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First we consider the simpler case in which the intensities of the Poisson processes depend only on time. Since we expect on average for a shift to an opposing regime to occur as the event approaches, we model the intensities to increase as the event date approaches. For all cases,

λ?

i j is assumed to be the maximum Poisson intensity that the process can ever reach. Thus we

have defined an upper bound on the intensity of our process. As previously statedκis the time intensity parameter andmi j(S,t) is the state-dependent market price of volatility risk. For the

discussion presented, assume that the state-dependent market prices of volatility risk take on constant values (i.e. mi j ≡mi j(S,t)).

7.2. EarningsReleaseFramework 127

We restrict our time intensity parameter to only take on strictly positive values such that

κ > 0. If κ = 0, this would insinuate that time has no impact on the switching intensity

and thus our framework reduces to that where the Poisson intensities take on constant fixed values. When the event occurs at maturity, we model the intensities to increase as the event date approaches. On the other hand, when the event date does not occur at option maturity, the intensities increase as we approach the event date and decrease after the event has occurred. In general our deterministic time-varying model is as follows:

λi j(t)= λ?i je −κ(τ?−t)sgn(τ?−t)_, (7.2) where sgn(τ?−t)=              −1 ift > τ? 0 ift =τ? 1 ift < τ? .

When the event occurs at maturity such that τ? = T, it follows that sgn(T −t) = 1 since

t≤ T for allt. The above result reduces to:

λi j(t)=λ?i je

−κ(T−t)_{. (7.3)}

The functione−κ(τ?−t)sgn(τ?−t) acts as a control on the Poisson intensity and due to its math- ematical properties, allows it to vary between zero andλ?_{i j}. Thus our deterministic intensity is a bounded function: 0 ≤λi j(t) ≤ λ?_{i j}. Ifκ(τ?−t) → ∞this would imply thatλi j(t) →0. Thus

if either the time to event or the chosen value for κ is very large, the switching effect would be turned offcompletely. This would result in the volatility never being able to switch out of its currently occupied regime. A more detailed investigation of the time-intensity parameter,κ, with respect to option maturity dates and event dates is presented in Section 7.3. For now, it is safe to assume thatκ(τ?−t) is finite such that the switching mechanism is enabled within our regime-switching framework. In other words, we only consider a finite time horizon.

For the remainder of this chapter, we will analyse numerical results pertaining to the time varying Poisson intensities in two separate cases. Case 1 describes a case in which the financial event occurs at option maturity, while Case 2 will refer to events that occur before maturity,

τ? _{< T. We will first analyse how our time varying intensities are modelled for both of these}

cases separately. Since our intensities are independent of stock price, Figures 7.6 and 7.7 plot the intensities against time alone.

In Figure 7.6, we can observe that the Poisson intensity increases as the maturity date of the option approaches, which coincides with the date of the financial event. Thus as we approach the financial event, it becomes more and more likely that our Poisson process will trigger a switch to the opposing regime. The speed at which the intensity changes with respect to time is controlled by the magnitude ofκand will be discussed later on.

Figure 7.7 illustrates the observable difference when the financial event occurs before op- tion maturity. As we approach the financial event at τ?, the Poisson intensity increases. It reaches it maximum intensity at the event datet = τ? and decreases as we move further away from the date and towards maturityτ? < t ≤ T. The intensity is modelled in this way as it is assumed that if the switch does not occur before or on the event that, it becomes highly unlikely that it will switch regimes afterwards. This is due to the fact that after the event has occurred

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3 4 5 6 7 8 9 10 Time λ (t)

Figure 7.6: Time varying Poisson intensity assuming the financial event occurs at maturity.

κ=1,λ? =10% (daily), andT =1 year.

and the information disclosed is now public, investors no longer have this unknown risk in their portfolio. Thus it is likely that if the volatility does not switch before or on the event date, that it will not switch at all. Once again, κcontrols the speed at which the time varying intensity grows and decays.

7.2. EarningsReleaseFramework 129 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4 5 6 7 8 9 10 Time λ (t)

Figure 7.7: Time varying Poisson intensity assuming the financial event occurs before maturity.

κ=1,λ? =10% (daily),τ?=9 months, andT =1 year.