Martingales

We analyze the price behavior of the main precious metals-gold, silver, platinum and palladium-before, during and in the aftermath of the 2007-08 financial crisis. Using the mildly explosive/multiple bubble technology developed by Phillips, Shi and Yu (2015, International Economic Review 56(4), 1043-1133), we find significant, short periods of mildly explosive behavior in the spot and futures prices of all four metals. Fewer periods are detected using exchange-rate adjusted prices, and almost none when deflated prices are used. We assess whether these findings are indicative of bubble behavior. Convenience yield is shown to have little efficacy in this regard, while other fundamental proxy variables and position data offer only very limited evidence against prices having been anything other than fundamentals-driven. Possible exceptions are in gold in the run-up to the highpoint of the financial crisis, and in silver and palladium around the launch of specific financial products. Some froth, however, is reported and discussed for each metal.

We consider a discrete-time financial market model with L 1 risky asset price process subject to proportional transaction costs. In this general setting, using a dual martingale representation we provide sufficient conditions for the super-replication cost to coincide with the replication cost. Next, we study the convergence problem in a stationary binomial model as the time step tends to zero, keeping the proportional transaction costs fixed. We derive lower and upper bounds for the limit of the super-replication cost. In the case of European call options and for a unit initial holding in the risky asset, the upper and lower bounds are equal. This result also holds for the replication cost of European call options. This is evidence (but not a proof) against the common opinion that the replication cost is infinite in a continuous-time model.

For d ≥ 1 and α ∈ (0, 2), consider the family of pseudo differential operators { ∆ + b ∆ α/2; b ∈ [0, 1]} on R d that evolves continuously from ∆ to ∆ + ∆ α/2. In this paper, we establish a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to ∆ + b ∆ α/2 (or equivalently, the sum of a Brownian motion and an independent symmetric α-stable process with constant multiple b 1/α) in C 1,1 open sets. Here a “uniform ” BHP means that the comparing constant in the BHP is independent of b ∈ [0, 1]. Along the way, a uniform Carleson type estimate is established for nonnegative functions which are harmonic with respect to ∆ + b ∆ α/2 in Lipschitz open sets. Our method employs a combination of probabilistic and analytic techniques. 1.

Let χ be a family of stochastic processes on a given filtered probability space (Ω, F, (Ft)t∈T, P) with T⊆R+. Under the assumption that the set Me of equivalent martingale measures for χ is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure that minimizes the relative entropy, with respect to P, in the class of martingale measures. We then provide the characterization of the density of the minimal entropy martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy.

In this section we give some crucial preliminaries from financial mathematics. The results are formulated in a nonrigorous way without stating the precise assumptions. We start by giving the solutions to two important stochastic differential equations. Theorem 3.1 (Linear stochastic differential equation with deterministic diffusion coefficient). The solution of the linear stochastic differential equation dX(t) = (α(t) + β(t)X(t))dt + γ(t)dW (t), where α, β, and γ are deterministic functions, is given by X(t) = X(s)e t s β(τ)dτ + t s α(u)e t u β(τ)dτ du + t s γ(u)e t u β(τ)dτ dW (u). Theorem 3.2 (Geometric Brownian motion). The solution of the lognormal linear stochastic differential equation dX(t) = μ(t)X(t)dt + σ(t)X(t)dW (t), where μ and σ are deterministic functions, is given by the generalized geometric Brownian motion X(t) = X(s) exp t s μ(τ) − σ 2 (τ) 2 dτ + t s σ(τ)dW (τ). In order to check solutions of stochastic differential equations, Itô's formula and the stochastic Leibniz rule as given next are useful. Theorem 3.3 (Itô's formula). dF (t, X(t)) = F t (t, X(t))dt + F x (t, X(t))dX(t) + 1 2 F xx (t, X(t))(dX(t)) 2 , 11

Statistical testing can be framed as a repetitive game between two players, Forecaster and Sceptic. In each round, Forecaster sets prices for various gambles, and Sceptic chooses which gambles to make. If Sceptic multiplies by a large factor the capital he puts at risk, he has evidence against Forecaster’s ability. His capital at the end of each round is a measure of his evidence against Forecaster so far. This can go up and then back down. If you report the maximum so far instead of the current value, you are exaggerating the evidence against Forecaster. In this article, we show how to remove the exaggeration. Removing it means systematically reducing the maximum in such a way that a rival to Sceptic can always play so as to obtain current evidence as good as Sceptic’s reduced maximum. We characterize the functions that can achieve such reductions. Because these functions may impose only modest reductions, we think of our result as a method of insuring against loss of evidence. In the context of an actual market, it is a method of insuring against the loss of what an investor has gained so far.