Probability and Finance Theory

The concept of a martingale is central in the pricing of contingent claims. Furthermore, the concept is crucial to obtain an elegant formulation of the fundamental theorems of asset pricing in a multi-period setting, thus generalizing the single-period versions of Theorems 6.19 and 6.20 we proved in Chapter 6. For a sound understanding of martingales we first have to talk about conditioning.

The Fundamental Theorem of Asset Pricing Without Probabilistic Prior Assumptions

‘Forward and Futures Prices of Contingent Claims’ applies the multi-period rational expectations model to analyse the pricing of forward and futures contracts and derives an expression for the difference between the prices of the two, which is commonly known as the convexity adjustment. These are illustrated using an example where the asset price, the pricing kernel and the bond prices are joint lognormal variables. The authors then consider the futures and forward prices of contingent claims, such as call options. Futures prices of options can be significantly different from the forward prices of the contingent claims, especially when the options are on interest-rate related assets such as bonds or swaps. Forward prices of long-term bond can be obtained by taking appropriate expected values of future spot pries under the risk-neutral measure.

A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time

In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbitrage is equivalent to the existence of an equivalent martingale measure. Such an equivalent measure can be derived as the normal unit vector of the hyperplane that separates the attainable gain subspace and the convex cone representing arbitrage opportunities. However, in two-price financial models (where there is a bid–ask price spread), the set of attainable gains is not a subspace anymore. We use convex optimization, and the conic property of this region to characterize the “no-arbitrage” principle in financial models with the bid–ask price spread present. This characterization will lead us to the generation of a set of price factor random variables. Under such a set, we can find the lower and upper bounds (supper-hedging and sub-hedging bounds) for the price of any future cash flow. We will show that for any given cash flow, for which the price is outside the above range, we can build a trading strategy that provides one with an arbitrage opportunity. We will generalize this structure to any two-price finite-period financial model.

In a continuous time market model we consider the problem of existence of an equivalent martingale measure with density lying within given lower and upper bounds and we characterize a necessary and sufficient condition for this. In this sense our main result can be regarded as a version of the fundamental theorem of asset pricing. In our approach we suggest an axiomatic description of prices on Lp-spaces (with p ∈[1, ∞)) and we rely on extension theorems for operators.

This paper provides a theoretical analysis on the impacts of using a suboptimal information set for the estimation of the pricing kernel and, more in general, for the validity of the fundamental theorems of asset pricing. While inferring the risk-neutral measure from options data provides a naturally forward-looking estimate, extracting the real world measure from historical returns is only partially informative, thus suboptimal with respect to investors’ future beliefs. As a consequence of this disalignment, the two measures no longer share the same nullset, thus distorting the investors’ risk premium and the validity of the pricing measure. From a probabilistic viewpoint, the missing beliefs are totally unaccessible stopping times on the coarser filtration set, so that an absolutely continuous strict local martingale, once projected on it, becomes continuous with jumps. Some empirical examples complete the paper.

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures. Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.

Contents: Acknowledgements General Introduction PART I HARRY M. MARKOWITZ Introduction to Part I Howard R. Vane and Chris Mulhearn 1. Harry Markowitz (1952a), 'Portfolio Selection' 2. Harry Markowitz (1952b), 'The Utility of Wealth' 3. H. Levy and H.M. Markowitz (1979), 'Approximating Expected Utility by a Function of Mean and Variance' 4. Harry M. Markowitz and Eric L. van Dijk (2003), 'Single-Period Mean-Variance Analysis in a Changing World' PART II MERTON H. MILLER Introduction to Part II Howard R. Vane and Chris Mulhearn 5. Franco Modigliani and Merton H. Miller (1958), 'The Cost of Capital, Corporation Finance and the Theory of Investment' 6. Franco Modigliani and Merton H. Miller (1959), 'The Cost of Capital, Corporation Finance and the Theory of Investment: Reply' 7. Merton H. Miller and Franco Modigliani (1961), 'Dividend Policy, Growth, and the Valuation of Shares' 8. Franco Modigliani and Merton H. Miller (1963), 'Corporate Income Taxes and the Cost of Capital: A Correction' 9. Merton H. Miller and Franco Modigliani (1966), 'Some Estimates of the Cost of Capital to the Electric Utility Industry, 1954-57' 10. Merton H. Miller (1977), 'Debt and Taxes' PART III WILLIAM F. SHARPE Introduction to Part III Howard R. Vane and Chris Mulhearn 11. William F. Sharpe (1963), 'A Simplified Model For Portfolio Analysis' 12. William F. Sharpe (1964), 'Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk' 13. William F. Sharpe (1966), 'Mutual Fund Performance' 14. William F. Sharpe (1978), 'Bank Capital Adequacy, Deposit Insurance, and Security Values' PART IV ROBERT C. MERTON Introduction to Part IV Howard R. Vane and Chris Mulhearn 15. Robert C. Merton (1969), 'Lifetime Portfolio Selection Under Uncertainty: The Continuous-Time Case' 16. Robert C. Merton (1971), 'Optimum Consumption and Portfolio Rules in a Continuous-Time Model' 17. Robert C. Merton (1973a), 'Theory of Rational Option Pricing' 18. Robert C. Merton (1973b), 'An Intertemporal Capital Asset Pricing Model' 19. Robert C. Merton (1974), 'On the Pricing of Corporate Debt: The Risk Structure of Interest Rates' 20. Robert C. Merton (1977), 'On the Pricing of Contingent Claims and the Modigliani-Miller Theorem' PART V MYRON S. SCHOLES Introduction to Part V Howard R. Vane and Chris Mulhearn 21. Fischer Black and Myron Scholes (1972), 'The Valuation of Option Contracts and a Test of Market Efficiency' 22. Fischer Black and Myron Scholes (1973), 'The Pricing of Options and Corporate Liabilities' 23. Fischer Black and Myron Scholes (1974), 'The Effects of Dividend Yield and Dividend Policy on Common Stock Prices and Returns' 24. Myron Scholes and Joseph Williams (1977), 'Estimating Betas from Nonsynchronous Data' 25. Merton H. Miller and Myron S. Scholes (1978), 'Dividends and Taxes' Name Index

The role of professional organizations in risk management is not a new concept. Many have been around for 50 or more years meeting the specialized needs of their members. What is new is the increasingly important role these organizations are now playing in assisting in the ongoing personal education of financial professionals and the development of the global regulatory environment within the financial services industry. This article explores the rationale behind the proliferation of professional organizations in the field of financial risk management, their historical role, what financial risk‐management professionals expect from these organizations, what the organizations can be reasonably expected to deliver, and the future role of professional risk‐management organizations of which the rapidly growing global discipline of financial risk management is a part. Two basic issues need to be addressed before dealing with the expectations for and the delivery of products and services from a professional risk‐management association. These issues present somewhat of a “chicken‐and‐egg” problem. The first involves defining financial risk management, and the second relates to simply being able to define the organization's role in attempting to realistically meet the expectations of an increasingly broad number of varied constituencies in which probably the most dynamic discipline is the financial services industry.

This paper considers the pricing of discretely sampled Asian and lookback options with floating and fixed strikes. In the modelling framework of Black and Scholes (1973), it is shown that a change of numeraire of the martingale measure can be used to reduce the dimension of these path-dependent option pricing problems to one in addition time. This means that the pricing problems can be solved by numerically solving onedimensional partial differential equations. The author demonstrates how a Crank‐ Nicolson scheme can be applied to the numerical solution. Finally, the methodology is extended to the case when the underlying stock exhibits discontinuous returns, and it is shown that in this case the Asian and lookback option pricing problems can be solved by numerically solving one-dimensional partial integrodifferential equations.

Introduction. This is a sequel to my paper [1]. The present developments are largely independent of the previous results except in so far as given in the Appendix. Theorem 1 shows a kind of solidarity among the states of a recurrent class; it generalizes a classical result due to Kolmogorov and permits a classification of recurrent states and classes. In ?2 some relations involving the mean recurrence and first passage times are given. In ??3-5 sequences of random variables associated in a natural way with a Markov chain are studied. Theorem 2 is a generalized ergodic theorem which applies to any recurrent class, positive or null. It turns out that in a null class there is a set of numbers which plays the role of stationary absolute probabilities. In the case of a recurrent random walk with independent, stationary steps these numbers are all equal to one and the result is particularly simple. Theorem 3 shows that the kind of solidarity exhibited in Theorem 1 persists in such a sequence; it leads to the clarification of certain conditions stated by Doblin(2) in connection with his central limit theorem. Using a fundamental idea due to Doblin, the weak and strong laws of large numbers, the central limit theorem, the law of the iterated logarithm, and the limit theorems for the maxima of the associated sequence are proved very simply. Owing to the great simplicity of the method it is the conditions of validity of these limit theorems that should deserve attention. Among other things, we shall show by an example that a certain set of conditions, attributed to Kolmogorov, is in reality not sufficient for the validity of the central limit theorem. Furthermore, conditions of validity for the strong limit theorems and the limit theorems for the maxima are obtained by a rather natural strengthening of corresponding conditions for the weak limit theorems. A word about the connection of these conditions with martingale theory closes the paper. 1. The sequence of random variables {Xn}, n=0, 1, 2, * , forms a denumerable Markov chain with stationary transition probabilities. The states will be denoted by the non-negative integers(3) 0, 1, 2, * * . The n-step transition probability from the state i to the state j will be denoted by P(n) (P(l) = Pij). Thus we have

This paper presents a unifying theory for valuing contingent claims under a stochastic term of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processeE for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results. IN RELATION TO the term of interest rates, arbitrage pricing theory has two purposes. The first, is to price all zero coupon (default free) bonds of varying maturities from a finite number of economic fundamentals, called state variables. The second, is to price all interest rate sensitive contingent claims, taking as given the prices of the zero coupon bonds. This paper presents a general theory and a unifying framework for understanding arbitrage pricing theory in this context, of which all existing arbitrage pricing models are special cases (in particular, Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), Ball and Torous (1983), Ho and Lee (1986), Schaefer and Schwartz (1987), and Artzner and Delbaen (1988)). The primary contribution of this paper, however, is a new methodology for solving the second problem, i.e., the pricing of interest rate sensitive contingent claims given the prices of all zero coupon bonds. The methodology is new because (i) it imposes its stochastic directly on the evolution of the forward rate curve, (ii) it does not require an inversion of the term structure to eliminate the market prices of risk from contingent claim values, and (iii) it has a stochastic spot rate process with multiple stochastic factors influencing the term structure. The model can be used to consistently price (and hedge) all contingent claims (American or European) on the term structure, and it is derived from necessary and (more importantly) sufficient conditions for the absence of arbitrage. The arbitrage pricing models of Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), and Artzner and Delbaen (1988) all require an IFormerly titled Bond Pricing and the Term Structure of Interest Rates: A New Methodology.

In the perfect and unrealistic Black and Scholes (J Polit Econ 81:637–659, 1973) world, the dynamics \((S_{t})_{t\in [0,T]}\) of the risky asset, under the historical probability \(\mathbb{P}\), is given by the following stochastic differential equation: $$\displaystyle{ dS_{t} =\mu S_{t}dt +\sigma S_{t}dW_{t} }$$ where \((W_{t})_{t\in [0,T]}\) is a standard Brownian motion under \(\mathbb{P}\). In this case, there is no ambiguity in the definition the arbitrage-free price of any European contingent claim with maturity T. In fact, in this complete market which is set in continuous time, this value is none other than the value of any replicating portfolio. Moreover, prices may be expressed in terms of conditional expectations under a unique equivalent martingale measure Q whose density with respect to the historical probability is given by the Girsanov theorem $$\displaystyle{ \frac{dQ} {d\mathbb{P}} = e^{-\frac{\mu -r} {\sigma } W_{T}-\left (\frac{\mu -r} {\sigma } \right )^{2} \frac{T} {2} } }$$ where r is the constant and continuously compound risk-free rate. Unfortunately, as we have seen in Sect. 2.1, the restrictive underlying hypotheses (constant volatility, independent increments, Gaussian log-returns, etc…) are questioned by many empirical studies and GARCH models appear as excellent alternative solutions to potentially overcome some well-documented systematic biases associated with the Black and Scholes model.

This paper develops the parametric restrictions imposed on diffusion state processes by the requirement of arbitrage-free asset pricing. Using the equivalent martingale measure as a starting point, the diffusion property is exploited to specify the shadow pricing func? tion, which takes conditional state variable probabilities under the reference measure into arbitrage-free contingent claim prices. The main results of the paper provide differential equations associated with the shadow price function that are used to identify restrictions on the parameters of assumed diffusion processes. The paper concludes with an application to the CIR model where the state variable, the instantaneous interest rate, is assumed to follow a square root process. Calculations are also provided for the parametric restrictions imposed on the Brownian bridge and OU state variable processes.