On the Validity of the Wiener Process Assumption in Option Pricing Models: Contradictory Evidence from Taiwan

This study tests the validity of the critical assumption underlying the option pricing model that the log form of the stock price movements follows the Wiener process, i.e., stock price movements follow a geometric Brownian motion. Using data compiled from the Taiwan Stock Exchange (TSE), this study's major empirical findings are as follows: first, the null hypothesis that the log of the stock prices is normally distributed is rejected; second, the null hypothesis that the stock price in log form has mean [ln P _s + (µ- ²)_t] and variance t is rejected; third, the null hypothesis that successive non-overlapping increments of the log of the stock price are independent from each other is also rejected. These empirical findings undermine the validity of the Wiener process assumption which is fundamental to many option pricing models.     

Nonparametric estimation for a stochastic volatility model

Consider discrete-time observations (X _{? δ} )_1≤?≤n+1 of the process X satisfying $dX_{t}=\sqrt{V_{t}}dB_{t}Consider discrete-time observations (X _{ℓ δ} )_{1≤ℓ≤n+1} of the process X satisfying dX_t=?{V_t}dB_tdX_{t}=\sqrt{V_{t}}dB_{t} , with V a one-dimensional positive diffusion process independent of the Brownian motion B. For both the drift and the diffusion coefficient of the unobserved diffusion V, we propose nonparametric least square estimators, and provide bounds for their risk. Estimators are chosen among a collection of functions belonging to a finite-dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method works.     

Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model

Abstract

In this paper we derive some results on the dividend payments prior to ruin in a Markovmodulated risk process in which the rate for the Poisson claim arrival process and the distribution of the claim sizes vary in time depending on the state of an underlying (external) Markov jump process {J(t); t ≥ 0}. The main feature of the model is the flexibility in modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate, and that the states of {J(t); t ≥ 0} could describe, for example, epidemic types in health insurance or weather conditions in car insurance. A system of integro-differential equations with boundary conditions satisfied by the nth moment of the present value of the total dividends prior to ruin, given the initial environment state, is derived and solved. We show that the probabilities that the surplus process attains a dividend barrier from the initial surplus without first falling below zero and the Laplace transforms of the time that the surplus process first hits a barrier without ruin occurring can be expressed in terms of the solution of the above-mentioned system of integro-differential equations. In the two-state model, explicit results are obtained when both claim amounts are exponentially distributed.     

Efficient estimation of drift parameters in stochastic volatility models

We study the parametric problem of estimating the drift coefficient in a stochastic volatility model , where Y is a log price process and V the volatility process. Assuming that one can recover the volatility, precisely enough, from the observation of the price process, we construct an efficient estimator for the drift parameter of the diffusion V. As an application we present the efficient estimation based on the discrete sampling with δ _n →0 and n δ _n →∞. We show that our setup is general enough to cover the case of ‘microstructure noise’ for the price process as well.      

A ruin model with a resampled environment

ABSTRACT

This paper considers a Cramér–Lundberg risk setting, where the components of the underlying model change over time. We allow the more general setting of the cumulative claim process being modeled as a spectrally positive Lévy process. We provide an intuitively appealing mechanism to create such parameter uncertainty: at Poisson epochs, we resample the model components from a finite number of d settings. It results in a setup that is particularly suited to describe situations in which the risk reserve dynamics are affected by external processes. We extend the classical Cramér–Lundberg approximation (asymptotically characterizing the all-time ruin probability in a light-tailed setting) to this more general setup. In addition, for the situation that the driving Lévy processes are sums of Brownian motions and compound Poisson processes, we find an explicit uniform bound on the ruin probability. In passing we propose an importance-sampling algorithm facilitating efficient estimation, and prove it has bounded relative error. In a series of numerical experiments we assess the accuracy of the asymptotics and bounds, and illustrate that neglecting the resampling can lead to substantial underestimation of the risk.     

Price fluctuations,market activity and trading volume

We investigate the relation between trading activity - measured by the number of trades [iopmath latex="$N_{Delta t}$"] N _t [/iopmath] - and the price change [iopmath latex="$G_{Delta t}$"] G _t [/iopmath] for a given stock over a time interval [iopmath latex="$[t,~t+Delta t]$"] [t, t + t] [/iopmath]. We relate the time-dependent standard deviation of price changes - volatility - to two microscopic quantities: the number of transactions [iopmath latex="$N_{Delta t}$"] N _t [/iopmath] in [iopmath latex="$Delta t$"] t [/iopmath] and the variance [iopmath latex="$W^2_{Delta t}$"] W ² _t [/iopmath] of the price changes for all transactions in [iopmath latex="$Delta t$"] t [/iopmath]. We find that [iopmath latex="$N_{Delta t}$"] N _t [/iopmath] displays power-law decaying time correlations whereas [iopmath latex="$W_{Delta t}$"] W _t [/iopmath] displays only weak time correlations, indicating that the long-range correlations previously found in [iopmath latex="$vert G_{Delta t} vert$"] |G _t| [/iopmath] are largely due to those of [iopmath latex="$N_{Delta t}$"] N _t [/iopmath]. Further, we analyse the distribution [iopmath latex="$P{N_{Delta t} gt x}$"] P{N _t>x} [/iopmath] and find an asymptotic behaviour consistent with a power-law decay. We then argue that the tail-exponent of [iopmath latex="$P{N_{Delta t} gt x}$"] P{N _t>x} [/iopmath] is insufficient to account for the tail-exponent of [iopmath latex="$P{G_{Delta t} gt x}$"] P{G _t>x} [/iopmath]. Since [iopmath latex="$N_{Delta t}$"] N _t [/iopmath] and [iopmath latex="$W_{Delta t}$"] W _t [/iopmath] display only weak interdependence, we argue that the fat tails of the distribution [iopmath latex="$P{G_{Delta t} gt x}$"] P{G _t>x} [/iopmath] arise from [iopmath latex="$W_{Delta t}$"] W _t [/iopmath], which has a distribution with power-law tail exponent consistent with our estimates for [iopmath latex="$G_{Delta t}$"] G _t [/iopmath]. Further, we analyse the statistical properties of the number of shares [iopmath latex="$Q_{Delta t}$"] Q _t [/iopmath] traded in [iopmath latex="$Delta t$"] t [/iopmath], and find that the distribution of [iopmath latex="$Q_{Delta t}$"] Q _t [/iopmath] is consistent with a Lévy-stable distribution. We also quantify the relationship between [iopmath latex="$Q_{Delta t}$"] Q _t [/iopmath] and [iopmath latex="$N_{Delta t}$"] N _t [/iopmath], which provides one explanation for the previously observed volume-volatility co-movement.     

A generalised Gerber–Shiu measure for Markov-additive risk processes with phase-type claims and capital injections

In this paper we consider a risk reserve process where the arrivals (either claims or capital injections) occur according to a Markovian point process. Both claim and capital injection sizes are phase-type distributed and the model allows for possible correlations between these and the inter-claim times. The premium income is modelled by a Markov-modulated Brownian motion which may depend on the underlying phases of the point arrival process. For this risk reserve model we derive a generalised Gerber–Shiu measure that is the joint distribution of the time to ruin, the surplus immediately before ruin, the deficit at ruin, the minimal risk reserve before ruin, and the time until this minimum is attained. Numeral examples illustrate the influence of the parameters on selected marginal distributions.     

On The Decomposition Of The Ruin Probability For A Jump-Diffusion Surplus Process Compounded By A Geometric Brownian Motion

Abstract

If one assumes that the surplus of an insurer follows a jump-diffusion process and the insurer would invest its surplus in a risky asset, whose prices are modeled by a geometric Brownian motion, the resulting surplus for the insurer is called a jump-diffusion surplus process compounded by a geometric Brownian motion. In this resulting surplus process, ruin may be caused by a claim or oscillation. We decompose the ruin probability in the resulting surplus process into the sum of two ruin probabilities: the probability that ruin is caused by a claim, and the probability that ruin is caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When claim sizes are exponentially distributed, asymptotical formulas of the ruin probabilities are derived from the integro-differential equations, and it is shown that all three ruin probabilities are asymptotical power functions with the same orders and that the orders of the power functions are determined by the drift and volatility parameters of the geometric Brownian motion. It is known that the ruin probability for a jump-diffusion surplus process is an asymptotical exponential function when claim sizes are exponentially distributed. The results of this paper further confirm that risky investments for an insurer are dangerous in the sense that either ruin is certain or the ruin probabilities are asymptotical power functions, not asymptotical exponential functions, when claim sizes are exponentially distributed.     

A note on a classical result in the collective risk theory

Abstract

In his paper “Über einige risikotheoretische Fragestellungen” (SAT 1942: 1–2, p. 43) C.-O. Segerdahl generalizes the theory of ruin probability ψ(u) to the case where interest is continuously added to the risk reserve u at the rate δ′.     

A numerical comment on an upper bound for ruin probabilities

Abstract

In a paper by de Vylder (1977) an upper bound for the probability of ruin is constructed. A numerical example is given for a Poisson-process with claim d.f.=l-e?y , the operational time T=100, the premium loading λ=0.05 (c= 1.05) and the initial reserve u=50. In this case the limit is found to be ψ(u, T)?00.0025.     

Ruin time and aggregate claim amount up to ruin time for the perturbed risk process

We consider the classical Sparre-Andersen risk process perturbed by a Wiener process, and study the joint distribution of the ruin time and the aggregate claim amounts until ruin by determining its Laplace transform. This is first done when the claim amounts follow respectively an exponential/Phase-type distribution, in which case we also compute the distribution of recovery time and study the case of a barrier dividend. Then the general distribution is considered when ruin occurs by oscillation, in which case a renewal equation is derived.     

Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift

Abstract

We extend the work of Browne (1995) and Schmidli (2001), in which they minimize the probability of ruin of an insurer facing a claim process modeled by a Brownian motion with drift. We consider two controls to minimize the probability of ruin: (1) investing in a risky asset and (2) purchasing quota-share reinsurance. We obtain an analytic expression for the minimum probability of ruin and the corresponding optimal controls, and we demonstrate our results with numerical examples.     

Small-time ruin for a financial process modulated by a Harris recurrent Markov chain

We consider a nonstandard ruin problem where: (i) the increments of the process are heavy-tailed and Markov-dependent, modulated by a general Harris recurrent Markov chain; (ii) ruin occurs when a positive boundary is attained within a sufficiently small time. Our main result provides sharp asymptotics for the small-time probability of ruin, viz., P(sup? _{n≤δ u} S _n ≥u), where {S _n } denotes the discrete partial sums of the process and δ∈(0,1/μ), where μ is the mean drift. We apply our results to obtain risk estimates which quantify, e.g., repetitive operational risk losses or the extremal behavior for a GARCH(1,1) process.     

Weighted V@R and its Properties

The paper deals with the study of a coherent risk measure, which we call Weighted V@R. It is a risk measure of the form where μ is a probability measure on [0,1] and TV@R stands for Tail V@R. After investigating some basic properties of this risk measure, we apply the obtained results to the financial problems of pricing, optimization, and capital allocation. It turns out that, under some regularity conditions on μ, Weighted V@R possesses some nice properties that are not shared by Tail V@R. To put it briefly, Weighted V@R is “smoother” than Tail V@R. This allows one to say that Weighted V@R is one of the most important classes (or maybe the most important class) of coherent risk measures.     

The Expected Discounted Penalty at Ruin for a Markov-Modulated Risk Process Perturbed by Diffusion

Abstract

A Markov-modulated risk process perturbed by diffusion is considered in this paper. In the model the frequencies and distributions of the claims and the variances of the Wiener process are influenced by an external Markovian environment process with a finite number of states. This model is motivated by the flexibility in modeling the claim arrival process, allowing that periods with very frequent arrivals and ones with very few arrivals may alternate. Given the initial surplus and the initial environment state, systems of integro-differential equations for the expected discounted penalty functions at ruin caused by a claim and oscillation are established, respectively; a generalized Lundberg’s equation is also obtained. In the two-state model, the expected discounted penalty functions at ruin due to a claim and oscillation are derived when both claim amount distributions are from the rational family. As an illustration, the explicit results are obtained for the ruin probability when claim sizes are exponentially distributed. A numerical example also is given for the case that two classes of claims are Erlang(2) distributed and of a mixture of two exponentials.     

Analysis of a threshold dividend strategy for a MAP risk model

We consider a class of Markovian risk models in which the insurer collects premiums at rate c₁(c₂) whenever the surplus level is below (above) a constant threshold level b. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the LST (with respect to time) of the joint distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin. By interpreting that the insurer pays dividends continuously at rate c₁?c₂ whenever the surplus level is above b, we also derive the expected discounted value of total dividend payments made prior to ruin. Our results are obtained by making use of an existing connection which links an insurer's surplus process to an embedded fluid flow process.     

Ruin probabilities and optimal capital allocation for heterogeneous life annuity portfolios

A large part of the actuarial literature is devoted to the derivation of ruin probabilities in various non-life insurance risk models. On the contrary, very few papers deal with ruin probabilities for life insurance portfolios. The difficulties arise from the dependence and non-stationarity of the annual payments made by the insurance company. This paper shows that the ruin probability in case of life annuity portfolios can be computed from algorithms derived by De Pril (1989) and Dhaene & Vandebroek (1995). Approximations for ruin probabilities are discussed. The present article complements the works of Frostig et al. (2003) who considered whole life, endowment, and temporary assurances, and of Denuit & Frostig (2008) who considered homogeneous life annuities portfolios. Here, heterogeneous portfolios (with respect to age and/or face amounts) are studied. Particular attention is paid to the capital allocation problem. The total amount of reserve is shared among the risk classes in order to minimize the ruin probability. It is then fair to charge a higher margin to the risk classes requiring more capital.