Real-world Probability Measure Research Articles

Martingale Pricing and Single Index Models: Unified Approach with Esscher and Minimal Relative Entropy Measures

In this paper, we explore the connection between a single index model under the real-world probability measure and martingale pricing via minimal relative entropy or Esscher transform, within the context of a one-period market model, possibly incomplete, with multiple risky assets and a single risk-free asset. The minimal relative entropy martingale measure and the Esscher martingale measure coincide in such a market, provided they both exist. From their Radon–Nikodym derivative, we derive a portfolio of risky assets in a natural way, termed portfolio G. Our analysis shows that pricing using the Esscher or minimal relative entropy martingale measure is equivalent to a single index model (SIM) incorporating portfolio G. In the special case of elliptical returns, portfolio G coincides with the classical tangency portfolio. Furthermore, in the case of jointly normal returns, Esscher or minimal relative entropy martingale measure pricing is equivalent to CAPM pricing.

Measures of Model Risk for Continuous-Time Finance Models

Abstract Measuring model risk is required by regulators in financial and insurance markets. We separate model risk into parameter estimation risk (PER) and model specification risk (MSR), and we propose expected shortfall type model risk measures applied to Lévy jump, affine jump-diffusion, and multifactor models. We investigate the impact of PER and MSR on the models’ ability to capture the joint dynamics of stock and option prices. Using Markov chain Monte Carlo techniques, we implement two methodologies to estimate parameters under the risk-neutral probability measure and the real-world probability measure jointly.

The impact of ambiguity on dynamic portfolio selection in the epsilon-contaminated binomial market model

We consider dynamic portfolio selection under ambiguity in the classical multi-period binomial market model. Ambiguity is incorporated in the real-world probability measure through an epsilon-contamination, that gives rise to a completely monotone capacity conveying a pessimistic investor’s ambiguous beliefs. The dynamic portfolio selection problem is formulated as a Choquet expected utility maximization problem on the final wealth. Then, the optimal final wealth is proved to be a function of the final stock price: this allows a dimension reduction of the problem, switching from an exponential to a linear size with respect to the number of periods. Finally, an explicit characterization of the optimal final wealth is given in the case of a constant relative risk aversion utility function and the interaction between the ambiguity and the relative risk aversion parameters is investigated.

Less-expensive long-term annuities linked to mortality, cash and equity

AbstractThis paper proposes a shift in the valuation and production of long-term annuities, away from the classical risk-neutral methodology towards a methodology using the real-world probability measure. The proposed production method is applied to three examples of annuity products, one having annual payments linked to a mortality index and the savings account and the others having annual payments linked to a mortality index and an equity index with a guarantee that is linked to the same mortality index and the savings account. Out-of-sample hedge simulations demonstrate the effectiveness of the proposed less-expensive production method. In contrast to classical risk-neutral production, which revolves around the savings account as reference unit, the long-term best-performing portfolio, the numéraire portfolio of the equity market, is employed as the fundamental reference unit in the production of the annuity. The numéraire portfolio is the strictly positive, tradable portfolio that when used as denominator or benchmark makes all benchmarked non-negative portfolios supermartingales. Under real-world valuation, the initial benchmarked value of a benchmarked contingent claim equals its real-world conditional expectation. The proposed real-world valuation and production can lead to significantly lower values of long-term annuities and their less-expensive production than suggested by the risk-neutral approach.

Measuring profitability of life insurance products under Solvency II

In this paper, we propose an enhanced method for the measurement of profitability of life insurance products. In contrast to most of the existing literature, we consider the development of the insurance contracts over their entire lifetime under the real-world probability measure and distinguish between different sources of capital. We study the pathwise realization of random variables describing shareholder profitability to obtain and analyze their distribution. These distributions are more versatile than single statistics such as expected values since they additionally allow for the analysis of extreme outcomes. Moreover, we specifically consider the strain on shareholders arising from the solvency capital requirement under Solvency II. We use a cost of capital approach based on the explicit computation of the solvency capital requirement and the interrelated capital required from shareholders for each year of the projection period. To demonstrate the feasibility of our profit measures, we provide a concrete application to products with interest rate guarantees including an internal model approach for market risks under Solvency II. Our numerical application shows that our proposed profit measures are particularly suitable for revealing the profitability of different life insurance products in today’s regulatory environment.

A Libor market model including credit risk under the real-world measure

We present a methodology to generate future scenarios of interest rates for different credit ratings under a real-world probability measure. More precisely, we explain how to perform simulations of the real-world forward rates for different rating classes by generalizing the multidimensional shifted lognormal London Interbank Offered Rate market model to account for credit ratings and a specification of the market prices of risk vector processes. The proposed methodology allows for the presence of negative interest rates, as currently observed in the markets, and guarantees the monotonicity of forward rates with respect to credit ratings.

Asymptotics of bond yields and volatilities for extended CIR models under the real-world measure

ABSTRACTThe Cox–Ingersoll–Ross CIR short rate model is a mean-reverting model of the short rate which, for suitably chosen parameters, permits closed-form valuation formulae of zero-coupon bonds and options on zero-coupon bonds. This article supplies proofs of the formulae for the expected present value of payoffs under the real-world probability measure, known as actuarial valuation. Importantly, we give formulae for asymptotic levels of bond yields and volatilities for extended CIR models when suitable conditions are imposed on the model parameters.

Mean-reverting additive energy forward curves in a Heath–Jarrow–Morton framework

In this paper, we make the traditional modeling approach of energy commodity forwards consistent with no-arbitrage. In fact, traditionally energy prices are modeled as mean-reverting processes under the real-world probability measure $$\mathbb {P}$$ , which is in apparent contradiction with the fact that they should be martingales under a risk-neutral measure $$\mathbb {Q}$$ . The key point here is that the two dynamics can coexist, provided a suitable change of measure is defined between $$\mathbb {P}$$ and $$\mathbb {Q}$$ . To this purpose, we design a Heath–Jarrow–Morton framework for an additive, mean-reverting, multicommodity market consisting of forward contracts of any delivery period. Even for relatively simple dynamics, we face the problem of finding a density between $$\mathbb {P}$$ and $$\mathbb {Q}$$ , such that the prices of traded assets like forward contracts are true martingales under $$\mathbb {Q}$$ and mean-reverting under $$\mathbb {P}$$ . Moreover, we are also able to treat the peculiar delivery mechanism of forward contracts in power and gas markets, where the seller of a forward contract commits to deliver, either physically or financially, over a certain period, while in other commodity, or stock, markets, a forward is usually settled on a maturity date. By assuming that forward prices can be represented as affine functions of a universal source of randomness, we can completely characterize the models which prevent arbitrage opportunities by formulating conditions under which the change of measure between $$\mathbb {P}$$ and $$\mathbb {Q}$$ is well defined. In this respect, we prove two results on the martingale property of stochastic exponentials. The first allows to validate measure changes made of two components: an Esscher-type density and a Girsanov transform with stochastic and unbounded kernel. The second uses a different approach and works for the case of continuous density. We show how this framework provides an explicit way to describe a variety of models by introducing, in particular, a generalized Lucia–Schwartz model and a cross-commodity cointegrated market.

Fair Pricing of Variable Annuities with Guarantees under the Benchmark Approach

In this paper we consider the pricing of variable annuities (VAs) with guaranteed minimum withdrawal benefits. We consider two pricing approaches, the classical risk-neutral approach and the benchmark approach, and we examine the associated static and optimal behaviors of both the investor and insurer. The first model considered is the so-called minimal market model, where pricing is achieved using the benchmark approach. The benchmark approach was introduced by Platen in 2001 and has received wide acceptance in the finance community. Under this approach, valuing an asset involves determining the minimum-valued replicating portfolio, with reference to the growth optimal portfolio under the real-world probability measure, and it both subsumes classical risk-neutral pricing as a particular case and extends it to situations where risk-neutral pricing is impossible. The second model is the Black-Scholes model for the equity index, where the pricing of contracts is performed within the risk-neutral framework. Crucially, we demonstrate that when the insurer prices and reserves using the Black-Scholes model, while the insured employs a dynamic withdrawal strategy based on the minimal market model, the insurer may be underestimating the value and associated reserves of the contract.

An extension of the Clark–Haussmann formula and applications

ABSTRACTThis work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.

Computing Risk Measures of Life Insurance Policies through the Cox–Ross–Rubinstein Model

The problem of computing risk measures of life insurance policies is complicated by the fact that two different probability measures, the real-world probability measure along the risk horizon and the risk-neutral one along the remaining time interval, have to be used. This implies that a straightforward application of the Monte Carlo method is not available and the need arises to resort to time consuming nested simulations or to the least squares Monte Carlo approach. We propose to compute common risk measures by using the celebrated binomial model of Cox, Ross, and Rubinstein (1979) (CRR). The main advantage of this approach is that the usual construction of the CRR model is not influenced by the change of measure and a unique lattice can be used along the whole policy duration. Numerical results highlight that the proposed algorithm computes highly accurate values.

Real-World Scenarios With Negative Interest Rates Based on the LIBOR Market Model

ABSTRACTIn this article, we present a methodology to simulate the evolution of interest rates under real-world probability measure. More precisely, using the multidimensional Shifted Lognormal LIBOR market model and a specification of the market price of risk vector process, we explain how to perform simulations of the real-world forward rates in the future, using the Euler‒Maruyama scheme with a predictor‒corrector strategy. The proposed methodology allows for the presence of negative interest rates as currently observed in the markets.

Quantization Under the Real-world Measure: Fast and Accurate Valuation of Long-dated Contracts

This paper provides a methodology for fast and accurate pricing of the long-dated contracts that arise as the building blocks of insurance and pension fund agreements. It applies the recursive marginal quantization (RMQ) and joint recursive marginal quantization (JRMQ) algorithms outside the framework of traditional risk-neutral methods by pricing options under the real-world probability measure, using the benchmark approach. The benchmark approach is reviewed, and the real-world pricing theorem is presented and applied to various long-dated claims to obtain less expensive prices than suggested by traditional risk-neutral valuation. The growth-optimal portfolio (GOP), the central object of the benchmark approach, is modelled using the time-dependent constant elasticity of variance model (TCEV). Analytic European option prices are derived and the RMQ algorithm is used to efficiently and accurately price Bermudan options on the GOP. The TCEV model is then combined with a $3/2$ stochastic short-rate model and RMQ is used to price zero-coupon bonds and zero-coupon bond options, highlighting the departure from risk-neutral pricing.

Hedging error estimate of the american put option problem in jump-diffusion processes

We consider discrete time hedging error of the American put option in case of brusque fluctuations in the price of assets. Since continuous time hedging is not possible in practice so we consider discrete time hedging process. We show that if the proportions of jump sizes in the asset price are identically distributed independent random variables having finite moments then the value process of the discrete time hedging uniformly approximates the value process of the corresponding continuous-time hedging in the sense of L1 and L2-norms under the real world probability measure.

Ross Recovery in Pricing Semigroups

Ross recovery specifies conditions under which it is possible to recover the real-world probability measure as well as the pricing kernel from a family of pricing operators. In this paper, we study Ross recovery theoretically within a semigroup framework using Perron-Frobenius operator theory. Thereby, we derive sufficient conditions guaranteeing existence and uniqueness of a transition independent pricing kernel consistent with the pricing semigroup. Our resultant theory provides a unified perspective on various existing recovery approaches. Building on our theoretical results, we also briefly address the estimation of a pricing semigroup and the corresponding (Markov) transition semigroup from a sample of option prices.

Detecting money market bubbles

The existence of a self-financing trading strategy that replicates the money market account at a fixed future date at a lower cost than the current value of this account constitutes a money market bubble (MMB). Understanding whether a market exhibits an MMB is crucial, in particular, for derivative pricing. An MMB precludes the existence of a risk-neutral probability measure. The benchmark approach allows to study MMBs and is formulated under the real world probability measure. It does not require the existence of a risk neutral probability measure. Using a range of well-known stochastic volatility models, we study the existence of an MMB in the US economy, and find that the US market exhibits an MMB for all models considered that allow it. This suggests that for derivative pricing and hedging care should be taken when making assumptions pertaining to the existence of a risk-neutral probability measure. Less expensive portfolios are likely to exist for a wide range of long-term derivatives, as typical for pensions.

An Approach to Measure the Expectation of Generic Functions of the Market Return

This paper proposes an approach that associates the risk-neutral probability measure with option prices and then computes the expectation of quantities under the real world probability measure, exploiting the form of the stochastic discount factor. This approach deviates from foundational approaches that embrace assumptions about preferences and economic primitives to propose formulas for prices and risk premiums. Our method is analytically tractable, absolved of distributional assumptions, and we use the approach to elaborate on empirical questions regarding disaster probabilities, conditional return moments, and conditional return asymmetries.

ASYMPTOTICS OF BOND YIELDS AND VOLATILITIES FOR EXTENDED VASICEK MODELS UNDER THE REAL-WORLD MEASURE

Vasicek's short rate model is a mean reverting model of the short rate which permits closed-form pricing formulae of zero coupon bonds and options on zero coupon bonds. This paper supplies proofs which are valid for any single factor mean reverting Gaussian short rate model having time-inhomogeneous parameters. The formulae are for the expected present value of payoffs under the real-world probability measure, known as actuarial pricing. Importantly, we give formulae for asymptotic levels of bond yields and volatilities for extended Vasicek models when suitable conditions are imposed on the model parameters.

Less-Expensive Valuation of Long Term Annuities Linked to Mortality, Cash and Equity

This paper proposes a paradigm shift in the valuation of long term annuities, away from classical no-arbitrage valuation towards valuation under the real world probability measure. Furthermore, we apply this valuation method to two examples of annuity products, one having annual payments linked to a mortality index and the savings account and the other having annual payments linked to a mortality index and an equity index with a guarantee that is linked to the same mortality index and the savings account. Out-of-sample hedge simulations demonstrate the effectiveness of real world valuation. In contrast to risk neutral valuation, which is a form of relative valuation, the long term average excess return of the equity market comes into play. Instead of the savings account, the numeraire portfolio is employed 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 nonnegative portfolios supermartingales. The benchmarked real world value of a benchmarked contingent claim equals its real world conditional expectation. This yields the minimal possible value for its hedgeable part and minimizes the fluctuations for its benchmarked hedge error. Under classical assumptions, actuarial and risk neutral valuation emerge as special cases of the proposed real world valuation. In long term liability and asset valuation, the proposed real world valuation can lead to signifi cantly lower values than suggested by classical approaches when an equivalent risk neutral probability measure does not exist.

Detecting Money Market Bubbles

Using a range of stochastic volatility models well-known in the nance literature, we study the existence of money market bubbles in the US economy. Money market bubbles preclude the existence of a risk-neutral pricing measure. Understanding whether markets exhibit money market bubbles is crucial from the point of view of derivative pricing since their existence implies the existence of a self-financing trading strategy that replicates the savings account's value at a fixed future date at a cheaper cost than the current value of the savings account. The benchmark approach is formulated under the real world probability measure and does not require the existence of a risk neutral probability measure. It hence emerges as the appropriate framework to study the potential existence of money market bubbles. Testing the existence of money market bubbles in the US economy we nd that for all models the US market exhibits a money market bubble. This conclusion suggests that for derivative pricing and hedging care should be taken when making assumptions pertaining to the existence of a risk-neutral probability measure. Less expensive hedge portfolios may exist for a wide range of derivatives.