Explicit solution simulation method for the 3/2 model
EEXPLICIT SOLUTION SIMULATION METHODFOR THE 3/2 MODEL
IRO REN´E KOUARFATE, MICHAEL A. KOURITZIN, AND ANNE MACKAY
Abstract.
An explicit weak solution for the 3/2 stochastic volatility model isobtained and used to develop a simulation algorithm for option pricing purposes.The 3/2 model is a non-affine stochastic volatility model whose variance process isthe inverse of a CIR process. This property is exploited here to obtain an explicitweak solution, similarly to Kouritzin (2018). A simulation algorithm based onthis solution is proposed and tested via numerical examples. The performance ofthe resulting pricing algorithm is comparable to that of other popular simulationalgorithms. Introduction
Recent work by Kouritzin (2018) shows that it is possible to obtain an explicitweak solution for the Heston model, and that this solution can be used to simulateasset prices efficiently. Exploiting the form of the weak solution, which naturallyleads to importance sampling, Kouritzin and MacKay (2020) suggest the use ofsequential sampling algorithms to reduce the variance of the estimator, inspired bythe particle filtering literature. Herein, we show that the main results of Kouritzin(2018) can easily be adapted to the 3/2 stochastic volatility model and thus be usedto develop an efficient simulation algorithm that can be used to price exotic options.The 3/2 model is a non-affine stochastic volatility model whose analytical tractabil-ity was studied in Heston (1997) and Lewis (2000). A similar process was usedin Ahn and Gao (1999) to model stochastic interest rates. Non-affine stochasticvolatility models have been shown to provide a good fit to empirical market data,sometimes better than some affine volatility models; see Bakshi et al. (2006) andthe references provided in the literature review section of Zheng and Zeng (2016).The 3/2 model in particular is preferred by Carr and Sun (2007) as it naturallyemerges from consistency requirements in their proposed framework, which modelsthe variance swap rate directly.As a result of the empirical evidence in its favor, and because of its analyticaltractability, the 3/2 model has gained traction in the academic literature over thepast decade. In particular, Itkin and Carr (2010) prices volatility swaps and optionon swaps for a class of Levy models with stochastic time change and uses the 3/2
Key words and phrases. a r X i v : . [ q -f i n . C P ] S e p I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY model as a particular case. The 3/2 model also allows for analytical expressionsfor price different volatility derivatives; see for example Drimus (2012), Goard andMazur (2013) and Yuen et al. (2015). Chan and Platen (2015) considers the 3/2model for pricing long-dated variance swaps under the real world measure. Zhengand Zeng (2016) obtains a closed-form partial transform of a relevant density anduses it to price variance swaps and timer options. In Grasselli (2017), the 3/2 modelis combined with the Heston model to create the new 4/2 model.For the 3/2 model’s growing popularity, there are very few papers that focus on itssimulation. One of them is Baldeaux (2012), who adapts the method of Broadie andKaya (2006) to the 3/2 model and suggests variance reduction techniques. The ca-pacity to simulate price and volatility paths from a given market model is necessaryin many situations, from pricing exotic derivatives to developing hedging strategiesand assessing risk. The relatively small size of the literature about simulating the3/2 model could be due to its similarity with the Heston model, which allows foreasy transfer of the methods developed for the Heston model to the 3/2 one. In-deed, the 3/2 model is closely linked to the Heston model; the stochastic processgoverning the variance of the asset price in the 3/2 is the inverse of a square-rootprocess, that is, the inverse of the variance process under Heston.This link between the Heston and the 3/2 model motivates the present work;Kouritzin (2018) mentions that his method cannot survive the spot volatility reach-ing 0. Since the volatility in the 3/2 model is given by the inverse of a “Hestonvolatility” (that is, the inverse of a square-root process), it is necessary to restrictthe volatility parameters in such a way that the Feller condition is met, in orderto keep the spot volatility from exploding. In other words, by definition of the3/2 model, the variance process satisfies the Feller condition, which makes it per-fectly suitable to the application of the explicit weak solution simulation methodsof Kouritzin (2018).It is also worthwhile to note that Kouritzin and MacKay (2020) notice that theresulting simulation algorithm performs better when the Heston parameters keepthe variance process further from 0. It is reasonable to expect that calibrating the3/2 model to market data give such parameters, since they would keep the varianceprocess (i.e. the inverse of the Heston variance) from reaching very high values.This insight further motivates our work, in which we adapt the method of Kouritzin(2018) to the 3/2 model.As stated above, many simulation methods for the Heston model can readily beapplied to the 3/2 model. Most of these methods can be divided into two categories;the first type of simulation schemes relies on discretizing the spot variance and thelog-price process. Such methods are typically fast, but the discretization induces abias which needs to be addressed, see Lord et al. (2010) for a good overview. Broadieand Kaya (2006) proposed an exact simulation scheme which relies on transitiondensity of the variance process and an inversion of the Fourier transform of theintegrated variance. While exact, this method is slow, and has thus prompted severalauthors to propose approximations and modifications to the original algorithm to
XPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 3 speed it up (see for example Andersen (2007)). B´egin et al. (2015) offers a goodreview of many existing simulation methods for the Heston model.The simulation scheme proposed by Kouritzin (2018) for the Heston model relieson an explicit weak solution for the stochastic differential equation (SDE) describingthe Heston model. This result leads to a simulation and option pricing algorithmwhich is akin to importance sampling. Each path is simulated using an artificialprobability measure, called the reference measure, under which exact simulation ispossible and fast. The importance sampling price estimator is calculated under thepricing measure by multiplying the appropriate payoff (a function of the simulatedasset price and volatility paths) by a likelihood, which weights each payoff pro-portionally to the likelihood that the associated path generated from the referencemeasure could have come from the pricing measure. The likelihood used as a weightin the importance sampling estimator is a deterministic function of the simulatedvariance process, and is thus easy to compute. The resulting pricing algorithm hasbeen shown to be fast and to avoid the problems resulting from discretization of thevariance process.In this paper, we develop a similar method for the 3/2 model by first obtaining aweak explicit solution for the two-dimensional SDE. We use this solution to developan option price importance estimator, as well as a simulation and option pricingalgorithm. Our numerical experiments show that our new algorithm performs atleast as well as other popular algorithms from the literature. We find that theparametrization of the model impacts the performance of the algorithm.The paper is organized as follows. Section 2 contains a detailed presentation ofthe 3/2 model as well as our main result. Our pricing algorithm is introduced inSection 3, in which we also outline existing simulation techniques, which we use inour numerical experiments. The results of these experiments are given in Section 4,and Section 5 concludes. 2.
Setting and main results
We consider a probability space (Ω , F , P ), where P denotes a pre-determined risk-neutral measure for the 3/2 model. The dynamics of the stock price under thischosen risk-neutral measure are represented by a two-dimensional process ( S, V ) = { ( S t , V t ) , t ≥ } satisfying (cid:40) d S t = rS t d t + √ V t S t ρ d W (1) t + √ V t S t (cid:112) − ρ d W (2) t d V t = κ V t ( θ − V t ) d t + εV / t d W (1) t , (2.1)with S = s > V = v >
0, and where W = { ( W (1) t , W (2) t ) , t ≥ } is a two-dimensional uncorrelated Brownian motion, r , κ , θ and ε are constants satisfying κ > − ε , and ρ ∈ [ − , r represents the risk-free rate and ρ represents the correlation between the stock price S and its volatility V . Since our goal in this work is to develop pricing algorithms, we only consider the risk-neutralmeasure used for pricing.
I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
The restriction κ > − ε imposed on the parameters keeps the variance processfrom exploding. This property becomes clear when studying the process U = { U t , t ≥ } defined by U t = V t for t ≥
0. Indeed, it follows from Itˆo’s lemma thatd U t = κθ (cid:18) κ + ε κθ U t (cid:19) d t − ε (cid:112) U t d W (1) t = ˜ κ (˜ θ − U t ) d t + ˜ ε (cid:112) U t d W (1) t where ˜ κ = κθ , ˜ θ = κ + ε κθ and ˜ ε = − ε . In other words, with the restriction κ > − ε , U is a square-root process satisfying the Feller condition ˜ κ ˜ θ > ˜ ε , so that P ( U t >
0) = 1for all t ≥ U , as follows (cid:40) d S t = rS t d t + (cid:112) U − t S t ρ d W (1) t + (cid:112) U − t S t (cid:112) − ρ d W (2) t d U t = ˜ κ (˜ θ − U t ) d t + ˜ ε √ U t d W (1) t , (2.2)with S = s and U = 1 /v .Although U is a square-root process, (2.2) is of course not equivalent to the Hestonmodel. Indeed, in the Heston model, it is the diffusion term of S , rather than itsinverse, that follows a square-root process. However, the ideas of Kouritzin (2018)can be exploited to obtain an explicit weak solution to (2.2), which will in turn beused to simulate the process.It is well-known (see for example Hanson (2010)) that if n := κ ˜ θ ˜ ε is an integer,the square-root process U is equal in distribution to the sum of n squared Ornstein-Uhlenbeck processes. Proposition 1 below relies on this result. Proposition 1.
Suppose that n = κ ˜ θ ˜ ε ∈ N + and let W (2) , Z (1) , . . . , Z ( n ) be indepen-dent standard Brownian motions on some probability space (Ω , F , P ) . For t ≥ ,define S t = s exp (cid:26) ρ ˜ ε log (cid:18) U t U (cid:19) + (cid:18) r + ρ ˜ κ ˜ ε (cid:19) t − (cid:18) ρ ˜ ε (cid:16) ˜ κ ˜ θ − ˜ ε / (cid:17) + 12 (cid:19) (cid:90) t U − s d s + (cid:112) − ρ (cid:90) t (cid:112) U − s d W (2) s (cid:27) ,U t = n (cid:88) i =1 (cid:16) Y ( i ) t (cid:17) where Y ( i ) t = ˜ ε (cid:90) t e − ˜ κ ( t − u ) d Z ( i ) u + e − ˜ κ t Y ( i )0 , with Y = (cid:112) U /n XPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 5 and W (1) t = n (cid:88) i =1 (cid:90) t Y ( i ) u (cid:113)(cid:80) nj =1 ( Y ( j ) u ) d Z ( i ) u . Let X = ( S, U ) , W = ( W (1) , W (2) ) and let {F t } t ≥ be the augmented filtrationgenerated by ( W (2) , Z (1) , . . . , Z ( n ) ) . Then, • W (1) is a standard Brownian motion, and • ( X, W ) , (Ω , F , P ) , {F t } t ≥ is a weak solution to (2.2) .Proof. We first observe that Y ( i ) , i ∈ { , . . . , n } , are independent Ornstein-Uhlenbeckprocesses, and that by L´evy’s characterization, W (1) is a Brownian motion. It followsfrom an application of Itˆo’s lemma that dU t = n (cid:88) i =1 (cid:18) ˜ ε − ˜ κ ( Y ( i ) t ) (cid:19) d t + ˜ εY ( i ) t d Z ( i ) t = (cid:32) n ˜ ε − ˜ κ n (cid:88) i =1 ( Y ( i ) t ) (cid:33) d t + ˜ ε n (cid:88) i =1 Y ( i ) t d Z ( i ) t = (cid:18) n ˜ ε − ˜ κU t (cid:19) d t + ˜ ε (cid:112) U t d W (1) t , (2.3)where the equality is obtained by multiplying and dividing the second term on theright-hand side by (cid:113)(cid:80) nj =1 ( Y ( i ) t ) . Here, since we work under the assumption that n = κ ˜ θ ˜ (cid:15) , (2.3) can be re-written as dU t = ˜ κ (cid:16) ˜ θ − U t (cid:17) d t + ˜ ε (cid:112) U t d W (1) t . An application of Itˆo’s lemma to S t completes the proof. (cid:3) An alternative, systematic way to verify the functional form for S t that avoidsour Itˆo-lemma-based guess and verify technique can be found in Kouritzin (2018).It is likely that for a given market calibration of the 3/2 model, n = κ ˜ θ ˜ (cid:15) is notan integer. For this reason, a more general result is needed to develop a simulationalgorithm based on an explicit weak solution.We generalize the definition of n and let n = max (cid:16) (cid:98) κ ˜ θ ˜ ε + (cid:99) , (cid:17) . We furtherdefine ˜ θ n by ˜ θ n = n ˜ ε κ . It follows that ˜ κ ˜ θ n = n ˜ ε .While U above cannot hit 0 under the Feller condition, it can get arbitrarilyclose, causing U − t to blow up. To go beyond the case κ ˜ θ ˜ (cid:15) ∈ N , we want to changemeasures, which is facilitated by stopping U from approaching zero. I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY
Given a filtered probability space (Ω , F , {F t } t ≥ , (cid:98) P ) with independent Brownianmotions Z (1) , . . . , Z ( n ) and W (2) , and a fixed δ >
0, we can define ( (cid:98) S, (cid:98) U ) = { ( (cid:98) S t , (cid:98) U t ) } t ≥ by (cid:98) S t = s exp (cid:26) ρ ˜ ε log( (cid:98) U t /U ) + (cid:18) r + ρ ˜ κ ˜ ε (cid:19) t − (cid:18) ρ ˜ ε (cid:16) ˜ κ ˜ θ − ˜ ε / (cid:17) + 12 (cid:19) (cid:90) t (cid:98) U − s d s + (cid:112) − ρ (cid:90) t (cid:98) U − / s d W (2) s (cid:27) , (2.4) (cid:98) U t = n (cid:88) i =1 ( Y ( i ) t ) , (2.5)and τ δ = inf { t ≥ (cid:98) U t ≤ δ } , where Y ( i ) t = ˜ ε (cid:90) t ∧ τ δ e − ˜ κ ( t − u ) d Z ( i ) u + e − ˜ κ t Y ( i )0 , with Y = (cid:112) U /n (2.6)for i ∈ { , . . . , n } .Theorem 1, to follow immediately, shows that it is possible to construct a prob-ability measure on (Ω , F ) under which ( (cid:98) S, (cid:98) U ) satisfies (2.2) until (cid:98) U drops below apre-determined threshold δ . Theorem 1.
Let (Ω , F , {F t } t ≥ , (cid:98) P ) be a filtered probability space on which Z (1) , . . . , Z ( n ) , W (2) are independent Brownian motions. Let ( (cid:98) S, (cid:98) U ) be defined as in (2.4) and (2.5) andlet τ δ = inf { t ≥ (cid:98) U t ≤ δ } for some δ ∈ (0 , . Define (cid:98) L ( δ ) t = exp − ˜ κ (˜ θ n − ˜ θ )˜ ε (cid:90) t (cid:98) U − / v d (cid:99) W (1) v − ˜ κ (cid:32) ˜ θ n − ˜ θ ˜ ε (cid:33) (cid:90) t (cid:98) U − v d v (2.7) with (cid:99) W (1) t = n (cid:88) i =1 (cid:90) t Y ( i ) u (cid:113)(cid:80) nj =1 ( Y ( j ) u ) d Z ( i ) u (2.8) and P δ ( A ) = (cid:98) E [1 A (cid:98) L ( δ ) T ] ∀ A ∈ F T for T > .Then, under the probability measure P δ , ( W (1) , W (2) ) , where W (1) t = (cid:99) W (1) t + ˜ κ ˜ θ − ˜ θ n ˜ ε (cid:90) t ∧ τ δ (cid:98) U − / s d s, XPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 7 are independent Brownian motions and ( (cid:98) S, (cid:98) U ) satisfies d (cid:98) S t = (cid:40) r (cid:98) S t d t + (cid:98) U − / t (cid:98) S t ρ d W (1) t + (cid:98) U − / t (cid:98) S t (cid:112) − ρ d W (2) t , t ≤ τ δ r δ (cid:98) S t d t + σ δ (cid:98) S t d W (2) t , t > τ δ , d (cid:98) U t = (cid:40) ˜ κ (˜ θ − (cid:98) U t ) d t + ˜ ε (cid:98) U / t d W (1) t , t ≤ τ δ , t > τ δ (2.9) on [0 , T ] , with r δ = r + ρ εδ (cid:0) κδ − κδ + ˜ ε − ρ ˜ ε (cid:1) ,σ δ = (cid:114) − ρ δ . Proof.
Let D = S ( R ), the rapidly decreasing functions. They separate pointsand are closed under multiplication so they separate Borel probability measures(see Blount and Kouritzin (2010)) and hence are a reasonable martingale problemdomain.To show that ( (cid:98) X, W ), (Ω , F , P δ ), { (cid:98) F t } t ≥ , with (cid:98) X = ( (cid:98) S, (cid:98) U ) and W = ( W (1) , W (2) ),is a solution to (2.9), we show that it solves the martingale problem associated withthe linear operator A t f ( s, u ) = (cid:18) rsf s ( s, u ) + ˜ κ (˜ θ − u ) f u ( s, u ) + 12 s u − f ss ( s, u ) + ρ ˜ εsf su ( s, u )+ 12 ε uf uu ( s, u ) (cid:19) [0 ,τ δ ] ( t ) + (cid:18) r δ sf s ( s, u ) + 1 − ρ δ f ss s (cid:19) [ τ δ ,T ] ( t )where f s = ∂f ( s,u ) ∂s , f u = ∂f ( s,u ) ∂u , f ss = ∂ f ( s,u ) ∂s , f uu = ∂ f ( s,u ) ∂u and f su = ∂ f ( s,u ) ∂s∂u . Thatis, we show that for any function f ∈ D , the process M t ( f ) = f ( (cid:98) S t , (cid:98) U t ) − f ( (cid:98) S , (cid:98) U ) − (cid:90) t ( A s f )( (cid:98) S v , (cid:98) U v ) d v, is a continuous, local martingale.First, we note by (2.4), (2.5), (2.6) as well as Itˆo’s lemma that ( (cid:98) S, (cid:98) U ) satisfies atwo-dimensional SDE similar to the 3/2 model (2.2), but with parameters κ , θ n , r δ and (cid:98) r t = r − ˜ κρ ˜ ε (˜ θ − ˜ θ n ) (cid:98) U − t . That is, (( (cid:98) S, (cid:98) U ) , (cid:99) W ), (Ω , F , (cid:98) P ), { (cid:98) F t } t ≥ , where { (cid:98) F t } t ≥ is the augmented filtration generated by ( Z , . . . , Z n , W (2) ), is a solution tod (cid:98) S t = (cid:40)(cid:98) r t (cid:98) S t d t + (cid:98) U − / t (cid:98) S t ρ d (cid:99) W (1) t + (cid:98) U − / t (cid:98) S t (cid:112) − ρ d W (2) t , t ≤ τ δ ,r δ (cid:98) S t d t + σ δ (cid:98) S t d W (2) t , t > τ δ , (2.10)d (cid:98) U t = (cid:40) ˜ κ (˜ θ n − (cid:98) U t ) d t + ˜ ε (cid:98) U / t d (cid:99) W (1) t , t ≤ τ δ , , t > τ δ . (2.11) I.R. KOUARFATE, M.A. KOURITZIN, AND A. MACKAY with (cid:98) S = s , (cid:98) U = 1 /v and (cid:99) W (1) defined by (2.8). It follows that for any function f ∈ D , df ( (cid:98) S t , (cid:98) U t ) = L t f ( (cid:98) S t , (cid:98) U t ) dt + (cid:16) ρ (cid:98) S t (cid:98) U − / t f s ( (cid:98) S t , (cid:98) U t ) + ˜ ε (cid:98) U / t f u ( (cid:98) S t , (cid:98) U t ) (cid:17) [0 ,τ δ ] ( t ) d (cid:99) W (1) t + (cid:16) (cid:98) U − / t [0 ,τ δ ] ( t ) + δ − / [ τ δ ,T ] (cid:17) (cid:112) − ρ (cid:98) S t f s ( (cid:98) S t , (cid:98) U t ) d W (2) t , (2.12)where the linear operator L is defined by L t f ( s, u ) = (cid:18)(cid:98) r t sf s ( s, u ) + ˜ κ (˜ θ n − u ) f u ( s, u ) + 12 s u − f ss ( s, u ) + ρ ˜ εsf su ( s, u )+ 12 ε uf uu ( s, u ) (cid:19) [0 ,τ δ ] ( t ) + (cid:18) r δ sf s ( s, u ) + 1 − ρ δ f ss s (cid:19) [ τ δ ,T ] ( t )(2.13)We observe that (cid:98) L ( δ ) t satisfies the Novikov condition, since by definition of (cid:98) U t , | ˜ κ (˜ θ n − ˜ θ ) | ˜ ε (cid:98) U t ≤ | ˜ κ (˜ θ n − ˜ θ ) | ˜ ε δ , (cid:98) P -a.s. for all t ≥
0. It follows that (cid:98) L ( δ ) t is a martingale and that P δ is a probabilitymeasure.We also have from (2.7) and (2.12) that for f ( s, u ) ∈ C ([0 , ∞ ] ), (cid:104)(cid:98) L ( δ ) , f ( (cid:98) S, (cid:98) U ) (cid:105) t = (cid:90) t ∧ τ δ (cid:98) L ( δ ) v (cid:16) ( r − (cid:98) r v ) (cid:98) S v f s ( (cid:98) S v , (cid:98) U v ) + ˜ κ (˜ θ − ˜ θ n ) f u ( (cid:98) S v , (cid:98) U v ) (cid:17) d v. (2.14)Next we define the process (cid:99) M ( f ) for any f ∈ D by (cid:99) M t ( f ) = (cid:98) L ( δ ) t f ( (cid:98) S t , (cid:98) U t ) − (cid:98) L ( δ )0 f ( (cid:98) S , (cid:98) U ) − (cid:90) t (cid:98) L ( δ ) v A v f ( (cid:98) S v , (cid:98) U v ) d v = (cid:98) L ( δ ) t f ( (cid:98) S t , (cid:98) U t ) − (cid:98) L ( δ )0 f ( (cid:98) S , (cid:98) U ) − (cid:104)(cid:98) L ( δ ) , f ( (cid:98) S, (cid:98) U ) (cid:105) t − (cid:90) t (cid:98) L ( δ ) v L ( δ ) v f ( (cid:98) S v , (cid:98) U v ) d v (2.15)Using integration by parts, we obtain (cid:99) M t ( f ) = (cid:90) t (cid:98) L ( δ ) v d f ( (cid:98) S v , (cid:98) U v ) d v + (cid:90) t f ( (cid:98) S v , (cid:98) U v ) d (cid:98) L ( δ ) v − (cid:90) t L ( δ ) v L ( δ ) v f ( (cid:98) S v , (cid:98) U v ) d v = (cid:90) t (cid:98) L ( δ ) v (cid:104) ˜ κ (˜ θ − ˜ θ n ) (cid:98) U − / v f ( (cid:98) S v , (cid:98) U v ) + (cid:16) ρ (cid:98) S v (cid:98) U − / v f s ( (cid:98) S v , (cid:98) U v )+˜ ε (cid:98) U / v f u ( (cid:98) S v , (cid:98) U v ) (cid:17) [0 ,τ δ ] ( v ) (cid:105) d (cid:99) W (1) v + (cid:90) t (cid:98) L ( δ ) v (cid:16) (cid:98) U − / v [0 ,τ δ ] ( v ) + δ − / [ τ δ ,T ] ( v ) (cid:17) (cid:112) − ρ (cid:98) S v f s ( (cid:98) S v , (cid:98) U v ) d W (2) v XPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 9 so (cid:99) M t ( f ) is a local martingale. However, since f is rapidly decreasing, sf s ( s, u ), uf u ( s, u ), sf su ( s, u ) and uf uu ( s, u ) are all bounded. We also have that (cid:98) U v ≥ δ and (cid:98) L ( δ ) v is integrable for all v . Hence, it follows by (2.13), (2.14), (2.15) and Tonelli that (cid:99) M ( f ) is a martingale.To finish the proof, it suffices to follow the remark on p.174 of Ethier and Kurtzand show that E (cid:34)(cid:18) f ( (cid:98) S t n +1 , (cid:98) U t n +1 ) − f ( (cid:98) S t n , (cid:98) U t n ) − (cid:90) t n +1 t n A v f ( (cid:98) S v , (cid:98) U v ) d v (cid:19) n (cid:89) k =1 h k ( (cid:98) S t k , (cid:98) U t k ) (cid:35) = 0 , (2.16)for 0 ≤ t < t < . . . < t n +1 , f ∈ D , h ∈ B ( R ) (the bounded, measurable functions)and where (cid:98) E [ · ] denotes the (cid:98) P -expectation. To do so, we re-write the left-hand sideof (2.16) as (cid:98) E (cid:34)(cid:98) L ( δ ) t n +1 (cid:18) f ( (cid:98) S t n +1 , (cid:98) U t n +1 ) − f ( (cid:98) S t n , (cid:98) U t n ) − (cid:90) t n +1 t n A v f ( (cid:98) S v , (cid:98) U v ) d v (cid:19) n (cid:89) k =1 h k ( (cid:98) S t k , (cid:98) U t k ) (cid:35) = (cid:98) E (cid:34)(cid:18)(cid:98) L ( δ ) t n +1 f ( (cid:98) S t n +1 , (cid:98) U t n +1 ) − (cid:98) L ( δ ) t n f ( (cid:98) S t n , (cid:98) U t n ) − (cid:90) t n +1 t n (cid:98) L ( δ ) v A v f ( (cid:98) S v , (cid:98) U v ) d v (cid:19) n (cid:89) k =1 h k ( (cid:98) S t k , (cid:98) U t k ) (cid:35) = (cid:98) E (cid:34)(cid:16) (cid:99) M t n +1 ( f ) − (cid:99) M t n ( f ) (cid:17) n (cid:89) k =1 h k ( (cid:98) S t k , (cid:98) U t k ) (cid:35) , which is equal to 0 since (cid:99) M ( f ) is a martingale. We can then conclude that ( (cid:98) S, (cid:98) U )solves the martingale problem for A with respect to (cid:98) P . (cid:3) Remark 1.
In Theorem 1, we indicate the dependence of the process (cid:98) L ( δ ) on thethreshold δ via the superscript. Indeed, (cid:98) L ( δ ) depends on δ through (cid:98) U . Going forward,for notational convenience, we drop the superscript, keeping in mind the dependenceof the likelihood process on δ . Pricing algorithm
In this section, we show how Theorem 1 can be exploited to price a financialoption in the 3/2 model. First, we justify that ( (cid:98) S, (cid:98) U ) defined in (2.4) and (2.5) canbe used to price an option in the 3/2 model, even if they satisfy (2.2) up to τ δ . Wealso present an algorithm to simulate paths of ( (cid:98) S, (cid:98) U ) under the 3/2 model as wellas the associated importance sampling estimator for the price of the option.3.1. Importance sampling estimator of the option price.
For the rest of thispaper, we consider an option with maturity T ∈ R whose payoff can depend onthe whole path of { ( S t , V t ) } t ∈ [0 ,T ] , or equivalently, { ( S t , U t ) } t ∈ [0 ,T ] . Indeed, since V t = U − t for all 0 ≤ t ≤ T and to simplify exposition, we will keep on workingin terms of the inverse of the variance process U going forward. We consider a payoff function φ T ( S, U ) with E [ | φ T ( S, U ) | ] < ∞ . We call π = E [ φ T ( S, U )] the price of the option and the function φ , its discounted payoff. For example, a calloption, which pays out the difference between the stock price at maturity, S T , and apre-determined exercise price K if this difference is positive, has discounted payofffunction e − rT max( S T − K,
0) and price E [ e − rT max( S T − K, Remark 2.
We work on a finite time horizon and the option payoff function φ T only depends on ( S, U ) up to T . We use the index T to indicate this restriction on ( S, U ) . The next proposition shows that it is possible to use ( (cid:98) S, (cid:98) U ), rather than ( S, U ),to price an option in the 3/2 model.
Proposition 2.
Suppose ( S, U ) is a solution to the 3/2 model (2.2) on probaiblityspace (Ω , F , P ) and τ δ = inf { t ≥ U t ≤ δ } . Define ( (cid:98) S, (cid:98) U ) , by (2.4) and (2.5) ,set (cid:98) τ δ = inf { t ≥ (cid:98) U t ≤ δ } for δ ∈ (0 , and let φ T ( S, U ) be a payoff functionsatisfying E [ | φ T ( S, U ) | ] < ∞ . Then, lim n →∞ E /n [ φ T ( (cid:98) S, (cid:98) U ) { τ /n >T } ] = E [ φ T ( S, U )] , where E δ [ · ] denotes the expectation under the measure P δ defined in Theorem 1.Proof. By Theorem 1, ( (cid:98) S, (cid:98) U ) satisfies (2.2) on [0 , τ /n ] under the measure P /n . Itfollows that E /n [ φ T ( (cid:98) S, (cid:98) U ) { (cid:98) τ /n >T } ] = E [ φ T ( S, U ) { τ /n >T } ] . Because U satisfies the Feller condition, lim n →∞ τ /n ≤ T = 0 , P -a.s. andlim n →∞ E /n [ φ T ( (cid:98) S, (cid:98) U ) { (cid:98) τ /n >T } ] = lim n →∞ E [ φ T ( S, U ) { τ /n >T } ] = E [ φ T ( S, U )]by the dominated convergence theorem. (cid:3)
We interpret Proposition 2 in the following manner: by choosing δ small enough,it is possible to approximate π by π ( δ )0 := E δ [ φ T ( (cid:98) S, (cid:98) U ) { τ δ >T } ], that is, using ( (cid:98) S, (cid:98) U )rather than ( S, U ). The advantage of estimating the price of an option via ( (cid:98) S, (cid:98) U )is that the trajectories can easily be simulated exactly under the reference mea-sure (cid:98) P defined in Theorem 1. In practice, we will show in Section 4 that for rea-sonable 3/2 model calibrations, it is usually possible to find δ small enough that E δ [ φ T ( (cid:98) S, (cid:98) U ) { τ δ >T } ] is almost undistinguishable from π .In the rest of this section, we explain how π ( δ )0 can be approximated with MonteCarlo simulation. As mentioned above, paths of ( (cid:98) S, (cid:98) U ) are easily simulated underthe reference measure (cid:98) P , not under P δ . It is therefore necessary to express π ( δ )0 usingTheorem 1 in the following manner π ( δ )0 = E δ [ φ T ( (cid:98) S, (cid:98) U ) { τ δ >T } ] = (cid:98) E [ (cid:98) L T φ T ( (cid:98) S, (cid:98) U ) { τ δ >T } ] . (3.1) XPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 11
From (3.1) and the strong law of large numbers, we can define (cid:98) π ( δ )0 , an importanceestimator for π ( δ )0 , by (cid:98) π ( δ )0 = (cid:80) Nj =1 φ T ( (cid:98) S ( j ) , (cid:98) U ( j ) ) (cid:98) L ( j ) T { τ ( j ) δ >T } (cid:80) Nj =1 (cid:98) L ( j ) T { τ ( j ) δ >T } , (3.2)where (cid:110) (cid:98) S ( j ) , (cid:98) U ( j ) , (cid:98) L ( j ) (cid:111) Nj =1 are N ∈ N simulated paths of ( (cid:98) S, (cid:98) V , (cid:98) L ).3.2. Simulating sample paths.
In light of Proposition 2, we now focus on thesimulation of ( (cid:98) S t , (cid:98) U t , (cid:98) L t ) t ≤ τ δ . Using (2.4) and (2.6), (cid:98) S and Y can easily be discretizedfor simulation purposes. To simplify the simulation of the process (cid:98) L , we write (2.7)as a deterministic function of (cid:98) U in Proposition 3 below. Proposition 3.
Let (cid:98) L t be defined as in Theorem 1, with (cid:98) U defined by (2.5) . Then,for t ≤ τ δ , (cid:98) L t can be written as (cid:98) L t = exp (cid:40) − (˜ κ ˜ θ n − ˜ κ ˜ θ )˜ ε (cid:34) log( (cid:98) U t / (cid:98) U ) + ˜ κt + ˜ κ ˜ θ − κ ˜ θ n + ˜ ε (cid:90) t (cid:98) U − s d s (cid:35)(cid:41) . (3.3) Proof.
An application of Itˆo’s lemma to log (cid:98) U t for t ≤ τ δ yieldslog( (cid:98) U t / (cid:98) U ) = (˜ κ ˜ θ n − ˜ ε / (cid:90) t (cid:98) U − s d s − ˜ κt + ˜ ε (cid:90) t (cid:98) U − / s d (cid:99) W (1) s . (3.4)Isolating (cid:82) t (cid:98) U − / s d (cid:99) W (1) s in (3.4) and replacing the resulting expression in (2.7) givesthe result. (cid:3) For t ∈ [0 , T ) and h ∈ (0 , T − t ), for simulation purposes, we can re-write (2.4),(2.6) and (3.3) in a recursive manner as (cid:98) S t + h = (cid:98) S t exp (cid:26) ρ ˜ ε log( (cid:98) U t + h / (cid:98) U t ) + ah − b (cid:90) t + ht (cid:98) U − s d s + (cid:112) − ρ (cid:90) t + ht (cid:98) U − / s d W (2) s , (cid:27) (3.5) Y ( i ) t + h = Y ( i ) t e − ˜ κ h + ˜ ε (cid:90) ( t + h ) ∧ τ δ t e − ˜ κ ( t + h − u ) d Z u , for i = 1 , . . . , n, (3.6)and (cid:98) L ( t + h ) ∧ τ δ = L t exp (cid:110) c (cid:16) log( (cid:98) U ( t + h ) ∧ τ δ / (cid:98) U t ) + ˜ κ ( h ∨ ( τ δ − t ))+ d (cid:90) ( t + h ) ∧ τ δ t (cid:98) U − s d s (cid:33)(cid:41) (3.7) where a = r + ρ ˜ κ ˜ ε b = ρ ˜ ε (˜ κ ˜ θ − ˜ ε / c = − ˜ κ ˜ θ n − ˜ κ ˜ θε d = ˜ κ ˜ θ − κ ˜ θ n + ˜ ε . We now discuss the simulation of ( (cid:98) S t + h , (cid:98) U t + h , (cid:98) L t + h ) given ( (cid:98) S t , (cid:98) U t , (cid:98) L t ), as well as { Y ( i ) t } ni =1 . Typically, h will be a small time interval, that is, we consider h (cid:28) T .It is easy to see from the above that given Y ( i ) t , Y ( i ) t + h follows a Normal distributionwith mean Y ( i ) t e − ˜ κ h and variance ˜ ε κ (1 − e − h ˜ κ ). The simulation of Y ( i ) t + h given Y ( i ) t isthus straightforward. (cid:98) U t + h can then be obtained by (2.5) as the sum of the squaresof each Y ( i ) t + h , for i = 1 , . . . , n .Given simulated values (cid:98) U t + h and (cid:98) U t , the term (cid:82) t + ht (cid:98) U − s d s , which appears in both (cid:98) S t + h and (cid:98) L t + h , can be approximated using the trapezoidal rule by letting (cid:90) t + ht (cid:98) U − s d s ≈ ( (cid:98) U t + (cid:98) U t + h )2 h. (3.8)More precise approximations to this integral can be obtained by simulating inter-mediate values (cid:98) U − t + ih for i ∈ (0 ,
1) and using other quadrature rules. In Kouritzin(2018) and Kouritzin and MacKay (2020), Simpson’s rule was preferred. In thissection, we use a trapezoidal rule only to simplify the exposition of the simulationalgorithm.Given that (cid:98) U t + h > δ and once an approximation for the deterministic integral (cid:82) t + ht (cid:98) U − s d s is calculated, (cid:98) L t + h can be simulated using (3.7). To generate a valuefor (cid:98) S t + h , it suffices to observe that conditionally on { (cid:98) U s } s ∈ [ t,t + h ] , (cid:82) t + ht (cid:98) U − / s d W (2) s follows a Normal distribution with mean 0 and variance (cid:82) t + ht (cid:98) U − s d s .The resulting algorithm produces N paths of ( (cid:98) S, (cid:98) U , (cid:98) L ) and the stopping times τ δ associated with each path; it is presented in Algorithm 1, in the appendix. Thesesimulated values are then used in (3.2) to obtain an estimate for the price of anoption. 4. Numerical experiment
Methods and parameters.
In this section, we assess the performance ofthe pricing algorithm derived from Theorem 1. To do so, we use Monte Carlosimulations to estimate the price of European call options. These Monte Carloestimates are compared with the exact price of the option, calculated with theanalytical expression available for vanilla options in the 3/2 model (see for exampleLewis (2000) and Carr and Sun (2007)). More precisely, we consider the payofffunction φ T ( S, U ) = max( S T − K,
0) for
K >
XPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 13 error by
M SE = E [( π − (cid:98) π ( δ )0 ) ]and the relative mean square error by RelM SE = E [( π − (cid:98) π ( δ )0 ) ] π , where π is the exact price of the option and (cid:98) π ( δ )0 is the estimate calculated with(3.2). The expectations above are approximated by calculating the estimates a largenumber of times and taking the mean over all runs.Throughout this section, we consider the five parameter sets presented in Table1. Parameter set 1 (PS1) was used in Baldeaux (2012). Parameter set 2 (PS2)was obtained by Drimus (2012) via the simultaneous fit of the 3/2 model to 3-month and 6-month S&P500 implied volatilities on July 31, 2009. The three otherparameter sets are modifications of PS2: PS3 was chosen so that κ ˜ θ ˜ ε ∈ N , and PS4and PS5 were selected to have a higher n . Recalling that n = max (cid:16) (cid:98) κ ˜ θ ˜ ε + (cid:99) , (cid:17) represents the number of Ornstein-Uhlenbeck processes necessary to simulate thevariance process, we have that n = 204 for PS1, n = 5 for PS2 and PS3 and n = 12for PS4 and PS5.Throughout the numerical experiments, the threshold we use is δ = 10 − . Forall parameter sets, the simulated process U never crossed below this threshold.Therefore, any δ below 10 − would have yielded the same results. Table 1.
Parameter sets S V κ θ ε ρ r κ ˜ θ/ ˜ ε PS1 1 1 2 1 . . − . . PS2 100 0 .
06 22 .
84 0 .
218 8 . − .
99 0 . . PS3 100 0 .
06 18 .
32 0 .
218 8 . − .
99 0 . . PS4 100 0 .
06 19 .
76 0 .
218 3 . − .
99 0 . . PS5 100 0 .
06 20 .
48 0 .
218 3 . − .
99 0 . . Results.
In this section, we present the results of our numerical experiments.We first test the sensitivity of our simulation algorithm to n , the number of Ornstein-Uhlenbeck processes to simulate. We then compare the performance of our algorithmto other popular ones in the literature.4.2.1. Sensitivity to n . We first test the impact of n on the precision of the algorithm.Such an impact was observed in Kouritzin and MacKay (2020) in the context of theHeston model. To verify whether this also holds for the 3/2 model, we consider thefirst three parameter sets and price at-the-money (that is, K = S ) European calloptions. For PS1, we follow Baldeaux (2012) and compute the price of a call optionwith maturity T = 1. The exact price of this option is 0.4431. PS2 and PS3 are used to obtain the price of at-the-money call options with T = 0 .
5, with respectiveexact prices 7.3864 and 7.0422. In all three cases, the length of the time step usedfor simulation is h = 0 . N ∈ { , , } simulations. The integral with respect to time (see step(3) of Algorithm 1) is approximated using M ∈ { , } sub-intervals and Simpson’s rule.The results of Table 2 show that the precision of the simulation algorithm seemto be affected by n . Indeed, as a percentage of the exact price, the MSE of the priceestimator is higher for PS1 than for the other parameter sets. This observationbecomes clearer as N increases.We recall that for PS3, κ ˜ θ ˜ ε is an integer, while this is not the case for PS2. Itfollows that for this latter parameter set, the weights (cid:98) L ( j ) T are all different, while theyare all equal to 1 for PS3. One could expect the estimator using uneven weightsto show a worse performance due to the possible great variance of the weights.However, in this case, both estimators show similar a performance; the algorithmdoes not seem to be affected by the use of uneven weights.Finally, Table 2 shows that increasing M may not significantly improve the pre-cision of the price estimator. Such an observation is important, since adding subin-tervals in the calculation of the time-integral slows down the algorithm. Keepingthe number of subintervals low reduces computational complexity of our algorithm,making it more attractive. Table 2.
Relative MSE as a percentage of π . N PS1 PS2 PS3 M = 2 M = 4 M = 2 M = 4 M = 2 M = 45000 0.271 0.316 0.183 0.225 0.239 0.21410000 0.203 0.158 0.111 0.112 0.172 0.14350000 0.158 0.135 0.085 0.083 0.067 0.0704.2.2. Comparison to other algorithms.
In this section, we compare the performanceof our new simulation algorithm for the 3/2 model to existing ones. The first bench-mark algorithm we consider is based on a Milstein-type discretization of the log-priceand variance process. The second one is based on the quadratic exponential schemeproposed by Andersen (2007) as a modification to the method of Broadie and Kaya(2006), which we adapted to the 3/2 model. These algorithms are outlined in theappendix.To assess the relative performance of the algorithms, we price in-the-money (
K/S =0 . K/S = 1) and out-of-the-money ( K/S = 1 .
05) call optionswith T = 1 year to maturity. The exact prices of the options, which are used tocalculate the MSE of the price estimates, are given in Table 3. We consider all XPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 15 parameter sets with the exception of PS1, since this parametrization requires thesimulation of 204 Ornstein-Uhlenbeck process, which makes our algorithm exces-sively slow.
Table 3.
Exact prices π of European call options. K/S P S P S P S P S κ ˜ θ ˜ ε / ∈ N , while the opposite is true for Figure 2.Overall, the precision of our weighted simulation algorithm is similar to that of theother two algorithms studied. However, certain parameter sets result in more preciseestimates. Figure 1 shows that the MSE is consistently larger with the weightedsimulation algorithms than with the benchmark ones for PS2. However, with PS4,the weighted algorithm performs as well as the other two algorithms, or better. Wenote that for PS2, n = 5 while for PS4, n = 12. It was observed in Kouritzin andMacKay (2020) in the case of the Heston model that as n increases, the weightedsimulation algorithm seems to perform better relatively to other algorithms. Thisobservation also seems to hold in the case of the 3/2 model.For parametrizations that satisfy κ ˜ θ ˜ ε ∈ N , such as in Figure 2, we observe thatthe weighted simulation algorithm is at least as precise, and often more, than theother algorithms. In this case, all the weights (cid:98) L T are even, which tends to decreasethe variance of the price estimator and thus, to decrease the relative MSE. It is alsointeresting to note that in the case of Figure 2, since κ ˜ θ ˜ ε ∈ N , it is not necessary tosimulate τ ( j ) δ and the trajectories (cid:98) L T . Indeed, in this case, it is possible to simplifythe algorithm using Proposition 1, which greatly speeds it up. When possible,parametrization that satisfy this condition should therefore be considered. N r m s e_ i t m (A) PS2,
K/S = 0 . N r m s e_a t m Algorithms
M QE WE WE2 (B)
PS2,
K/S = 1 N r m s e_o t m (C) PS2,
K/S = 1 . N r m s e_ i t m (D) PS4,
K/S = 0 . N r m s e_a t m (E) PS2,
K/S = 1 N r m s e_o t m (F) PS4,
K/S = 1 . Figure 1.
Relative MSE as a function of N , PS2 and PS4, algo-rithms: Milstein (dot), quadratic exponential (triangle), weighted, M = 2 (square), weighted, M = 4 (cross). XPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 17 N r m s e_ i t m (A) PS3,
K/S = 0 . N r m s e_a t m Algorithms
M QE WE WE2 (B)
PS3,
K/S = 1 N r m s e_o t m (C) PS3,
K/S = 1 . N r m s e_ i t m (D) PS5,
K/S = 0 . N r m s e_a t m (E) PS5,
K/S = 1 N r m s e_o t m (F) PS5,
K/S = 1 . Figure 2.
Relative MSE as a function of N , PS3 and PS5, algo-rithms: Milstein (dot), quadratic exponential (triangle), weighted, M = 2 (square), weighted, M = 4 (cross).5. Conclusion
In this paper, we present a weak explicit solution to the 3/2 model, up untilthe inverse of the variance process drops below a given threshold. We develop asimulation algorithm based on this solution and show that it can be used to priceoptions in the 3/2 model, since in practice, the inverse variance process stays awayfrom 0. Numerical examples show that our simulation algorithm performs at leastas well as popular algorithms presented in the literature.
It is important to note that the method that we present in this paper couldbe significantly sped up by the use of sequential resampling, as implemented inKouritzin and MacKay (2020) for the Heston model. Such improvements, left forfuture work, could give a significant advantage to our weighted simulation algorithmfor the 3/2 model.
Appendix
This section presents the simulation algorithms used to produce the numericalexamples in Section 4. Algorithm 1 stems from the results we present Theorem 1.Algorithm 2 is a Milstein-type algorithm applied to the 3/2 model. Algorithm 3 isAndersen (2007)’s approximation to the algorithm proposed by Broadie and Kaya(2006), modified for the 3/2 model, since the original algorithm was developed forthe Heston model. Algorithms 2 and 3 are considered for comparison purposes.For all algorithms, we consider a partition { , h, h, . . . , mh } , with mh = T of thetime interval [0 , T ], and outline the simulation of N paths of ( (cid:98) S, (cid:98) U , (cid:98) L ), as well asthe associated stopping times τ δ .To simplify the exposition of Algorithm 1, we define the following constants: α h = e − ˜ κ h , σ h = ˜ ε κ (cid:0) − e − h ˜ κ (cid:1) . We also drop the hats to simplify the notation.
Algorithm 1 (Weighted explicit simulation) . I. Initialize:
Set the starting values for each simulated path: { ( S ( j )0 , L ( j )0 , τ ( j ) δ ) = ( S , , T + h ) } Nj =1 , { Y ( l,j )0 = (cid:112) U /n } n,Nl,j =1 II. Loop on time: for i = 1 , . . . , m Loop on particles: for j = 1 , . . . , N , do(1) For l = 1 , . . . , n , generate Y ( l,j ) ih using Y ( l,j ) ih ∼ N (cid:16) α h Y ( l,j )( i − h , σ h (cid:17) . (2) Set U ( j ) ih = (cid:80) nl =1 ( Y ( l,j ) ih ) .(3) Let IntU ( j ) ih ≈ (cid:82) ih ( i − h ( U ( j ) s ) − d s using (3.8) (or another quadrature rule).(4) Generate S ( j ) ih from S ( j )( i − h using (2.4) , with (cid:82) ( i − hih (cid:98) ( U ( j ) s ) − / dW (2) s ∼ N (0 , IntU ( j ) ih ) .(5) If ih ≤ τ ( j ) δ ,(i) If U ( j ) ih > δ , generate L ( j ) ih from L ( j )( i − h using (3.7) .(ii) Otherwise, set τ ( j ) δ = t . Algorithm 2.
I. Initialize:
XPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 19
Set the starting values for each simulated path: { ( S ( j )0 , U ( j )0 ) = ( S , U } Nj =1 II. Loop on time: for i = 1 , . . . , m Loop on particles: for j = 1 , . . . , N , do(1) Correct for possible negative values: ¯ u ( j ) = max( U (( i − h ) , (2) Generate U ( j ) ih from U ( j )( i − h : U ( j ) ih = U ( j )( i − h + ˜ κ (˜ θ − ¯ u ( j ) ) h + ˜ (cid:15) √ ¯ u ( j ) hZ ( j )1 + 14 ˜ ε (( Z ( j )1 ) − h, with Z ( j )1 ∼ N (0 , .(3) Generate S ( j ) ih from S ( j )( i − h : S ih = S ( i − h exp (cid:40)(cid:18) r − u (cid:19) h + (cid:114) h ¯ u Z ( j )2 (cid:41) , with Z ( j )2 ∼ N (0 , . Algorithm 3.
I. Initialize: (1) Set the starting values for each simulated path: { ( S ( j )0 , U ( j )0 ) = ( S , U ) } Nj =1 (2) Fix the constant φ c ∈ [1 , . II. Loop on time: for i = 1 , . . . , m Loop on particles: for j = 1 , . . . , N , do(1) Set the variables m i,j and s i,j : m i,j = ˜ θ + ( U ( j )( i − h − ˜ θ ) e − ˜ κh s i,j = U ( i − h ˜ ε e − ˜ κh ˜ κ (1 − e − ˜ κh ) + ˜ θ ˜ ε κ (1 − e − ˜ κh ) (2) Set φ i,j = s i,j m i,j .(3) If φ i,j < φ c ,Generate U ( j ) ih from U ( j )( i − h : U ( j ) ih = a i,j ( b i,j + Z ( j ) ) , where Z ( j ) ∼ N (0 , and b i,j = 2 φ − i,j − (cid:113) φ − i,j (cid:113) φ − i,j − a i,j = m i,j b i,j (4) If φ i,j ≥ φ c ,Generate U ( j ) ih from U ( j )( i − h : U ( j ) ih = 1 β log (cid:18) − p i,j − X ( j ) (cid:19) , where X ( j ) ∼ U nif orm (0 , and p i,j = ψ i,j − ψ i,j + 1 , β i,j = 1 − p i,j m i,j (5) Let IntU ( j ) ih ≈ (cid:82) ih ( i − h ( U ( j ) s ) − d s using (3.8) .(6) Generate S ( j ) ih from S ( j )( i − h using (2.4) , with (cid:82) ( i − hih (cid:98) ( U ( j ) s ) − / dW (2) s ∼ N (0 , IntU ( j ) ih ) . References
D.-H. Ahn and B. Gao. A parametric nonlinear model of term structure dynamics.
The Review of Financial Studies , 12(4):721–762, 1999.L. B. Andersen. Efficient simulation of the Heston stochastic volatility model.
Jour-nal of computational finance , 11(3):1–42, 2007.G. Bakshi, N. Ju, and H. Ou-Yang. Estimation of continuous-time models with anapplication to equity volatility dynamics.
Journal of Financial Economics , 82(1):227–249, 2006.J. Baldeaux. Exact simulation of the 3/2 model.
International Journal of Theoreticaland Applied Finance , 15(05):1250032, 2012.J.-F. B´egin, M. B´edard, and P. Gaillardetz. Simulating from the heston model: Agamma approximation scheme.
Monte Carlo Methods and Applications , 21(3):205–231, 2015.D. Blount and M. A. Kouritzin. On convergence determining and separating classesof functions.
Stochastic processes and their applications , 120(10):1898–1907, 2010.M. Broadie and O. Kaya. Exact simulation of stochastic volatility and other affinejump diffusion processes.
Operation Research , 54(2):217–231, 2006.P. Carr and J. Sun. A new approach for option pricing under stochastic volatility.
Review of Derivatives Research , 10(2):87–150, 2007.L. Chan and E. Platen. Pricing and hedging of long dated variance swaps under a3/2 volatility model.
Journal of Computational and Applied Mathematics , 278:181–196, 2015.G. G. Drimus. Options on realized variance by transform methods: a non-affinestochastic volatility model.
Quantitative Finance , 12(11):1679–1694, 2012.S. Ethier and T. G. Kurtz. Markov processes: Characterization and convergence,2005.J. Goard and M. Mazur. Stochastic volatility models and the pricing of vix options.
Mathematical Finance: An International Journal of Mathematics, Statistics andFinancial Economics , 23(3):439–458, 2013.M. Grasselli. The 4/2 stochastic volatility model: a unified approach for the hestonand the 3/2 model.
Mathematical Finance , 27(4):1013–1034, 2017.
XPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL 21
F. B. Hanson. Stochastic calculus of heston’s stochastic-volatility model. In
Proceed-ings of the 19th International Symposium on Mathematical Theory of Networksand Systems–MTNS , volume 5, 2010.S. L. Heston. A simple new formula for options with stochastic volatility. 1997.A. Itkin and P. Carr. Pricing swaps and options on quadratic variation understochastic time change models—discrete observations case.
Review of DerivativesResearch , 13(2):141–176, 2010.M. A. Kouritzin. Explicit Heston solutions and stochastic approximation for path-dependent option pricing.
International Journal of Theoretical and Applied Fi-nance , 21(01):1850006, 2018.M. A. Kouritzin and A. MacKay. Branching particle pricers with heston examples.
International Journal of Theoretical and Applied Finance , 2020.A. Lewis. Option valuation under stochastic volatility with mathematica code, 2000.R. Lord, R. Koekkoek, and D. V. Dijk. A comparison of biased simulation schemesfor stochastic volatility models.
Quantitative Finance , 10(2):177–194, 2010.C. H. Yuen, W. Zheng, and Y. K. Kwok. Pricing exotic discrete variance swapsunder the 3/2-stochastic volatility models.
Applied Mathematical Finance , 22(5):421–449, 2015.W. Zheng and P. Zeng. Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 model.
Applied mathematical finance , 23(5):344–373, 2016.
Department of Mathematics, Universit´e du Qu´ebec `a Montr´eal, Montreal, QC,H3C 3P8 Canada
E-mail address : kouarfate.iro [email protected] Department of Mathematical and Statistical Sciences, University of Alberta,Edmonton, AB, T6G 2G1 Canada
E-mail address : [email protected] Department of Mathematics, Universit´e du Qu´ebec `a Montr´eal, Montreal, QC,H3C 3P8 Canada
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