown that all semimartingale strong Markov processes, up to a change of time and space, have such a representation.

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Article

In this article, we consider a specific optimal execution problem associated
to accelerated share repurchase contracts. When firms want to repurchase their
own shares, they often enter such a contract with a bank. The bank buys the
shares for the firm and is paid the average market price over the execution
period, the length of the period being decided upon by the bank during the
buying process. Mathematically, the problem is new and related to both option
pricing (Asian and Bermudan options) and optimal execution. We provide a model,
along with associated numerical methods, to determine the optimal stopping time
and the optimal buying strategy of the bank.

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Article

In the framework of the displaced-diffusion LIBOR market model, we derive the pathwise adjoint method for the iterative predictor-corrector and one of the Glasserman–Zhao drift approximations in the spot measure. This allows us to compute fast deltas and vegas under these schemes. We compare the discretisation bias obtained when computing Greeks with these methods to those obtained under the log-Euler and predictor-corrector approximations by performing tests with interest rate caplets and cancellable receiver swaps. The two predictor-corrector type methods were the most accurate by far. In particular, we found the iterative predictor-corrector method to be more accurate and slightly faster than the predictor-corrector method, the Glasserman–Zhao method, used, to be relatively fast but highly inconsistent, and the log-Euler method to be reasonably accurate but only at low volatilities. Standard errors were not significantly different across all four discretisations.

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Article

In this paper we present a theoretical framework for determining dynamic ask
and bid prices of derivatives using the theory of dynamic coherent
acceptability indices in discrete time. We prove a version of the First
Fundamental Theorem of Asset Pricing using the dynamic coherent risk measures.
We introduce the dynamic ask and bid prices of a derivative contract in markets
with transaction costs. Based on these results, we derive a representation
theorem for the dynamic bid and ask prices in terms of dynamically consistent
sequence of sets of probability measures and risk-neutral measures. To
illustrate our results, we compute the ask and bid prices of some
path-dependent options using the dynamic Gain-Loss Ratio.

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Article

This work compares the accuracy of different measures of Value at Risk (VaR) of fixed income portfolios calculated on the basis of different multi-factor empirical models of the term structure of interest rates (TSIR). There are three models included in the comparison: (1) regression models, (2) principal component models, and (3) parametric models. In addition, the cartography system used by Riskmetrics is included. Since calculation of a VaR estimate with any of these models requires the use of a volatility measurement, this work uses three types of measurements: exponential moving averages, equal weight moving averages, and GARCH models. Consequently, the comparison of the accuracy of VaR estimates has two dimensions: the multi-factor model and the volatility measurement. With respect to multi-factor models, the presented evidence indicates that the Riskmetrics model or cartography system is the most accurate model when VaR estimates are calculated at a 5% confidence level. On the contrary, at a 1% confidence level, the parametric model (Nelson and Siegel model) is the one that yields more accurate VaR estimates. With respect to the volatility measurements, the results indicate that, as a general rule, no measurement works systematically better than the rest. All the results obtained are independent of the time horizon for which VaR is calculated, i.e. either one or ten days.

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Article

We present a fast method to price and hedge CMS spread options in the displaced-diffusion co-initial swap market model. Numerical tests demonstrate that we are able to obtain sufficiently accurate prices and Greeks with computational times measured in milliseconds. Further, we find that CMS spread options are weakly dependent on the at-the-money Black implied volatility skews.

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Article

In this work we improve the algorithm of Han and Wu [SIAM J. Numer. Anal. 41 (2003), 2081–2095] for American Options with respect to stability, accuracy and order of computational effort. We derive an exact discrete artificial boundary condition (ABC) for the Crank–Nicolson scheme for solving the Black–Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level.
Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions.
Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples and compare it to previously obtained discretized ABCs of Mayfield and Han and Wu.

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Article

This paper proposes a simple scheme for static hedging of defaultable contingent claims. It generalizes the techniques developed by Carr and Chou (1997), Carr and Madan (1998), and Takahashi and Yamazaki (2009a) to credit-equity models. Our scheme provides a hedging strategy across credit and equity markets, where suitable defaultable contingent claims are accurately replicated by a feasible number of plain vanilla equity options. Another point is that shorter maturity options are available to hedge longer maturity defaultable contingent claims. Through numerical examples, it is shown that the scheme is applicable to both structural and intensity-based models.

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Article

This paper derives the multi-period fair actuarial values for six deductible insurance policies offered in today's insurance markets. The loss in any given period is generated by the Weibull distribution with a known shape parameter but an unknown scale parameter. The insurer is assumed to be a Bayesian decision maker, in the sense that he/she learns sequentially about the unknown scale parameter by observing the realizations of the filed claims. It is shown that the insurer's underlying predictive loss distributions belong to the Burr family, and the multi-period actuarially fair policy value can be derived. With a proper loading, an insurance premium can be quoted. Our major contribution is the analytical derivations of the fair actuarial values for deductible insurance policies in the presence of parameter uncertainty and Bayesian learning.

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Article

The paper introduces a limit version of multiple stopping options such that
the holder selects dynamically a weight function that control the distribution
of the payments (benefits) over time. In applications for commodities and
energy trading, a control process can represent the quantity that can be
purchased by a fixed price at current time. In another example, the control
represents the weight of the integral in a modification of the Asian option.
The pricing for these options requires to solve a stochastic control problem.
Some existence results and pricing rules are obtained via modifications of
parabolic Bellman equations.

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Article

In this paper we analyze the mean-variance hedging approach in an incomplete market under the assumption of additional market information, which is represented by a given, finite set of observed prices of non-attainable contingent claims. Due to no-arbitrage arguments, our set of investment opportunities increases and the set of possible equivalent martingale measures shrinks. Therefore, we obtain a modified mean-variance hedging problem, which takes into account the observed additional market information. Solving this we obtain an explicit description of the optimal hedging strategy and an admissible, constrained variance-optimal signed martingale measure, that generates both the approximation price and the observed option prices. Key Words: option pricing, mean variance hedging, incomplete markets, varianceoptimal martingale measure.

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Article

In this paper, we develop an algorithm to calculate the prices and Greeks of barrier options in a hyper-exponential additive model with piecewise constant parameters. We obtain an explicit semi-analytical expression for the first-passage probability. The solution rests on a randomization and an explicit matrix Wiener-Hopf factorization. Employing this result we derive explicit expressions for the Laplace-Fourier transforms of the prices and Greeks of barrier options. As a numerical illustration, the prices and Greeks of down-and-in digital and down-and-in call options are calculated for a set of parameters obtained by a simultaneous calibration to Stoxx50E call options across strikes and four different maturities. By comparing the results with Monte-Carlo simulations, we show that the method is fast, accurate, and stable.

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Article

Islamic banks do not pay interest on customers' deposit accounts. Instead, customers' funds are placed in profit-sharing investment accounts (PSIA). Under this arrangement, the returns to the bank's customers are their pro-rata shares of the returns on the assets in which their funds are invested, and if these returns are negative so are the returns to the customers. The bank is entitled to a contractually agreed share of positive returns (profits) as remuneration for its work as asset manager; however, if the returns are zero or negative, the bank receives no remuneration but does not share in any loss.
In the case of Unrestricted PSIA, the investment account holders' funds are invested (i.e., commingled) in the bank's asset pool together with the bank's shareholders' own funds and the funds of current account holders. In that case, the bank's own funds that are invested in the asset pool are treated the same as those of Unrestricted PSIA holders for profit and loss sharing purposes; however, the shareholders also receive as part of their profit the remuneration earned by the bank as asset manager (less certain expenses not chargeable to the PSIA holders). This remuneration (management fees) represents an important source of revenue and profits for Islamic banks.
From a capital market perspective, this arrangement presents an apparent anomaly, as follows: shareholders and Unrestricted PSIA holders share the same asset risk on the commingled funds, but shareholders enjoy higher returns because of the management fees. On the other hand, competitive pressure may induce the bank to forgo some of its management fees in order to pay a competitive return to its PSIA holders. In this way, some of the PSIA holders' asset risk is absorbed by the shareholders. This phenomenon has been termed "displaced commercial risk" [2].
This paper analyzes this phenomenon. We argue that, in principle, displaced commercial risk is potentially an efficient and value-creating means of sharing risks between two classes of investor with different risk diversification capabilities and preferences: wealthy shareholders who are potentially well diversified, and less wealthy PSIA holders who are not. In practice, however, Islamic banks set up reserves with the intention of minimizing any need to forgo management fees.

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Article

We generalize the arbitrage-free valuation framework for counterparty credit risk (CCR) adjustments when credit triggers are allowed in the contract. The settlement of the deal for the investor could be either obliged or optional to execute when the counterparty hits the credit trigger before any default events from the two parties. General formulas for credit value adjustment (CVA) are given for all four cases: obliged unilateral, obliged bilateral, optional unilateral and optional bilateral. The unilateral CVA with an optional credit trigger is found to be the same as the unilateral CVA with an analogous obliged credit trigger. We show that adding credit triggers will decrease the unilateral CVA for both obliged and optional cases, which are in line with the motivation of investors to reduce CCR. However, adding credit triggers may not necessarily reduce bilateral CVA. Counter-intuitively, we show that the bilateral CVA may actually increase by adding credit triggers. Moreover, the increased amount of bilateral CVA due to credit triggers for one party is exactly the same amount of bilateral CVA reduced for the other party. The CVA calculation is subjected to large uncertainty of model risks, mostly due to the lack of data for calibrating jump-to-default probabilities. Some explicit models for obliged unilateral CVA are discussed with special caveats on the model assumptions. Numerical examples are also given to illustrate the model risk of CVA calculation due to the uncertainty of jump sizes, even though pure jump models are assumed.

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Article

We propose three analytic approximation methods for numerical valuation of the guaranteed annuity options in deferred annuity pension policies. The approximation methods include the stochastic duration approach, Edgeworth expansion, and analytic approximation in affine diffusions. The payoff structure in the annuity policies is similar to a quanto call option written on a coupon-bearing bond. To circumvent the limitations of the one-factor interest rate model, we model the interest rate dynamics by a two-factor affine interest rate term structure model. The numerical accuracy and the computational efficiency of these approximation methods are analyzed. We also investigate the value sensitivity of the guaranteed annuity option with respect to different parameters in the pricing model.

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Article

We consider a stochastic factor financial model where the asset price process
and the process for the stochastic factor depend on an observable Markov chain
and exhibit an affine structure. We are faced with a finite time investment
horizon and derive optimal dynamic investment strategies that maximize the
investor's expected utility from terminal wealth. To this aim we apply Merton's
approach, as we are dealing with an incomplete market. Based on the
semimartingale characterization of Markov chains we first derive the HJB
equations, which in our case correspond to a system of coupled non-linear PDEs.
Exploiting the affine structure of the model, we derive simple expressions for
the solution in the case with no leverage, i.e. no correlation between the
Brownian motions driving the asset price and the stochastic factor. In the
presence of leverage we propose a separable ansatz, which leads to explicit
solutions in this case as well. General verification results are also proved.
The results are illustrated for the special case of a Markov modulated Heston
model.

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Article

In this paper we propose a new finite element method for pricing of bond options under time inhomogeneous one-factor affine models of short interest rates: the Hull–White model and the extended CIR model. The stability and weak convergence are established. Numerical results are presented to examine the method and to compare the calibrated models.

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Article

We consider a slight perturbation of the Hull-White short rate model and the resulting modified forward rate equation. We identify the model coefficients by using the martingale property of the normalized bond price. The forward rate and the system parameters are then estimated by using the maximum likelihood method.

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Article

Multivariate Affine term structure models have been increasingly used for pricing derivatives in fixed income markets. In these models, uncertainty of the term structure is driven by a state vector, while the short rate is an affine function of this vector. The model is characterized by a specific form for the stochastic differential equation (SDE) for the evolution of the state vector. This SDE presents restrictions on its drift term which rule out arbitrages in the market. In this paper we solve the following inverse problem: Suppose the term structure of interest rates is modelled by a linear combination of Legendre polynomials with random coefficients. Is there any SDE for these coefficients which rules out arbitrages? This problem is of particular empirical interest because the Legendre model is an example of factor model with clear interpretation for each factor, in which regards movements of the term structure. Moreover, the Affine structure of the Legendre model implies knowledge of its conditional characteristic function. From the econometric perspective, we propose arbitrage-free Legendre models to describe the evolution of the term structure. From the pricing perspective, we follow Duffie et al. [22] in exploring their conditional characteristic functions to obtain a computational tractable method to price fixed income derivatives.

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Article

We study the simplest discrete-time finite-maturity model in which default arises when the firm is not able to pay its debt obligation using the current cash-flow plus the corporate liquidity. An important distinction is made between liquidity and solvency of the firm. The corporate financial policy is simultaneously defined by the dividend policy, and the leverage policy (the coupon and the principal of the bond). When the corporate financial policy implies no default risk and no taxes, we show that the corporate financial policy is irrelevant and this irrelevance result holds for any probability measure. When the corporate financial policy implies now some default risk, we show that the value of the firm is a piecewise decreasing function of the dividend policy for any leverage policy, so that dividend policy affects the value of the firm. However, shareholders may not always have the incentives to implement this optimal dividend policy. We show that when the value of the assets is low, shareholders have an incentive to deviate from this optimal dividend policy, and we also study the resulting agency costs. We finally compare the resulting quantities of our model to the base case suggested by Huang and Huang (2003).

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Article

We continue an investigation into a class of agent-based market models that are motivated by a psychologically-plausible form of bounded rationality. Some of the agents in an otherwise efficient hypothetical market are endowed with differing tolerances to the tension caused by being in the minority. This herding tendency may be due to purely psychological effects, momentum-trading strategies, or the rational response to perverse marketplace incentives.
The resulting model has the important properties of being both very simple and insensitive to its small number of fundamental parameters. While it is most certainly a caricature market, with only boundedly rational traders and the globally available information stream being modeled directly, other market participants and effects are indirectly replicated. We show that all of the most important "stylized facts" of real market statistics are reproduced by this model.
Another useful aspect of the model is that, for certain parameter values, it reduces to a standard efficient-market system. This allows us to isolate and observe the effects of particular kinds of non-rationality. To this end, we consider the effects of different asymmetries in agent behavior and show that one in particular leads to skew statistics consistent with those seen in some real financial markets.

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Article

We pursue an inverse approach to utility theory and consumption & investment
problems. Instead of specifying an agent's utility function and deriving her
actions, we assume we observe her actions (i.e. her consumption and investment
strategies) and ask if it is possible to derive a utility function for which
the observed behaviour is optimal. We work in continuous time both in a
deterministic and stochastic setting. In the deterministic setup, we find that
there are infinitely many utility functions generating a given consumption
pattern. In the stochastic setting of the Black-Scholes complete market it
turns out that the consumption and investment strategies have to satisfy a
consistency condition (PDE) if they are to come from a classical utility
maximisation problem. We show further that important characteristics of the
agent such as her attitude towards risk (e.g. DARA) can be deduced directly
from her consumption/investment choices.

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Article

We determine the optimal dynamic investment policy for a mean quadratic variation objective function by numerical solution of a nonlinear Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). We compare the efficient frontiers and optimal investment policies for three mean variance like strategies: pre-commitment mean variance, time-consistent mean variance, and mean quadratic variation, assuming realistic investment constraints (e.g. no bankruptcy, finite shorting, borrowing). When the investment policy is constrained, the efficient frontiers for all three objective functions are similar, but the optimal policies are quite different.

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Article

The paper offers a new perspective on optimal portfolio choice by investigating how and to what extent knowledge of an investor's desirable initial investment choice can be used to determine his future optimal portfolio allocations. Optimality of investment decisions is built on the so-called forward investment performance criteria and, in particular, on the time-monotone ones. It is shown that for this class of forward criteria the desired initial allocations completely characterize the future optimal investment strategies. The analysis uses the connection between a nonlinear equation, satisfied by the local risk tolerance, and the backward heat equation. Complete solutions are provided as well as various examples.

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Article

With an alternative choice of risk criterion, we solve the HJB equation explicitly to find a closed-form solution for the optimal trade execution strategy in the Almgren–Chriss framework assuming the underlying unaffected stock price process is geometric Brownian motion.

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Article

We deal with discretization schemes for the simulation of the Heston stochastic volatility model. These simulation methods yield a popular and flexible pricing alternative for pricing and managing a book of exotic derivatives which cannot be valued using closed-form expressions. For the Heston dynamics an exact simulation method was developed by Broadie and Kaya (2006), however we argue why its practical use is limited. Instead we focus on efficient approximations of the exact scheme, aimed to resolve the disadvantages of this method; one of the main bottlenecks in the exact scheme is the simulation of the Non-central Chi-squared distributed variance process, for which we suggest an efficient caching technique. At first sight the creation of a cache containing the inverses of this distribution might seem straightforward, however as the parameter space of the inverse Non-central Chi-squared distribution is three-dimensional, the design of such a direct cache is rather complicated, as pointed out by Broadie and Andersen. Nonetheless, for the case of the Heston model we are able to tackle this dimensionality problem and show that the three-dimensional inverse of the non-central chi-squared distribution can effectively be reduced to a one dimensional cache. The performed analysis hence leads to the development of three new efficient simulation methods (the NCI, NCI-QE and BK-DI scheme). Finally, we conclude with a comprehensive numerical study of these new schemes and the exact scheme of Broadie and Kaya, the almost exact scheme of Smith, the Kahl-Jäckel scheme, the FT scheme of Lord et al. and the QE-M scheme of Andersen. From these results, we find that the QE-M scheme is the most efficient, followed closely by the NCI-M, NCI-QE-M and BK-DI-M schemes, whilst we observe that all other considered schemes perform a factor 6 to 70 times less efficient than the latter four methods.

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Article

In this paper, we analyze whether model risk/asset-specific ambiguity is an issue for institutional investors. For this purpose, we first show how model risk (which turns out to be equivalent to special cases of ambiguity) affects optimal portfolio allocation. Using average portfolio holdings for traditional and alternative asset classes of 119 institutional investors, we then calibrate our model to implicitly determine the ambiguity factors of different asset classes. We find that institutional investors are strongly ambiguity-averse, as documented by a Sharpe ratio that is only 60 percent that of an (unambiguous) efficient portfolio. In line with intuition, we document that equity and bond portfolios have a rather low ambiguity, while alternative investments such as real estate, private equity, and hedge fund investments exhibit a very high ambiguity. These results are robust with regard to the size of the expected returns supposed by the investors.

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Article

This paper proposes an alternative approach to the modeling of the interest rate term structure. It suggests that the total market price for risk is an important factor that has to be modeled carefully. The growth optimal portfolio, which is characterized by this factor, is used as reference unit or benchmark for obtaining a consistent price system. Benchmarked derivative prices are taken as conditional expectations of future benchmarked prices under the real world probability measure. The inverse of the squared total market price for risk is modeled as a square root process and shown to influence the medium and long term forward rates. With constant parameters and constant short rate the model already generates a hump shaped mean for the forward rate curve and other empirical features typically observed.

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Article

We study the perpetual American call option pricing problem in a model of a financial market in which the firm issuing a traded asset can regulate the dividend rate by switching it between two constant values. The firm dividend policy is unknown for small investors, who can only observe the prices available from the market. The asset price dynamics are described by a geometric Brownian motion with a random drift rate modeled by a continuous time Markov chain with two states. The optimal exercise time of the option for small investors is found as the first time at which the asset price hits a boundary depending on the current state of the filtering dividend rate estimate. The proof is based on an embedding of the initial problem into a two-dimensional optimal stopping problem and the analysis of the associated parabolic-type free-boundary problem. We also provide closed form estimates for the rational option price and the optimal exercise boundary.

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Article

This article develops a numerical method to price American-style Asian option in the context of the generalized autoregressive conditional heteroscedasticity (GARCH) asset return process. The development is based on dynamic programming coupled with the replacement of the normally distributed variable with a binomial one and the whole procedure is under the locally risk-neutral valuation relationship (LRNVR). We investigate the computational and implementation issues of this method and compare them with those of a candidate procedure which involves piecewise-polynomial approximation of the value function. Complexity analysis and computational results suggest that our method is superior to the candidate one and the generated GARCH option prices are capable of reflecting the changes in the conditional volatility of underlying asset.

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Article

This paper describes a fast, flexible numerical technique to price American options and generate their value surface through time. The method runs faster and more accurately than the standard CRR binomial method in practical cases and calculates options on a considerably broader family of new, useful underlying asset processes. The technique relies on the Fast Fourier Transform (FFT) to convolve a transition function for the underlying asset process. The method allows the underlying asset process to be quite general; the previously known standard geometric Brownian motion and the Variance Gamma process [8], and a novel, purely empirical transition function are compared by computing their respective American put value surface and the exercise boundaries.

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Article

Efficient numerical methods for pricing American options using Heston's stochastic volatility model are proposed. Based on this model the price of a European option can be obtained by solving a two-dimensional parabolic partial differential equation. For an American option the early exercise possibility leads to a lower bound for the price of the option. This price can be computed by solving a linear complementarity problem. The idea of operator splitting methods is to divide each time step into fractional time steps with simpler operators. This paper proposes componentwise splitting methods for solving the linear complementarity problem. The basic componentwise splitting decomposes the discretized problem into three linear complementarity problems with tridiagonal matrices. These problems can be efficiently solved using the Brennan and Schwartz algorithm, which was originally introduced for American options under the Black and Scholes model. The accuracy of the componentwise splitting method is increased by applying the Strang symmetrization. The good accuracy and the computational efficiency of the proposed symmetrized splitting method are demonstrated by numerical experiments.

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Article

Accurately as well as efficiently calculating the early exercise boundary is the key to the highly nonlinear problem of pricing American options. Many analytical approximations have been proposed in the past, aiming at improving the computational efficiency and the easiness of using the formula, while maintaining a reasonable numerical accuracy at the same time. In this paper, we shall present an approximation formula based on Bunch and Johnson's work [6]. After clearly pointing out some errors in Bunch and Johnson's paper [6], we will propose an improved approximation formula that can significantly enhance the computational accuracy, particularly for options of long lifetime.

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Article

This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston (1993), and by a Poisson jump process of the type originally introduced by Merton (1976). We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer (1998) for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen & Toivanen (2007). The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation which is taken as the benchmark. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.

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Article

The aim of this work is to present a modification of the standard binomial method which allows to price American barrier options improving the efficiency of the trinomial methods. Our approach is based on a suitable interpolation of binomial values and allows to price and hedge such options also in the critical case of near barriers. All the different types of single barrier options are considered, in the case of knock-in barriers a new implementation of the binomial method is provided.

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Article

In this paper, a new analytical formula as an approximation to the value of American put options and their optimal exercise boundary is presented. A transform is first introduced to better deal with the terminal condition and, most importantly, the optimal exercise price which is an unknown moving boundary and the key reason that valuing American options is much harder than valuing its European counterparts. The pseudo-steady-state approximation is then used in the performance of the Laplace transform, to convert the systems of partial differential equations to systems of ordinary differential equations in the Laplace space. A simple and elegant formula is found for the optimal exercise boundary as well as the option price of the American put with constant interest rate and volatility. Other hedge parameters as the derivatives of this solution are also presented.

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Article

In this paper, we investigate the generalization of the Call-Put duality equality obtained in Alfonsi and Jourdain (preprint, 2006, available at ) for perpetual American options when the Call-Put payoff (y - x)+ is replaced by ϕ(x,y). It turns out that the duality still holds under monotonicity and concavity assumptions on ϕ. The specific analytical form of the Call-Put payoff only makes calculations easier but is not crucial unlike in the derivation of the Call-Put duality equality for European options. Last, we give some examples for which the optimal strategy is known explicitly.

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Article

American options in a multi-asset market model with proportional transaction
costs are studied in the case when the holder of an option is able to exercise
it gradually at a so-called mixed (randomised) stopping time. The introduction
of gradual exercise leads to tighter bounds on the option price when compared
to the case studied in the existing literature, where the standard assumption
is that the option can only be exercised instantly at an ordinary stopping
time. Algorithmic constructions for the bid and ask prices and the associated
superhedging strategies and optimal mixed stoping times for an American option
with gradual exercise are developed and implemented, and dual representations
are established.

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Article

We develop a new approach for pricing both continuous-time and discrete-time American options which is based on the fact that any American option is equivalent to a European one with a consumption process involved. This approach admits the construction of an upper bound (a lower bound) on the true price using some lower bound (an upper bound) by Monte Carlo simulation. A number of effective estimators of upper and lower bounds with the reduced variance are proposed. The method is supported by numerical experiments which look promising.

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Article

In this paper, we present a correction to Merton (1973)'s well-known classical case of pricing perpetual American put options by considering the same pricing problem under a stochastic volatility model with the assumption that the volatility is slowly varying. Two analytic formulae for the option price and the optimal exercise price of a perpetual American put option are derived, respectively. Upon comparing the results obtained from our analytic approximations with those calculated by a spectral collocation method, it is shown that our current approximation formulae provide fast and reasonably accurate numerical values of both option price and the optimal exercise price of a perpetual American put option, within the validity of the assumption we have made for the asymptotic expansion. We shall also show that the range of applicability of our formulae is remarkably wider than it was initially aimed for, after the original assumption on the order of the "volatility of volatility" being somewhat relaxed. Based on the newly-derived formulae, the quantitative effect of the stochastic volatility on the optimal exercise strategy of a perpetual American put option has also been discussed. A most noticeable and interesting result is that there is a special cut-off value for the spot variance, below which a perpetual American put option priced under the Heston model should be held longer than the case of the same option priced under the traditional Black-Scholes model, when the price of the underlying is falling.

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In this paper, we investigate the price interdependence between seven international stock markets, namely Irish, UK, Portuguese, US, Brazilian, Japanese and Hong Kong, using a new testing method, based on the wavelet transform to reconstruct the data series, as suggested by Lee [11]. We find evidence of intra-European (Irish, UK and Portuguese) market co-movements with the US market also weakly influencing the Irish market. We also find co-movement between the US and Brazilian markets and similar intra-Asian co-movements (Japanese and Hong Kong). Finally, we conclude that the circle of impact is that of the European markets (Irish, UK and Portuguese) on both American markets (US and Brazilian), with these in turn impacting on the Asian markets (Japanese and Hong Kong) which in turn influence the European markets. In summary, we find evidence for intra-continental relationships and an increase in importance of international spillover effects since the mid 1990s, while the importance of historical transmissions has decreased since the beginning of this century.

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In this paper, we present a generic method for the Monte-Carlo pricing of (generalized) auto-callable products (aka. trigger products), i.e., products for which the payout function features a discontinuity with a (possibly) stochastic location (the trigger) and value (the payout).
The Monte-Carlo pricing of products with discontinuous payout is known to come with a high Monte-Carlo error. The numerical calculation of sensitivities (i.e., partial derivatives) of such prices by finite differences gives very noisy results, since the Monte-Carlo approximation (being a finite sum of discontinuous functions) is not smooth. Additionally, the Monte-Carlo error of the finite-difference approximation explodes as the shift size tends to zero.
Our method combines a product specific modification of the underlying numerical scheme, which is to some extent similar to an importance sampling and/or partial proxy simulation scheme and a reformulation of the payoff function into an equivalent smooth payout.
From the financial product we merely require that hitting of the stochastic trigger will result in an conditionally analytic value. Many complex derivatives can be written in this form. A class of products where this property is usually encountered are the so called auto-callables, where a trigger hit results in cancellation of all future payments except for one redemption payment, which can be valued analytically, conditionally on the trigger hit.
From the model we require that its numerical implementation allows for a calculation of the transition probability of survival (i.e., non-trigger hit). Many models allows this, e.g., Euler schemes of Itô processes, where the trigger is a model primitive.
The method presented is effective across a large range of cases where other methods fail, e.g. small finite difference shift sizes or short time to trigger reset (approaching maturity); this means that a practitioner can use this method and be confident that it will work consistently.
The method itself can be viewed as a generalization of the method proposed by Glasserman and Staum (2001), both with respect to the type (and shape) of the boundaries, as well as, with respect to the class of products considered. In addition we explicitly consider the calculation of sensitivities.

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This paper develops a unified formulation and a new computational methodology for the entire class of the multi-factor Markovian interest rate models. The early exercise premium representation for general American options is derived for all Markovian models. The option cash flow functions are decomposed into fast and slowly varying components. The fast varying components have the same expression for all options within a model. They are calculated analytically. Only the slowly varying components are option specific. Their backward induction for a finite time interval is carried out from Taylor expansion expressions. The small coefficient of the expansion is the product of the variance and the width of the time interval. The option price is calculated by dividing its time horizon into smaller intervals and numerically iterating the Taylor expansion expressions of one time interval. Other new results include: (i) The derivation of a new "almost" Markovian LIBOR market model and its related Markovian short-rate model; (ii) the universal form of the critical boundary near the maturity for the American options in the one-factor Markovian models; and (iii) approximate analytic expressions for the entire critical boundary of the American put stock option. The put price calculated from the boundary has relative precision better than 10-5.

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This article derives a series of analytic formulae for various contingent claims under the real-world probability measure using the stylised minimal market model (SMMM). This model provides realistic dynamics for the growth optimal portfolio (GOP) as a well-diversified equity index. It captures both leptokurtic returns with correct tail properties and the leverage effect. Under the SMMM, the discounted GOP takes the form of a time-transformed squared Bessel process of dimension four. From this property, one finds that the SMMM possesses a special and interesting relationship to non-central chi-square random variables with zero degrees of freedom. The analytic formulae derived under the SMMM include options on the GOP, options on exchange prices and options on zero-coupon bonds. For options on zero-coupon bonds, analytic prices facilitate efficient calculation of interest rate caps and floors.

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We use a path integral approach for solving the stochastic equations underlying the financial markets, and we show the equivalence between the path integral and the usual SDE and PDE methods. We analyze both the one-dimensional and the multi-dimensional cases, with point dependent drift and volatility, and describe a covariant formulation which allows general changes of variables. Finally we apply the method to some economic models with analytical solutions. In particular, we evaluate the expectation value of functionals which correspond to quantities of financial interest.

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