The Czech Treasury Yield Curve from 1999 to the Present

The author estimates the Czech Treasury yield curve at a daily frequency from 1999 to the present. He uses the parsimonious yield curve model of Nelson and Siegel (1987), for which he suggests a parameter restriction that avoids abrupt changes in parameter estimates and thus allows for the economic interpretation of the model to hold. The estimation of the model parameters is based on market prices of Czech government bonds. The Nelson-Siegel model is shown to fit the Czech bond price data well without being over-parameterized. Thus, the model provides an accurate and consistent picture of the Czech Treasury yield curve evolution. The estimated parameters can be used to calculate spot rates and hence par rates, forward rates or the discount function for practically any maturity. To eh author´s knowledge, consistent time series of spot rates are not available for the Czech economy.yield curve, spot rates, treasury market, Nelson-Siegel

Executive stock option exercise with full and partial information on a drift change point

We analyse the optimal exercise of an executive stock option (ESO) written on a stock whose drift parameter falls to a lower value at a change point, an exponentially distributed random time independent of the Brownian motion driving the stock. Two agents, who do not trade the stock, have differing information on the change point, and seek to optimally exercise the option by maximising its discounted payoff under the physical measure. The first agent has full information, and observes the change point. The second agent has partial information and filters the change point from price observations. This scenario is designed to mimic the positions of two employees of varying seniority, a fully informed executive and a partially informed less senior employee, each of whom receives an ESO. The partial information scenario yields a model under the observation filtration $\widehat{\mathbb{F}}$ in which the stock drift becomes a diffusion driven by the innovations process, an $\widehat{\mathbb{F}}$-Brownian motion also driving the stock under $\widehat{\mathbb{F}}$, and the partial information optimal stopping value function has two spatial dimensions. We rigorously characterise the free boundary PDEs for both agents, establish shape and regularity properties of the associated optimal exercise boundaries, and prove the smooth pasting property in both information scenarios, exploiting some stochastic flow ideas to do so in the partial information case. We develop finite difference algorithms to numerically solve both agents' exercise and valuation problems and illustrate that the additional information of the fully informed agent can result in exercise patterns which exploit the information on the change point, lending credence to empirical studies which suggest that privileged information of bad news is a factor leading to early exercise of ESOs prior to poor stock price performance.Comment: 48 pages, final version, accepted for publication in SIAM Journal on Financial Mathematic

Modely finančních časových řad a jejich aplikace

I study, develop and implement selected interest rate models. I begin with a simple categorization of interest rate models and with an explanation why interest rate models are useful. I explain and discuss the notion of arbitrage. I use Oldrich Vasicek's seminal model (Vasicek; 1977) to develop the idea of no-arbitrage term structure modeling. I introduce both the partial di erential equation and the risk-neutral approach to zero-coupon bond pricing. I briefly comment on affine term structure models, a general equilibrium term structure model, and HJM framework. I present the Czech Treasury yield curve estimates at a daily frequency from 1999 to the present. I use the parsimonious Nelson-Siegel model (Nelson and Siegel; 1987), for which I suggest a parameter restriction that avoids abrupt changes in parameter estimates and thus allows for the economic interpretation of the model to hold. The Nelson-Siegel model is shown to fit the Czech bond price data well without being over-parameterized. Thus, the model provides an accurate and consistent picture of the Czech Treasury yield curve evolution. The estimated parameters can be used to calculate spot rates and hence par rates, forward rates or discount function for practically any maturity. To my knowledge, consistent time series of spot rates are not available for the Czech economy. I introduce two estimation techniques of the short-rate process. I begin with the maximum likelihood estimator of a square root diff usion. A square root di usion serves as the short rate process in the famous CIR model (Cox, Ingersoll and Ross; 1985b). I develop and analyze two Matlab implementations of the estimation routine and test them on a three-month PRIBOR time series. A square root diff usion is a restricted version of, so called, CKLS di ffusion (Chan, Karolyi, Longsta and Sanders; 1992). I use the CKLS short-rate process to introduce the General Method of Moments as the second estimation technique. I discuss the numerical implementation of this method. I show the importance of the estimator of the GMM weighting matrix and question the famous empirical result about the volatility speci cation of the short-rate process. Finally, I develop a novel yield curve model, which is based on principal component analysis and nonlinear stochastic di erential equations. The model, which is not a no-arbitrage model, can be used in areas, where quantification of interest rate dynamics is needed. Examples, of such areas, are interest rate risk management, or the pro tability and risk evaluation of interest rate contingent claims, or di erent investment strategies. The model is validated by Monte Carlo simulations

Mean-variance hedging of contingent claims with random maturity

We study the mean-variance hedging of an American-type contingent claim that is exer- cised at a random time in a Markovian setting. This problem is motivated by applications in the areas of employee stock option valuation, credit risk or equity-linked life insurance policies with an underlying risky asset value guarantee. Our analysis is based on dynamic programming and uses PDE techniques. In particular, we prove that the complete solution to the problem can be expressed in terms of the solution to a system of one quasi-linear parabolic PDE and two linear parabolic PDEs. Using a suitable iterative scheme involving linear parabolic PDEs and Schauder's interior estimates for parabolic PDEs, we show that each of these PDEs has a classical C1;2 solution. Using these results, we express the claim's mean-variance hedging value that we derive as its expected discounted payoff with respect to an equivalent martingale measure that does not coincide with the minimal martingale measure, which, in the context that we consider, identifies with the minimum entropy martingale measure as well as the variance-optimal martingale measure. Furthermore, we present a numerical study that illustrates aspects of our theoretical results