Simulating the Evolution of the Implied Distribution

Motivated by the implied stochastic volatility literature (Britten–Jones and Neuberger, forthcoming; Derman and Kani, 1997; Ledoit and Santa–Clara, 1998) this paper proposes a new and general method for constructing smile–consistent stochastic volatility models. The method is developed by recognising that option pricing and hedging can be accomplished via the simulation of the implied risk neutral distribution. We devise an algorithm for the simulation of the implied distribution, when the first two moments change over time. The algorithm can be implemented easily, and it is based on an economic interpretation of the concept of mixture of distributions. It can also be generalised to cases where more complicated forms for the mixture are assumed.     

Futures Options and the Volatility of Futures Prices

Assuming nonstochastic interest rates, European futures options are shown to be European options written on a particular asset referred to as a futures bond. Consequently, standard option pricing results may be invoked and standard option pricing techniques may be employed in the case of European futures options. Additional arbitrage restrictions on American futures options are derived. The efficiency of a number of futures option markets is examined. Assuming that at-the-money American futures options are priced accurately by Black's European futures option pricing model, the relationship between market participants' ex ante assessment of futures price volatility and the term to maturity of the underlying futures contract is also investigated empirically.     

The Term Structure of Interest Rates in a Partially Observable Economy

This paper investigates the term structure of interest rates in a multiperiod production and exchange economy with incomplete information. Unable to observe their stochastic investment opportunities, investors engage in dynamic Bayesian inference. This results in the endogenous identification of a more complex production function which generates a richer term structure, resembling the one that actual market prices imply. In addition, this paper introduces a characteristic function of the term structure and demonstrates that, in contrast with a fully observable economy, the widely investigated expectations hypothesis holds true only if interest rates are nonstochastic.     

A Complete Markovian Stochastic Volatility Model in the HJM Framework

This paper considers a stochastic volatility version of the Heath, Jarrow and and Morton (1992) term structure model. Market completeness is obtained by adapting the Hobson and Rogers (1998) complete stochastic volatility stock market model to the interest rate setting. Numerical simulation for a special case is used to compare the stochastic volatility model against the traditional Vasicek (1977) model.     

The principle of not feeling the boundary for the SABR model

The stochastic alpha–beta–rho (SABR) model is widely used in fixed income and foreign exchange markets as a benchmark. The underlying process may hit zero with a positive probability and therefore an absorbing boundary at zero should be specified to avoid arbitrage opportunities. However, a variety of numerical methods choose to ignore the boundary condition to maintain the tractability. This paper develops a new principle of not feeling the boundary to quantify the impact of this boundary condition on the distribution of underlying prices. It shows that the probability of the SABR hitting zero decays to 0 exponentially as the time horizon shrinks. Applying this principle, we further show that conditional on the volatility process, the distribution of the underlying process can be approximated by that of a time-changed Bessel process with an exponentially negligible error. This discovery provides a theoretical justification for many almost exact simulation algorithms for the SABR model in the literature. Numerical experiments are also presented to support our results.     

Computerised detection and stochastic forecast of age, period and cohort effects

In this study, we adapt and apply suitable methods of image processing and computer vision to actuarial science. In particular, we design a multistage algorithm based on the well known Canny operator with a view to detecting abrupt changes in incremental mortality development factors by age over time. These edges indicate the boundaries between areas of higher and lower mortality improvements. The computerised detection algorithm allows for a more objective judgement concerning the existence of specific mortality patterns. Furthermore, we propose a stochastic mortality forecast model that may be viewed as a Lévy process. First, objectively identified mortality patterns are removed from the matrix of mortality improvement rates. We then forecast residual mortality development factors by applying a non-parametric block bootstrap simulation. Finally, future age, period and cohort effects are superimposed on a simulation basis. Notably, our stochastic mortality model is capable of incorporating specific stress scenarios such as mortality shocks.     

Pension ratios as “correlates” of municipal pension underfunding

Unlike the corporate sector, detailed estimates of unfunded pension liabilities for most local governments are not available. Thus, prior research on the association between unfunded pension liabilities and municipal creditor decisions (Copeland and Ingram 1983; Marks and Raman 1985) has implicitly assumed that certain pension ratios are good surrogates for municipal pension underfunding. In this paper, we rely on a theoretical model by Ehrenberg (1980) to test empirically the appropriateness of pension ratios as “correlates” of municipal pension underfunding. These ratios were found to be correlated with pension underfunding, although they accounted for only about 30 percent of the variance in the underfunding variable.     

Model-free hedge ratios and scale-invariant models

A price process is scale-invariant if and only if the returns distribution is independent of the price measurement scale. We show that most stochastic processes used for pricing options on financial assets have this property and that many models not previously recognised as scale-invariant are indeed so. We also prove that price hedge ratios for a wide class of contingent claims under a wide class of pricing models are model-free. In particular, previous results on model-free price hedge ratios of vanilla options based on scale-invariant models are extended to any contingent claim with homogeneous pay-off, including complex, path-dependent options. However, model-free hedge ratios only have the minimum variance property in scale-invariant stochastic volatility models when price–volatility correlation is zero. In other stochastic volatility models and in scale-invariant local volatility models, model-free hedge ratios are not minimum variance ratios and our empirical results demonstrate that they are less efficient than minimum variance hedge ratios.     

Pricing exotic options using MSL-MC

Today, better numerical approximations are required for multi-dimensional SDEs to improve on the poor performance of the standard Monte Carlo pricing method. With this aim in mind, this paper presents a method (MSL-MC) to price exotic options using multi-dimensional SDEs (e.g. stochastic volatility models). Usually, it is the weak convergence property of numerical discretizations that is most important, because, in financial applications, one is mostly concerned with the accurate estimation of expected payoffs. However, in the recently developed Multilevel Monte Carlo path simulation method (ML-MC), the strong convergence property plays a crucial role. We present a modification to the ML-MC algorithm that can be used to achieve better savings. To illustrate these, various examples of exotic options are given using a wide variety of payoffs, stochastic volatility models and the new Multischeme Multilevel Monte Carlo method (MSL-MC). For standard payoffs, both European and Digital options are presented. Examples are also given for complex payoffs, such as combinations of European options (Butterfly Spread, Strip and Strap options). Finally, for path-dependent payoffs, both Asian and Variance Swap options are demonstrated. This research shows how the use of stochastic volatility models and the θ scheme can improve the convergence of the MSL-MC so that the computational cost to achieve an accuracy of O(ε) is reduced from O(ε^?3) to O(ε^?2) for a payoff under global and non-global Lipschitz conditions.     

Pricing via recursive quantization in stochastic volatility models

We provide the first recursive quantization-based approach for pricing options in the presence of stochastic volatility. This method can be applied to any model for which an Euler scheme is available for the underlying price process and it allows one to price vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein [Rev. Financ. Stud. 1991, (4), 727–752] model and the SABR model introduced in Hagan et al. [Wilmott Mag., 2002, 84–108]. A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation.     

A Statistical Basis for Claims Experience Monitoring

Abstract

By claims experience monitoring is meant the systematic comparison of the forecasts from a claims model with claims experience as it emerges subsequently. In the event that the stochastic properties of the forecasts are known, the comparison can be represented as a collection of probabilistic statements. This is stochastic monitoring. This paper defines this process rigorously in terms of statistical hypothesis testing. If the model is a regression model (which is the case for most stochastic claims models), then the natural form of hypothesis test is a number of likelihood ratio tests, one for each parameter in the valuation model. Such testing is shown to be very easily implemented by means of generalized linear modeling software. This tests the formal structure of the claims model and is referred to as microtesting. There may be other quantities (e.g., amount of claim payments in a defined interval) that require testing for practical reasons. This sort of testing is referred to as macrotesting, and its formulation is also discussed.     

Structural models of credit risk are useful: Evidence from hedge ratios on corporate bonds

Structural models of credit risk provide poor predictions of bond prices. We show that, despite this, they provide quite accurate predictions of the sensitivity of corporate bond returns to changes in the value of equity (hedge ratios). This is important since it suggests that the poor performance of structural models may have more to do with the influence of non-credit factors rather than their failure to capture the credit exposure of corporate debt. The main result of this paper is that even the simplest of the structural models [Merton, R., 1974. On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance 29, 449–470] produces hedge ratios that are not rejected in time-series tests. However, we find that the Merton model (with or without stochastic interest rates) does not capture the interest rate sensitivity of corporate debt, which is substantially lower than would be expected from conventional duration measures. The paper also shows that corporate bond prices are related to a number of market-wide factors such as the Fama-French SMB (small minus big) factor in a way that is not predicted by structural models.     

Stochastic duration and fast coupon bond option pricing in multi-factor models

Generalizing Cox, Ingersoll, and Ross (1979), this paper defines the stochastic duration of a bond in a general multi-factor diffusion model as the time to maturity of the zero-coupon bond with the same relative volatility as the bond. Important general properties of the stochastic duration measure are derived analytically, and the stochastic duration is studied in detail in various well-known models. It is also demonstrated by analytical arguments and numerical examples that the price of a European option on a coupon bond (and, hence, of a European swaption) can be approximated very accurately by a multiple of the price of a European option on a zero-coupon bond with a time to maturity equal to the stochastic duration of the coupon bond. This revised version was published online in June 2006 with corrections to the Cover Date.     

Financial Ratio Adjustment: Industry-Wide Effects or Strategic Management

This paper proposes an alternative model for analyzing financial ratio behavior. The model postulates that (1) firms' financial ratios reflect unexpected changes in industry conditions; and (2) managers attempt to move their financial ratio toward the long-run desirable target. This model is employed to assess the relative weights of financial ratio movement that are associated with these two forces. The results show that changes in financial ratios can be due to both external shocks and strategic adjustment by management. The amount of financial ratio smoothing due to strategic adjustment appears to be substantial. Furthermore, the speed of convergence toward the optimal targets varies across industries and firms of different size.     

An Induced Theory of Financial Ratios

In recent years, a considerable volume of research has emerged on the analytical, empirical and statistical properties of financial ratios. In the present paper the methods of continuous time stochastic calculus are applied to the modelling of financial ratios. Two standard stochastic processes are examined and applied, namely the geometric Brownian motion and the elastic random walk. The properties of aggregate or ‘industry ratios’ and the reasonableness or otherwise of the ‘proportionality assumption’ are also examined. Typically, the approach of this paper implies that accounting ratios will be non-linear functions of time and normality will be the exception rather than the rule.     

The Use of Dynamic Financial Analysis to Determine Whether an Optimal Growth Rate Exists for a Property-Liability Insurer

Prior research on the aging phenomenon has demonstrated that new business for property‐liability (P‐L) insurers generates high loss ratios that gradually decline as a book of business goes through successive renewal cycles. Although the experience on new business is initially unprofitable, the renewal book of business eventually becomes profitable over time. Within this context, insurers need to manage their exposure growth in order to maximize long run profitability. Dynamic financial analysis (DFA), a relatively new tool for P‐L insurers, utilizes Monte Carlo simulation to generate the overall financial results for an insurer under a large number of scenarios. This article uses a publicly available DFA model—along with the estimated market value of an insurer, based on 1990–2001 data for stock P‐L insurers and underlying financial variables—to determine optimal growth rates of a P‐L insurer based on mean–variance analysis, stochastic dominance, and constraints on leverage.     

A displaced-diffusion stochastic volatility LIBOR market model: motivation,definition and implementation

Abstract

We present an extension of the LIBOR market model which allows for stochastic instantaneous volatilities of the forward rates in a displaced-diffusion setting. We show that virtually all the powerful and important approximations that apply in the deterministic setting can be successfully and naturally extended to the stochastic volatility case. In particular we show that (i) the caplet market can still be efficiently and accurately fit; (ii) that the drift approximations that allow the evolution of the forward rates over time steps as long as several years are still valid; (iii) that in the new setting the European swaption matrix implied by a given choice of volatility parameters can be efficiently approximated with a closed-form expression without having to carry out a Monte Carlo simulation for the forward rate process; and (iv) that it is still possible to calibrate the model virtually perfectly via simply matrix manipulations so that the prices of the co-terminal swaptions underlying a given Bermudan swaption will be exactly recovered, while retaining a desirable behaviour for the evolution of the term structure of volatilities.     

Government Deposit Insurance and the Diamond-Dybvig Model

The apparent banking market failure modeled by Diamond and Dybvig [1983] rests on their inconsistently applying their sequential servicing constraint to private banks but not to their government deposit insurance agency. Without this inconsistency, banks can provide optimal risk-sharing without tax-based deposit insurance, even when the number of type 1 agents is stochastic, by employing a contingent bonus contract. The threat of disintermediation noted by Jacklin [1987] in the nonstochastic case is still present but can be blocked by contractual trading restrictions. This article complements Wallace [1988], who considers an alternative resolution of this inconsistency.     

On the ‘Steady State’ Properties of Financial Ratios

This paper uses the methods of continuous time stochastic calculus to investigate the ‘steady state’ properties of financial ratios. Basing our analysis on previous work in the area, we show that, if a financial ratio can be characterised as a diffusion process which possesses an asymptotic equilibrium, then the Fokker-Kolmogorov-Planck forward equation may be used to ‘retrieve’ its probability density. The approach is ‘flexible’ enough to incorporate a wide variety of density functions, many of which have not been investigated in the literature. We demonstrate the procedures which may be used to derive both the cross-sectional and time series tests implied by these distributions. The paper also includes a section dealing with the methods which may be used for parameter estimation, once the underlying distribution has been determined.