In this section we elaborate on the transformations that reconnect the model equations employed in Chapter3and Chapter4 to conventional models in literature.
We start by noticing that in Eq. (3.1) we use the linearised version of the geometric model, that is
x= logS, wherebyS is conventionally the equity price or the index level. In strictly financial terms, we
are looking at the instantaneous intensities of growthxof the continuous compounding lawS. Considering
the SDE, we notice we perform a first modification, for strictly statistical reasons. Specifically, if we have 4It should be noticed that, when no measurement error is included, the update equation delivers exactlyxat the fist component of the observableY.
A.4. MODEL TRANSFORMATIONS 168 the stochastic volatility model,
dS =S√vdW
wherev is adapted, Markovian, mean reverting and positive almost certainly. The absence of a drift or
jump does not affect the result. Indeed, if we consider thelogtransform of the geometric motion, which
is characterised by the SDE
dx=−1 2vdt+
√
vdW
we proceed with the computation of the first lag autocovariance of the processx. In order to do so, we
define theksteps forward return over the period∆ computed att, that is
∆X(k) =−1 2 k+1 Z k v(t+ ∆τ) dτ + k+1 Z k p v(t+ ∆τ) dW(t+ ∆τ), k∈N0
First fact we notice is that the expectation of the forward return, that is
E[∆X(k)|Ft] =−12 k+1
Z
k
E[v(t+ ∆τ)|Ft] dτ
entailing that the transformation of the initial martingale model of the price generates a model of in- stantaneous returns that are expected to have a negative drift. Now, this inconvenience is not major as it can be corrected by modifying the model for x with a positive constant, assuming that the original
price model is trending upward in correspondence to empirical evidence. The problem with the statistical analysis of the logarithmic transform of the initial model is the following. Elaborating from the price model ofSwe obtain an SDE that is characterised by non zero autocovariance. In fact, considering that v is Markovian and mean reverting, the mixed moment of the return with lagk becomes
E[∆X(0)∆X(k)|Ft] = 14 k+1
Z
k
E[v(t+ ∆)v(t+ ∆τ)|Ft] dτ, k∈N
and because the joint distribution of∆X(0)and∆X(k)is non factorisable, the autocovariance function
will be different from0, implying the presence of autocorrelation in the model for x, which cannot be
reconciled with the observed behaviour of market returns. As for in Chapter 3 we are performing the statistical analysis of the return process, we modify the SDE in order to obtain a weekly stationary model of the first order differences. Therefore, with respect to models formulated upon the price level and struc- tured as a geometric process like the referenced models in Tab. 14, the stochastic model in the main equation of Chapter3 is the result of the application of the logarithmic transform, but by concurrently dropping the Jensen term arising from the transformation and that would otherwise generate dynamic features that do not fit the empirical evidence5. Nonetheless, this solution necessitates further investi-
gation, as we notice that the formulation of Eq. (3.1) entails the presence of the volatility component in the drift of the price levelS, which can be reconciled with the empirical evidence but that has been
5On the other hand, in the case of the stochastic hazard class of Chapter3, the latter consideration is unnecessary as the logarithmic transform of jump-diffusions such asMerton(1976) andKou(2002) do not produce a mean reversion term in the drift ofx.
analysed inBenzoni(2002) and appears to show a non significant factor loading. For future research we plan to develop a likelihood model of the price level, in order to avoid this modelling issue and further align the historical and risk-neutral measure approach.
Another transformation that is necessary to reconcile the model version exploited in this thesis with the standard versions presented in literature is that of the constant elasticity of variance model. In literature, when referring to the CEV model for equity stock we retrieve two versions, a single factor and a two factor model, both modelling the price level with a geometric process. In the first version, the plain BS model whereby the stochastic diffusion factor is determined by a multiple of the level of price, is modified by exponentiating the price level factor in the diffusion component, in order to modulate the response to oscillation in the value ofS. In the formulation ofBeckers (1980) andMacbeth and Merville(1980),
disregarding the drift, the price SDE is the following
dS=σS1+γdW
In order to obtain the formulation of this thesis, we might in principle reshape the diffusion asσSuγdW,
whereby in the original model the volatility factoruis perfectly correlated with price. This dependence
is loosen by introducing the auxiliary factorv of the Eq. (3.1) and (4.1)−(4.2). Another model that in
literature is referred to as the CEV model, is the stochastic volatility model employed inJones(2003) and Aït-Sahalia and Kimmel(2007), whereby the latent factor v is formulated as theCox and Ross (1976)
model, which originally applied the SDE to model the dynamics of the short term interest rate. The bivariate model is
dS = S√udW0
du = (a−bu) dt+σu1+γdW1
whereby the exponentiation factor1 +γin the diffusion component ofuproduces acceleration or deceler-
ation of the volatility that increases or reduces the kurtosis of the return distribution, let the remaining parameters be constant. Now, taking into consideration the difficulty of the parameter estimation exer- cise, see for instanceDai and Singleton (2000), Aït-Sahalia and Kimmel(2007), Collin-Dufresne et al. (2008), amplified by the latency ofuand the second order action ofγ, we apply a transformation to move
theγ parameter onto the observablex. Thus, with respect to the latter CEV formulation, we consider
only the model witha= 0and introduce the transformation
V =u−2γ ⇔u=V2γ
obtaining, after redefinition of some coefficients
dS = SVγdW0
dV = (˜a−˜bV) dt+ ˜σ
√
VdW1
A.4. MODEL TRANSFORMATIONS 170 the stochastic volatility factor or standardise its diffusion coefficient. The latter two transformations are respectively used in Chapter3and Chapter4and determine the repositioning of the parameter from the latent factor to the diffusion of the observable.
Finally, for the sake of precision, we mention that with respect to the log-normal volatility model ofScott (1987) in Chapter 4 the equation is presented as a Gaussian mean reversion factor that is used as the stochastic exponent in the diffusion ofS, whereas in the original paper the SDE of the diffusion is written
as a geometric mean reversion. A straightforward application of stochastic calculus yields the version of this thesis. Furthermore, we also scale the latent factor in order to obtain a standardised diffusion. As a consequence, the parameterγ0 in the log-normal volatility model in Eq. (4.1)−(4.3) acts as a rescaling
ofx, although the parameterθ produces the same effect upon the stochastic diffusion. The latter con-
siderations entails the structural redundancy of the LEV factors that are affected by uncertainty in the context of optimisation, whereby they tend to be shrunk towards output values not significantly different from0.5, or even lower, but at the same time transferring variability to the jump component.
Pseudo-Codes
In this section, we present several algorithms used in the analysis, arranged in pseudo code. The main routines which are called have significant names that allow to deduce the functions they embed. However, the codes do not follow any standard syntax and might not necessarily compile or run even if the called function or the several variables’ and objects’ definition were provided. The following routines are intended primarily to exemplify and clarify the calculation steps of the program, presenting the main variables and loops.
B.1. THE AML ALGORITHM 172
B.1 The AML algorithm
The likelihood function of the process described by the Eq. (3.1) and illustrated in Appendix A.1 is presented in this section in a pseudo-code snippet. Disregarding consistency check functions and ancillary procedures and implying that some of the output variables are global and hence not all of them are passed to the subsequent procedures. In general, we assume that the necessary data are progressively produced and then used. The main steps are summarised by the sub-functions that, in the order of presentation, determine the range of variation for the variablev, practically fixing the upper bound in relation to the
parametric value; solve the PIDE for each initial condition in the vectorv0, assuming the initial condition
forxis always0 (the solver determines the solution grid according to the projected variance); merge all
the solution grids and adjusting the functions by interpolation in order to have an individual solution grid for each initial condition; eventually integrate out thev dimension and finally the initial condition,
by weighting for the stationary distribution ofv. Once obtained the marginal individual likelihood, the xdata are exploited to determine the sample likelihoodL. Formally,
vector v0 = v_domain ( theta ) ;
i n t k = v . s i z e 1 ( ) ; mesh2 X0( k ) ; mesh2 V0( k ) ; mesh2 F0( k ) ; f o r( i n t i =0; i<k ; i ++){ T = jd_solver ( theta , v0 ( i ) , t ) ; X0 . data2 ( i ) = get <0>(T) ; V0 . data2 ( i ) = get <1>(T) ; F0 . data2 ( i ) = get <2>(T) ; } T = merge (X0 , V0 , F0 ) ; matrix X( get <0>(T) ) ; matrix V( get <1>(T) ) ; mesh2 F( get <2>(T) ) ; T = stationary_v ( theta ) ; vector vs ( get <0>(T) ) ; vector f s ( get <1>(T) ) ; T = jd_marginalise_v1 (X,V, F ) ; vector x = get <0>(T) ; matrix G = get <1>(T) ;
vector l = jd_marginalise_v0 ( v0 ,G) ;