Algunas características sociales de los estudiantes de la generación

We shall provide some simulations of price processes. In the figures below, different colours represent different numbers of information sources available to the market. Hence, each colour is associated with a different volatility process and a Brownian motion governing the price process. One may view each colour as a different economic regime, suggesting that each jump is a regime switch. We shall develop a more general regime-switching framework in Chapter 4.

Figure 3.1: A price process with four jumps. Different colours represent different economic regimes: Blue regime, red regime, green regime and etc. The price process is governed by a different Brownian motion and a stochastic volatility process during each regime. Cash flow: XT = 1. Parameters: T = 5, rt= 0, κi= 1/T

and ρi_{= 0.5. Stopping times are uniformly distributed on [0, T ].}

Figure 3.2: A price process with two jumps. There are three different regimes. Cash flow: XT = 0.

Figure 3.3: A price process with five jumps. There are six different regimes. Cash flow: XT = 1.

Parameters: T = 5, rt= 0, κi= 1/T and ρi= 0.5. Stopping times are uniformly distributed on [0, T ].

Figure 3.4: A price process with four jumps. There are five different regimes. Cash flow: XT = 0.

Chapter 4

Random Deactivation-Reactivation of

Information and Regime Switches

The main aim of this chapter is to develop an information-based framework to model regime switches in a given economy. In a way, in Chapter 3, we have already presented an approach for modelling regime switches. More precisely, one may argue that there is a bicausal re- lationship between appearances of new information sources (or public announcements) and passing from one economic regime to another. From the results presented in Chapter 3, this suggests that every regime switch coincides with a jump in the price process. However, we believe that it is still a rather restrictive viewpoint to expect a price jump at every regime switch. Therefore, we would like to adapt a more elaborate information-based standpoint in our approach. In general terms, we prefer to view regime switches as events that coincide with changes in the sources of information in the market. By changes of information sources, we do not neccesarily mean appearances of new information sources. It may as well be that a source of market information stops flowing for a random period of time before it is active again.

There is a vast stream of mathematical literature on regime switches. For example, the continuous-time version of the stochastic regime-switching model of Hamilton (1989) (also see Hamilton, 1996) implies that asset prices switch between two states where the switches are governed by a Markov point process, and prices are continuous during each state. In a given economic regime, the continuous changes of a price process are governed by a diffusion process with its own volatility. Diffusion processes together with Markov point processes can be analysed under Hidden Markov models, which have a wide spectrum of applications in mathematics (see, for example, Elliott, Aggoun, and Moore, 1997). Cecchetti, Lang and Mark (1990), and Driffill and Sola (1998) model dividends using two-state Markov- switching models to represent the US stock market. Kim, Piger and Startz (2005) discuss the estimation of Markov regime switch models where the switches are endogenous. Driffill,

Kenc and Sola (2002) price perpetual American call options when the underlying prices are modelled as regime-switching processes which have stochastic dividends that switch between two economic states characterised by different volatilities. Naik (1993), Bollen, Gray and Whalley (2000) and Chourdakis and Tzavalis (2000) are few other examples of option pricing under regime-switching economies.

In the literature, it is common to start with a model of a price process that has the char- acteristics to represent regime switches in an economy. This motivates us to ask whether it is possible to reverse this approach. More precisely, we start by specifying the flow of in- formation first, and derive price processes under regime-switching economies, where regime switches are events that coincide with changes in the sources of information in the market. In addition, we would still like our price processes to exhibit similar behaviour as assumed in the current literature. For example, price processes are usually assumed to have differ- ent volatilities during different economic regimes. In this respect, the material presented in Chapter 3 can be interpreted with a regime switch perspective, since we have seen that price processes are governed by different diffusion and volatility processes in between stop- ping times. Then, each time interval between the stopping times (say, between important newscasts) can be interpreted as a different economic state. Our aim is to further develop an information-based framework that allows us to derive a rich class of price dynamics under regime-switching economies, and which potentially sheds light on our understanding of how regime switches may arise in a given economy.

This chapter is organised as follows: Section 2 provides a brief mathematical setting. Section 3 is the pricing of financial derivatives when new sources of information appear at stopping times. This section includes European options and few examples of credit-based products. In Section 4, we stop the flow of information. Section 5 presents the random deactivation-reactivation of information sources. We generalise the deactivation-reactivation setting to the multiple market factor scenerio. In addition, as a special example, we introduce a market filtration where each stopping time induces a switch from one information source to another.

4.1

Mathematical Setting

The mathematical setting in this chapter is almost exactly the same as the one in Chapter 3. To save space, we do not restate everything that we have already stated, and refer the reader to Chapter 3.2.

We let (Ω, F , Q) be the probability space equipped with {Ft}0≤t≤∞, where Q is the

pricing measure. We assume that all filtrations are right-continuous and complete, and we fix a finite time horizon [0, T ]. The minor difference with respect to the previous chapter arises in our view of the cash flow XT. In this chapter, XT is not neccessarily continuous.

If XT is discrete, we denote its probability mass function by p(xj) > 0 (i.e., Q(XT = xj))

for some index j where xj ∈ X, and its conditional mass function at time t by φit(xj):

φi_t(xj) = p(xj|Fξ

i

t ) = p(xj|ξti), (4.1)

given that {ξ_ti} is a Brownian information process for i = 1, 2, . . .. There may be countably infinite information processes. We have

φi_t(xj) = exph_{(T −t)}T (κi_x jξti− 1 2(κ i_x j)2t) i p(xj) P Xexp h T (T −t)(κixjξ i t−12(κixj)2t) i p(xj)