Robust pricing and hedging under trading restrictions and the emergence of local martingale models

We pursue the robust approach to pricing and hedging in which no probability measure is fixed, but call or put options with different maturities and strikes can be traded initially at their market prices. We allow the inclusion of robust modelling assumptions by specifying a set of feasible paths on which (super)hedging arguments are required to work. In a discrete-time setup with no short selling, we characterise absence of arbitrage and show that if call options are traded, then the usual pricing–hedging duality is preserved. In contrast, if only put options are traded, a duality gap may appear. Embedding the results into a continuous-time framework, we show that the duality gap may be interpreted as a financial bubble and link it to strict local martingales. This provides an intrinsic justification of strict local martingales as models for financial bubbles arising from a combination of trading restrictions and current market prices.     

Stock option use by Australian IPOs

This paper examines the use of stock options by Australian IPOs. Stock options in this setting exhibit heterogeneity of ‘in the moneyness’ attributes. We can therefore identify incentive ‘out of the money’ options and reward ‘at or in the money’ options and study their usage. The setting also features options granted to employees (CEO, executives, non-executive directors, and managers) and external parties (e.g., underwriters, lenders, seed capitalists, and promoters). We examine the motives for granting stock options with different ‘in the moneyness’ attributes to employees and external parties, as well as the performance outcomes. Our results suggest ‘in the moneyness’ attributes of options vary according to uncertainties about growth options, general operating risks, and/or agency risks relating to the use of IPO proceeds; however, these relations do not translate into robust links from the option grants to future performance.     

Valuing debt options: Empirical evidence

A two-factor model using the instantaneous rate of interest and the return on a consol bond to describe the term structure of interest rates — the Brennan-Schwartz model — is used to derive theoretical prices for American call and put options on US government bonds and treasury bills. These model prices are then compared with market prices. The theoretical model used to value the dept options also provides hedge ratios which may be used to construct zero-investment portfolios which, in theory, are perfectly riskless. Several trading strategies based on these ‘riskless’ portfolios are examined.     

The options market response to accounting earnings announcements

We examine the reaction of the equity options market to accounting earnings announcements over the period 1996–2008 using changes in implied volatility to measure the options market response to earnings news. We find that positive earnings surprises and positive profit announcements produce a larger uncertainty resolution than negative earnings surprises and loss announcements. We demonstrate an inverse relation between the change in implied volatility and earnings news in a three-day window immediately after an earnings announcement. We refer to the magnitude of this relation as the ‘options market earnings response coefficient’. This ‘options market earnings response coefficient’ is stronger for both bad news announcements and positive profit announcements. We do not find any significant relation between changes in implied volatility and earnings news in the pre- or post-announcement periods. We conclude that the options market efficiently absorbs earnings information.     

No-arbitrage up to random horizon for quasi-left-continuous models

This paper studies the impact, on no-arbitrage conditions, of stopping the price process at an arbitrary random time. As price processes, we consider the class of quasi-left-continuous semimartingales, i.e., semimartingales that do not jump at predictable stopping times. We focus on the condition of no unbounded profit with bounded risk (called NUPBR), also known in the literature as no arbitrage of the first kind. The first principal result describes all the pairs of quasi-left-continuous market models and random times for which the resulting stopped model fulfils NUPBR. Furthermore, for a subclass of quasi-left-continuous local martingales, we construct explicitly martingale deflators, i.e., strictly positive local martingales whose product with the price process stopped at a random time is a local martingale. The second principal result characterises the random times that preserve NUPBR under stopping for any quasi-left-continuous model. The analysis carried out in the paper is based on new stochastic developments in the theory of progressive enlargements of filtrations.     

A pure martingale dual for multiple stopping

In this paper, we present a dual representation for the multiple stopping problem, hence multiple exercise options. As such, it is a natural generalization of the method in Rogers (Math. Finance 12:271–286, 2002) and Haugh and Kogan (Oper. Res. 52:258–270, 2004) for the standard stopping problem for American options. We term this representation a ‘pure martingale’ dual as it is solely expressed in terms of an infimum over martingales rather than an infimum over martingales and stopping times as in Meinshausen and Hambly (Math. Finance 14:557–583, 2004). For the multiple dual representation, we propose Monte Carlo simulation methods which require only one degree of nesting.     

Pricing a class of exotic commodity options in a multi-factor jump-diffusion model

In a recent paper, Crosby introduced a multi-factor jump-diffusion model which would allow futures (or forward) commodity prices to be modelled in a way which captured empirically observed features of the commodity and commodity options markets. However, the model focused on modelling a single individual underlying commodity. In this paper, we investigate an extension of this model which would allow the prices of multiple commodities to be modelled simultaneously in a simple but realistic fashion. We then price a class of simple exotic options whose payoff depends on the difference (or ratio) between the prices of two different commodities (for example, spread options), or between the prices of two different (i.e. with different tenors) futures contracts on the same underlying commodity, or between the prices of a single futures contract as observed at two different calendar times (for example, forward start or cliquet options). We show that it is possible, using a Fourier transform-based algorithm, to derive a single unifying form for the prices of all these aforementioned exotic options and some of their generalizations. Although we focus on pricing options within the model of Crosby, most of our results would be applicable to other models where the relevant ‘extended’ characteristic function is available in analytical form.     

Cutting to the core? A reflection upon recent education policy debates within the Institute of Chartered Accountants in England and Wales

Knowledge is widely regarded as a characteristic of professions, but given the ever-increasing knowledge base, professional accountancy bodies have begun to question whether professional examinations can continue to cover all areas relevant to the work of accountants. At the heart of the debate about how the professional knowledge base is to be defined lie questions about how chartered accountants are ‘made’. This paper discusses the introduction of a ‘core and options’ model for professional accountancy education as a possible means of ‘making the chartered accountant’. Using Porter's (1981) theory of historical explanation, it discusses an episode in the recent history of education policy at The Institute of Chartered Accountants in England and Wales (ICAEW) which illuminates aspects of the core and options model. The paper concludes that, while the introduction of core curricula in accounting education can be justified on educational grounds, the rejection of core curricula by ICAEW suggests that educational debates were strongly influenced by the wider political, economic, social and professional environments and the resultant educational policy can be viewed as the product of a variety of competing agendas.     

Forward-neutral valuation relationships for options on zero coupon bonds

This paper extends the literature on Risk-Neutral Valuation Relationships (RNVRs) to derive valuation formulae for options on zero coupon bonds when interest rates are stochastic. We develop Forward-Neutral Valuation Relationships (FNVRs) for the transformed-bounded random walk class. Our transformed-bounded random walk family of forward bond price processes implies that (i) the prices of the zero coupon bonds are bounded below at zero and above at one, and (ii) negative continuously compounded interest rates are ruled out. FNVRs are frameworks for option pricing, where the forward prices of the options are martingales independent of the market prices of risk. We illustrate the generality and flexibility of our approach with models that yield several new closed-form solutions for call and put options on discount bonds.     

Implied volatility skews and stock return skewness and kurtosis implied by stock option prices

The Black-Scholes* option pricing model is commonly applied to value a wide range of option contracts. However, the model often inconsistently prices deep in-the-money and deep out-of-the-money options. Options professionals refer to this well-known phenomenon as a volatility ‘skew’ or ‘smile’. In this paper, we examine an extension of the Black-Scholes model developed by Corrado and Su that suggests skewness and kurtosis in the option-implied distributions of stock returns as the source of volatility skews. Adapting their methodology, we estimate option-implied coefficients of skewness and kurtosis for four actively traded stock options. We find significantly nonnormal skewness and kurtosis in the option-implied distributions of stock returns.     

Bivariate normal mixture spread option valuation

We discuss the pricing and hedging of European spread options on correlated assets when the marginal distribution of each asset return is assumed to be a mixture of normal distributions. Being a straightforward two-dimensional generalization of a normal mixture diffusion model, the prices and hedge ratios have a firm behavioural and theoretical foundation. In this ‘bivariate normal mixture’ (BNM) model no-arbitrage option values are just weighted sums of different ‘2GBM’ option values that are based on the assumption of two correlated lognormal diffusions, and likewise for their sensitivities. The main advantage of this approach is that BNM option values are consistent with both volatility smiles and with the implied correlation ‘frown’. No other ‘frown consistent’ spread option valuation model has such straightforward implementation. We apply analytic approximations to compare BNM valuations of European spread options with those based on the 2GBM assumption and explain the differences between the two as a weighted sum of six second-order 2GBM sensitivities. We also examine BNM option sensitivities, finding that these, like the option values, can sometimes differ substantially from those obtained under the 2GBM model. Finally, we show how the correlation frown that is implied by the BNM model is affected as we change (a) the correlation structure and (b) the tail probabilities in the joint density of the asset returns.     

Unspanned stochastic volatility and fixed income derivatives pricing

We propose a parsimonious ‘unspanned stochastic volatility’ model of the term structure and study its implications for fixed-income option prices. The drift and quadratic variation of the short rate are affine in three state variables (the short rate, its long-term mean and variance) which follow a joint Markov (vector) process. Yet, bond prices are exponential affine functions of only two state variables, independent of the current interest rate volatility level. Because this result holds for an arbitrary volatility process, such a process can be calibrated to match fixed income derivative prices. Furthermore, this model can be ‘extended’ (by relaxing the time-homogeneity) to fit any arbitrary term structure. In its ‘HJM’ form, this model nests the analogous stochastic equity volatility model of Heston (1993) [Heston, S.L., 1993. A closed form solution for options with stochastic volatility. Review of Financial Studies 6, 327–343]. In particular, if the volatility process is specified to be affine, closed-form solutions for interest rate options obtain. We propose an efficient algorithm to compute these prices. An application using data on caps and floors shows that the model can capture very well the implied Black spot volatility surface, while simultaneously fitting the observed term structure.     

A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options

Vanilla (standard European) options are actively traded on many underlying asset classes, such as equities, commodities and foreign exchange (FX). The market quotes for these options are typically used by exotic options traders to calibrate the parameters of the (risk-neutral) stochastic process for the underlying asset. Barrier options, of many different types, are also widely traded in all these markets but one important feature of the FX options markets is that barrier options, especially double-no-touch (DNT) options, are now so actively traded that they are no longer considered, in any way, exotic options. Instead, traders would, in principle, like to use them as instruments to which they can calibrate their model. The desirability of doing this has been highlighted by talks at practitioner conferences but, to our best knowledge (at least within the realm of the published literature), there have been no models which are specifically designed to cater for this. In this paper, we introduce such a model. It allows for calibration in a two-stage process. The first stage fits to DNT options (or other types of double barrier options). The second stage fits to vanilla options. The key to this is to assume that the dynamics of the spot FX rate are of one type before the first exit time from a ‘corridor’ region but are allowed to be of a different type after the first exit time. The model allows for jumps (either finite activity or infinite activity) and also for stochastic volatility. Hence, not only can it give a good fit to the market prices of options, it can also allow for realistic dynamics of the underlying FX rate and realistic future volatility smiles and skews. En route, we significantly extend existing results in the literature by providing closed-form (up to Laplace inversion) expressions for the prices of several types of barrier options as well as results related to the distribution of first passage times and of the ‘overshoot’.     

New analytical option pricing models with Weyl–Titchmarsh theory

Abstract

In this paper we consider the valuation of Bermudan callable derivatives with multiple exercise rights. We present in this context a new primal–dual linear Monte Carlo algorithm that allows for efficient simulation of the lower and upper price bounds without using nested simulations (hence the terminology). The algorithm is essentially an extension of the primal–dual Monte Carlo algorithm for standard Bermudan options proposed by Schoenmakers et al. [SIAM J. Finance Math., 2013, 4, 86–116] to the case of multiple exercise rights. In particular, the algorithm constructs upwardly a system of dual martingales to be plugged into the dual representation of Schoenmakers. At each level, the respective martingale is constructed via a backward regression procedure starting at the last exercise date. The thus constructed martingales are finally used to compute an upper price bound. The algorithm also provides approximate continuation functions that may be used to construct a price lower bound. The algorithm is applied to the pricing of flexible caps in a Hull and White model setup. The simple model choice allows for comparison of the computed price bounds with the exact price obtained by means of a trinomial tree implementation. As a result, we obtain tight price bounds for the considered application. Moreover, the algorithm is generically designed for multi-dimensional problems and is tractable to implement.     

Portfolio sensitivity to changes in the maximum and the maximum drawdown

In this article, we define new ‘Greeks’ for financial derivatives: sensitivities to the running maximum and the running maximum drawdown of an underlying asset. Some types of portfolios, such as the net asset value of a hedge fund or performance fees, are sensitive to these parameters. In order to illustrate the concept of the new ‘Greeks’, we derive probabilistic representations of sensitivities for two classes of financial contracts: forwards on the maximum drawdown and lookback options. These results allow us to interpret the delta-hedge of the contracts in a novel way.     

Unifying Black–Scholes Type Formulae Which Involve Brownian Last Passage Times up to a Finite Horizon

The authors recently discovered some interesting relations between the Black–Scholes formula and last passage times of the Brownian exponential martingales, which invites one to seek analogous results for last passage times up to a finite horizon. This is achieved in the present paper, where Yuri’s formula, as originally presented in Akahori et al. (On the pricing of options written on the last exit time, 2008), is also derived.     

Investment timing and optimal capital structure under liquidity risk

Deterioration in debt market liquidity reduces debt values and affects firms' decisions. Considering such risk, we develop an investment timing model and obtain analytic solutions. We carry out a comprehensive analysis in optimal financing, default, and investment strategies, and stockholder–bondholder conflicts. Our model explains stylized facts and replicates empirical findings in credit spreads. We obtain six new insights for decision makers. We propose a ‘new trade-off theory’ for optimal capital structure, a new tax effect, and new explanations of ‘debt conservatism puzzle’ and ‘zero-leverage puzzle’. Failure in recognizing liquidity risk results in substantially over-leveraging, early bankruptcy or investment, overpriced options, and undervalued coupons and credit spreads. In addition, agency costs are surprisingly small for a high liquidity risk or a low project risk. Interestingly, the risk shifting incentive and debt overhang problem decrease with liquidity risk under moderate tax rates while they increase under high tax rates.     

Multilevel dual approach for pricing American style derivatives

In this article we propose a novel approach to reduce the computational complexity of the dual method for pricing American options. We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation into a sum of representations that correspond to different levels of approximation to the target martingale. By next replacing in each representation true conditional expectations with their Monte Carlo estimates, we arrive at what one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm reveals that the computational complexity of getting the corresponding target upper bound, due to the target martingale, can be significantly reduced. In particular, it turns out that using our new approach, we may construct a multilevel version of the well-known nested Monte Carlo algorithm of Andersen and Broadie (Manag. Sci. 50:1222–1234, 2004) that is, regarding complexity, virtually equivalent to a non-nested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example.     

The Perceived Importance of an Ethical Issue as a Determinant of Ethical Decision-making for Accounting Students in an Academic Setting

In this study, we investigate the impact of an ethical decision-making construct, the ‘perceived importance of an ethical issue’ (PIE) for accounting students in an academic setting. One objective was to test the measure of this construct for generalizability in gauging ethical decision- making. The other objective was to draw inferences on the PIE construct for pedagogical purposes. Using tests of validity, the measure reasonably captured the ‘perceived importance of an ethical issue’ construct. In addition, the PIE construct was significantly related to both ethical judgment and behavioural intention. Implications for accounting educators and ethics training in business are discussed.     

The Impact of Irreversibility,Uncertainty and Timing Options on Deprival Valuations and the Detection of Monopoly Profits

This paper has two purposes. The first is to derive rules identifying the deprival value of an asset (i) which is irreversible to one extent or another; (ii) the benefit stream of which is subject to continuing uncertainty; and (iii) for which an option to wait exists as to when to reacquire should the owner be deprived of it. The second is to consider whether accounting rates of return employing these deprival value rules can be developed to help in the detection of monopoly profits in circumstances where investment decision-making takes place in the presence of irreversibility, uncertainty and the existence of timing options. The ‘new’ deprival value rules for an asset differ from the ‘conventional’ ones in that present value less the value of the option to reinvest in the asset appears in the ‘new’ rules wherever present value appears in the ‘conventional’ rules. Examples are provided which suggest that ‘new’ and ‘conventional’ deprival value rules can differ materially. A further result is that accounting rates of return can be developed using the ‘new’ deprival value rules that are, in principle, useful in the detection of monopoly profits. Nonetheless, in practice such use requires a level of information that renders the result superfluous in the sense that the provision of replacement cost balance sheet data, combined with the level of information needed, is sufficient to reveal the presence of monopoly profits directly.