A model of returns for the post-credit-crunch reality: hybrid Brownian motion with price feedback

Recent market events have reinvigorated the search for realistic return models that capture greater likelihoods of extreme movements. In this paper we model the medium-term log-return dynamics in a market with both fundamental and technical traders. This is based on a trade arrival model with variable size orders and a general arrival-time distribution. With simplifications we are led in the jump-free case to a local volatility model defined by a hybrid SDE mixing both arithmetic and geometric or CIR Brownian motions, whose solution in the geometric case is given by a class of integrals of exponentials of one Brownian motion against another, in forms considered by Yor and collaborators. The reduction of the hybrid SDE to a single Brownian motion leads to an SDE of the form considered by Nagahara, which is a type of ‘Pearson diffusion’, or, equivalently, a hyperbolic OU SDE. Various dynamics and equilibria are possible depending on the balance of trades. Under mean-reverting circumstances we arrive naturally at an equilibrium fat-tailed return distribution with a Student or Pearson Type~IV form. Under less-restrictive assumptions, richer dynamics are possible, including time-dependent Johnson-SU distributions and bimodal structures. The phenomenon of variance explosion is identified that gives rise to much larger price movements that might have a priori been expected, so that ‘25σ’ events are significantly more probable. We exhibit simple example solutions of the Fokker–Planck equation that shows how such variance explosion can hide beneath a standard Gaussian facade. These are elementary members of an extended class of distributions with a rich and varied structure, capable of describing a wide range of market behaviors. Several approaches to the density function are possible, and an example of the computation of a hyperbolic VaR is given. The model also suggests generalizations of the Bougerol identity. We touch briefly on the extent to which such a model is consistent with the dynamics of a ‘flash-crash’ event, and briefly explore the statistical evidence for our model.     

Estimation and Prediction of a Non-Constant Volatility

In this paper we study volatility functions. Our main assumption is that the volatility is a function of time and is either deterministic, or stochastic but driven by a Brownian motion independent of the stock. Our approach is based on estimation of an unknown function when it is observed in the presence of additive noise. The set up is that the prices are observed over a time interval [0, t], with no observations over (t, T), however there is a value for volatility at T. This value is may be inferred from options, or provided by an expert opinion. We propose a forecasting/interpolating method for such a situation. One of the main technical assumptions is that the volatility is a continuous function, with derivative satisfying some smoothness conditions. Depending on the degree of smoothness there are two estimates, called filters, the first one tracks the unknown volatility function and the second one tracks the volatility function and its derivative. Further, in the proposed model the price of option is given by the Black–Scholes formula with the averaged future volatility. This enables us to compare the implied volatility with the averaged estimated historical volatility. This comparison is done for three companies and has shown that the two estimates of volatility have a weak statistical relation.     

A maximum likelihood approach to volatility estimation for a Brownian motion using high,low and close price data

Abstract

Volatility plays an important role in derivatives pricing, asset allocation, and risk management, to name but a few areas. It is therefore crucial to make the utmost use of the scant information typically available in short time windows when estimating the volatility. We propose a volatility estimator using the high and the low information in addition to the close price, all of which are typically available to investors. The proposed estimator is based on a maximum likelihood approach. We present explicit formulae for the likelihood of the drift and volatility parameters when the underlying asset is assumed to follow a Brownian motion with constant drift and volatility. Our approach is to then maximize this likelihood to obtain the estimator of the volatility. While we present the method in the context of a Brownian motion, the general methodology is applicable whenever one can obtain the likelihood of the volatility parameter given the high, low and close information. We present simulations which indicate that our estimator achieves consistently better performance than existing estimators (that use the same information and assumptions) for simulated data. In addition, our simulations using real price data demonstrate that our method produces more stable estimates. We also consider the effects of quantized prices and discretized time.     

Stochastic Life Annuities

Abstract

This paper gives analytic approximations for the distribution of a stochastic life annuity. It is assumed that returns follow a geometric Brownian motion. The distribution of the stochastic annuity may be used to answer questions such as “What is the probability that an amount F is sufficient to fund a pension with annual amount y to a pensioner aged x?” The main idea is to approximate the future lifetime distribution with a combination of exponentials, and then apply a known formula (due to Marc Yor) related to the integral of geometric Brownian motion. The approximations are very accurate in the cases studied.     

Skew Brownian Motion and Pricing European Options

Abstract

The volatility smile and systematic mispricing of the Black–Scholes option pricing model are the typical motivation for examining stochastic processes other than geometric Brownian motion to describe the underlying stock price. In this paper a new stochastic process is presented, which is a special case of the skew-Brownian motion of Itô and McKean. The process in question is the sum of a standard Brownian motion and an independent reflecting Brownian motion that is similar in construction to the stochastic representation of a skew-normal random variable. This stochastic process is taken in its exponential form to price European options. The derived option price nests the Black–Scholes equation as a special case and is flexible enough to accommodate stochastic volatility as well as stochastic skewness.     

Optimal dividends under a drawdown constraint and a curious square-root rule

Finance and Stochastics - In this paper, we address the problem of optimal dividend payout strategies from a surplus process governed by Brownian motion with drift under a drawdown constraint,...     

On Pricing Exponential Square Root Barrier Knockout European Options

A barrier option is one of the most popular exotic options which is designedto give a protection against unexpected wild fluctuation of stock prices.Protection is given to both the writer and holder of such an option.Kunitomo and Ikeda (1992) analytically obtained a pricing formula forexponential double barrier knockout options. Since the logarithm of theirproposed barriers for the stock price process S(t), whichisassumed to be geometric Brownian motion, are nothing but straight lineboundaries, the protection provided by them is not uniform over time. Toremedy this problem, we propose square root curved boundaries±btfor the underlying Brownian motion process W(t). Since thestandarddeviation of Brownian motion is proportional to t, theseboundaries(after transformation) can be made to provide more uniform protectionthroughout the life time of the option. We will apply asymptoticexpansions of certain conditional probabilities obtained by Morimoto (1999)to approximate pricing formulae for exponential square root double barrierknockout European call options. These formulae allow us to computenumerical values in a very short time (t < 10^–6sec), whereas it takesmuch longer to perform Monte Carlo simulations to determine optionpremiums.     

Geometric Asian options: valuation and calibration with stochastic volatility

The scaling properties of two alternative fractal models recently proposed to characterize the dynamics of stock market prices are compared. The former is the Multifractal Model of Asset Return (MMAR) introduced in 1997 by Mandelbrot, Calvet and Fisher in three companion papers. The latter is the multifractional Brownian motion (mBm), defined in 1995 by Péltier and Lévy Véhel as an extension of the very well-known fractional Brownian motion (fBm).

We argue that, when fitted on financial time series, the partition function as well as the scaling function of the mBm, i.e. of a generally non-multifractal process, behave as those of a genuine multifractal process. The analysis, which concerns the daily closing prices of eight major stock indexes, suggests to evaluate prudently the recent findings about the multifractal behaviour in finance and economics.     

Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model

A drawdown constraint forces the current wealth to remain above a given function of its maximum to date. We consider the portfolio optimisation problem of maximising the long-term growth rate of the expected utility of wealth subject to a drawdown constraint, as in the original setup of Grossman and Zhou (Math. Finance 3:241–276, 1993). We work in an abstract semimartingale financial market model with a general class of utility functions and drawdown constraints. We solve the problem by showing that it is in fact equivalent to an unconstrained problem with a suitably modified utility function. Both the value function and the optimal investment policy for the drawdown problem are given explicitly in terms of their counterparts in the unconstrained problem.     

A new closed-form formula for pricing European options under a skew Brownian motion

In this paper, we present a new pricing formula based on a modified Black–Scholes (B-S) model with the standard Brownian motion being replaced by a particular process constructed with a special type of skew Brownian motions. Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options.” The European Journal of Finance 13 (6): 523–544] have worked on this model, the results they obtained are incorrect. In this paper, not only do we identify precisely where the errors in Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options”. The European Journal of Finance 13 (6): 523–544] are, we also present a new closed-form pricing formula based on a newly proposed equivalent martingale measure, called ‘endogenous risk neutral measure’, by which only endogenous risks should and can be fully hedged. The newly derived option pricing formula takes the B-S formula as a special case and it does not induce any significant additional burden in terms of numerically computing option values, compared with the effort involved in computing the B-S formula.     

Barrier present value maximization for a diffusion model of insurance surplus

In this paper, we study a barrier present value (BPV) maximization problem for an insurance entity whose surplus process follows an arithmetic Brownian motion. The BPV is defined as the expected discounted value of a payment made at the time when the surplus process reaches a high barrier level. The insurance entity buys proportional reinsurance and invests in a Black–Scholes market to maximize the BPV. We show that the maximal BPV function is a classical solution to the corresponding Hamilton–Jacobi–Bellman equation and is three times continuously differentiable using a novel operator. Its associated optimal reinsurance-investment control policy is determined by verification techniques.     

Double boundary crossing result for the brownian motion

Abstract

In a recent paper a general bounded crossing result for the Brownian motion is obtained for a linear upper boundary based on the method of Kac (1951). Based on his main results we are able to develop in the present paper some simple expressions for crossing probabilities in case of a lower and an upper linear boundary. We will consider a Brownian motion process for the surplus of an insurance portfolio. This surplus must stay between two given bounds. If the surplus will cross the upper boundary, we can pay a dividend. The lower boundary can reflect the influence of control authorities and regulation measurements.     

American option valuation using first-passage densities

This paper explores the advantages of pricing American options using the first-passage density of a Brownian motion to a curved barrier. First, we demonstrate that, under this approach, the exact computation of the optimal boundary becomes secondary. Consequently, a simple approximation to the optimal boundary suffices to obtain accurate prices. Moreover, the first-passage approach tends to give more accurate prices than the early-exercise-premium integral representation. We present two ways of implementing the approach. The first is based on an exact representation of the first-passage density. The second exploits the method of images, which gives us a family of barriers with first-passage densities given in closed form. Both methods are very easy to implement and give accurate prices. In particular, the images-based method is extremely accurate.     

Estimation and pricing under long-memory stochastic volatility

We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein–Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.     

An Empirical Comparison of Two Stochastic Volatility Models using Indian Market Data

We conduct an empirical comparison of hedging strategies for two different stochastic volatility models proposed in the literature. One is an asymptotic expansion approach and the other is the risk-minimizing approach applied to a Markov-switched geometric Brownian motion. We also compare these with the Black–Scholes delta hedging strategies using historical and implied volatilities. The derivatives we consider are European call options on the NIFTY index of the Indian National Stock Exchange. We compare a few cases with profit and loss data from a trading desk. We find that for the cases that we analyzed, by far the better results are obtained for the Markov-switched geometric Brownian motion.     

Correlation estimation using components of Japanese candlesticks

Using the wick’s difference from the classical Japanese candlestick representation of daily open, high, low, close prices brings efficiency when estimating the correlation in a bivariate Brownian motion. An interpretation of the correlation estimator given in [Rogers, L.C.G. and Zhou, F., Estimating correlation from high, low, opening and closing prices. Ann. Appl. Probab., 2008, 18(2), 813–823] in the light of wicks’ difference allows us to suggest modifications, which lead to an increased efficiency and robustness over the baseline model. An empirical study of four major financial markets confirms the advantages of the modified estimator.     

Pricing growth-rate risk

We characterize the compensation demanded by investors in equilibrium for incremental exposure to growth-rate risk. Given an underlying Markov diffusion that governs the state variables in the economy, the economic model implies a stochastic discount factor process S. We also consider a reference growth process G that may represent the growth in the payoff of a single asset or of the macroeconomy. Both S and G are modeled conveniently as multiplicative functionals of a multidimensional Brownian motion. We consider the pricing implications of parametrized family of growth processes G ^ε , with G ⁰=G, as ε is made small. This parametrization defines a direction of growth-rate risk exposure that is priced using the stochastic discount factor S. By changing the investment horizon, we trace a term structure of risk prices that shows how the valuation of risky cash flows depends on the investment horizon. Using methods of Hansen and Scheinkman (Econometrica 77:177–234, 2009), we characterize the limiting behavior of the risk prices as the investment horizon is made arbitrarily long.