“La teoría perciana se presenta como una semiótica (teoría general de los signos) cognoscitiva, como una disciplina filosófica que pretende la
4. PROPUESTA PARA LA INTERPRETACIÓN VISUAL DEL ORGANÓN
It is now necessary to discuss how this solution is different from other known results and why it was necessary to derive it from the beginning. The same methodology is described in Liu et al. (2003) for a variation of the SVCJ model. They parametrise by adapting the Pan (2002) values for stocks and options. In fact the model is widely used in the options related literature, while papers using equity data rely on the EJP model.
dSt St = (r + ηVt− µλVt)dt + p VtdWtY + Ξ Y t dNt (5.12) dVt = (α0 − β0Vt− κ0λVt)dt + σV p VtdWtV + ΞVt dNt (5.13)
where St is the price, not the return or the log price as previously, and Vt is volatility,
r is the constant risk-free rate, ηVt is the volatility premium which is insignificant for
stock prices (Eraker et al. (2003)), ΞS,V are jump sizes with means µ and κ respectively,
stochastic arrival intensity is λVt of Poisson process N, µλVt is returns jump premium
and κ0λVt is the volatility jump premium.
The most important difference is the use of a stochastic arrival intensity that is linearly related to volatility. The process for the Poisson parameter is a Cox process of the general form λ = λ0+ λ1Vt. Setting λ0 = 0 leads to the specification in (5.12)
and (5.13) while setting λ1 = 0 leads to the EJP specification of constant rate in (5.1)
and (5.2). A less important difference is the inclusion of the volatility premium, which has some importance for options, and the correction in the mean for the jump premium µλVt. The drift of the diffusion thus includes one additional term that compensates for
jumps. Since these changes are introduced linearly, the solution is not greatly affected. The solution for φ is
φ = η − µλ γ + ρσVB(t) γ + λE(ξY)(1 − φE(ξY))−γ γ exp(B(t)E(ξ V )) (5.14)
where B(t) solves the differential equation B0(t) − 1 2γφ 2(1 − γ) + 1 2σ 2 VB 2(t) + (σ Vφρ(1 − γ) − κλ − β)B(t)
+(η − µλ)(1 − γ)φ + λE(ξY)(1 − φE(ξY))−γexp(B(t)E(ξV)) = 0 (5.15) with initial conditions
A(T ) = 0, B(T ) = 0
The reason why stochastic arrival intensity and a Cox process were selected is because they allow the time-varying V terms to be eliminated from the denominators. An
CHAPTER 5. MODEL ESTIMATION AND OPTIMAL PORTFOLIO WEIGHTS
additional term λVt appears in the numerators of each fraction which leads to (5.14) and
(5.15). The ODE for B(t) is not a Riccati equation and can only be solved numerically. Setting λ1 = 0 on the other hand (or equivalently defining the arrival rate as λ) turns
the ODE for B(t) into a Riccati equation which can be solved in closed form Branger and Hansis (2015). However, in the EJP model V can not be eliminated: the absence of the jump premium in the drift excludes λVt from appearing and the last jumps related
term is multiplied by λ only.
The Liu et al model comes with a set of drawbacks. The selection of a Cox process does, indeed, produce a numerical solution for portfolio weights and provides a time variable jump arrival intensity. Nevertheless, estimations of that model in the literature prove to be extremely rare. Apart from the GMM estimation of Pan (2002), the only other paper to the knowledge of the thesis that provides parameters for that exact model is Eraker (2004) using MCMC estimation largely taken from Eraker et al. (2003). In all the following literature (Branger, Brag, Schneider Hansis etc) the Pan parameters are the only one used (based on a dataset ending in 1999), with the Eraker parameters not only having been ignored but also having never been updated. During the PhD an attempt was made to adapt the code to fit to its stochastic arrival intensity counterpart and estimate it, but after 100 − 200 repetitions the software would crash due to memory overload issues. On the contrary, the EJP model has been estimated multiple times in different markets and over different periods, thus providing a verified and tested history of application and results. In addition, the parameters in Eraker et al. (2003) and Broadie et al. (2007) have been calibrated in numerous cases (e.g Branger and Hansis (2012, 2015)) to fit the Liu et al. model when needed. At the very least, this demonstrates a difficulty in getting parameter estimates via MCMC for the stochastic arrival intensity variation of SVCJ and, given the need to take the financial crisis into account, existing estimates are inadequate.
A second consideration is the Cox process itself. Even in the early jumps literature there was evidence for misspecification under stochastic arrival intensity with only marginal improvements in accuracy at the cost of complexity (Bates (2000)). The main advancements took place with a constant λ and where time variability was introduced the model tended to be simpler (e.g constant volatility, as in Wachter (2013). These remarks highlight analytical and estimation based issues. Also, the (seemingly positive) relationship between jump frequency and volatility may be intuitive but not necessarily linear. It is included int he Liu et a model because it leads to a tractable solution for portfolio weights combined with an ODE for a numerical solution, which translates into
5.3. OPTIMAL PORTFOLIO WEIGHTS
reverse-engineering convenience. No further argument is offered. Other processes that can provide similar, or more intuitive, structures are infinite-jump Levy processes, who are better at capturing small jumps Li et al. (2006) or Hawkes processes (Fiˇcura and Witzany (2015)), who introduce a self-exciting (-feeding) component. The latter is also related to persistent jumps whose effects dissipate over time. Cox processes are still instantaneous, are able to attribute a higher jump probability when volatility is higher but pose estimation difficulties and,most importantly, do not perform better than models with constant arrival intensity.