A Simple Approach to Pricing Options with Jumps

Merton has provided a formula for the price of a European call option on a single stock where the stock price process contains a continuous Poisson jump component, in addition to a continuous log-normally distributed component. In Merton's analysis, the jump-risk is not priced. Thus the distribution of the jump-arrivals and the jump-sizes do not change under the change of measure. We go onto introduce a Radon-Nikodym derivative process that induces the change of measure from the market measure to an equivalent martingale measure. The choice of parameters in the Radon-Nikodym derivative allows us to price the option under different financial-economic scenarios. We introduce a hedging argument that eliminates the jump-risk in some sort of averaged sense, and derive an integro-partial differential equation of the option price that is related to the one obtained by Merton.

Portfolio Optimization with Ambiguous Correlation and Stochastic Volatilities

This study was conducted to examine the properties of a portfolio selection model when securities returns, background asset returns, and the corporate social responsibility index are all normal fuzzy numbers, and to propose a hybrid fruit fly optimisation algorithm (FOA) for solving a dynamic multi-objective optimisation model. A nondominated sorting hybrid FOA derived concurrently from NSGA-II and the FOA is presented here. The kurtosis of a portfolio without background risk was a constant value when risky asset returns were fuzzy normal numbers, and the kurtosis of a portfolio increased when the variance of back-ground risk increased. The main contribution of this study is the establishment of a dynamic multi-objective fuzzy portfolio model that considers corporate social responsibility and background risk. An analytic expression of portfolio kurtosis including background risk was deduced when the return of risky assets and background risk were fuzzy normal numbers.

Portfolio adjusting optimization under credibility measures

An experiment is conducted in which investment advice is passed from one cohort to the next. Participants make asset allocation decisions for thirty years to a safe and risky asset and provide annual forecasts (beliefs) on the return on the risky asset. Risky asset returns are drawn from the price returns on the S&P 500 from 1921-2010; Cohorts 1/2/3 received the actual returns in the time period from 1921-1950/1951-1980/1981-2010 respectively. Results show that negative investment advice passed from Cohort 1 to Cohort 2 leads to: 1) significantly lower allocations to the risky asset in Cohort 2 compared to Cohort 1; and 2) a 19% difference in allocations between cohorts who received either positive advice or negative advice in Cohort 2. A second experiment examines the effect on Cohort 3 from receiving consistent and mixed advice from Cohorts 1 and 2. The results from the first experiment are replicated showing that positive (negative) advice received from prior cohorts leads to higher (lower) investment beliefs and portfolio allocations to a riskier asset. Statistical analyses are conducted to determine if the large differences in allocations are driven by changes in beliefs about future returns or changes in risk attitude. The primary takeaway is that experiential cohort advice has a significant impact on subsequent cohorts even when it has little informational value.

The aim of this thesis is to develop efficient valuation methods for financial contracts under models with jumps and stochastic volatility, and to present their rigorous mathematical underpinning. For efficient risk management, large books of exotic options need to be priced and hedged under models that are flexible enough to describe the observed option prices at speeds close to real time. To do so, hundreds of vanilla options, which are quoted in terms of implied volatility, need to be calibrated to market prices quickly and accurately on a regular basis. With this in mind we develop efficient methods for the evaluation of (i) vanilla options, (ii) implied volatility and (iii) common path-dependent options. Firstly, we derive a new numerical method for the classical problem of pricing vanilla options quickly in time-changed Brownian motion models. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier option pricing method with respect to speed appears to favour our newly developed method in the cases considered. Secondly, we use this method to derive a procedure to compute, for a given set of arbitrage-free European call option prices, the corresponding Black-Scholes implied volatility surface. In order to achieve this, rational function approximations of the inverse of the Black-Scholes formula are used. We are thus able to work out implied volatilities more efficiently than is possible using other common methods. Error estimates are presented for a wide range of parameters. Thirdly, we develop a new Monte Carlo variance reduction method to estimate the expectations of path-dependent functionals, such as first-passage times and occupation times, under a class of stochastic volatility models with jumps. The method is based on a recursive approximation of the first-passage time probabilities and expected occupation times of Levy bridge processes that relies in part on a randomisation of the timeparameter. We derive the explicit form of the recursive approximation in the case of bridge processes corresponding to the class of Levy processes with mixed-exponential jumps, and

In this article, futures and commodity options are analyzed in the context of Merton's (1987) model of capital market equilibrium with incomplete information. First, following Dusak (1973) and Black (1976), the conditions under which Merton's model can be applied to the valuation of forward and futures contracts are proposed. Then an application to futures markets is given. We provide a partial differential equation and the formulas for European commodity options, futures contracts, and American options in the same context. The models are simulated and compared to standard models with no information costs. We find that model prices are not significantly different from standard model prices. However, our models correct for some pricing biases in standard models. In particular, they reduce the overvaluation bias for European and American commodity options. It seems that the costs of gathering and processing information regarding the option and its underlying asset play a role in explaining the biases observed in standard models. This work can be applied to other futures markets. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 645–664, 1999

This paper studies how wealth and aging affect portfolio choices in a life‐cycle model with ambiguity aversion. Ambiguity aversion implies wealthier and older agents are endogenously more optimistic about risky asset returns, relative to poorer/younger agents. As life expectancy grows, old agents become even more optimistic, while young agents become more pessimistic, amplifying age gaps in portfolio composition. We find evidence for the mechanism in survey data on portfolios and subjective life expectancy. In a quantitative extension of the model, plausible life expectancy projections imply a 26% increase in the age gradient of conditional risky asset shares between 2019 and 2100.

On Dynamic Measures of Risk

Despite decades of studies, there is still no consensus on what type of serial dependence, if any, might be present in risky asset returns. This manuscript provides an empirical study of the prices of energy commodities, gold and copper in the futures markets and demonstrates that, for these assets, the level of asymmetry of asset returns varies through time and can be forecast using past returns. A regime switching model is used to construct a managed futures trading strategy that provides returns that are statistically significant. It is also demonstrated how such model can be used to make probabilistic predictions of commodity prices in futures markets, which can be used to drive value-at-risk and potential future exposure metrics or guide dynamic hedging strategies of commodity price risk.

Investment in Education and Strategic Asset Allocation

Artificial bee colony algorithm for constrained possibilistic portfolio optimization problem

An optimization model of the portfolio adjusting problem with fuzzy return and a SMO algorithm

Adaptive online portfolio selection with transaction costs

A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M 2. For each μ∈M 2, a real valued measure 〈μ〉 on the predictable σ-algebra ℘ is constructed. The stochastic integral of a random function \(\mathfrak{h} \in L^2 \left( {\left\langle \mu \right\rangle } \right)\) with respect to μ is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I μis an isometry from L 2(〈μ〉) to a stable subspace of M 2, its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well.