A quickest detection problem with an observation cost

In the classical quickest detection problem, one must detect as quickly as possible when a Brownian motion without drift "changes" into a Brownian motion with positive drift. The change occurs at an unknown "disorder" time with exponential distribution. There is a penalty for declaring too early that the change has occurred, and a cost for late detection proportional to the time between occurrence of the change and the time when the change is declared. Here, we consider the case where there is also a cost for observing the process. This stochastic control problem can be formulated using either the notion of strong solution or of weak solution of the s.d.e. that defines the observation process. We show that the value function is the same in both cases, even though no optimal strategy exists in the strong formulation. We determine the optimal strategy in the weak formulation and show, using a form of the "principle of smooth fit" and under natural hypotheses on the parameters of the problem, that the optimal strategy takes the form of a two-threshold policy: observe only when the posterior probability that the change has already occurred, given the observations, is larger than a threshold $A\geq0$, and declare that the disorder time has occurred when this posterior probability exceeds a threshold $B\geq A$. The constants $A$ and $B$ are determined explicitly from the parameters of the problem.Comment: Published at http://dx.doi.org/10.1214/14-AAP1028 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Esscher transform and the duality principle for multidimensional semimartingales

The duality principle in option pricing aims at simplifying valuation problems that depend on several variables by associating them to the corresponding dual option pricing problem. Here, we analyze the duality principle for options that depend on several assets. The asset price processes are driven by general semimartingales, and the dual measures are constructed via an Esscher transformation. As an application, we can relate swap and quanto options to standard call and put options. Explicit calculations for jump models are also provided.Comment: Published in at http://dx.doi.org/10.1214/09-AAP600 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Esscher Transforms in Discrete Finance Models

The object of our study is where each Sn is a m-dimensional stochastic (real valued) vector, i.e. denned on a probability space (Ω, , P) and adapted to a filtration ( n )0≤n≤N with 0 being the σ-algebra consisting of all null sets and their complements. In this paper we interpret as the value of some financial asset k at time n. Remark: If the asset generates dividends or coupon payments, think of as to include these payments (cum dividend process). Think of dividends as being reinvested immediately at the ex-dividend price. Definition 1 (a) A sequence of random vectors where is called a trading strategy. Since our time horizon ends at time N we must always have ϑN ≡ 0. The interpretation is obvious: stands for the number of shares of asset k you hold in the time interval [n,n + 1). You must choose ϑn at time n. (b) The sequence of random variables where Sn stands for the payment stream generated by ϑ (set ϑ−1 ≡ 0

When to sell Apple and the NASDAQ? Trading bubbles with a stochastic disorder model

In this paper, the authors apply a continuous time stochastic process model developed by Shiryaev and Zhutlukhin for optimal stopping of random price processes that appear to be bubbles. By a bubble we mean the rising price is largely based on the expectation of higher and higher future prices. Futures traders such as George Soros attempt to trade such markets. The idea is to exit near the peak from a starting long position. The model applies equally well on the short side, that is when to enter and exit a short position. In this paper we test the model in two technology markets. These include the price of Apple computer stock AAPL from various times in 2009-2012 after the local low of March 6, 2009; plus a market where it is known that the generally very successful bubble trader George Soros lost money by shorting the NASDAQ-100 stock index too soon in 2000. The Shiryaev-Zhitlukhin model provides good exit points in both situations that would have been profitable to speculators following the model

Land and stock bubbles, crashes and exit strategies in Japan circa 1990 and in 2013

We study the land and stock markets in Japan circa 1990 and in 2013. While the Nikkei stock average in the late 1980s and its -48% crash in 1990 is generally recognized as a financial market bubble, a bigger bubble and crash was in the land market. The crash in the Nikkei which started on the first trading day of 1990 was predictable in April 1989 using the bond-stock earnings yield model which signaled a crash but not when. We show that it was possible to use the change point detection model based solely on price movements for profitable exits of long positions both circa 1990 and in 2013