Generalized Mean-Reverting 4/2 Factor Model

Journal of Risk and Financial Management

Article Generalized Mean-Reverting 4/2 Factor Model

Yuyang Cheng †, Marcos Escobar-Anel *,† and Zhenxian Gong † Department of Statistical and Actuarial Sciences, Western University, London, ON N6A 3K7, Canada; [email protected] (Y.C.); [email protected] (Z.G.) * Correspondence: [email protected] † These authors contributed equally to this work.

Received: 13 September 2019; Accepted: 26 September 2019; Published: 8 October 2019

Abstract: This paper proposes and investigates a multivariate 4/2 Factor Model. The name 4/2 comes from the superposition of a CIR term and a 3/2-model component. Our model goes multidimensional along the lines of a principal component and factor covariance decomposition. We find conditions for well-defined changes of measure and we also find two key characteristic functions in closed-form, which help with pricing and risk measure calculations. In a numerical example, we demonstrate the significant impact of the newly added 3/2 component (parameter b) and the common factor (a), both with respect to changes on the implied volatility surface (up to 100%) and on two risk measures: value at risk and expected shortfall where an increase of up to 29% was detected.

Keywords: stochastic covariance; 4/2 model; option pricing; risk measures

1. Introduction Continuous-time stochastic covariance models are crucial in capturing many stylized facts in financial data, from heteroscedasticity and fat tails to changing correlations and leverage effects. Early work in this field focused on discrete time models in the form of generalized autoregressive conditional heteroskedasticity (GARCH) models (see Engle 2002). The best-known representatives in continuous time, are the stochastic Wishart family (see Da Fonseca et al. 2007; Gourieroux 2006) and the Ornstein–Uhlenbeck (OU) family (see Muhle-Karbe et al. 2012) of models, as well as general linear-quadratic jump-diffusions (see Cheng and Scaillet 2007). These approaches are more realistic than the classical Black–Scholes lognormal model, but they quickly become intractable as dimensions increase in terms of the number of parameters and simulation paths, commonly known as the “curse of dimensionality”. Recent papers (see De Col et al. 2013; Escobar 2018) have presented models built from linear combination of tractable one-dimensional counterparts. These models involve fewer parameters than Wishart- or OU-type approaches, owing to a reduction in dimensionality while providing a closed-form solution to financial problems. In this paper, we introduce a multivariate mean-reverting stochastic volatility factor model that combines 1/2 (Heston-type, Heston 1993) and 3/2 processes (Platen 1997) for the modeling of volatility. Such underlying volatility processes were coined 4/2 by Grasselli(2017). Our paper takes advantage of the factor structure in asset prices and allows for a mean-reverting structure on the assets thereby aiming at capturing either multivariate commodity behavior or multiple volatility indexes (see Gnoatto et al. 2018 for an alternative multivariate non-mean-reverting generalization based on a pairwise-structure applied to the exchange-rate market). In particular, our setting reduces the dimension of the parametric space which is a way of controlling the “curse of dimensionality” making parameters identifiable and popular estimation methods feasible. Secondly, the presence of independent common and intrinsic factors, each with its own stochastic volatility, enables an elegant separable structure for characteristic functions (c.f.s) and captures several stylized facts, such

J. Risk Financial Manag. 2019, 12, 159; doi:10.3390/jrfm12040159 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2019, 12, 159 2 of 21 as: stochastic volatility, stochastic correlation among stocks (see Engle 2002), co-movements in the variances (see Diebold and Nerlove 1989), multiple factors in the volatilities (see Heston et al. 2009) and stock correlations (see Da Fonseca et al. 2007). Thirdly, a factor representation is compatible with economical interpretations, where common factors are exogenous variables explaining financial markets, and intrinsic factors relate to companies’ intrinsic risks. Lastly, closed-form expressions are available for joint c.f.s; this is useful for derivative pricing and risk management calculations via Fourier transformations, and it makes c.f.-based estimations methods feasible (see Carr and Madan 1999; Caldana and Fusai 2013; Fusai et al. 2018). The rationale for a 4/2 volatility process rather than a 1/2 or 3/2 model is masterly presented in Grasselli(2017) for a one-dimensional structure. For instance, as observed by the author, the 1/2 process predicts that the implied volatility skew will ﬂatten when the instantaneous volatility increases (crises), while the 3/2 model predicts steepening skews. The empirical violation of the Feller condition in the 1/2 model is also noted, which makes volatility paths stay closer to 0 for a longer period than empirically supported, while the 3/2 model admits extreme paths with spikes in instantaneous volatility. The two processes complement each other as they imply very different dynamics for the evolution of the implied volatility surface. It stands to reason that such a convenient underlying drive for multidimensional structures should be used to improve not only marginal volatility behavior, but also the dependence structure. We obtain an analytical representation for the c.f. of the vector of asset prices, which is in closed-form for non-mean-reverting nested cases. This type of c.f. is helpful for derivative pricing purposes. We also produce a second conditional c.f. that can be used for exact simulations of the non-mean reverting assets given the terminal volatilities, where the latter can be simulated exactly via chi-squares. We identify a set of conditions that not only produces well-deﬁned changes of measure, but also avoids local martingales; hence, it can be used for risk-neutral pricing purposes. Our results were applied numerically to parameters inspired by commodity prices1. There is a vast literature on commodity modeling (see, for instance, Chiarella et al. 2013; Schwartz 1997, and more recently Schneider and Tavin 2018). In our numerical study, we investigated the impact of the new parameters (b, the weight of 3/2 in the overall instantaneous volatility) on the shape of the implied volatility surface and the values of two risk measures: VaR and expected shortfall.

2. Model Description

Next, we define the model in a filtered probability space (Ω, , , F) where 0 contains all F P F subsets of the ( ) null sets of and F is right-continuous. We first provide the processes under P− F the historical measure , then followed by the processes under a (conveniently chosen) risk-neutral P measure . Suppose that X(t) = (X (t), ... , X (t)) is a vector of asset prices with the following Q 1 n 0 -measure representation: P

1 This can also be applied to volatility indexes, such as those reported by the Chicago Board Options Exchange (CBOE), which are clearly a mean-reverting asset class with stochastic volatility. J. Risk Financial Manag. 2019, 12, 159 3 of 21

2 2 p n ˜ dXi(t) 2 bj bi = Li + ci ∑ aij vj(t) + ∑ βij ln(Xj(t)) + c˜i v˜i(t) +  dt Xi(t)   − v˜ (t) !  j=1 q vj(t) j=1 q i   q  p  p ˜   bj bi  + ∑ aij vj(t) + dWj(t) + v˜i(t) + dWi(t)   v˜ (t) ! j=1 q vj(t) q i  q  p e dv (t) = α (θ v (t))dt + ξ v (t)dBP(t), j = 1, . . . , p j j j − j j j j q dv˜ (t) = α˜ (θ˜ v˜ (t))dt + ξ˜ v˜ (t)dB˜ P(t), i = 1, . . . , n i i i − i i i i q P P The quadratic variation structure is dBj (t), dWj(t) = ρjdt, dBi (t), dWi(t) = ρidt and zero otherwise. In the language of factor analysis,D aij is the Eijth entryD of the matrix ofE factor loadings e e (A) that captures the correlations among assets. The communalities are representede by Vj(t) = 2 bj 1/2 vj(t) + (in matrix form, Λnxp = A diag V (t) ) and the intrinsic residual variance √vj(t) q 2 ˜ bi is Ψ = diag(V(t)), with Vj(t) = v˜i(t) + . This leads to a factors decomposition of the √v˜ (t) i quadratic variation of asset prices asp follows: e e

Σ(t)dt = ΛΛ0 + Ψ dt = A diag(V(t))A0 + diag(V(t)) dt Whenever necessary, we assume n = p and A = (aij)n p to be ane orthogonal matrix. In this × setting, ci and c˜i represent risk premiums of asset Xi(t) associated with the common and intrinsic factors, respectively. β = (βij)n n is an invertible matrix, which captures the spillover at the expected × return level Xi(t) on asset Xj(t). In other words, it represents the impact from other assets on the long term average price of the current one. P P Based on the quadratic variation relationship deﬁned in this model, if we assume that Bj , Bj (t)⊥, B˜ P(t), B˜ P(t) are independent Brownian motions with 1 ρ 1 and 1 ρ˜ 1. Then, i i ⊥ − ≤ j ≤ − ≤ i ≤

P 2 P dW (t) = ρ dB (t) + 1 ρ dB (t)⊥ j j j − j j q P 2 P dW (t) = ρ˜ dB˜ (t) + 1 ρ˜ dB˜ (t)⊥. i i i − i i q vj j = 1, ... , n and v˜i i =e1, ... , n follow standard CIR processes, hence αj, θj, and ξj are ξ2 positive constants satisfying α θ j (the Feller condition). Similarly, α˜ , θ˜ , and ξ˜ are positive j j ≥ 2 i i i ˜ 2 constants satisfying α˜ θ˜ ξi . Note that the Feller condition in CIR model guarantees that the process i i ≥ 2 remains positive. The transformation Y = ln X would create a multivariate Ornstein–Uhlenbeck process with a 4/2 stochastic factor structure:

2 n n ˜ 2 1 2 bj 1 bi dYi(t) = Li + ci ∑ aij vj(t) + ∑ βijYj(t) + c˜i v˜i(t) +  dt − 2 =  ( )  − = − 2 v˜i(t) !  j 1 q vj t j 1 q   q  p  n ˜   bj bi  + ∑ aij vj(t) + dWj(t) + v˜i(t) + dWi(t) =  ( )  v˜i(t) ! j 1 q vj t q  q  p e J. Risk Financial Manag. 2019, 12, 159 4 of 21

This general model includes a notable particular case, which is a direct generalization of Grasselli(2017) to a factor setting when βij = 0, i, j = 1, ... , n ; this case is studied in more detail and it is named FG, given its analytical ﬂexibility.

3. Results This section describes viable changes of measure and two key characteristic functions of the targeted multivariate process; one for pricing and the other for simulations. The proofs are presented in AppendixA.

3.1. Change of Measure Here, we explore the topic of creating a risk-neutral measure for pricing purposes. As noted Q by Grasselli(2017); Platen and Heath(2010) and Baldeaux et al.(2015) among others, a risk-neutral measure may not be supported by data in the presence of a 3/2 model (e.g., 1 ), as the parametric √v(t) constraints needed for the discounted asset price process to be a - martingale are violated with real Q data; hence, we can only produce a strict -local martingale (i.e., would be absolute continuous Q Q but not equivalent to ). In such situation, the standard risk-neutral pricing methodology would fail P (biased prices), and we have to turn to the benchmark approach for pricing (see Baldeaux et al. 2015).

The next proposition entertains the following changes of measure with constant λj, λ⊥j , λi and λ⊥ (see Escobar and Gong 2019 for other types of changes of measures) then identiﬁes the parametric i e conditions needed for the existence of a valid risk-neutral measure . e Q b ˜ Q j P ˜ Q bi ˜ P dBj (t) = λj vj(t) + dt + dBj (t), dBi (t) = λi v˜i(t) + dt + dBi (t)   v˜ (t) ! q vj(t) q i  q  e p b ˜ Q j P ˜ Q bi ˜ P dBj (t)⊥ = λ⊥j vj(t) + dt + dBj (t)⊥, dBi (t)⊥ = λi⊥ v˜i(t) + dt + dBi (t)⊥   v˜ (t) ! q vj(t) q i  q  e p Proposition 1. The change of measure is well-deﬁned for pricing purposes under the following four conditions:

2 ξ 2α θ 2ξ max b λ , λ⊥b , b a ρ ,..., b a ρ (1) j ≤ j j − j j j j j j 1j j j nj j 2 n o ξ 2α θ 2ξ max b λ , λ⊥b , b ρ (2) i ≤ i i − i i i i i i i n o α e e e e e e e e e e j max λj , λ⊥j < (3) ξj n o αi max λi , λ⊥ < (4) i ξ n o i e e e Moreover, if βij = 0 for i, j = 1, . . . , n, then the following muste also be satisﬁed:

n 2 2 L = r, c = a ρ λ + 1 ρ λ⊥ , c = ρ λ + 1 ρ λ⊥ (5) i i ∑ ij j j − j j i i i − i i j=1 q q e e e e e Proof is included in AppendixA. J. Risk Financial Manag. 2019, 12, 159 5 of 21

3.2. Characteristic Function This section aims at obtaining an analytical representation for the c.f. If Z(t) = eβtY(t) is deﬁned βt such that e is a matrix exponential, then Zi(t) is represented as:

2 2 n n ˜ βt 1 2 bk 1 bj dZi(t) = e Li + cj ajk vk(t) + + c˜j v˜j(t) + dt (6) ∑ ij  ∑    j=1 − 2 k=1 vk(t) ! − 2 ˜ ( )  q q vj t  p  q  n  n ˜  βt  bk bj  + e ajk vk(t) + dWk(t) + v˜j(t) + dWj(t) ∑ ij  ∑    j=1 k=1 vk(t) ! ˜ ( )  q q vj t  p  q  e   βt βt For convenience, we use e ij as the ij component of the matrix e . Note that Zi(t) is no longer a mean-reverting process although it accounts for time dependent coefficients. Next, we find the conditional c.f. for the increments of Z, defined as

Φ (T, ω) = E exp iω0(Z(T) Z(t)) Z(t) = z , v(t) = v (7) Z(t),v(t) − | t t Under a risk neutral measure, this c.f. can be used for pricing some ﬁnancial products, given the integrability conditions (a discussion of the generalized c.f. as per Grasselli 2017 is beyond the scope of this paper.). For convenience, we formulate it as v(t) = (v1(t),..., vn(t), v1(t),..., vn(t)).

Proposition 2. Let (Z(t))t 0 evolve according to the model in Equation (6). Thee c.f. ΦZ(t),ev(t) is then given ≥ as follows:

Φ (T, ω) = E exp iω0(Z(T) Z(t)) Z(t) = y , v(t) = v Z(t),v(t) − | t t n = ∏ ΦGG T, 1; Lk(ω), hk(ω), gk(ω), κk, θk, ξk, ρk, bk, ck, vk,t, Sk∗,t k=1 n j Φ T, 1; 0, L (ω), h (ω), g (ω), κ , θ , ξ , ρ , b , c , v , S∗ × ∏ GG j j j j j j j j k j,t t j=1 e e e e e e e where ΦGG is a one-dimensional generalization of the c.f. from Grasselli(2017) provided in Lemma A1.

Proof is provided in AppendixA. The c.f. above involves single expected values with respect to Brownian motion B(t). In each term, ΦGG (i.e., the second set of Brownian W(t)) is eliminated, hence this is a drastic simpliﬁcation compared to the original 2n dimensional joint expectation. A particular, fully solvable case is the FG model (βij = 0, i, j = 1, . . . , n).

Corollary 1. Let (Z(t))t 0 evolve according to the FG model (βij = 0, i, j = 1, ... , n). The c.f. ΦZ(t),v(t) is ≥ subsequently presented as follows:

Φ (T, ω) = E exp iω0(Z(T) Z(t)) Z(t) = y , v(t) = v Z(t),v(t) − | t t n = ∏ ΦG T, 1; Lk(ω), hk(ω), gk(ω), κk, θk, ξk, ρk, bk, ck, vk,t, Sk∗,t k=1 n j Φ T, 1; 0, L (ω), h (ω), g (ω), κ , θ , ξ , ρ , b , c , v , S∗ × ∏ G j j j j j j j j k j,t t j=1 e e e e e e e where ΦG is the one-dimensional c.f provided by Grasselli(2017) in Proposition 3.1 and given in the AppendixB for completeness. J. Risk Financial Manag. 2019, 12, 159 6 of 21

See AppendixA for proof. Next, we turn to the conditional c.f. of the increments of Z given the terminal value of the CIR processes. This is deﬁned as follows:

Φ (τ, ω) = E exp ω0(Z(T) Z(t)) Z(t) = z , v(T) = v (8) Z(t),v(T) − | t T The above is useful when we need to work with the joint distribution of (Z(T), v(T)) given (Z(t), v(t)). For such cases, we can try to rely on a convenient simulation scheme combining the distribution of Z(T) given (Z(t), v(T)) (via Equation (8)) with that of v(T) given v(t), the latter is known to be non-centered chi-squared. In this way, we can avoid usual discretization algorithms such as the Euler–Maruyama or Milstein schemes, which are generally not suitable for the CIR process (due to failure of the Lipschitz condition at 0). In this vein, when working with the non mean-reverting factor model (βij = 0, i, j = 1, ... , n), we can easily adapt the procedures in Grasselli(2017) to provide an exact simulation scheme for the model given the vector of the independent CIR process at maturity T (i.e., v(T)). This requires only the c.f. provided next:

Corollary 2. Let (Z(t))t 0 evolve according to the FG model (βij = 0, i, j = 1, ... , n). Then, the c.f. ΦZ(t),v(T) ≥ is then given as follows:

n ΦZ(t),v(T)(T, ω) = ∏ ΦG,T T, φ; L, hj, gj, κj, θj, ξj, ρj, bj, cj, vj,T, S∗j,t j=1 n i Φ T, 1; 0, h , g , κ , θ , ξ , ρ , b , c , v , S∗ × ∏ G,T i i i i i i i i i,T t i=1 e e e e e e e where ΦG,T is the one-dimensional c.f provided by Grasselli(2017) in Proposition 4.1 and given in the AppendixB for completeness.

Proof of this result is provided in AppendixA. Unsurprisingly, the previous result cannot be extended to the mean-reverting case, due to the absence of closed formulas for the object:

T T 1 T E exp u B(s)ν(s)ds + C(s) ds + D(s) ln(ν(s))ds ν(T) ν(s) | Zt Zt Zt which is not solvable even when two of the three deterministic functions B(s), C(s) and D(s) are zero.

4. Discussion: One Common Factor in Two Dimensions

We assume two assets, i.e., X1(t) and X2(t), with one common volatility component, and one intrinsic factor each. The asset prices thereby follow the system of SDE for i = 1, 2:

dY (t) = (L β Y (t)) dt i i − i i ˜ 1 2 b1 2 1 bi 2 + (ci )[a ( v1(t) + ) ] + (c˜i )( v˜i(t) + ) dt − 2 i v (t) − 2 v˜ (t) q 1 q i ! b1 p bi p +ai v1(t) + dW1(t) + vi(t) + dWi(t) v1(t) ! vi(t) ! q q e p e p e e dv (t) = α (θ v (t))dt + ξ v (t)dB (t) 1 1 1 − 1 1 1 1 q dv˜ (t) = α˜ (θ˜ v˜ (t))dt + ξ˜ v˜ (t)dB˜ (t) i i i − i i i i q J. Risk Financial Manag. 2019, 12, 159 7 of 21

with dBj(t), dWj(t) = ρjdt, dBi(t), dWi(t) = ρidt for j = 1; i = 1, 2. The following table (TableD 1) gives a baselineE parameter set for the one-factor, two-dimensional 4/2 factor model used in the subsequente e sections.e The choice of parameters in Scenario A was made by combining the seminal works of Schwartz(1997) (see Oil and Copper in Tables IV and V) and Heston(1993) . Scenario B combines Schwartz(1997) (see Oil and Copper, Tables IV and V) with Heston et al.(2009). In both cases, we assume a simple structure for the market price of risk 2 (c1 = c2 = c˜1 = c˜2 = 0) . ˜ The θi, i = 1, 2 in the table are set to match the long term volatilities as estimated in Schwartz(1997), which are 0.334 (Oil, Table IV) and 0.233 (Copper, Table V):

2 2 ˜ 2 b1 b1 E a v1(t) + + v˜1(t) + 1 √v (t) √v˜ (t) " 1 1 # (9) 2αp2 2α˜ p˜2 2 1b1 1b1 ˜ ˜ 2 = a 2 + 2b1 + θ1 + 2 + 2b1 + θ1 = (0.334) 1 2α1θ1 ξ 2α˜ 1θ˜1 ξ˜ − 1 − 1 ˜ This explains the values of θi in the table.

Table 1. Toy parametric values.

Initial Values

X1(0) = 18, X2(0) = 100 v1(0) = θ1, v˜1(0) = θ˜1, v˜2(0) = θ˜2 Commodity Drift, Schwartz(1997) β11 = 0.301, β12 = 0, β21 = 0, β22 = 0.369 L1 = 3.09β11 = 0.93, L2 = 4.85β22 = 1.79 Commodity St. Volatility, Heston(1993); Schwartz(1997). Scenario A α1 = α˜ 1 = α˜ 2 = 2 θ1 = 0.01, θ˜1 = 0.0753, θ˜2 = 0.0124 ξ1 = ξ˜1 = ξ˜2 = 0.1 ρ = ρ = ρ = 0.5 1 1 2 − Commodity St. Volatility, Heston et al.(2009); Schwartz(1997). Scenario B α1 = αe˜ 1 = eα˜ 2 = 0.2098 θ1 = 0.1633, θ˜1 = 0.0685, θ˜2 = 0.0689 ξ1 = ξ˜1 = ξ˜2 = 0.1706 ρ = ρ = ρ = 0.9 1 1 2 − New parameters. c1 = ce2 = ce˜1 = c˜2 = 0 a1 = a2 = 0.75 b1 = b1 = b2 = 0.008

e e The present section considers two independent cases. First, we study the impact of the parameters b1, b1 and b2 on implied volatility surfaces and on two risk measures for a portfolio of underlyings. We then assess the impact of the commonalities a1 and a2 on these same targets, i.e., implied volatilities ande risk measures.e To ensure that the cases lead to reasonable assets behavior, we report the expected return, variance of return for each asset, as well as the correlation between two assets and the leverage effects in Tables2 and3 under Scenarios A and B, respectively. We simulated 500,000 paths with dt = 0.1 and considered the following scenarios for b: b1 = 0.008, b˜1 = b˜2 = 0; b1 = 0, b˜1 = b˜2 = 0.008; b1 = b˜1 = b˜2 = 0 and b1 = b˜1 = b˜2 = 0.008.

2 Variations on c will be studied in future research as part of a calibration exercise (see Medvedev and Scaillet 2007 for viable approaches and Escobar and Gschnaidtner 2016 for some pitfalls). J. Risk Financial Manag. 2019, 12, 159 8 of 21

Table 2. First four moments for scenarios on 3/2 component (b), Scenario A.

˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ b1 = 0.008, b1 = b2 = 0 b1 = 0, b1 = b2 = 0.008 b1 = b1 = b2 = 0 b1 = b1 = b2 = 0.008 X (T) X (0) [ 1 − 1 ] 0.0494 0.0489 0.0503 0.0480 E X1(0) X (T) X (0) [ 2 − 2 ] 0.0760 0.0754 0.0764 0.0750 E X2(0) X (T) X (0) [ 1 − 1 ] 0.0663 0.0680 0.0618 0.0729 V X1(0) X (T) X (0) [ 2 − 2 ] 0.0367 0.0390 0.0323 0.0445 V X2(0) Corr(ln X1(T) , ln X2(T)) 0.3194 0.0896 0.1060 0.2799 Corr(ln X (T) , < ln X (T) >) 0.4287 0.4520 0.4443 0.4406 1 1 − − − − Corr(ln X (T) , < ln X (T) >) -0.4148 0.4511 0.4420 0.4280 2 2 − − −

Table 3. First four moments for scenarios on 3/2 component (b), Scenario B.

˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ b1 = 0.008, b1 = b2 = 0 b1 = 0, b1 = b2 = 0.008 b1 = b1 = b2 = 0 b1 = b1 = b2 = 0.008 X (T) X (0) [ 1 − 1 ] 0.0514 0.0504 0.0527 0.0499 E X1(0) X (T) X (0) [ 2 − 2 ] 0.0774 0.0775 0.0787 0.0754 E X2(0) X (T) X (0) [ 1 − 1 ] 0.0360 0.0606 0.0247 0.1022 V X1(0) X (T) X (0) [ 2 − 2 ] 0.0359 0.1508 0.0248 0.0723 V X2(0) Corr(ln X1(T) , ln X2(T)) 0.7533 0.0099 0.4698 0.0156 Corr(ln X (T) , < ln X (T) >) 0.4509 0.2031 0.5560 0.0273 1 1 − − − − Corr(ln X (T) , < ln X (T) >) 0.4496 0.0398 0.5517 0.2444 2 2 − − − −

Similarly, we considered the following scenarios for a: a1 = a2 = 0; a1 = 0.75, a2 = 0; a1 = 0, a2 = 0.75 and a1 = a2 = 0.75. Tables4 and5 present key statistics for the returns under Scenarios A and B, respectively.

Table 4. First four moments for scenarios on commonalities (a), Scenario A.

a1 = a2 = 0 a1 = 0.75, a2 = 0 a1 = 0, a2 = 0.75 a1 = a2 = 0.75 X (T) X (0) [ 1 − 1 ] 0.0491 0.0492 0.0493 0.0492 E X1(0) X (T) X (0) [ 2 − 2 ] 0.0761 0.0757 0.0746 0.0759 E X2(0) X (T) X (0) [ 1 − 1 ] 0.0658 0.0727 0.0660 0.0732 V X1(0) X (T) X (0) [ 2 − 2 ] 0.0371 0.0370 0.0444 0.0447 V X2(0) Corr(lnX (T) , lnX (T)) 0.0000 0.0011 0.0004 0.2841 1 2 − Corr(lnX (T) , < lnX (T) >) 0.4523 0.4400 0.4529 0.4430 1 1 − − − − Corr(lnX (T) , < lnX (T) >) 0.4507 0.4521 0.4285 0.4279 2 2 − − − −

Table 5. First four moments for scenarios on commonalities (a), Scenario B.

a1 = a2 = 0 a1 = 0.75, a2 = 0 a1 = 0, a2 = 0.75 a1 = a2 = 0.75 X (T) X (0) [ 1 − 1 ] 0.0514 0.0496 0.0512 0.0500 E X1(0) X (T) X (0) [ 2 − 2 ] 0.0768 0.0772 0.0756 0.0752 E X2(0) X (T) X (0) [ 1 − 1 ] 0.0420 0.0733 0.0420 0.0857 V X1(0) X (T) X (0) [ 2 − 2 ] 0.0507 0.0564 0.0746 0.0719 V X2(0) Corr(lnX (T) , lnX (T)) 0.0004 0.0000 0.0005 0.0039 1 2 − − Corr(lnX (T) , < lnX (T) >) 0.1126 0.0047 0.3203 0.2634 1 1 − − − − Corr(lnX (T) , < lnX (T) >) 0.1600 0.2544 0.0164 0.0111 2 2 − − − −

4.1. Pricing Option

The section prices European call option on the asset X1 based on our 4/2 generalized factor model. It explores the implied volatility surface in a three-dimensional plot with strike prices as the x-axis, time to maturity as the y-axis, and corresponding implied volatility as z-axis. We take the strike prices J. Risk Financial Manag. 2019, 12, 159 9 of 21

K to be 15, 16.4, 17.8, 19.2, 20.6, and 22 and the expiry dates T are 0.2, 0.36, 0.52, 0.68, 0.84 and 1.0. By choosing these strike prices, we account for the in-the-money, at-the-money, and out-of-the-money options, given the initial asset price 18. Subsequently, for each strike price and expiry date, we can obtain a simulated call price as follows

rT Q + c(T, K) = e− E [(X1(T) K) ], − where X1(T) is approximated using the Euler method. We extract the implied volatility by matching the Black–Scholes option price formula with simulated call prices and solve for the volatility parameter. Hence, we can treat the dynamics of Y(t) as an O-U process such that:

2 dY(t) = (L 0.5σ βY(t))dt + σdW∗(t). 1 − − Next, we consider the two cases described above. The ﬁrst one studies the impact of b, which represents the size of the 3/2 component on the covariance; and the second examines the impact of a, the size of the commonality. In the case of b, we ﬁrst extract the implied volatility surface by matching the standard BS formula for changes on b and b˜ respectively (see Figures1 and2).

˜ ˜ (a) b1=0, b1 between (0, 0.008) (b) b1=0.008, b1 between (0, 0.008)

Figure 1. Impact of b1 (common factor, 3/2 component) on implied volatility, Scenario A.

˜ ˜ (a) b1=0, b1 between (0, 0.008) (b) b1=0.008, b1 between (0, 0.008)

Figure 2. Impact of b1 (common factor, 3/2 component) on implied volatility, Scenario B. J. Risk Financial Manag. 2019, 12, 159 10 of 21

For Scenario A, Figure1a,b illustrates that even small changes in ( b1) the common factor 3/2 component (from 0 to 0.008) can lead to a 7% difference in implied volatility (from 0.275 to 0.295, or 0.285 to 0.305). The joint effect of the common and intrinsic 3/2 components (b1 and b˜1) can be obtained by combining those two figures leading to a 11% change (from 0.275 to 0.305) in the presence of relatively small values of b. For Scenario B, we observe that the impact of intrinsic factor on volatility surface is more significant than in Scenario A through a comparison of Figure2a,b. The effect of b1 on implied volatility increase by approximately 31% (0.145 to 0.19), as shown in Figure2a, when only the common factor is present. In Figure2b, we observe a volatility “smile" with the difference of approximately 12.2% (0.245 to 0.275). The joint effect of the common and intrinsic 3/2 components in this case is 100% (0.145 to 0.29). Figures1 and2 jointly demonstrate that, given different underlying process for common and intrinsic factors, the impact of the 3/2 component can be crucial. Next, we study a, the weight of the common factor (commonality). We again extract the implied volatility surface from matching the standard BS formula for changes on a. Figure3a,b displays the significant increase in implied volatility due to the commonality of the asset with the market (a1). The change in implied volatility can increase up to 12.5% (from 0.28 to 0.315) in Scenario A and up to 30% (from 0.22 to 0.32) in Scenario B.

(a) a1 between (0, 1). Scenario A (b) a1 between (0, 1). Scenario B

Figure 3. Impact of commonality (a1) on implied volatility.

4.2. Risk Measures This section examines the impact of b and a on important risk measures, in particular the value at risk (VaR) and the expected shortfall (ES). For clarity and calculation purposes, these measures are deﬁned as follows:

α = P (X(T) VaR ) (10) ≤ − α 1 α ES = VaR dγ (11) α γ − Z0 where X(T) = ω (X (T) X (0)) + ω (X (T) X (0)) is the profit and loss portfolio with equal 1 1 − 1 2 2 − 2 weights (w1 = w2 = 1/2) (see DeMiguel et al. 2007 for a rationale and support of this simple strategy). We let α vary from 0.001 to 0.2 with a discretization size of 200. We first study the impact of b1, b˜1 and b˜2 on VaR and ES for a fixed value of α = 0.01. Figure4a,b illustrates a substantial increase in VaR, from 16 (when all b values are set to zero) to 19.5 (all b set to 0.008), which is a 21% increase (α = 0.01) due to the presence of b. In other words, an investor would have to set aside 21% more capital in the presence of 3/2 components. Similarly, ES increases from VersionVersion September September 23, 23, 2019 2019 submitted submitted to J. to RiskJ. Risk Financial Financial Manag. Manag. 11 of11 21 of 21

182182 FiguresFigures 3a 3a and and 3b 3b displays displays the the signiﬁcant signiﬁcant increase increase in in implied implied volatility volatility due due to to the the commonality commonality of of 183 183 thethe asset asset with with the the market market (a1 ().a1 The). The change change in in implied implied volatility volatility can can increase increase up up to to12.5% 12.5% (from (from 0.28 0.28 to to 184184 0.315)0.315) in in scenario scenarioA Aandand up up to to 30% 30% (from (from 0.22 0.22 to to 0.32) 0.32) in inscenario scenarioB. B.

185185 4.2.4.2. Risk Risk Measures. Measures.

186186 ThisThis section section examines examines the the impact impact of ofb andb anda ona on important important risk risk measures, measures, in inparticular particular the the value value at at 187187 riskrisk (VaR) (VaR) and and the the expected expected shortfall shortfall (ES). (ES). For For clarity clarity and and calculation calculation purposes, purposes, these these measures measures are are 188188 deﬁneddeﬁned as as follows: follows:

α α= =P (PX((XT()T) VaRVaR) α) (11)(11) ≤≤ − − α 1 1 α α ESES= = VaRVaRγdγdγ (12)(12) α− α 0 − Z0 Z

189189 where,where,X(XT()T =) =ω ω(X(X(T()T) X X(0))(0)) + +ω ω(X(X(T()T) X X(0))(0))is theis the proﬁt proﬁt and and loss loss portfolio portfolio with with 1 1 1 1 − −1 1 2 2 2 2 − −2 2 190190 equalequal weights weights (w (1w=1 =w2w=2 =1/12/)2 (see) (see (10 (10) for) for a rationale a rationale and and support support of ofthis this simple simple strategy). strategy). We We will will 191191 letletα variesα varies from from 0.001 0.001 to to 0.2 0.2 with with a discretization a discretization size size of of200. 200.

192192 ˜ ˜ WeWe first first study study the the impact impact of ofb1,bb˜11, band1 andb˜2 bon2 on VaR VaR and and ES ES for for a fixed a fixed value value of ofα =α 0.01= 0.01. Figures. Figures 4a 4a 193193 andand 4b 4b illustrate illustrate a substantial a substantial increase increase in in VaR, VaR, from from 16 16 (when (when all allb valuesb values are are set set to tozero) zero) to to19.5 19.5 (all (allb b J. Risk Financial Manag. 2019, 12, 159 11 of 21 194194 setset to to 0.008), 0.008), this this is ais 21% a 21% increase increase (α (=α =0.010.01) due) due toto the the presence presence of ofb. Inb. In other other words words an an investor investor would would 195195 havehave to to set set aside aside 21% 21% more more capital capital in in the the presence presence of of3/2 3/2 components. components. Similarly Similarly ES ES increases increases from from21 21 − − 196196 inin the the presence presence of of 3/2 3/2 components components to to18.518.5inin their their absence, absence, which which constitutes constitutes a 12% a 12% increase increase in thein the 21 in the presence of 3/2 components− − to 18.5 in their absence, which constitutes a 12% increase in 197197 − − averageaveragethe VaR. averageVaR. VaR.

(a) Value at Risk vs. α, various b (b) Expected Shortfall vs. α, various b (a)(a)ValueValue at atRisk Risk vs. vs.α, variousα, various b b (b)(b)ExpectedExpected Shortfall Shortfall vs. vs.α, variousα, various b b FigureFigure 4.Figure 4.ImpactImpact 4. ofImpact of3/2 3/2 components of components 3/2 components (b) ( onb) on Risk ( Riskb) measures. on measures. Risk measures, Scenario Scenario ScenarioA A A.

198198 FiguresFiguresFigure 5a 5a and and5a,b 5b 5b also also display displays display aa substantial a substantial substantial increase increase in in VaR,in VaR, VaR, from from from 17.5 17.5 17.5 (all (all (allb setb setb toset tozero) to zero) zero) to 22.5to to 22.5 ( 22.5b (b (b set 199199 setset to to 0.008),to 0.008), 0.008), which which which represents represents represents a 28.6% a 28.6% a 28.6% increase increase increase (α (=α =0.01 (α0.01=) due0.01) due to) due tob. Thisb. to Thisb means. This means 28.6% means 28.6% more 28.6% more capital more capital is capital is is 200200 requiredVersionrequiredrequired September in in the the presence in 23, presence the 2019 presence submitted of of 3/2 3/2 components. of to components. 3/2J. Risk components. Financial Similarly, Similarly, Manag. Similarly, the the ES ES increases the increases ES increases from from27 from27withwith 3/227 3/2with components components 3/212 components of 21 Version September 23, 2019 submitted to J. Risk Financial Manag. − − − 12 of 21 201201 toto 1919to without without19 without them, them, representing them, representing representing a 29.6% a 29.6% a increase 29.6% increase increase in inthe the average in average the average VaR. VaR. VaR. −− −

202202 AA similar similar analysis analysis was was performed performed with with respect respect to tothe the commonality commonalitya, ina, inthe the presence presence of stochasticof stochastic 203203 volatilityvolatility (in (in the the common common factor) factor) versus versus constant constant volatility; volatility; in in other other words, words, we we assessed assessed the the impact impact 204204 ofofa pera per se se and and that that of of stochastic stochastic correlation correlation produced produced by by the the 4/2 4/2 model. model. Figure Figure 6a 6ademonstrates demonstrates an an 205205 increaseincrease in in VaR, VaR, from from 16 16 to to 18.5, 18.5, a 15.5% a 15.5% increase increase (α (=α =0.010.01); Figure); Figure 6b 6bshows shows that that the the VaR VaR jumps jumps from from 206206 1717 to to 23, 23, a 35% a 35% growth growth due due to to the the increase increase in ina. a.

(a) Value at Risk vs. α, various b (b) Expected Shortfall vs. α, various b (a)(a)ValueValue at at Risk Risk vs. vs.α,α various, various b b (b)(b)ExpectedExpected Shortfall Shortfall vs. vs.α,α various, various b b Figure 5. Impact of 3/2 components (b) on Risk measures, Scenario B. FigureFigure 5. 5.ImpactImpact of of 3/2 3/2 components components (b ()b on) on Risk Risk measures. measures. Scenario ScenarioBB A similar analysis was performed with respect to the commonality a, in the presence of stochastic volatility (in the common factor) versus constant volatility; in other words, we assessed the impact of a per se and that of stochastic correlation produced by the 4/2 model. Figure6a demonstrates an increase in VaR, from 16 to 18.5, a 15.5% increase (α = 0.01). Figure6b shows that the VaR jumps from 17 to 23, a 35% growth due to the increase in a.

(a)(a)ValueValue at at Risk Risk vs. vs.α,α various, various a. a. Scenario ScenarioAA (b)(b)ValueValue at at Risk Risk vs. vs.α,α various, various a. a. Scenario ScenarioBB FigureFigure 6. 6.ImpactImpact of of commonality commonality (a ()a on) on Value Value at at Risk. Risk.

207 207 5.5. Conclusions Conclusions

208 208 TheThe generalized generalized men-reverting men-reverting 4/2 4/2 factor factor model model was was proposed proposed and and studied studied in in this this paper. paper. We We 209 209 providedprovided analytical analytical expressions expressions for for key key characteristic characteristic functions functions and and conditions conditions for for well-deﬁned well-deﬁned changes changes 210 210 ofof measures. measures. We We also also explored explored the the impact impact of ofb,b i.e., i.e. the the 3/2 3/2 component component of of the the model model in in the the volatility volatility 211 211 process,process, and anda,a the, the commonalities commonalities in in the the absence absence and and presence presence of of stochastic stochastic volatility volatility on on the the common common 212 212 factor.factor. These These impacts impacts were were measured measured with with respect respect to to implied implied volatility volatility surfaces surfaces and and two two important important 213 213 riskrisk measures. measures. The The results results demonstrate demonstrate that that even even small small values values of of the the 3/2 3/2 component component (b ()b) can can lead lead to to a a 214 214 100%100% change change in in the the implied implied volatility volatility surface, surface, as as well well as as up up to to 28%, 28%, 29% 29% increases increases in in the the VaR VaR and and ES ES 215 215 measuresmeasures respectively. respectively.

216 216 AuthorAuthor Contributions: Contributions:AllAll authors authors contributed contributed equally. equally.

217 217 ConflictsConflicts of of Interest: Interest:nono conflict conflict of of interest. interest. VersionVersion September September 23, 23, 2019 2019 submitted submitted to toJ.J. Risk Risk Financial Financial Manag. Manag. 1212 of of 21 21

J.(a) Risk(a)ValueValue Financial at at Risk Risk Manag. vs. vs.2019αα,, various various, 12, 159 b b (b)(b)ExpectedExpected Shortfall Shortfall vs. vs.α,α various, various b b 12 of 21 FigureFigure 5. 5.ImpactImpact of of 3/2 3/2 components components (b ()b) on on Risk Risk measures. measures. Scenario ScenarioBB

(a) Scenario A: Value at Risk vs. α for different (b) Scenario B: Value at Risk vs. α for different (a)(a)ValueValue at at Risk Risk vs. vs.αα,, various various a. a. Scenario ScenarioAA (b)(b)ValueValue at at Risk Risk vs. vs.αα, various, various a. a. Scenario ScenarioBB commonality (a) values. commonality (a) values. FigureFigure 6. 6.ImpactImpact of of commonality commonality (a ()a) on on Value Value at at Risk. Risk. Figure 6. Impact of commonality (a) on Value at Risk under Scenario A and Scenario B.

207207 5.5. Conclusions Conclusions 5. Conclusions 208208 TheThe generalized generalized men-reverting men-reverting 4/2 4/2 factor factor model model was was proposed proposed and and studied studied in in this this paper. paper. We We The generalized men-reverting 4/2 factor model is proposed and studied in this paper. We provide 209209 providedprovided analytical analytical expressions expressions for for key key characteristic characteristic functions functions and and conditions conditions for for well-defined well-defined changes changes analytical expressions for key characteristic functions and conditions for well-defined changes of 210210 ofof measures. measures. We We also also explored explored the the impact impact of ofbb,, i.e. i.e. the the 3/2 3/2 component component of of the the model model in in the the volatility volatility measures. We also explore the impact of b, i.e., the 3/2 component of the model in the volatility 211211 process,process, and andaa,, the the commonalities commonalities in in the the absence absence and and presence presence of of stochastic stochastic volatility volatility on on the the common common process, and a, the commonalities in the absence and presence of stochastic volatility on the common 212212 factor.factor. These These impacts impacts were were measured measured with with respect respect to to implied implied volatility volatility surfaces surfaces and and two two important important factor. These impacts were measured with respect to implied volatility surfaces and two important 213213 riskrisk measures. measures. The The results results demonstrate demonstrate that that even even small small values values of of the the 3/2 3/2 component component (b ()b) can can lead lead to to a a risk measures. The results demonstrate that even small values of the 3/2 component (b) can lead to a 214214 100%100% change change in in the the implied implied volatility volatility surface, surface, as as well well as as up up to to 28%, 28%, 29% 29% increases increases in in the the VaR VaR and and ES ES 100% change in the implied volatility surface, as well as up to 28% and 29% increases in the VaR and 215215 measuresmeasures respectively. respectively. ES measures, respectively.

Author Contributions: All authors contributed equally. 216216 AuthorAuthor Contributions: Contributions:AllAll authors authors contributed contributed equally. equally.

217 Funding: This research received no external funding. 217 ConflictsConflicts of of Interest: Interest:nono conflict conflict of of interest. interest. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A. Proofs Proof. Proof of Proposition1. The ﬁrst step is to ensure the change of measure is well-deﬁned and for this we use Novikov’s condition, i.e., generically

2 2 2 2 1 T b 2 λ T λ b T 1 E exp λ2 ν(t) + ds = eλ bTE exp ν(s)ds + ds < ∞.   2 0 ν(t)  2 0 2 0 ν(s) Z q ! Z Z   p  From Grasselli, in order for this expectation to exist, we need two conditions:

λ2 α2 α > = λ < (A1) − 2 − 2ξ2 ⇒ | | ξ

and λ2b2 (2αθ ξ2)2 2αθ ξ2 − = λ − = ξ2 2αθ 2 λ b ξ (A2) − 2 ≥ − 8ξ2 ⇒ | | ≤ 2 b ξ ⇒ ≤ − | || | | | The latter condition in Equation (A2) implies, in particular, that our volatility processes satisfy Feller’s condition under and ; in other words, it ensures all our CIR processes stay away from zero P Q under both measures. J. Risk Financial Manag. 2019, 12, 159 13 of 21

Applying Equation (A2) to our setting leads to (i, j = 1, . . . , n):

2 ξ 2α θ 2ξ max λ b , λ⊥b (A3) j ≤ j j − j j j j j 2 n o ξ 2α θ 2ξ max λ b , λ⊥b (A4) i ≤ i i − i i i i i n o

Now, we apply Equatione (A1) producinge e twoe extra e sete of conditionse e (i, j = 1, . . . , n):

αj max λj , λ⊥j < (A5) ξj n o αi max λi , λ⊥ < (A6) i ξ n o i e e e The second step applies to the case βij = 0 for i, j = 1, ...e , n and it is to ensure the drift of the asset price equal the short rate:

n 2 2 L = r, c = a ρ λ + 1 ρ λ⊥ , c = ρ λ + 1 ρ λ⊥ i i ∑ ij j j − j j i i i − i i j=1 q q e e e e e For the most general case (β = 0 for some i or j), the second step should be adapted to any ij 6 particular prescribed drift structure under the -measure. Q The third step is to ensure the drift-less asset price process is a true -martingale and not just a Q local -martingale: Q n ˜ dXi(t) bj Q bi Q = (.) dt + ∑ aij vj(t) + dWj (t) + v˜i(t) + dWi (t) Xi(t)   v˜ (t) ! j=1 q vj(t) q i  q  p e Here, we test the martingale property using the Feller nonexplosion test for volatilities, hence considering the following n2 + n changes of Brownian motion for the volatility processes and checking the processes do not reach zero under the various measures:

b ˜ Q j P ˜ Q bi ˜ P dBij (t) = aijρj vj(t) + dt + dBj (t), dBi (t) = ρi v˜i(t) + dt + dBi (t)   v˜ (t) ! q vj(t) q i  q  e p This leads to the following conditions:

ξ2 2α θ 2 a ρ b ξ , i, j = 1, . . . , n (A7) j ≤ j j − ij j j j ξ2 2α θ 2 λ ρ b ξ , i = 1, . . . , n (A8) i ≤ i i − i i i i

We can combine the ﬁrste and thirde e steps e ine Equationse e (A3), (A7), (A4) and (2) into the ﬁnal conditions. J. Risk Financial Manag. 2019, 12, 159 14 of 21

Proof. Proof of Proposition2. ij We start by deﬁning new processes dS and dS with i, j, k = 1, 2, . . . , n and t 0. ik,t t ≥

n ˜ ˜ βt 1 bj bj dZi(t) = e cj v˜j(t) + dt + v˜j(t) + dWj ∑ ij j=1  − 2  ˜ ( )   ˜ ( )   q vj t q vj t  e  q   q  e 2 n n n βt Lj βt 1 2 bk + ∑ ∑ e dt + ∑ e ck ajk vk(t) + dt  ij n ! ij − 2 ! v (t) !  k=1 j=1 j=1 q k n  n p  βt bk + ∑ ∑ e ajk vk(t) + dWk ij ! v (t) ! k=1 j=1 q k n n ij p = ∑ dSt + ∑ dSik,t j=1 k=1

By the dependence structure implied by the model, it follows that all S are independent for a ﬁx i, hence we can transform the characteristic function using the processes S as follows:

n n .j .j ΦZ(t),v(t)(T, ω) = ∏ E exp iω0 (S k,T S k,t) St, v(t) ∏ E exp iωi ST St St, v(t) k=1 { · − · } | j=1 { − } | h i n For each factor j = 1, 2, ... , n we deﬁne S∗ = ω0S k,t = ∑i=1 ωiSik,t; the dynamics of S∗ can be k,t · k,t expressed as

dSk∗,t = ω0dS k,t · 2 bk bk = L(ω, t) + hk(ω, t) vk(t) + dt + gk (ω, t) vk(t) + dWk,t  v (t)  v (t) q k ! q k !  p  p L where h (ω, t) = ∑n c 1 a2 f (ω, t), L(ω, t) = ∑n j f (ω, t) and g (ω, t) = ∑n a f (ω, t) k j=1 k − 2 jk j j=1 n j k j=1 jk j n βt and fj(ω, t) = ∑i=1 ωi e ij. These three functions are deterministic, linear combinations of fj(ω, t). Next, we ﬁnd the characteristic function for the increments of S : k∗,t

E exp iφ S∗ S∗ S∗ , v (t) = v = Φ T, φ; L(ω), h (ω), g (ω), κ , θ , ξ , ρ , b , c , v , S∗ . { k,T − k,t } | k,t k k,t GG k k k k k k k k k,t k,t h i The generic function ΦGG is provided in Lemma A1. It follows similarly for idiosyncratic factors:

j j dSt∗ = ω0dSt· n n ˜ βt 1 bj = ωi e cj v˜j(t) + dt ∑ ∑ ij i=1 j=1 − 2 !!  ˜ ( )  q vj t e  q  n n ˜ βt bj + ωi e v˜j(t) + dWj,t ∑ ∑ ij i=1 j=1 !  ˜ ( )  q vj t  2 q  e ˜ ˜ bj bj = h (ω, t) v˜ (t) + dt + g (ω, t) v˜ (t) + dW j  j  j  j  j,t q v˜j(t) q v˜j(t)  q   q  e where h (ω, t) = ∑n ω ∑n eβt c 1 and g (ω, t) = ∑n ω ∑n eβt . j i=1 i j=1 ij j − 2 j i=1 i j=1 ij e J. Risk Financial Manag. 2019, 12, 159 15 of 21

Combining all pieces together, we obtain:

n n j j ΦZ(t),v(t)(T, ω) = E exp iω0 (S k,T S k,t) St, v(t) E exp iω0 S· St· St, v(t) ∏ { · − · } | ∏ { T − } | k=1 j=1 h i n = ∏ ΦGG T, 1; Lk(ω), hk(ω), gk(ω), κk, θk, ξk, ρk, bk, ck, vk,t, Sk∗,t k=1 n j Φ T, 1; 0, h (ω), g (ω), κ , θ , ξ , ρ , b , c , v , S∗ × ∏ GG j j j j j j j j j,t t j=1 e e e e e e e

Lemma A1. Let the generic process be:

2 b b dZ(t) = L(t) + h(t) v(t) + dt + g (t) v(t) + dWt  v(t)  v(t) q ! q !  p  p dv(t) = α(θ v(t))dt + ξ v(t)dB(t) − dB(t), dW(t) = ρdt q h i with g (t) differentiable, then

T bρ u g(t) g (t) ν(t) Φ (T, φ; L, h, g, κ, θ, ξ, ρ, b, c, v , Z ) = exp u A (s) ds ν(t) ξ exp uρ GG t t − ξ Zt bρ T T T u g(T) 1 g (T) ν(T) E ν(T) ξ exp u B(s)ν(s)ds + C(s) ds + D(s) ln(ν(s))ds + ρ t × ν(s) ξ | F Zt Zt Zt

where A, B, C and D are provided in the proof.

Proof. Proof of Lemma A1. Let the generic process be:

2 b b dZ(t) = L(t) + h(t) v(t) + dt + g (t) v(t) + dWt  v(t)  v(t) q ! q !  p  p dv(t) = α(θ v(t))dt + ξ v(t)dB(t) − dB(t), dW(t) = ρdt q h i We want to ﬁnd

uZ(T) uZ(t) E e t = e ΦGG (T, φ; L, h, g, κ, θ, ξ, ρ, b, c, vt, Zt) |F h i Letting ν˜(t) = g (t) ν(t) and νˆ(t) = g (t) ln(ν(t)), we have following:

dν˜(t) = αθg (t) dt + g0 (t) αg (t) ν(t)dt + g (t) ξ ν(t)dB(t), (A9) − q and

g (t) ∂νˆ(t) 1 ∂2νˆ(t) dνˆ(t) = 0 νˆ(t)dt + d ln(ν(t)) + < d ln(ν(t)) > (A10) g (t) ∂ ln(ν(t)) 2 ∂ ln(ν(t))2 g (t) g (t) ξ αθ ξ2 = 0 νˆ(t)dt + dB(t) + g (t) ( α)dt g (t) dt g (t) ν(t) ν(t) − − 2ν(t) p J. Risk Financial Manag. 2019, 12, 159 16 of 21

( ) From Equations (A9) and (A10), we solve for T g (s) ν(s)dB(s) and T g s dB(s): t t √ν(s) R p R T g(T)ν(T) g(t)ν(t) αθ T 1 T g (s) ν(s)dB(s) = − g (s) ds (g (s) αg (s)) ν(s)ds, (A11) t ξ − ξ t − ξ t 0 − R p R R T g(s) 1 ν(T)g(T) α T 1 ξ2 T g(s) 1 T dB(s) = ln + g (s) ds + αθ ds g0 (s) ln(ν(s))ds. (A12) t √ν(s) ξ ν(t)g(t) ξ t ξ 2 − t ν(s) − ξ t R R R R Split W(t) into B(t) and its orthogonal part B(t)⊥:

2 T T b Z(T) = Z(t) + L (s) ds + h(s) ν(s) + ds t t ν(s) Z Z q ! T p b 2 + g (s) ν(s) + (ρdB(s) + 1 ρ dB(s)⊥), t ν(s) − Z q ! q p then substitute Equation (A11) and (A12) to eliminate dB(t). Z(t) can be rewritten now as:

T T 2 T b 2 b Z(T) = Z(t) + L (s) ds + h(s) ν(s) + 2b + ds + 1 ρ g(s) a ν(s) + dB(s)⊥ t t ν(s) − t ν(s) Z Z q Z q ! g (T) ν(T) g (t) ν(t) αθρ T ρ T p + ρ − g (s) ds g0 (s) αg (s) ν(s)ds ξ ξ ξ − Zt − Zt − bρ ν(T)g(T) αbρ T bρ ξ2 T g (s) bρ T + ln + g (s) ds + αθ ds g0 (s) ln(ν(s))ds ξ g(t) ξ ξ 2 − ν(s) − ξ ν(t) Zt Zt Zt Grouping conveniently, we obtain:

T T T 1 T Z(T) = Z(t) + A (s) ds + B(s)ν(s)ds + C(s) ds + D(s) ln(ν(s))ds ν( ) Zt Zt Zt s Zt g(T) T bρ ν(T) g (T) ν(T) g (t) ν(t) 2 b + ln + ρ − + 1 ρ g(s) a ν(s) + dB(s)⊥ ξ ν(t)g(t) ξ − t ν(s) q Z q ! p where

αbρ αθρ A (s) = L (s) + 2bh(s) + g (s) ξ − ξ ρ B(s) = h(s) g0 (s) αg (s) − ξ − bρ ξ2 C(s) = b2h(s) + αθ g(s) ξ 2 − bρ D(s) = g0 (s) − ξ

Let ( t)t 0 denote the ﬁltration generated by ν(t), t 0. Using iterated expectation and G ≥ ≥ independence, we can write the conditional moment generating function of Z(T) as:

T bρ uZ(T) uZ(T) u g(t) g (t) ν(t) E e t = E E e t t t = exp u Z(t) + A (s) ds ν(t) ξ exp uρ |F | F G | F − ξ Zt h i h h T [ i T i T 1 bρ ( ) g (T) ν(T) E[ exp u B(s)ν(s)ds + C(s) ds + D(s) ln(ν(s))ds + ln ν(T)g T + ρ × ν(s) ξ ξ Zt Zt Zt T 2 b E exp u 1 ρ g(s) ν(s) + dB(s)⊥ t t t] × " ( − t ν(s) ! ) | F G # | F q Z q [ p J. Risk Financial Manag. 2019, 12, 159 17 of 21

The inner expectation, conditioned on t t, leads to a normal random variable with mean 0 F G 2 and variance (Ito’s Isometry) u2(1 ρ2) T g2(s)(ν(s) + b + 2b)ds. Putting all together: − t S ν(s) R T bρ uZ(T) u g(t) g (t) ν(t) E e t = exp u Z(t) + A (s) ds ν(t) ξ exp uρ |F t − ξ h i Z bρ T T T u g(T) 1 g (T) ν(T) E ν(T) ξ exp u B(s)ν(s)ds + C(s) ds + D(s) ln(ν(s))ds + ρ t × ν(s) ξ | F Zt Zt Zt where

A(s) = A(s) + u(1 ρ2)bg2(s) − 1 B(s) = B(s) + u2(1 ρ2)g2(s) 2 − 1 C(s) = C(s) + u2(1 ρ2)b2g2(s) 2 −

Proof. Proof of Corollary1. i The proof starts similarly to Proposition2. We start by deﬁning new processes dSij,t and dSt with i, j = 1, 2, . . . , n and t 0. ≥ ˜ ˜ 1 bi bi dZi(t) = ci v˜i(t) + dt + v˜i(t) + dWi − 2 v˜ (t) v˜ (t) " q i ! q i ! # 2 e n e p p L 1 bj bj + i dt + c a2 v (t) + dt + a v (t) + dW ∑  n j − 2 ij  j  ij  j  j j=1 q vj(t) q vj(t)    n  q   q   i = dSt + ∑ dSij,t j=1

By the dependence structure implied by the model, it follows that all S are independent for a ﬁx i, hence we can transform the characteristic function using the processes S as follows:

n n i i ΦZ(t),v(t)(T, ω) = ∏ E exp iω0 S j,T S j,t St, v(t) ∏ E exp iωi ST St St, v(t) j=1 { · − · } | i=1 { − } | h i n For each factor j = 1, 2, ... , n we deﬁne S∗ = ω0S j,t = ∑i=1 ωiSij,t, the dynamics of S∗ can be j,t · j,t expressed as

dS∗j,t = ω0dS j,t · 2 bj b = L(ω) + h (ω) v (t) + dt + g (ω) v (t) + k dW  j  j   j  j  j,t q vj(t) q vj(t)     q    q  where h (ω) = ∑n ω c 1 a2 , L(ω) = ∑n ω Li and g (ω) = ∑n ω a . Next, we ﬁnd the j i=1 i i − 2 ij i=1 i n j i=1 i ij characteristic function for the increments of S∗j,t:

E exp iφ S∗ S∗ S∗ , v (t) = v = Φ T, φ; L, h , g , κ , θ , ξ , ρ , b , c , v , S∗ . { j,T − j,t } | j,t j j,t G j j j j j j j j j,t j,t h i The generic function ΦG is provided in the AppendixB. J. Risk Financial Manag. 2019, 12, 159 18 of 21

i Similarly for the idiosyncratic factors dSt with i = 1, 2, . . . , n:

˜ 2 ˜ i i bi bi dSt∗ = ωidSt = hi(ω) v˜i(t) + dt + gi (ω) v˜i(t) + dWi,t v˜ (t) v˜ (t) q i ! q i ! p p e where h (ω) = ω c 1 and g (ω) = ω . i i i − 2 i i Combining all pieces together, we obtain: e n n i i ΦZ(t),v(t)(T, ω) = E exp iω0 S j,T S j,t St, v(t) E exp iωi ST St St, v(t) ∏ { · − · } | ∏ { − } | j=1 i=1 h i n n i = Φ T, φ; L, h , g , κ , θ , ξ , ρ , b , c , v , S∗ Φ T, 1; 0, h , g , κ , θ , ξ , ρ , b , c , v , S∗ ∏ G j j j j j j j j j,t j,t × ∏ G i i i i i i i i i,t t j=1 i=1 e e e e e e e

Proof. Proof of Corollary2. The proof uses Corollary1, where we express the joint c.f. as the product of one dimensional c.f.s of 4/2 type.

n n i i ΦZ(t),v(t)(T, ω) = E exp iω0 S j,T S j,t St, v(t) E exp iωi ST St St, v(t) ∏ { · − · } | ∏ { − } | j=1 i=1 h i n n i = Φ T, φ; L, h , g , κ , θ , ξ , ρ , b , c , v , S∗ Φ T, 1; 0, h , g , κ , θ , ξ , ρ , b , c , v , S∗ ∏ G j j j j j j j j j,t j,t × ∏ G i i i i i i i i i,t t j=1 i=1 e e e e e e e uZ(T) Hence, every one of these functions (ΦG (T, φ; L, h, g, κ, θ, ξ, ρ, b, c, vt, Zt) = E e t ) | F capture the c.f. of a process of the type: h i

2 b b dZ(t) = L + h v(t) + dt + g v(t) + dWt  v(t)  v(t) q ! q !  p  p dv(t) = α(θ v(t))dt + ξ v(t)dB(t) − dB(t), dW(t) = ρdt q h i It is not difﬁcult to realize therefore that the c.f. given v(T) can be similarly computed for every one of those processes, hence one can infer:

n ΦZ(t),v(T)(T, ω) = ∏ ΦG,T T, φ; L, hj, gj, κj, θj, ξj, ρj, bj, cj, vj,T, S∗j,t j=1 n i Φ T, 1; 0, h , g , κ , θ , ξ , ρ , b , c , v , S∗ × ∏ G,T i i i i i i i i i,T t i=1 e e e e e e e uZ(T) where ΦG,T (T, φ; L, h, g, κ, θ, ξ, ρ, b, c, vT, Zt) = E e t v(T) is provided next in AppendixB. | F ∪ h i J. Risk Financial Manag. 2019, 12, 159 19 of 21

Appendix B. Helpful Results Given the 4/2 process, the following c.f. are used in this paper:

2 b b dZ(t) = L + h v(t) + dt + g v(t) + dWt  v(t)  v(t) q ! q !  p  p dv(t) = α(θ v(t))dt + ξ v(t)dB(t) − dB(t), dW(t) = ρdt q h i

• uZ(T) Φ (T, u; L, h, g, α, θ, ξ, ρ, b, c, vt, Zt) = E e t G |F α2θ h 1 i gραθ gbρα = exp uZ(t) + (T t) + u r + 2(h )g2b + (T t) + u2(1 ρ2)g2b(T t) ξ2 − − 2 − ξ ξ − − − m +1 u 1 + mu + αθ + ugbρ 1 mu αθ ugbρ 2 2 ξ2 ξ √Au 2 + 2 2 ξ ugρ − ν(t) − ξ − Ku(T) ×  2 √Au  − ξ ξ sinh 2 t   Γ 1 + mu + αθ + ugbρ ν(t) √Au(T t) 2 2 ξ2 ξ exp 2 Au coth − + α ugρξ × ξ − 2 − Γ(mu + 1) p 1 mu αθ ugbρ Auν(t) 1 F1 + + + , mu + 1, ,  2 ξ ugρ 2 √Au(T t)  × 2 2 ξ ξ4(K (T) ) sinh − u − ξ 2   with

gρα 1 1 A = α2 2ξ2 u + (h )g2 + u2(1 ρ2)g2 , u − ξ − 2 2 − 2 2 2 2 ξ 2 gbρ ξ 1 2 2 1 2 2 2 2 mu = αθ 2ξ u αθ + (h )g b + u (1 ρ )g b , ξ2 s − 2 − ξ 2 − − 2 2 − 1 √A (T t) K (T) = A coth u − + α u ξ2 u 2 p • uZ(T) ΦG,T (T, u; L, h, g, κ, θ, ξ, ρ, b, c, vt, Zt) = E e t v(T) |F ∪ 1 aραθh bρα i = exp uZ(t) + u r + 2(h )g2b + (T t) + u2(1 ρ2)g2b(T t) − 2 − ξ ξ − − − ugρ ugbρ ν(T) exp (ν(T) ν(t)) + log × ξ − ξ ν(t) α(T t) √A sinh − u 2 ν(T) + ν(t) α(T t) √Au(T t) exp α coth − Au coth − √A (T t) 2 × α sinh u − ξ 2 − 2 2 p 2√Auν(T)ν(t) I 2 √A (T t) 2 ξ2 2 u + 2 ξ sinh 2 − 2 αθ 2 2ξ Bu ξ r − , × 2α√ν(T)ν(t) I 2αθ α(T t) 2 1 ξ2sinh − ξ − 2 with gbρ ξ2 1 1 B = u αθ + (h )g2b2 + u2(1 ρ2)g2b2, u ξ 2 − − 2 2 − J. Risk Financial Manag. 2019, 12, 159 20 of 21

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