1. Contingent Claim Pricing using Probability Distortion Operators: Methods from Insurance Risk Pricing and their Relationship to Financial Theory
Refereed paper published in the journal:
Applied Mathematical Finance, Volume 10, Issue 1 March 2003 , pages 19 - 47
ABSTRACT
This paper considers the pricing of contingent claims using an approach developed and used in insurance pricing. The approach is of interest and significance because of the increased integration of insurance and financial markets and also because insurance-related risks are trading in financial markets as a result of securitization and new contracts on futures exchanges. This approach uses probability distortion functions as the dual of the utility functions used in financial theory. The pricing formula is the same as the Black-Scholes formula for contingent claims when the underlying asset price is log-normal. The paper compares the probability distortion function approach with that based on financial theory. The theory underlying the approaches is set out and limitations on the use of the insurance-based approach are illustrated. The probability distortion approach is extended to the pricing of contingent claims for more general assumptions than those used for Black-Scholes option pricing.
KEYWORDS
Contingent Claim Pricing; Probability Distortion Functions; Non-expected Utility; Insurance Pricing; Black And Sholes.
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2. Dynamic Portfolio Allocation, the Dual Theory of Choice and Probability Distortion Functions
Refereed paper published in the journal:
ASTIN Bulletin International Actuarial Association - 2006: Vol. 36, No. 1, pp:
187-217
http://www.casact.org/library/astin/vol36no1/187.pdf
ABSTRACT
Standard optimal portfolio choice models assume that investors maximise the
expected utility of their future outcomes. However, behaviour which is inconsistent
with the expected utility theory has often been observed.
In a discrete time setting, we provide a formal treatment of risk measures
based on distortion functions that are consistent with Yaari’s dual (non-expected
utility) theory of choice (1987), and set out a general layout for portfolio optimisation in this non-expected utility framework using the risk neutral computational
approach.
As an application, we consider two particular risk measures. The first one
is based on the PH-transform and treats the upside and downside of the risk
differently. The second one, introduced by Wang (2000) uses a probability distortion
operator based on the cumulative normal distribution function. Both
risk measures rank-order prospects and apply a distortion function to the entire
vector of probabilities.
KEYWORDS
Portfolio allocation, dual theory, probability distortion, equilibrium pricing.
Comments(34) 22.04.
3. CAPM and Option Pricing With Elliptically Contoured Distributions
Refereed paper published in the journal:
Journal of Risk & Insurance, Volume 75 Issue 2, Pages 387 - 409, 5 May 2008
ABSTRACT
This article offers an alternative proof of the capital asset pricing model (CAPM) when asset returns follow a multivariate elliptical distribution. Empirical studies continue to demonstrate the inappropriateness of the normality assumption for modeling asset returns. The class of elliptically contoured distributions, which includes the more familiar Normal distribution, provides flexibility in modeling the thickness of tails associated with the possibility that asset returns take extreme values with nonnegligible probabilities. As summarized in this article, this class preserves several properties of the Normal distribution. Within this framework, we prove a new version of Stein's lemma for this class of distributions and use this result to derive the CAPM when returns are elliptical. Furthermore, using the probability distortion function approach based on the dual utility theory of choice under uncertainty, we also derive an explicit form solution to call option prices when the underlying is log-elliptically distributed. The Black–Scholes call option price is a special case of this general result when the underlying is log-normally distributed.
Comments(34) 22.04.
4. Dynamic Portfolio Optimization and Asset Pricing
Martingale Methods and Probablity Distortion Functions
Book published by Verlag Dr. Muller ISBN: 978-3-639-11058-6 (6 January 2009)
This monograph consists of three contributions to financial and insurance mathematics.
The first part considers numerical methods for dynamic portfolio optimization in the expected utility model. The aim is to compare the risk-neutral computational approach (RNCA) also known as the martingale approach to stochastic dynamic programming (SDP) in a discrete-time setting. The main idea of the RNCA is to use the completeness and the arbitrage free properties of the market to compute the optimal consumption rules and then determine the trading strategy that finances this optimal consumption. In contrast, SDP solves for the optimal consumption and investment rules simultaneously using backward recursion and the principle of optimality. The setting that we consider is a discrete time and state space lattice. We provide some new theoretical results relating to the Hyperbolic Absolute Risk Aversion class of utility functions as well as propose a straightforward implementation of RNCA in binomial and trinomial lattices. Moreover, instead of discretizing the Hamilton-Jacobi-Bellman equation with possibly more than one state variable, we use symbolic algorithms to implement stochastic dynamic programming. This new approach provides a simpler numerical procedure for computing optimal consumption-investment policies. A comparison of the RNCA with SDP demonstrates the superiority of the RNCA in terms of computation.
The second part considers the pricing of contingent claims using an approach developed and applied in insurance. This approach utilizes probability distortion functions as the dual of the utility functions used in financial theory. The main idea of the dual theory is to distort the subjective probabilities rather than outcomes to express the investor's risk aversion. In the first part, the RNCA for asset allocation uses the same principle as risk-neutral valuation for derivative pricing. The idea of the second part of this research is to show that the risk-neutral valuation can be recovered from the probability distortion function approach, thereby establishing consistency between the insurance and the financial approaches. We prove that pricing contingent claims under the real world probability measure using an appropriate distortion operator produces arbitrage-free prices when the underlying asset prices are log-normal. We investigate cases when the insurance-based approach fails to produce arbitrage-free prices and determine the appropriate distortion operator under more general assumptions than those used in Black-Scholes option pricing.
In the third part we introduce dynamic portfolio optimization with risk measures based on probability distortion functions and provide a formal treatment of this class of risk measures. We employ the RNCA to study the consumption-investment problem in discrete time with preferences consistent with Yaari's dual (non-expected utility) theory of choice. As an application, we first consider risk measures based on the Proportional Hazard Transform that treats the upside and downside of the risk differently and secondly a risk measure based on the standard Normal cumulative distribution function. When the objective is to maximize a dual utility of wealth, and the underlying security returns are normal, the efficient frontier is found to be the same as in the mean-variance portfolio problem for an equivalent risk tolerance. When the objective is to maximize a dual utility of consumption, then ``plunging'' behavior occurs (investing everything is the risky asset). Other properties of the optimal consumption-investment policies in the dual theory are also investigated and discussed.
Published Monograph (6 January 2009):
- Paperback: 244 pages
- Publisher: VDM Verlag Dr. Müller
- Language: English
- ISBN-10: 3639110587
- ISBN-13: 978-3639110586
- Product Dimensions: 8.7 x 5.9 x 0.6 inches
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5. Pricing Electricity Forwards using the Real Option Theory
Refereed paper forthcoming in the journal of Energy Markets
ABSTRACT
Electricity Markets are liberalised in many countries throughout the
world. Liberalised in the sense that the wholesale electricity prices
are no longer fixed by government or a regulatory body, but rather
determined by the law of supply and demand. Although the electricity
market is new, the contracts traded over the counter and on exchange
range from vanilla swaps to exotic options and other more complex
contracts compared to capital markets. Nevertheless, there is no
standard methodology for pricing such contracts as is the case of
products in financial markets, where closed-form solutions of Black
and Scholes type are commonly used.
There has been a quite substantial research aiming to produce a
benchmark pricing methodology for forward electricity contracts.
To-date, there are three main schools of thought. The first one
essentially applies the risk-neutral valuation approach (with some
adjustment) to price electricity contracts. The second one uses simple
actuarial principles based on expectation of future prices and the
concept of certainty equivalent of contingent claims. The third one
considers an equilibrium approach where market players are maximising
the expected utility of their consumption through time, subject to
constraints involving conversion of energy sources into electricity.
From this perspective, the forward pricing problem based on spot
electricity prices remains unsolved because the underlying asset
(electricity) is not storable (economically) and cannot be traded from
one period to the next. The theory of trading claims on non-tradeable
assets (real options) is seen to have a natural application to
electricity.
In this paper, we first examine the characteristics of the electricity
pool prices and look at the relationship between the spot price
average and the forward price. We then apply the theory of trading
claims on non-tradeable assets (real options) to electricity. The
resulting forward price depends on the volume underlying the contract,
the spot price weekly averages for previous periods, the market price
of risk and the interest rate.
In conclusion, this paper sets up a new methodology for pricing
electricity forwards, where the resulting prices are more intuitive in
capturing the market parameters as well as the characteristics of the
underlying contract.
KEYWORDS
Expected utility, fair pricing, energy derivatives, real option theory, electricity forwards.
6. The Compass Rose Pattern in Electricity Prices
Refereed paper published in the journal:
CHAOS 19, 043106 - October 2009
ABSTRACT
The “compass rose pattern” is known to appear in the phase portraits, or scatter diagrams,
of the high-frequency returns of financial time series. We first show that this pattern is
also present in the returns of spot electricity prices. Early researchers investigating these
phenomena hoped that these patterns signalled the presence of rich dynamics, possibly
chaotic, or fractal in nature. Although there is a definite autoregressive and conditional
heteroscedasticity structure in electricity returns, we find that after simple filtering no
pattern remains. The lack of a compass rose pattern is consistent with the short-term
deterministic component in returns being successfully removed, although higher order
weekly and seasonal cycles may remain. Though the series are non-normal in terms of
their distribution, statistical tests fail to identify significant chaotic, or fractal structures,
in price returns. The phase diagram of the filtered returns provides a useful visual check
on independence, a property necessary for pricing and trading derivatives and portfolio
construction, as well as providing useful insights into the market dynamics.
KEYWORDS
chaos; compass rose; electricity prices; fractal; high-frequency data; Hurst.
7. A probabilistic Approach for Evaluating Alternatives to Reduce Minimum Send Out Rate at LNG Regasification Terminal
Forthcoming in LNG Journal
ABSTRACT
This paper presents a multi-scenario stochastic valuation model to select the most attractive Minimum Send Out (MSO) option for an LNG terminal. The model utilizes the Poisson and the exponential distributions to model the number of days the MSO equipment would operate conditional on the probability of securing LNG contract. These distributions feed into a cash flow model that produces an NPV distribution for a particular MSO option. Forward simulations use Crystal Ball with one million scenarios. As an input, the model takes CAPEX and OPEX data provided by the LNG terminal operator and user data including sourcing success and LNG-Gas (e.g. TTF) spread. Optional input may include preferred start up date, installation lead time and permitting. The final result is a set of NPV distributions corresponding to all MSO options which form the basis for comparison / selection of the preferred MSO alternative that has the highest risk-adjusted return. Optionality includes cargo diversion and market flexibility offered by each equipment alternative. We -at Essent Trading- have used the model in real life to make our selection and decision for a preferred MSO option to be installed at Gate LNG terminal and we expect that our choice would be implemented by the start of commercial operations in late 2011.
KEYWORDS
LNG Terminal; Regasificaiton; Liquefaction; Decision making under uncertainty; Minmum send out rate; Simulations.
8. Fair Pricing of Energy Derivatives - A Comparative Study
BY Mahmoud Hamada
Energy Risk Management - EnergyAustralia
May 7, 2004
World Energy Congress proceedings, 2004
ABSTRACT
The Australian National Electricity Market (NEM) is a highly volatile
market in which price setting is driven by the law of supply and demand. The volatile nature of this market presents significant risk to market participants and as such stabilisation of price has implications for the prevention of financial distress.
To alleviate the risk associated with this volatility, over the counter - and eventually exchange traded - derivatives allow hedge positions to be
established, thus resulting in a less volatile hedged exposure to the under
lying. With the importance of derivative trading for market participants, consistent methods of derivative pricing become important.
This paper is a comparative study of two existing quantitative approaches for pricing energy derivatives, namely: a cash-flow at risk model
based on Monte Carlo simulation of the underlying spot price, combined
with extreme value theory, and the SDE approach based on no-arbitrage
theory utilising convenience yields and cost of carry to address the inability to store electricity.
The paper provides a presentation of the underlying theory of the two
approaches with an empirical comparative study, contrasting calibration
methods to quoted brokerage premiums and statistical methods of estimation from historical data. Advantages and disadvantages of both methods
are discussed in light of the results obtained.
KEYWORDS
Fair pricing, energy derivatives, ARIMA models, extreme value theory, Hill estimator, convenience yield
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