Application of Maximum Likelihood Estimation to Stochastic Short Rate Models

2020

ASTIN Bull.

Variable annuities are products offered by pension funds and life offices that provide periodic future payments to the investor and often have ancillary benefits that guarantee survival benefits or sums insured on death. This paper extends the benchmark approach to value and hedge long-dated variable annuities using a combination of cash, bonds and equities under a variety of market models, allowing for dependence between financial and insurance markets. Under a simplified case of independence, the results show that when the discounted index is modelled as a time-transformed squared Bessel process, less-expensive valuation and reserving is achieved regardless of the short rate model or the mortality model.

Section: Data and Parameter Estimationmentioning

confidence: 99%

Less-Expensive Valuation and Reserving of Long-Dated Variable Annuities When Interest Rates and Mortality Rates Are Stochastic

2020

ASTIN Bull.

“…Here 1 X>K denotes the indicator function, equalling one if the random variable X exceeds the value K, and zero otherwise. The availability of explicit formulae for the transition density function of the short rate makes it not only possible to provide explicit formulae for the above prices but also to fit the model to historical data using maximum likelihood estimation, as demonstrated in Fergusson and Platen (2015). In determining the formulae for ZCBs and ZCB options we derive also a formula for the moment generating function of log (B T =B t ), which can be used for the approximation of other prices.…”

Section: Asymptotics Of Bond Yields and Volatilities For Extended Vasmentioning

confidence: 99%

Asymptotics of Bond Yields and Volatilities for Extended Vasicek Models Under the Real-World Measure

2017

Ann. Finan. Econ.

Self Cite

Vasicek's short rate model is a mean reverting model of the short rate which permits closedform pricing formulae of zero coupon bonds and options on zero coupon bonds. This paper supplies proofs which are valid for any single factor mean reverting Gaussian short rate model having time-inhomogeneous parameters. The formulae are for the expected present value of payoffs under the real-world probability measure, known as actuarial pricing. Importantly, we give formulae for asymptotic levels of bond yields and volatilities for extended Vasicek models when suitable conditions are imposed on the model parameters.

“…For example, in Poulsen (1999) a lognormal approximation is employed. In Fergusson and Platen (2015) the parameters of the CIR short rate model were estimated using the maximum likelihood method using the Newton-Raphson iterative method applied to both the log-likelihood function and an approximate log-likelihood function based on the lognormal distribution. In Fergusson and Platen (2015) the parameters of the 3/2 short rate model were also estimated using a gradient ascent method.…”

Section: Introductionmentioning

confidence: 99%

Fast maximum likelihood estimation of parameters for square root and Bessel processes

Studies in Nonlinear Dynamics &Amp; Econometrics

2020

Explicit formulae for maximum likelihood estimates of the parameters of square root processes and Bessel processes and first and second order approximate sufficient statistics are supplied. Applications of the estimation formulae to simulated interest rate and index time series are supplied, demonstrating the accuracy of the approximations and the extreme speed-up in estimation time. This significantly improved run time for parameter estimation has many applications where ex-ante forecasts are required frequently and immediately, such as in hedging interest rate, index and volatility derivatives based on such models, as well as modelling credit risk, mortality rates, population size and voting behaviour.