PRICING DISCRETELY MONITORED BARRIER OPTIONS AND DEFAULTABLE BONDS IN LÉVY PROCESS MODELS: A FAST HILBERT TRANSFORM APPROACH

This paper presents a novel method to price discretely monitored single- and double-barrier options in Lévy process-based models. The method involves a sequential evaluation of Hilbert transforms of the product of the Fourier transform of the value function at the previous barrier monitoring date and the characteristic function of the (Esscher transformed) Lévy process. A discrete approximation with exponentially decaying errors is developed based on the Whittaker cardinal series (Sinc expansion) in Hardy spaces of functions analytic in a strip. An efficient computational algorithm is developed based on the fast Hilbert transform that, in turn, relies on the FFT-based Toeplitz matrix–vector multiplication. Our method also provides a natural framework for credit risk applications, where the firm value follows an exponential Lévy process and default occurs at the first time the firm value is below the default barrier on one of a discrete set of monitoring dates.     

TIME‐CHANGED MARKOV PROCESSES IN UNIFIED CREDIT‐EQUITY MODELING

This paper develops a novel class of hybrid credit‐equity models with state‐dependent jumps, local‐stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time‐changed Markov diffusion process with state‐dependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with state‐dependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has local‐stochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state‐dependent jumps, local‐stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time‐changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump‐to‐default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local‐stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.     

New forms of labour contract and labour flexibility in Russia

This paper assesses the claims that employment in the new Russian private sector relies heavily on informal and unregistered labour agreements and that the violation of existing labour law by new private employers is driven by their need for more flexible working arrangements.
The paper shows that these claims are unsupported on every count. The new private sector does not rely heavily on informal or illegal forms of employment and there is no evidence that it uses labour more flexibly, in any of the usual senses of the term, than the traditional sectors of the economy. The conclusion is that there is no economic justification for the systematic violation of the existing labour legislation. It is suggested, therefore, that enforcement of the existing law is a precondition for its effective reform in those areas in which it might be a barrier to restructuring.     

Higher-order improvements of the parametric bootstrap for long-memory Gaussian processes

This paper determines coverage probability errors of both delta method and parametric bootstrap confidence intervals (CIs) for the covariance parameters of stationary long-memory Gaussian time series. CIs for the long-memory parameter _d0$d_0$ are included. The results establish that the bootstrap provides higher-order improvements over the delta method. Analogous results are given for tests. The CIs and tests are based on one or other of two approximate maximum likelihood estimators. The first estimator solves the first-order conditions with respect to the covariance parameters of a “plug-in” log-likelihood function that has the unknown mean replaced by the sample mean. The second estimator does likewise for a plug-in Whittle log-likelihood.     

Nonlinearity,nonstationarity, and spurious forecasts

Implications of nonlinearity, nonstationarity, and misspecification are considered from a forecasting perspective. Our model allows for small departures from the martingale difference sequence hypothesis by including a nonlinear component, formulated as a general, integrable transformation of the I(1)$I(1)$ predictor. We assume that the true generating mechanism is unknown to the econometrician and he is therefore forced to use some approximating functions. It is shown that in this framework the linear regression techniques lead to spurious forecasts. Improvements of the forecast accuracy are possible with properly chosen nonlinear transformations of the predictor. The paper derives the limiting distribution of the forecasts’ mean squared error (MSE). In the case of square integrable approximants, it depends on the _L2$L_2$-distance between the nonlinear component and approximating function. Optimal forecasts are available for a given class of approximants.     

Computing exponential moments of the discrete maximum of a Lévy process and lookback options

We present a fast and accurate method to compute exponential moments of the discretely observed maximum of a Lévy process. The method involves a sequential evaluation of Hilbert transforms of expressions involving the characteristic function of the (Esscher-transformed) Lévy process. It can be discretized with exponentially decaying errors of the form O(exp (−aM ^b )) for some a,b>0, where M is the number of discrete points used to compute the Hilbert transform. The discrete approximation can be efficiently implemented using the Toeplitz matrix–vector multiplication algorithm based on the fast Fourier transform, with total computational cost of O(NMlog (M)), where N is the number of observations of the maximum. The method is applied to the valuation of European-style discretely monitored floating strike, fixed strike, forward start and partial lookback options (both newly written and seasoned) in exponential Lévy models. This research was supported by the National Science Foundation under grant DMI-0422937.     

On the first hitting time density for a reducible diffusion process

In this paper, we study the classical problem of the first hitting time density to a moving boundary for a diffusion process, which satisfies the Cherkasov condition, and hence, can be reduced to a standard Wiener process. We give two complementary (forward and backward) formulations of this problem and provide semi-analytical solutions for both. By using the method of heat potentials, we show how to reduce these problems to linear Volterra integral equations of the second kind. For small values of t, we solve these equations analytically by using Abel equation approximation; for larger t we solve them numerically. We illustrate our method with representative examples, including Ornstein–Uhlenbeck processes with both constant and time-dependent coefficients. We provide a comparison with other known methods for finding the hitting density of interest, and argue that our method has considerable advantages and provides additional valuable insights. We also show applications of the problem and our method in various areas of financial mathematics.     

IS DIVERSIFICATION A JOB SAFETY NET FOR SELL-SIDE ANALYSTS?

We study sell-side analysts’ motives to diversify their portfolios across industries. Despite the negative association between diversification and accuracy, more than 60% of analysts cover multiple industries. We argue that analysts’ choice to diversify is rooted in concerns about future job security. We find that more diversified analysts are less likely to experience job turnover and leave the profession but are not more likely to advance their careers. For experienced and all-star analysts, diversification does not improve career outcomes. We conclude that industry diversification is a safety mechanism for inexperienced and unranked analysts who are concerned about job security.