Description: This is a stochastic model used to find the distribution of the discounted value of all future cash-flows.

Description: Several authors have used Fourier inversion to compute prices of puts and calls, some using Parseval’s theorem. The expected value of max[S K,0] also arises in excess-of-loss of stop-loss insurance. This is the abstract of a paper that shows that Fourier methods may be used to compute them. The authors take the idea of using Parseval’s theorem further: 1. formulas requiring weaker assumptions, 2. relationship with classical inversion theorems for probability distributions, 3. formulas for pay-offs which occur in insurance. Numerical examples are provided.

Description: This paper presents a solution to the problem of discounting cash flows when the cash flows and discount factors are random variables with given distributions.

Description: This paper focuses on two aspects of SFAS 87. The first is the availability of the discount rate, and its consequences. The second is the corridor approach to gains and losses amortization. From Actuarial Research Clearing House 1993, VOL. 2.

Description: In the classical risk theoretic model with Poisson claim arrival and exponential claims, consider the discounted value of claims nos. 3, 6, 9, and so on. [To risk theory specialists, considering every third claim in a Poisson arrival process is the same as assuming waiting times are Erlang[3] in a Sparre-Andersen model.] The distribution of the discounted value of future claims can be found. It turns out that this is the same as the distribution of the product of two independent variables, one having a gamma distribution, the other a 'complex-parameter beta product distribution'. The latter will be defined, it involves extending the usual beta distribution.

Description: This presentation considers the pricing of European derivatives in a Black-Scholes model with stochastic volatility. The presentation shows how Parseval’s theorem may be used to express those prices as Fourier integrals. This is a significant improvement over Monte Carlo simulation in many cases. The main ingredient in this presentation's method is the Laplace transform of the ordinary [constant volatility] price of a put or call in the Black-Scholes model, where the transform is taken with respect to maturity [T]. The presentation derives these formulas and then applies them to the case where volatility is modelled as a Markov chain in continuous time, the so-called 'Markov regime switching model'. This model has been used previously in stochastic volatility modelling, but mostly with only N=2 states. The presentation shows how to use N=3 states without difficulty, and how larger number of states can be handled. Numerical illustrations are given, including the volatility smile in a three-state model.

Description: The problem of finding the distribution of sums of lognormally distributed random variables is discussed. References going back to the 1930’s are given, as well as some possible solutions. A formula for the characteristic function of one lognormal is stated, and then the moments and distribution of the logarithm of sums of lognormals are considered.

Description: Feature article discussing the use of stochastic volatility models in the pricing of investments and options.