A Benchmark Framework for Risk Management

The paper describes a general framework for contingent claim valuation for finance, insurance and general risk management. It considers security prices and portfolios with finite expected returns, where the growth optimal portfolio is taken as numeraire or benchmark. Benchmarked nonnegative wealth processes are shown to be supermartingales. Fair benchmarked values are conditional expectations of future benchmarked prices under the real world probability measure. Standard risk neutral and actuarial pricing formulas are obtained as special cases of fair pricing. The proposed benchmark framework covers the infinite time horizon and does not require the existence of an equivalent risk neutral pricing measure.benchmark model; growth optimal portfolio; fair pricing; risk neutral pricing; actuarial pricing

A Benchmark Approach to Finance

This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling and risk measurement. It is based on the natural assumption that investors prefer more or less, in the sense that the higher drift is preferred. Each such investor is shown to hold an efficient portfolio in the sense of Markowitz with units in the market portfolio and the savings account of his or her home currency. If the market portfolio is diversified or monetary authorities aim to maximize the growth rates of the portfolios of their market participants through corresponding interest policies, then the market portfolio is the growth optimal portfolio (GOP). In this setup the capital asset pricing model follows without the use of expected utility functions or equilibrium assumptions. The expected increase of the discounted value of GOP is shown to coincide with the expected increase of its discounted underlying value. The discounted GOP has the dynamics of a time transformed squared Bessel process of dimension four. The time transformation is given by the discounted underlying value of the GOP. The squared volatility of the GOP equals the discounted GOP drift, when expressed in units of the discounted GOP. Risk neutral derivative pricing and actuarial pricing are generalized by the fair pricing concept, which uses the GOP as numeraire and the real world probability measure as pricing measure. An equivalent risk neutral martingale measure does not exist under the derived minimal market model.benchmark model; market portfolio; growth optimal portfolio; efficient frontier; captal asset pricing model; fair pricing; stochastic volatility; minimal market model

Pricing and Hedging for Incomplete Jump Diffusion Benchmark Models

This paper considers a class of incomplete financial market models with security price processes that exhibit intensity based jumps. The benchmark or numeraire is chosen to be the growth optimal portfolio. Portfolio values, when expressed in units of the benchmark, are local martingales. In general, an equivalent risk neutral martingale measure need not exist in the proposed framework. Benchmarked fair derivative prices are defined as conditional expectations of future benchmarked prices under the real world probability measure. This concept of fair pricing generalizes classical risk neutral pricing. The pricing under incompleteness is modeled by the choice of the market prices for risk. The hedging is performed under minimization of profit and loss fluctuations.benchmark model; jump diffusions; incomplete market; growth optimal portfolio; fair pricing; hedge error minimization

On the Pricing and Hedging of Long Dated Zero Coupon Bonds

The pricing and hedging of long dated derivative contracts is a challenging area of research. As a result of utility indifference pricing for general payoffs the growth optimal portfolio turns out to be the appropriate numeraire or benchmark with the real world probability measure as corresponding pricing measure. This concept of real world pricing can be applied for valuing long dated derivatives. An equivalent risk neutral probability measure does not need to exist under this benchmark approach. This paper develops a parsimonious model for a stock index dynamics, which is based on a time transformed squared Bessel process. It uses a diversified world stock index as proxy for the growth optimal portfolio. Surprisingly low prices result for long dated zero coupon bonds that can be replicated using historical data. Such prices and hedges are difficult to explain under the prevailing risk neutral approach.growth optimal portfolio; benckmark approach; real world pricing; expected utility maximization; utility indifference pricing; long dated zero coupon bonds; minimal market model

The Law of Minimum Price

This paper introduces a realistic, generalized market modeling framework for which the Law of One Price no longer holds. Instead the Law of the Minimal Price will be derived, which for contingent claims with long term to maturity may provide significantly lower prices than suggested under the currently prevailing approach. This new law only requires the existence of the numeraire portfolio, which turns out to be the portfolio that maximizes expected logarithmic utility. In several ways it will be shown that the numeraire portfolio cannot be outperformed by any nonnegative portfolio. The new Law of the Minimal Price leads directly to the real world pricing formula, which uses the numeraire portfolio as numeraire and the real world probability for calculating conditional expectations. The cost efficient pricing and hedging of extreme maturity zero coupon bonds illustrates the new law in the context of the US market.law of one price; law of the minimal price; derivative pricing; real world pricing; numeraire portfolio; growth optimal portfolio; strong arbitrage; extreme maturity bond

A Unifying Approach to Asset Pricing

This paper introduces a general market modeling framework under which the Law of One Price no longer holds. A contingent claim can have in this setting several self-financing, replicating portfolios. The new Law of the Minimal Price identifies the lowest replicating price process for a given contingent claim. The proposed unifying asset pricing methodology is model independent and only requires the existence of a tradable numeraire portfolio, which turns out to be the growth optimal portfolio that maximizes expected logarithmic utility. By the Law of the Minimal Price the inverse of the numeraire portfolio becomes the stochastic discount factor. This allows pricing in extremely general settings and avoids the restrictive assumptions of risk neutral pricing. In several ways the numeraire portfolio is the “best” performing portfolio and cannot be outperformed by any other nonnegative portfolio. Several classical pricing rules are recovered under this unifying approach. The paper explains that pricing by classical no-arbitrage arguments is, in general, not unique and may lead to overpricing. In an example, a surprisingly low price of a zero coupon bond with extreme maturity illustrates one of the new effects that can be captured under the proposed benchmark approach, where the numeraire portfolio represents the benchmark.law of one price; law of the minimal price; benchmark approach; derivative pricing; numeraire portfolio; asset pricing; arbitrage

Real World Pricing of Long Term Contracts

Long dated contingent claims are relevant in insurance, pension fund management and derivative pricing. This paper proposes a paradigm shift in the valuation of long term contracts, away from classical no-arbitrage pricing towards pricing under the real world probability measure. In contrast to risk neutral pricing, the long term excess return of the equity market, known as the equity premium, is taken into account. Further, instead of the savings account, the numeraire portfolio isused, as the fundamental unit of value in the analysis. The numeraire portfolio is the strictly positive, tradable portfolio that when used as benchmark makes all benchmarked non negative portfolios supermartingales, which means intuitively that these are downward trending or at least trendless. Furthermore, the benchmarked real world price of a benchmarked claimis defined to be its real world conditional expectation. This yields the minimal possible price for its hedgable part and minimizes the variance of the benchmarked hedge error. The pooled total benchmarked replication error of a large insurance company or bank essentially vanishes due to diversification. Interestingly, in long terml iability and asset valuation, real world pricing can lead to significantly lower prices than suggested by classical no-arbitragea rguments. Moreover, since the existence of some equivalent risk neutral probability measure is no longer required, a wider and more realistic modeling framework is available for exploration. Classical actuarial and risk neutral pricing emerge as special cases of real world pricing.long term pricing; real world pricing; risk neutral pricing; numeraire portfolio; law of the minimal price; strong arbitrage; hedges imulation; diversification; liquidity premium

Modeling the Volatility and Expected Value of a Diversified World Index

This paper considers a diversified world stock index in a continuous financial market with the growth optimal portfolio (GOP) as the reference unit or benchmark. Diversified broadly based portfolios, which include major world stock market indices, are shown to approximate the GOP. It is demonstrated that a key financial quantity is the drift of the discounted GOP, which can be expressed explicitly using a certain quadratic variation term. Using real market approximations for the discounted GOP it is shown that its drift does not vary greatly in the long term. For a diversified world index this leads to a natural model where the discounted index is a time transformed squared Bessel process of dimension four. The inverse of the squared GOP volatility then follows a square root process of dimension four.world index; volatility; benchmark model; growth optimal portfolio; bessel process; square root process

Investments for the Short and Long Run

This paper aims to discuss the optimal selection of investments for the short and long run in a continuous time financial market setting. First it documents the almost sure pathwise long run outperformance of all positive portfolios by the growth optimal portfolio. Secondly it assumes that every investor prefers more rather than less wealth and keeps the freedom to adjust his or her risk aversion at any time. In a general continuous market, a two fund separation result is derived which yields optimal portfolios located on the Markowitz efficient frontier. An optimal portfolio is shown to have a fraction of its wealth invested in the growth optimal portfolio and the remaining fraction in the savings account. The risk aversion of the investor at a given time determines the volatility of her or his optimal portfolio. It is pointed out that it is usually not rational to reduce risk aversion further than is necessary to achieve the maximum growth rate. Assuming an optimal dynamics for a global market, the market portfolio turns out to be growth optimal. The discounted market portfolio is shown to follow a particular time transformed diffusion process with explicitly known transition density. Assuming that the transformed time growth exponentially, a parsimonious and realistic model for the market portfolio dynamics results. It allows for efficient portfolio optimization and derivative pricing.growth optimal portfolio; portfolio selection; risk aversion; minimal market model

Capital Asset Pricing for Markets with Intensity Based Jumps

This paper proposes a unified framework for portfolio optimization, derivative pricing, modeling and risk measurement in financial markets with security price processes that exhibit intensity based jumps. It is based on the natural assumption that investors prefer more for less, in the sense that for two given portfolios with the same variance of its increments, the one with the higher expected increment is preferred. If one additionally assumes that the market together with its monetary authority acts to maximize the long term growth of the market portfolio, then this portfolio exhibits a very particular dynamics. In a market without jumps the resulting dynamics equals that of the growth optimal portfolio (GOP). Conditions are formulated under which the well-known capital asset pricing model is generalized for markets with intensity based jumps. Furthermore, the Markowitz efficient frontier and the Sharpe ratio are recovered in this continuous time setting. In this paper the numeraire for derivative pricing is chosen to be the GOP. Primary security account prices, when expressed in units of the GOP, turn out to be supermartingales. In the proposed framework an equivalent risk neutral martingale measure need not exist. Fair derivative prices are obtained as conditional expectations of future payoff structures under the real world probability measure. The concept of fair pricing is shown to generalize the classical risk neutral and the actuarial net present value pricing methodologies.benchmark model; jump diffusions; growth optimal portfolio; market portfolio; effiient frontier; Sharpe ratio; fair pricing; actuarial pricing