Variance and interest rate risk in unit-linked insurance policies
VVariance and interest rate riskin unit-linked insurance policies
David Baños , Marc Lagunas-Merino , and Salvador Ortiz-Latorre Department of Mathematics, University of Oslo.June 29, 2020
Abstract
One of the risks derived from selling long term policies that any insurance company has,arises from interest rates. In this paper we consider a general class of stochastic volatilitymodels written in forward variance form. We also deal with stochastic interest rates to obtainthe risk-free price for unit-linked life insurance contracts, as well as providing a perfect hedgingstrategy by completing the market. We conclude with a simulation experiment, where we priceunit-linked policies using Norwegian mortality rates. In addition we compare prices for theclassical Black-Scholes model against the Heston stochastic volatility model with a Vasicekinterest rate model.
Keywords: unit-linked policies; pure endowment; term insurance; stochastic volatility models;stochastic interest rates.
MSC classification:
A unit-linked insurance policy is a product offered by insurance companies. Such contractspecifies an event under which the insured of the contract obtains a fixed amount. Typically, thepayoff of such contract is the maximum value between some prescribed quantity, the guarantee,and some quantity depending on the performance of a stock or fund. For instance, if G is somepositive constant amount, and S is the value of some equity or stock at the time of expiration ofthe contract, then a unit-linked contract pays H = max { G, f ( S ) } , where f is some suitable function of S . Here, the payoff H is always larger than G , hence being G a minimum guaranteed amount that the insured will receive. Naturally, the price of such contractdepends on the age of the insured at the moment of entering the contract and the time of expiration,likewise, it also depends on the event that the insured is alive at the time of expiration.The risk of such contracts depends on the risk of the financial instruments used to hedge theclaim H , and there are many ways to model it. The most classical one is considering the evolutionof S under a Black-Scholes model, this is for instance the case in [7] or [1], where the authors derivepricing and reserving formulas for unit-linked contracts in such setting. One can also consider amore general class of models. For example, it is empirically known that the driving volatility of S is, in general, not constant. One could then take a stochastic model for the volatility, as it is donein [13], where the authors carry out pricing and hedging under stochastic volatility. Since there ismore randomness in the model, complete hedging is no longer possible, the authors in [13] providethe so-called local risk minimizing strategies.In this paper instead, we look at the problem from two different perspectives. On the one hand,we also consider stochastic volatility, as market evidence shows. Nonetheless, there are available1 a r X i v : . [ q -f i n . P R ] J un nstruments in the market for hedging against volatility risk, the so-called forward variance swaps.Such products are contracts on the future performance of the volatility of the stock. In sucha way, we want to price unit-linked contracts taking into account that the insurance companycan trade these instruments, as well. On the other hand, it is known that unit-linked productsshare similarities with European call options. For example, authors in [7] recognize the payoff ofunit-linked products as European call options plus some certain amount. However, European calloptions have very short maturities, typically between the same day of the contract up to two years,while it is not uncommon to have unit-linked insurance contracts that last for up to 40 years. Forthis reason, there is an inherent risk in the interest rate driving the intrinsic value of money. Inthis paper we take such long-term risk into account, as well. Classically, most of the literatureabout equity-linked policies assume deterministic interest rates. Nevertheless, some research onstochastic interest rates has also been carried. For example, in [5] the authors consider stochasticinterest rates under the Heath-Jarrow-Morton framework and study different types of premiumpayments. In addition, a comparison with the classical Black-Scholes model is offered in [5]. Alsoin [4], the Vasicek and Cox-Ingersoll-Ross model is considered for the interest rate. In this paperwe consider a general framework including both cases.While many results in the literature deal with the construction of risk minimizing strategies inincomplete markets, in this paper instead, inspired by [12], we complete the market by allowingfor the possibility to trade other instruments that one can find in the market. On the one hand,we introduce zero-coupon bonds to hedge against interest rate risk and, on the other hand, weintroduce variance swaps to hedge against volatility risk.This paper is organized as follows. First, we introduce in Section 2 our insurance and economicframework with the specific models for the money account, stock and volatility. Then, in Section3, we complete the market by incorporating zero-coupon bonds and variance swaps in the market.We derive the dynamics of the new instruments used to hedge and apply the risk-neutral theory toprice insurance contracts subject to the performance of an equity or fund with stochastic interestand volatility. In Section 4 we take a particular model; the Vasicek model for the interest rate anda Heston model written in forward variance form. We implement the model and do a comparisonstudy with the classical Black-Scholes model in Section 5, where we generate price surfaces underNorwegian mortality rates and different maturities. We conclude Section 5 with a Monte-Carlosimulation of the price distributions. The two basic elements needed in order to build a financial model robust enough to be able toprice unit-linked policies, are a financial market and a group of individuals to write insurance on.We consider a finite time horizon
T > and a given probability space (Ω , A , P ) where Ω is the setof all possible states of the world, A be a σ -algebra of subsets of Ω and P be a probability measureon (Ω , A ) . We model the information flow at each given time with a filtration F = {F t , t ∈ [0 , T ] } given by a collection of increasing σ -algebras, i.e. F s ⊂ F t ⊂ A for t ≥ s . We will also assumethat F contains all the sets of probability zero and that the filtration is right continuous. Wealso take A = F T . The information flow F comes from two sources; the financial market and thestates of the insured that are relevant in the policy. The market information available at time t will be denoted by G t and the information regarding to the state of the insured by H t . We willassume throughout the paper that the σ -algebras G t and H t are independent for all t , which impliesthat the value of the market assets is independent of the health condition of the insured. We alsoassume that F t = G t ∨ H t , for all t , where G t ∨ H t is the σ -algebra generated by the union of G t and H t . This can be understood as the total amount of information available in the economy attime t , that is the information one can get by recording the values of market assets and the healthstate of the insured from time 0 to time t . The market information G will be modeled using the filtration generated by three independentstandard Brownian motions, W t , W t and W t . These three Brownian motions represent the sources2f risk in our model. We will consider a market formed by assets of two different natures. A bankaccount, considered of riskless nature and stock or bond prices, which are of risky nature.We start by defining the bank account, whose price process is denoted by B = { B t } t ∈ [0 ,T ] , suchthat B = 1 . We will assume the asset evolves according to the following differential equation dB t = r t B t dt, t ∈ [0 , T ] , (1)where r t is the instantaneous spot rate and it is assumed to have integrable trajectories. Actually,we will assume that this rate evolves under the physical measure P , according to the following SDE dr t = µ ( t, r t ) dt + σ ( t, r t ) dW t , r > , t ∈ [0 , T ] , (2)where µ, σ : [0 , T ] × R → R are Borel measurable functions such that, for every t ∈ [0 , T ] and x ∈ R | µ ( t, x ) | + | σ ( t, x ) | ≤ C (1 + | x | ) , for some positive constant C , and such that for every t ∈ [0 , T ] and x, y ∈ R | µ ( t, x ) − µ ( t, y ) | + | σ ( t, x ) − σ ( t, x ) | ≤ L | x − y | , for some constant L > . We will also assume there exists (cid:15) > , such that σ ( t, x ) ≥ (cid:15) > forevery ( t, x ) ∈ R + × R . These conditions are sufficient to guarantee a unique global strong solutionof (2) , weaker conditions may be imposed, see e.g. [11, Chapter IX, Theorem 3.5].One of the risky assets will be the stock. We describe the stock price process S = { S t } t ∈ [0 ,T ] by a general mean-reverting stochastic volatility model. Specifically, we will consider the followingSDEs dS t S t = b ( t, S t ) dt + a ( t, S t ) f ( ν t ) dW t , S > , (3) dν t = − κ ( ν t − ¯ ν ) dt + h ( ν t ) dW t , ν > , (4)for t ∈ [0 , T ] . Here a, b are uniformly Lipschitz continuous and bounded functions, such that a ( t, x ) > for all ( t, x ) ∈ [0 , T ] × R . The function f is assumed to be continuous with linear growthand strictly positive. We assume that h is a non-negative, linear growth, invertible function suchthat, | h ( x ) − h ( y ) | ≤ (cid:96) ( | x − y | ) , for some function (cid:96) defined on (0 , ∞ ) such that (cid:90) (cid:15) dz(cid:96) ( z ) = ∞ , for any (cid:15) > . Then, [11, Chapter IX, Theorem 3.5(ii)] guarantees the existence of a pathwise unique solution ofequation (3) . We call ν t , the instantaneous variance.Due to the fact that neither ν nor r are tradable, our market model is highly incomplete. Inthe forthcoming section, we will complete the market by introducing financial instruments in orderto hedge against the risk derived from instantaneous variance and interest rates.We introduce the numéraire, with respect to which we will discount our cashflows Definition 2.1.
The (stochastic) discount factor D t,T between two time intervals t and T , ≤ t ≤ T, is the amount at time t that is equivalent to one unit of currency payable at time T , and isgiven by D t,T = B t B T = exp (cid:32) − (cid:90) Tt r s ds (cid:33) . (5) In what follows, we introduce our insurance model. More specifically, we want to model theinsurance information H as the one generated by a regular Markov chain X = { X t , t ∈ [0 , T ] } S which regulates the states of the insured at each time t ∈ [0 , T ] . Forinstance, in an endowment insurance, the state S = {∗ , †} consists of the two states, ∗ = "alive"and † = "deceased". In a disability insurance we have three states, S = {∗ , (cid:5) , †} where (cid:5) stands for"disabled". Observe that X is right-continuous with left limits and, in particular, H satisfies theusual conditions.Introduce the following processes: I Xi ( t ) = (cid:40) , if X t = i, , if X t (cid:54) = i , i ∈ S ,N Xij ( t ) = { s ∈ (0 , t ) : X t − = i, X t = j } , i, j ∈ S , i (cid:54) = j. Here, denotes the counting measure and X t − (cid:44) lim u → tu The price of a zero-coupon bond, P t,T is given by P t,T = E Q [ D t,T | G t ] = E Q (cid:20) B t B T | G t (cid:21) = E Q (cid:34) exp (cid:40) − (cid:90) Tt r s ds (cid:41) | G t (cid:35) , (10) where Q is the equivalent martingale measure given by (9) . See [8, Definition 4.1. in Section4.3.1 and Section 5.1] for definitions. The next classical result gives a connection between the bond price in (10) and the solution toa linear PDE, see e.g. [8]. Lemma 3.3. Assume that for any T > , F T ∈ C , ([0 , T ] × R ) is a solution to the boundaryproblem on [0 , T ] × R given by ∂ t F T ( t, x ) + µ ( t, x ) ∂ x F T ( t, x ) + 12 σ ( t, x ) ∂ x F T ( t, x ) − xF T ( t, x ) = 0 ,F T ( T, x ) = 1 . Then M t (cid:44) F T ( t, r t ) e − (cid:82) t r u du , t ∈ [0 , T ] , is a local martingale. If in addition either: (a) E Q (cid:20)(cid:82) T (cid:12)(cid:12)(cid:12) ∂ x F T ( t, r t ) e − (cid:82) t r u du σ ( t, r t ) (cid:12)(cid:12)(cid:12) dt (cid:21) < ∞ , or(b) M is uniformly bounded,then M is a martingale, and F T ( t, r t ) = E Q (cid:104) e − (cid:82) Tt r u du | G t (cid:105) , t ∈ [0 , T ] . (11)The dynamics of the zero-coupon bond P in terms of the function F T are given by dP t,T = L P ( F T ) ( t, r t ) dt + ∂ x F T ( t, r t ) σ ( t, r t ) dW t , (12)where L P (cid:44) ∂ t + µ ( t, x ) ∂ x + σ ( t, x ) ∂ x − x .We turn now to the definition of the forward variance process. The forward variance ξ t,u , for ≤ t ≤ u , is by definition the conditional expectation of the future instantaneous variance, seee.g. [2], that is, ξ t,u (cid:44) E Q [ ν u | G t ] , ≤ t ≤ u, (13)where Q is the risk-neutral pricing measure defined in (9). Following [6], one can easily rewritethe general stochastic volatility model, given by equations (3) and (4) in forward variance form.This is achieved by taking conditional expectation of equation (4) , which yields d E Q [ ν u | G t ] = − κ (cid:16) E Q [ ν u | G t ] − ¯ ν (cid:17) du, u > t, Solving the previous linear ODE, by integrating on [ t, u ] , we have ξ t,u = ¯ ν + e − κ ( u − t ) ( ν t − ¯ ν ) . (14)There are two things to notice at this point. The first is that, by construction ν t = ξ t,t , for every t ∈ [0 , T ] . Second is that differentiating the previous equation, we can characterize the dynamicswith respect to t for the forward variance as follows dξ t,u = e − κ ( u − t ) h ( ν t ) dW t . (15)6olving equation (14) for ν t , yields ν t = ¯ ν + e κ ( u − t ) ( ξ t,u − ¯ ν ) (cid:44) ψ ( t, u, ξ t,u ) . Usually, the dynamics of the forward variance in any forward variance model, are given throughthe following SDE, dξ t,u = λ ( t, u, ξ t,u ) dW t . (16)As a consequence of the previous result and in our case, the function λ in equation (16) is fullycharacterized by λ ( t, u, ξ t,u ) (cid:44) e − κ ( u − t ) ( h ◦ ψ ) ( t, u, ξ t,u ) . (17)Note that any finite-dimensional Markovian stochastic volatility model can be rewritten inforward variance form. Since we will only be interested in the fixed case u = T , we will drop thedependence on T for ξ t,T and write instead ξ t = ξ t,T .We will show how to form a portfolio with a perfect hedge. The financial instruments neededin order to build a riskless portfolio are the underlying asset, a variance swap and the zero-couponbond.From now on, we will assume that the function F T , solution to the PDE in Lemma 3.3 isinvertible in the space variable, for every t ∈ [0 , T ] , e.g. this is the case if r t , t ∈ [0 , T ] is given bythe Vasicek model. Introduce the notation, G T ( t, x ) (cid:44) ∂ x F T ( t, x ) , (18)then ∂ x F T ( t, r t ) = G T (cid:0) t, F − T ( t, P t,T ) (cid:1) , where r t = F − T ( t, P t,T ) . Let Π = { Π t } t ∈ [0 ,T ] be a stochastic process representing the value of a portfolio consistingof a long position on an option with price V t , where V t = V ( t, S t , ξ t , P t,T ) , and respective shortpositions on ∆ t units of the underlying asset, Σ t units of a variance swap, and Ψ t units of azero-coupon bond. Therefore, we can characterize the process Π as Π t = V ( t, S t , ξ t , P t,T ) − ∆ t S t − Σ t ξ t − Ψ t P t,T , t ∈ [0 , T ] . (19) Definition 3.4. We say that the portfolio Π is self-financing if, and only if, d Π t = dV ( t, S t , ξ t , P t,T ) − ∆ t dS t − Σ t dξ t − Ψ t dP t,T , for every t ∈ [0 , T ] . Definition 3.5. We say that the portfolio Π is perfectly hedged, or risk-neutral, if it is self-financingand Π T = 0 . From now on, and throughout the rest of the paper, we will only differentiate between timederivative ∂ t V and space derivatives ∂ x V , ∂ y V , ∂ z V , to write the partial derivatives of V = V ( t, x, y, z ) . We will also denote second order spatial partial derivatives of V with respect to S t , ξ t , P t,T , respectively by ∂ x V , ∂ y V , ∂ z V and the second order crossed derivatives as ∂ x ∂ y V , ∂ x ∂ z V , ∂ y ∂ z V . In order to simplify the notation in the following results, we shall define Ξ T ( t, x ) (cid:44) G T (cid:0) t, F − T ( t, x ) (cid:1) · σ (cid:0) t, F − T ( t, x ) (cid:1) , where recall that G T is given in (18). 7 heorem 3.6. Let Π be a portfolio defined as in (19) , and assume V ∈ C , (cid:0) [0 , T ] × R (cid:1) . If Π isa replicating portfolio, then V fulfils, ∂ t V + 12 (cid:16) x a ( t, x ) f ( ψ ( t, T, y )) ∂ x V + λ ( t, T, y ) ∂ y V + Ξ T ( t, z ) ∂ z V (cid:17) (20) − r t ( V − x∂ x V − y∂ y V − z∂ z V ) = 0 , for every t ∈ [0 , T ] and V ( T, S T , ξ T , P T,T ) = max ( S T , G ) . (21) Proof. It is important to notice that we will use the notation V t to refer to the process V ( t, S t , ξ t , P t,T ) ,and similarly for the partial derivatives. For instance, ∂ x V t = ∂ x V ( t, S t , ξ t , P t,T ) . By means ofItô’s lemma, we are able to write the change in our portfolio { V t } t ∈ [0 ,T ] as follows d Π t = ∂ t V t dt + ∂ x V t dS t + ∂ y V t dξ t + ∂ z V t dP t,T + 12 ∂ x V t d [ S, S ] t + 12 ∂ y V t d [ ξ, ξ ] t + 12 ∂ z V t d [ P, P ] t + ∂ x ∂ y V t d [ S, ξ ] t + ∂ x ∂ z V t d [ S, P ] t + ∂ y ∂ z V t d [ ξ, P ] t − ∆ t dS ( t ) − Σ t dξ t − Ψ t dP t,T = ∂ t V t dt + { ∂ x V t − ∆ t } dS t + { ∂ y V t − Σ t } dξ t + { ∂ z V t − Ψ t } dP t,T + 12 ∂ x V t d [ S, S ] t + 12 ∂ y V t d [ ξ, ξ ] t + 12 ∂ z V t d [ P, P ] t + ∂ x ∂ y V t d [ S, ξ ] t + ∂ x ∂ z V t d [ S, P ] t + ∂ y ∂ z V t d [ ξ, P ] t . Using the dynamics for dS t , dξ t , dP t,T and the quadratic covariations, given by d [ S, S ] t = S t a ( t, S t ) f ( ψ ( t, T, ξ t )) dt,d [ ξ, ξ ] t = λ ( t, T, ξ t ) dt,d [ P, P ] t = Ξ T ( t, P t,T ) dt,d [ S, ξ ] t = 0 ,d [ S, P ] t = 0 ,d [ ξ, P ] t = 0 , we obtain d Π t = ∂ t V t dt + { ∂ x V t − ∆ t } dS t + { ∂ y V t − Σ t } dξ t + { ∂ z V t − Ψ t } dP t,T + 12 S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t dt + 12 λ ( t, T, ξ t ) ∂ y V t dt + 12 Ξ T ( t, P t,T ) ∂ z V t dt = (cid:40) ∂ t V t + 12 (cid:16) S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t + λ ( t, T, ξ t ) ∂ y V t + Ξ T ( t, P t,T ) ∂ z V t (cid:17)(cid:41) dt + { ∂ x V t − ∆ t } dS t + { ∂ y V t − Σ t } dξ t + { ∂ z V t − Ψ t } dP t,T . Now, in order to make the portfolio instantaneously risk-free, we must impose that the returnon our portfolio equals the risk-free rate r t , i.e. d Π t = r t Π t dt = r t ( V t − ∆ t S t − Σ t ξ t − Ψ t P t,T ) dt ,and force the coefficients in front of dS t , dξ t and dP t,T to be zero, i.e. ∆ t = ∂ x V t , Σ t = ∂ y V t , Ψ t = ∂ z V t . This implies that d Π t = (cid:40) ∂ t V t + 12 (cid:16) S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t + λ ( t, T, ξ t ) ∂ y V t + Ξ T ( t, P t,T ) ∂ z V t (cid:17)(cid:41) dt. ∆ t = ∂ x V t , Σ t = ∂ y V t , Ψ t = ∂ z V t , we have the PDE for the unit-linked product, endingthe proof.From now on, in order to ease the notation, we will define the differential operator in (20) as L V (cid:44) ∂ t + 12 (cid:16) x a ( t, x ) f ( ψ ( t, T, y )) ∂ x + λ ( t, T, y ) ∂ y + Ξ T ( t, z ) ∂ z (cid:17) (22) − r t (1 − x∂ x − y∂ y − z∂ z ) . We will now prove that the discounted option price is a martingale. Theorem 3.7. Let V be the solution to the PDE given by equation (20) with terminal condition (21) . Then B − t V ( t, S t , ξ t , P t,T ) = E Q (cid:2) B − T V ( T, S T , ξ T , P T,T ) | G t (cid:3) , where Q indicates the risk-neutral measure.Proof. We start by imposing that the discounted price process, ˜ S t = B − t S t , the discountedvariance swap ˜ ξ t , and the discounted zero-coupon bond price ˜ P t,T are Q − martingales, where dB t = r t B t dt and dB − t = − r t B − t dt . To do so, we will also make use of the relationship be-tween the Brownian motions and their Q -measure counterparts, given by (8) . d ˜ S t = dB − t S t + B − t dS t = − r t B − t S t dt + B − t (cid:2) b ( t, S t ) S t dt + a ( t, S t ) f ( ψ ( t, T, ξ t )) S t dW t (cid:3) = ˜ S t (cid:104) ( b ( t, S t ) − r t ) dt + a ( t, S t ) f ( ψ ( t, T, ξ t )) (cid:104) dW Q , t + γ t dt (cid:105)(cid:105) = ˜ S t (cid:2) b ( t, S t ) − r t + a ( t, S t ) f ( ψ ( t, T, ξ t )) γ t (cid:3) dt + ˜ S t a ( t, S t ) f ( ν t ) dW Q , t . Now, the discounted price process ˜ S t is a Q − martingale if, and only if γ t = r t − b ( t, S t ) a ( t, S t ) f ( ψ ( t, T, ξ t )) . (23)We do the same for the discounted forward variance process, d ˜ ξ t = dB − t ξ t + B − t dξ t = − r t B − t ξ t dt + B − t λ ( t, T, ξ t ) dW t = B − t (cid:2) λ ( t, T, ξ t ) γ t − r t ξ t (cid:3) dt + B − t λ ( t, T, ξ t ) dW Q , t . Therefore, the discounted variance swap is a Q − martingale if, and only if γ t = r t ξ t λ ( t, T, ξ t ) . (24)Finally, we impose that the discounted zero-coupon bond price process is a Q -martingale analo-gously d ˜ P t,T = dB − t P t,T + B − t dP t,T = − r t B − t P t,T dt + B − t (cid:20) ∂ t F T ( t, r t ) + µ ( t, r t ) ∂ x F T ( t, r t )+ 12 σ ( t, r t ) ∂ x F T ( t, r t ) − r t F T ( t, r t ) (cid:21) dt + B − t ∂ x F T ( t, r t ) σ ( t, r t ) dW t = B − t (cid:20) ∂ t F T ( t, r t ) + (cid:0) µ ( t, r t ) + σ ( t, r t ) γ t (cid:1) ∂ x F T ( t, r t ) σ ( t, r t ) ∂ x F T ( t, r t ) − r t ( F T ( t, r t ) + P t,T ) (cid:21) dt + B − t Ξ T ( t, P t,T ) dW Q , t . Therefore, the discounted zero-coupon bond is a Q -martingale if, and only if F T ( t, r t ) + P t,T (cid:20) ∂ t F T ( t, r t ) + (cid:0) µ ( t, r t ) + σ ( t, r t ) γ t (cid:1) ∂ x F T ( t, r t ) + 12 Ξ T ( t, P t,T ) (cid:21) = r t . (25)Now, we are able to characterize γ i , for all i ∈ { , , } , by solving the linear system given byequations (23) , (24) and (25) .Therefore, we will apply Itô’s lemma to the discounted price of the option, d (cid:2) B − t V ( t, S t , ξ t , P t,T ) (cid:3) = dB − t V ( t, S t , ξ t , P t,T ) + B − t dV ( t, S t , ξ t , P t,T ) . In order to relax the notation we will drop the dependencies of V , allowing us to rewrite theprevious expression as d (cid:2) B − t V t (cid:3) = dB − t V t + B − t dV t = − r t B − t V t dt + B − t [ ∂ t V t dt + ∂ x V t dS t + ∂ y V t dξ t + ∂ z V t dP t,T ]+ B − t (cid:20) ∂ x V t d [ S, S ] t + 12 ∂ y V t d [ ξ, ξ ] t + 12 ∂ z V t d [ P, P ] t (cid:21) + B − t [ ∂ x ∂ y V t d [ S, ξ ] t + ∂ x ∂ z V t d [ S, P ] t + ∂ y ∂ z V t d [ ξ, P ] t ] . Furthermore, d (cid:2) B − t V t (cid:3) = B − t ( ∂ t V t − r t V t ) dt + B − t (cid:0) ∂ x V t (cid:2) b ( t, S t ) S t dt + S t a ( t, S t ) f ( ψ ( t, T, ξ t )) dW t (cid:3) + ∂ y V t (cid:2) λ ( t, T, ξ t ) dW t (cid:3)(cid:1) + B − t ∂ z V t (cid:2) L P ( F T ( t, r t )) dt + Ξ T ( t, P t,T ) dW t (cid:3) + B − t (cid:20) S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t + 12 λ ( t, T, ξ t ) ∂ y V t + 12 Ξ T ( t, P t,T ) ∂ z V t (cid:21) dt = B − t (cid:20) ∂ t V t − r t V t + b ( t, S t ) S t ∂ x V t + L P ( F T ( t, r t )) ∂ z V t + 12 (cid:16) S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t + λ ( t, T, ξ t ) ∂ y V t + Ξ T ( t, P t,T ) ∂ z V t (cid:17)(cid:21) dt + B − t (cid:2) S t f ( ψ ( t, T, ξ t )) ∂ x V t dW t + λ ( t, T, ξ t ) ∂ y V t dW t + Ξ T ( t, P t,T ) ∂ z V t dW t (cid:3) . If we replace the Brownian motions under the P -measure by the ones under the Q -measuregiven by equation (8) , we can rewrite the previous expression as follows d (cid:2) B − t V t (cid:3) == B − t (cid:20) ∂ t V t − r t V t + b ( t, S t ) S t ∂ x V t + L P ( F T ( t, r t )) ∂ z V t + 12 (cid:16) S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t + λ ( t, T, ξ t ) ∂ y V t + Ξ T ( t, P t,T ) ∂ z V t (cid:17)(cid:21) dt + B − t S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t (cid:104) dW Q , t + γ t dt (cid:105) + B − t λ ( t, T, ξ t ) ∂ y V t (cid:104) dW Q , t + γ t dt (cid:105) + B − t Ξ T ( t, P t,T ) ∂ z V t (cid:104) dW Q , t + γ t dt (cid:105) = B − t (cid:20) ∂ t V t − r t V t + S t (cid:2) b ( t, S t ) + γ t a ( t, S t ) f ( ψ ( t, T, ξ t )) (cid:3) ∂ x V t γ t λ ( t, T, ξ t ) ∂ y V t + (cid:2) L P ( F T ( t, r t )) + γ t Ξ T ( t, P t,T ) (cid:3) ∂ z V t + 12 (cid:16) S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t + λ ( t, T, ξ t ) ∂ y V t + Ξ T ( t, P t,T ) ∂ z V t (cid:17)(cid:21) dt + B − t (cid:104) S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t dW Q , t + λ ( t, T, ξ t ) ∂ y V t dW Q , t + Ξ T ( t, P t,T )) ∂ z V t dW Q , t (cid:105) . Applying equations (23) , (24) , (25) and reorganizing the terms in the previous equation, we have d (cid:2) B − t V t (cid:3) = B − t (cid:20) ∂ t V t + r t ( S t ∂ x V t + ξ t ∂ y V t + P t,T ∂ z V t − V t )+ 12 (cid:16) S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t + λ ( t, T, ξ t ) ∂ y V t + Ξ T ( t, P t,T ) ∂ z V t (cid:17)(cid:21) dt + B − t (cid:104) S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V t dW Q , t + λ ( t, T, ξ t ) ∂ y V t dW Q , t + Ξ T ( t, P t,T ) ∂ z V t dW Q , t (cid:105) . Now, noticing that the dt term in the previous equation is the differential operator (22) applied to V , we can write the following d (cid:2) B − t V ( t, S t , ξ t , P t,T ) (cid:3) = B − t L V V ( t, S t , ξ t , P t,T ) dt + B − t S t a ( t, S t ) f ( ψ ( t, T, ξ t )) ∂ x V ( t, S t , ξ t , P t,T ) dW Q , t + B − t λ ( t, u, ξ t ) ∂ y V ( t, S t , ξ t , P t,T ) dW Q , t + B − t Ξ T ( t, P t,T ) ∂ z V ( t, S t , ξ t , P t,T ) dW Q , t . Next, integrating on the interval [ s, t ] , with s ≤ t, we can write the previous equation in integralform as B − t V ( t, S t , ξ t , P t,T ) = V ( s, S s , ξ s , P s,T ) + (cid:90) ts B − τ L V V ( τ, S τ , ξ τ , P τ,T ) dτ + (cid:90) ts B − τ S τ a ( τ, S τ ) f ( ψ ( τ, T, ξ τ )) ∂ x V ( τ, S τ , ξ τ , P τ,T ) dW Q , τ + (cid:90) ts B − τ λ ( τ, u, ξ τ ) ∂ y V ( τ, S τ , ξ τ , P τ,T ) dW Q , τ + (cid:90) ts B − τ Ξ T ( t, P t,T ) ∂ z V ( τ, S τ , ξ τ , P τ,T ) dW Q , τ . Taking the conditional expectation with respect to the risk neutral measure, we have that E Q (cid:2) B − t V ( t, S t , ξ t , P t,T ) | G s (cid:3) = V s + E Q (cid:20)(cid:90) ts B − τ L V V ( τ, S τ , ξ τ , P τ,T ) dτ | G s (cid:21) . Notice that the previous expression is a martingale if, and only if, L V V ( t, S t , ξ t , P t,T ) ≡ , for all t ∈ [0 , T ] . This section is devoted to providing the reader with a particular model. We will assume thatthe evolution of the short-term rate is given by a Vasicek model and consider a Heston model forthe risky asset written in forward variance form.Let us consider the following SDE for the short-term rate given by the Vasicek model. dr t = k ( θ − r t ) dt + σdW t , r > , t ∈ [0 , T ] , (26)and the Heston model for the risky asset, given by dS t = µ t S t dt + S t √ ν t dW t , S > , t ∈ [0 , T ] , (27)11 ν t = − κ ( ν t − ¯ ν ) dt + η √ ν t dW t , ν > , t ∈ [0 , T ] . (28)It is well known that the SDE (26) admits the following closed expression. r T = e − k ( T − t ) r t + θ (cid:16) − e − k ( T − t ) (cid:17) + σ (cid:90) Tt e − k ( T − s ) dW s . Now, we know that r T , conditional on G t , is normally distributed with mean and variance E [ r T | G t ] = e − k ( T − t ) r t + θ (cid:16) − e − k ( T − t ) (cid:17) , Var [ r T | G t ] = σ k (cid:16) − e − k ( T − t ) (cid:17) . One can show, see, e.g. [10] that the price of the zero-coupon bond under the dynamics givenin (26) is P t,T = A ( t, T ) e − B ( t,T ) r t , where B ( t, T ) (cid:44) k (cid:0) − e − k ( T − t ) (cid:1) and A ( t, T ) (cid:44) exp (cid:16)(cid:16) θ − σ k (cid:17) ( B ( t, T ) + t − T ) − σ k B ( t, T ) (cid:17) . If we now apply Itô’s Lemma to f ( t, r t ) = A ( t, T ) e − B ( t,T ) r t , we have dP t,T = ∂ t f ( t, r t ) dt + ∂ r f ( t, r t ) dr t + 12 ∂ rr f ( t, r t ) d [ r, r ] t = ∂ t P t,T dt − A ( t, T ) B ( t, T ) e − B ( t,T ) r t dr t + 12 A ( t, T ) B ( t, T ) e − B ( t,T ) r t d [ r, r ] t = ∂ t P t,T dt + P t,T (cid:18) − B ( t, T ) dr t + 12 B ( t, T ) d [ r, r ] t (cid:19) . Replacing the term dr t in the previous equation by its SDE (26) , we have dP t,T P t,T = − (cid:18) B ( t, T ) k ( θ − r t ) − B ( t, T ) σ (cid:19) dt − σB ( t, T ) dW t . (29)The forward variance in this case has the following dynamics dξ t,u = ηe − κ ( u − t ) (cid:112) ξ t,t dW t . (30)The Heston model, as any Markovian model, can be rewritten in forward variance form by meansof equations (27) and (30) . The following corollary gives the specific risk-neutral measure for theVasicek-Heston model, that will be useful for simulation purposes in the next section. Corollary 4.1. The risk-neutral measure under the Vasicek-Heston model is given by the measurein (7) with γ t = 1Θ ( t, ν t ) (cid:104) η √ ν t (cid:16) − B ( t, T ) k ) r t + B ( t, T ) (cid:0) B ( t, T ) σ − kθ (cid:1) (cid:17)(cid:105) ,γ t = − 1Θ ( t, ν t ) (cid:104) B ( t, T ) ση ( µ t − r t ) (cid:105) ,γ t = 1Θ ( t, ν t ) (cid:104) B ( t, T ) e κ ( T − t ) r t σξ t (cid:105) , where Θ ( t, x ) (cid:44) B ( t, T ) ση √ x. Proof. We will proceed similarly as in Theorem . . We have to impose that the discounted priceprocess, ˜ S t , the discounted variance swap ˜ ξ t , and the discounted zero-coupon bond price ˜ P t,T are Q − martingales. d ˜ S t = dB − t S t + B − t dS t − r t B − t S t dt + B − t (cid:2) µ t S t dt + S t √ ν t dW t (cid:3) = ˜ S t (cid:16)(cid:2) µ t − r t + √ ν t γ t (cid:3) dt + √ ν t dW Q , t (cid:17) , now the discounted price process ˜ S t is a Q − martingale if and only if γ t = r t − µ t √ ν t . (31)We do the same for the discounted forward variance, hence we obtain d ˜ ξ t = dB − t ξ t + B − t dξ t = − r t B − t ξ t dt + B − t ηe − κ ( T − t ) √ ν t dW t = B − t (cid:16) − r t ξ t dt + ηe − κ ( T − t ) √ ν t (cid:104) dW Q , t + γ t dt (cid:105)(cid:17) = B − t (cid:16)(cid:104) ηe − κ ( T − t ) √ ν t γ t − r t ξ t (cid:105) dt + ηe − κ ( T − t ) √ ν t dW Q , t (cid:17) , therefore the discounted variance swap ˜ ξ t , is a Q − martingale if and only if γ t = ξ t r t ηe − κ ( T − t ) √ ν t . (32)Finally, we impose that the discounted zero-coupon bond price process is a Q -martingale in ananalogous computation, d ˜ P t,T = dB − t P t,T + B − t dP t,T = − r t B − t P t,T + B − t P t,T (cid:20) − (cid:18) B ( t, T ) k ( θ − r t ) − B ( t, T ) σ (cid:19) dt − σB ( t, T ) dW t (cid:21) = − ˜ P t,T (cid:18)(cid:18) r t + B ( t, T ) k ( θ − r t ) − B ( t, T ) σ (cid:19) dt + σB ( t, T ) dW t (cid:19) = − ˜ P t,T (cid:18) r t + σB ( t, T ) γ t + B ( t, T ) k ( θ − r t ) − B ( t, T ) σ (cid:19) dt − ˜ P t,T (cid:16) σB ( t, T ) dW Q , t (cid:17) , therefore the discounted zero-coupon bond is a Q -martingale if and only if − − B ( t, T ) k (cid:18) σB ( t, T ) γ t + B ( t, T ) kθ − B ( t, T ) σ (cid:19) = r t . (33)The result follows, solving the linear system formed by equations (31) , (32) and (33) . In this section, we present an implementation of the Heston model written in forward variancetogether with a Vasicek model for the interest rates, in order to price numerically a unit-linkedproduct. We will implement a Monte Carlo scheme for simulating prices under this model andcompare it against a classical Black-Scholes model. The Heston process will be simulated using afull-truncation scheme [3] in the Euler discretization on both models. We first show the discretizedversions of the SDE’s for each model and the result of the model comparison given some initialconditions.Let N ∈ N be the number of time steps in which the interval [0 , T ] is equally divided. Thenconsider the uniform time grid t k (cid:44) ( kT ) /N, for all k = 1 , . . . , N of length ∆ t = T /N. We presentthe following Euler schemes for each model.1. Classical Black Scholes S t k +1 = S t k exp (cid:18)(cid:18) r − 12 ¯ ν (cid:19) ∆ t + ¯ ν √ ∆ t (cid:16) W Q ( t k +1 ) − W Q ( t k ) (cid:17)(cid:19) , where the parameters for the simulation are ( S , r, ¯ ν ) , given by: S = 100 , r = 0 . , ¯ ν = 0 . . 13igure 1: Pricing comparative between the Black-Scholes model and a Vasicek-Heston model writ-ten in forward variance with the mentioned initial conditions and T ∈ { , , , } .2. Vasicek-Heston Model written in forward variance r t k +1 = r t k + (cid:104) k (cid:16) θ − ( r t k ) + (cid:17) + σγ ( t k ) (cid:105) ∆ t + σ √ ∆ t (cid:16) W Q ( t k +1 ) − W Q ( t k ) (cid:17) ,ξ t k +1 ( t N ) = ξ t k ( t N ) + r t k ξ t k ( t N ) ∆ t + ηe − κ ( t N − t k ) (cid:113) ( ν t k ) + ∆ t (cid:16) W Q ( t k +1 ) − W Q ( t k ) (cid:17) ,ν t k +1 = ¯ ν + e κ ( t N − t k +1 ) (cid:0) ξ t k +1 ( t N ) − ¯ ν (cid:1) ,S t k +1 = S t k + r t k S t k ∆ t + S t k (cid:113) ( ν t k ) + ∆ t (cid:16) W Q ( t k +1 ) − W Q ( t k ) (cid:17) ,P ( t k +1 , t N ) = P ( t k , t N ) + P ( t k , t N ) (cid:104) r t k ∆ t − σB ( t k , t N ) √ ∆ t (cid:16) W Q ( t k +1 ) − W Q ( t k ) (cid:17)(cid:105) . where the parameters are ( S , µ, ν , ¯ ν, κ, η, r , θ, k, σ ) and were set as S = 100 , µ = 0 . , ¯ ν = 0 . , ν = 0 . , κ = 10 − , η = 0 . , θ = r = 0 . , k = 0 . , and σ = 0 . .For simulation purposes, the Monte Carlo scheme was implemented using 5,000 simulations. Thefollowing graphs in Figure 1, results from the implementation of the previous models with thementioned initial conditions, and for T = { , , , } . 14s shown in Theorem . and Corollary . , in order to properly price a unit-linked product,it only remains to multiply the value of the derivative priced using the Monte Carlo scheme, timesthe probability that an x -year old insured survives during the life of the product ( T years). To doso, we have used Norwegian mortality from 2018 extracted from Statistics Norway.Age Men Women Total4 50 45 959 7 2 914 10 3 1319 26 13 3924 33 6 3929 63 24 8734 72 27 9939 93 43 13644 109 68 17749 156 111 26754 258 177 43559 454 310 76464 737 495 123269 1206 824 203074 1990 1331 332179 3602 2447 604984 6626 4628 1125489 12469 9053 21522 ≥ 90 21 909 24 230 46139Table 1: Norwegian mortality in 2018, per 100 000 inhabitants. Data from Statistics Norway,table: 05381As it is usual, mortality among men is higher, we consider however, the aggregated mortalityfor simplicity. To model the mortality given in Table 1 we use the Gompertz-Makeham law ofmortality which states that the death rate is the sum of an age-dependent component, whichincreases exponentially with age, and an age-independent component, i.e. µ ∗† ( t ) = a + be ct , t ∈ [0 , T ] . This law of mortality describes the age dynamics of human mortality rather accuratelyin the age window from about 30 to 80 years of age, which is good enough for our purposes. Forthis reason, we excluded the very first and last observations from the table. We then find the bestfit for µ ∗† in the class of functions C = { f ( t ) = a + be ct , t ∈ [0 , T ] , a, b, c ∈ R } . As stated previously,since the stochastic process X = { X t } t ∈ [0 ,T ] , which regulates the states of the insured, is a regularMarkov chain, then the survival probability of an x -year old individual during the next T years is T p x = ¯ p ∗∗ ( x, x + T ) = exp (cid:32) − (cid:90) x + Tx µ ∗† ( τ ) dτ (cid:33) . Figure 2 shows the fitted Gompertz-Makeham law based on the mortality data from Table 1.Now, using the Vasicek-Heston model written in forward variance, we can compute a unit-linked price surface in terms of the guarantee, or strike price, and the age of the insured given aterminal time for the product T > . In particular, the graphs below show the price surfaces forfixed T = { , , , } .From the plots in Figure 3, we can observe that the longer time to maturity is, the lower theunit-linked price is, since the less probable it is that the insured survives. This effect has greaterimpact in the price, than the effect of future volatility, or uncertainty arising from the stochasticityin interest rates. This behavior is easily observed by noting how the price surface collapses to zeroas the contract’s time to maturity increases, as well as the age of the insured when entering thecontract. Hence, we can say that time to maturity has a cancelling effect on price, i.e. on one handit increases price as the stock or fund pays longer performance, but on the other hand it decreasesprice due to a lower probability of surviving during the time to maturity of the unit-linked contract.15igure 2: Joint plot of the mortality data given in Table 1, together with the fitted curve usingthe Gompertz-Makeham law of mortality.Figure 3: unit-linked price surfaces under a Vasicek-Heston model written in forward variance fordifferent policy maturities, in terms of the guaranteed amount desired by the insured and his ageat the time of acquisition. 16he following plots in Figures 4 and 5, are aimed at providing the reader with an overview ofthe distributional properties of the price process at a constant survival rate equal to one. The firstthing that comes to sight, is how the variance and time to maturity are directly proportional. Also,the longer the time to maturity of the unit-linked product is, the more leptokurtic the distributionof the insurance product price is. This is an important thing to take into account in the modelingof prices due to the impact in the hedging of such insurance products.Figure 4: unit-linked price histograms with constant survival rate equal to 1.Figure 5: QQ-Plot between the unit-linked price input data and the Standard Normal Distributionfor maturities T = { , , , } years. 17 .1 Pure endowment Consider an endowment for a life aged x with maturity T > . The policy pays the amount E T := max { G e , S T } if the insured survives by time T where G e > is a guaranteed amount and S T is the value of a fund at the expiration time. This policy is entirely determined by the policyfunction a ∗ ( t ) = (cid:40) E T if t ≥ T else . In view of (6) and the above function, the value of this insurance at time t given that theinsured is still alive is then given by V + ∗ ( t, A ) = E Q (cid:34)(cid:90) Tt B t B s p ∗∗ ( x + t, x + s ) da ∗ ( s ) (cid:12)(cid:12)(cid:12) G t (cid:35) = E Q (cid:20) B t B T E T (cid:12)(cid:12)(cid:12) G t (cid:21) p ∗∗ ( x + t, x + T ) , (34)The above quantity corresponds to the formula in Theorem 3.7.Observe that, the payoff of an endowment can be written as max { G e , S T } = ( G e − S T ) + + G e , where ( x ) + (cid:44) max { x, } , which corresponds to a call option with strike price G e plus G e . In thecase that S is modelled by the Black-Scholes model (with constant interest rate) we know that theprice at time t of a call option with strike G e and maturity T is given by BS ( t, T, S t , G e ) (cid:44) Φ( d ( t, T )) S t − Φ( d ( t, T )) G e e − r ( T − t ) , where Φ denotes the distribution function of a standard normally distributed random variable and d ( t, T ) (cid:44) log( S t /G e ) + (cid:0) r + σ (cid:1) ( T − t ) σ √ T − t , d ( t, T ) (cid:44) d ( t, T ) − σ √ T − t. Then we have that the unit-linked pure endowment under the Black-Scholes model has theprice BSE ( t, T, S t , G e ) (cid:44) Φ( d ( t, T )) S t + G e e − r ( T − t ) Φ( − d ( t, T )) . (35)The single premium at the beginning of this contract under the Black-Scholes model is then π BS (cid:44) BSE (0 , T, S , G e ) . It is also possible to compute yearly premiums by introducing payment of yearly premiums π BS in the policy function a ∗ , i.e. a ∗ ( t ) = − π BS t if t ∈ [0 , T ) and a ∗ ( t ) = − π BS T + E T if t ≥ T , thenthe value of the insurance at any given time t ≥ with yearly premiums, denoted by V π ∗ , becomes − π BS (cid:90) Tt e − r ( s − t ) p ∗∗ ( x + t, x + s ) ds + BSE ( t, T, S t , G e ) . We choose the premiums in accordance with the equivalence principle, i.e. such that the valuetoday is , π BS = BSE (0 , T, S , G e ) (cid:82) T e − rs p ∗∗ ( x, x + s ) ds . Under the Vasicek-Heston model instead, the value of policy at time t ≥ with yearly premiums π V H is V + ∗ ( t, A ) = E Q (cid:34)(cid:90) Tt B t B s p ∗∗ ( x + t, x + s ) da ∗ ( s ) (cid:12)(cid:12)(cid:12) G t (cid:35) = − π V H (cid:90) Tt E Q (cid:20) B t B s (cid:12)(cid:12)(cid:12) G t (cid:21) p ∗∗ ( x + t, x + s ) ds + E Q (cid:20) B t B T E T (cid:12)(cid:12)(cid:12) G t (cid:21) p ∗∗ ( x + t, x + T ) . monetary unit, using theclassical Black-Scholes with constant interest r = 1% and µ = 1 . , σ = 4% and a Vasicek-Hestonmodel with parameters S = 1 , µ = 1 . , ¯ ν = 1% , ν = 4% , κ = 10 − , η = 10 − , θ = r = 1% ,k = 0 . , σ = 2% .A single premium payment π V H corresponds to V + ∗ (0 , A ) , i.e. π V H = E Q (cid:20) E T B T (cid:21) p ∗∗ ( x, x + T ) and the yearly ones correspond to π V H = V + ∗ (0 , A ) (cid:82) T E Q (cid:104) B s (cid:105) p ∗∗ ( x, x + s ) ds . In Figure 6 we compare the single premiums using the classical Black-Scholes unit-linked modelin contrast to the Vasicek-Heston model proposed for different maturities T with parameters S = 1 , G e = 1 , r = 1% and µ = 1 . , σ = 4% for the Black-Scholes model, and S = 1 , , G e = 1 , µ = 1 . , ¯ ν = 1% , ν = 4% , κ = 10 − , η = 10 − , θ = r = 1% , k = 0 . , σ = 2% for the Vasicek-Hestonmodel. Consider now an endowment for a life aged x with maturity T > that pays, in addition, adeath benefit in case the insured dies within the period of the contract. That is the policy pays theamount E T := max { G e , S T } if the insured survives by time T as before and, in addition, a deathbenefit of D t := max { G d , S t } if t ∈ [0 , T ) . This policy is entirely determined by the two policyfunctions a ∗ ( t ) = (cid:40) E T if t ≥ T else , a ∗† ( t ) = (cid:40) D t if t ∈ [0 , T )0 else . In view of (6) and the above functions, the value of this insurance at time t given that theinsured is still alive is then given by V + ∗ ( t, A ) = E Q (cid:20) B t B T E T (cid:12)(cid:12)(cid:12) G t (cid:21) p ∗∗ ( x + t, x + T ) + (cid:90) Tt E Q (cid:20) B t B s D s (cid:12)(cid:12)(cid:12) G t (cid:21) p ∗∗ ( x + t, x + s ) µ ∗† ( x + s ) ds. (36)19ollowing similar arguments as in the case of a pure endowment, by adding the function a ∗† inthe computations, we obtain that the single premiums π BS and π V H for the Black-Scholes modeland Vasicek-Heston model, respectively, are given by. π BS = BSE (0 , T, S , G e ) + (cid:90) T e − rs BSE (0 , s, S , G d ) p ∗∗ ( x, x + s ) µ ∗† ( x + s ) ds, where the function BSE is given in (35), and π V H = E Q (cid:20) E T B T (cid:21) p ∗∗ ( x, x + T ) + (cid:90) T E Q (cid:20) D s B s (cid:21) p ∗∗ ( x, x + s ) µ ∗† ( x + s ) ds.. In Figure 7 we compare the single premiums using the classical Black-Scholes unit-linked modelin contrast to the Vasicek-Heston model proposed for different maturities T with parameters S = 1 , G e = G d = 1 , r = 1% and µ = 1 . , σ = 4% for the Black-Scholes model, and S = 1 , , G e = 1 , µ = 1 . , ¯ ν = 1% , ν = 4% , κ = 10 − , η = 10 − , θ = r = 1% , k = 0 . , σ = 2% for theVasicek-Heston model.Figure 7: Single premiums for an endowment with benefits equal to monetary unit, using theclassical Black-Scholes with constant interest r = 1% and µ = 1 . , σ = 4% and a Vasicek-Hestonmodel with parameters S = 1 , µ = 1 . , ¯ ν = 1% , ν = 4% , κ = 10 − , η = 10 − , θ = r = 1% ,k = 0 . , σ = 2% . References [1] K. K. Aase and S.-A. Persson. Pricing of unit-linked life insurance policies. ScandinavianActuarial Journal , 1994(1):26–52, 1994.[2] S.M. Ould Aly. Forward variance dynamics: Bergomi’s model revisited. Appl. Math. Finance ,21(1):84–107, 2014.[3] L. Andersen. 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