Fourier-Cosine Method for Pricing and Hedging Insurance Derivatives

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1 Theoretcal Economcs Letters, 218, 8, ISSN Onlne: ISSN Prnt: Fourer-Cosne Method for Prcng and Hedgng Insurance Dervatves Ludovc Goudenège 1, Andrea Molent 2, Xao We 3, Antonno Zanette 4 1 Fédératon de Mathématques de CentraleSupélec, Unversté Pars-Saclay, Pars, France 2 Dpartmento d Management, Unverstà Poltecnca delle Marche, Ancona, Italy 3 Chna Insttute for Actuaral Scence, Central Unversty of Fnance and Economcs, Bejng, Chna 4 Dpartmento d Scenze Economche e Statstche, Unverstà d Udne, Udne, Italy How to cte ths paper: Goudenège, L., Molent, A., We, X. and Zanette, A. (218) Fourer-Cosne Method for Prcng and Hedgng Insurance Dervatves. Theoretcal Economcs Letters, 8, Receved: December 8, 217 Accepted: February 6, 218 Publshed: February 9, 218 Abstract We ntroduce the Fourer-Cosne method for prcng and hedgng nsurance dervatves. We mplement ths method for a partcular problem of varable annutes under the Black-Scholes model for the nvestment account. The numercal results show the relablty of the Fourer-Cosne method for prcng and hedgng nsurance dervatves. Keywords Copyrght 218 by authors and Fourer-Cosne, Prcng, Hedgng, Insurance, Dervatves ScentfcResearch Publshng Inc. Ths work s lcensed under the CreatveCommons Attrbuton Internatonal 1. Introducton Lcense (CC BY 4.). Lfe nsurance products wth guarantees dependent on stock market ndcator Open Access are sold n nsurance markets of dfferent countres. For nstance we can cte unted-lnked contracts sold n the Unted Kngdom snce the late 196s, the varable annutes n the Unted States whch form a major part of the sells n annutes market, the segregated fund contracts n Canada snce late 199s and the varable annutes startng to sell from 211 n Chna. Ths knd of nsurance products offer the opportunty to earn money from the bull fnancal market, whle provdng the protecton aganst bear fnancal market. The opportunty to earn money and the protecton from losng money become an deal management tool for fnancng the post-retrement ncome of the polcyholder, ths leads to the popularty of varable annutes n many nsurance markets. Concernng the populaton agng problem n the world and the potental nflaton rsk n economc, the demand of ths knd of products wll contnue to grow. DOI: /tel Feb. 9, Theoretcal Economcs Letters

2 As far as the fnancal market s consdered, the nsurer of the equty-lnked nsurance (see Hardy [1]) faces the fnancal rsk n addton to mortalty rsk and lapse rsk. Thus the valuaton of ths knd of nsurance contracts corresponds to fnd hedgng aganst the rsk of the contracts. The fnancal rsk stems from the guaranteed benefts, t can be treated as an embed opton n the contract. Properly evaluate the fnancal rsk s essental to the nsurer. Ths knd of contracts caused neffcent fnancng to ther provders durng the fnancal crss n 28, forcng some nsurers to reconsder the desgn and prcng of these products. As an nsurance rsk, the mortalty rsk s usually hedged by dversfcaton, whch means that we can assume that the mortalty rate s to be determnstc when there are a suffcently large number of contracts. In some cases, the lapse rsk s also treated as dversfable, thus t s also assumed as a determnstc probablty of the contracts n force at maturty, however for the contract provded guarantees of wthdrawal beneft, the lapse rsk depends on the guaranteed benefts, then t wll be consdered n the prcng of the embed opton. In order to prce the equty-lnked nsurance t s essental to evaluate the guaranteed beneft as an embed opton whle consderng the lapse rsk. The research about the valuaton of equty-lnked nsurance began wth the work of Boyle and Brennan, Schwartz n [2] and [3]. They assumed both mortalty and lapse rsk to be dversfable and the stock ndcator to follow the Black-Scholes models, thus the embed guaranteed benefts can be prced by applyng the famous Black-Scholes formulas n opton prcng. More recently a general framework to prce dfferent guaranteed benefts has been proposed n [4] usng Monte Carlo smulaton, where the close formula s not avalable due to the complexty of some guaranteed benefts, especally when the lapse rsk cannot be treated as dversfable, the guaranteed benefts wth the lapse rsk s a path dependent opton. Other authors use Monte Carlo methods. Bacnello et al. [5] evaluate varable annutes (ncludng GLWBs) are prced usng a Monte Carlo approach. Holz et al. [6] and Klng et al. [7] used a Monte Carlo approach to prce varables annutes products. As Monte Carlo smulaton s tme consumng for path dependent opton, other methods for opton prcng would help to ncrease the computaton effcency of valuaton of the guaranteed benefts. PDE methods for these types of guarantees are proposed n Chen et al. [8], Forsyth et al. [9] and Belanger et al. [1]. The Fourer-Cosne method, whch was frst ntroduced to opton prcng by Fang and Oosterlee n [11], s based on the rsk-neutral opton valuaton formula (dscounted expected payoff approach). Classcally the opton value of a fnancal product could be known wthout the knowledge of the transtonal densty functon. But ths densty can be recovered from ts characterstc functon by a truncated Fourer-Cosne expanson. Thus ths approach permts to prce approxmately the opton value, by computng ts Fourer-Cosne coeffcents. Here we propose to ntroduce ths Fourer-Cosne method for prcng and hedgng the guaranteed benefts, and to prove ts effcency wth a comparson wth already known nu- DOI: /tel Theoretcal Economcs Letters

3 mercal methods. In order to compare wth already known results, we present the mplementaton of ths methodology for the valuaton of Guaranteed Lfelong Wthdrawal Benefts (GLWB) fully descrbed n [9] for the Black-Scholes model and n [12] for Black-Scholes model wth stochastc nterest rates. The paper s organzed as follows. In Secton 1 we descrbe the Fourer-Cosne methodology, n Secton 2 the treated nsurance dervatve, and n Secton 3 we propose the numercal results. Fnally we gve some concluson and remarks about all the methodology and perspectves n the last secton. 2. Fourer-Cosne Methodology Let us gve a bref ntroducton of the COS opton prcng method. The concept behnd COS prcng method s to recover the condtonal densty by ts characterstc functon through Fourer-Cosne expansons. It can be appled for all processes as soon as the characterstc functon s avalable, whch ncludes all affne processes. The method performs mpressvely, especally when the underlyng follows a Lévy processes. Everythng starts from the rsk-neutral valuaton formula r t ( ) ( ) ( ) v xt, = e v yt, f y x d, y where v( xt, ) s the opton value, and xy, can be any ncreasng functons of the underlyng at t and T respectvely. In order to numercally evaluate ths ntegral, we truncate the ntegraton range, amng at r t b ( ) ( ) ( ) v xt, e v yt, f y x d. y (2.1) The ntegraton range has to been chosen such that the condton b ( ) ( ) a f y x d y f y x dy < TOL a s satsfed, where TOL s a pre-specfed tolerance value An error analyss of the varous approxmatons can be found n [11] [13]. The condtonal densty functon of the underlyng s then approxmated by means of the characterstc functon va a truncated Fourer-Cosne expanson of order N, as follows: 2 N 1 kπ akπ y a ( ) ' f y x R φ ; x exp cos kπ, b a k= b a b a b a (2.2) where R means takng the real part of the expresson n brackets, and φ( ω ; x) s the characterstc functon of f ( y x ) defned as: ω y φ( ω; x) e x. = (2.3) The prme at the sum symbol n (2.2) ndcates that the frst term n the expanson s multpled by one-half. Replacng f ( y x ) n (2.1) by ts approxmaton (2.2) and nterchangng ntegraton and summaton, we obtan the COS algorthm to approxmate the value of a European opton: DOI: /tel Theoretcal Economcs Letters

4 where r t N 1 kπ akπ (, ) e ' v xt = R φ ; x exp V, k= k b a b a (2.4) 2 b y a Vk = v( yt, ) cos kπ dy b a a b a (2.5) s the Fourer-Cosne coeffcent of v( yt, ), whch s avalable n closed form for several payoff functons of European optons. whose condtonal densty ( ) The COS algorthm exhbts an exponental convergence rate for all processes f y x s n C (( ab, ) ). The sze of the nte- ab can be determned wth help of the cumulants [11]. graton nterval [, ] 3. Insurance Dervatves In ths secton, dvded nto two subsectons, we present the Guaranteed Lfelong Wthdrawal Benefts contract descrpton. In the frst subsecton we descrbe the dynamcs of the proposed nsurance product between montorng dates, and n the second subsecton we descrbe the events occurrng at montorng dates Guaranteed Lfelong Wthdrawal Benefts Model In ths secton we wll consder the ndex Ŝ whch follows a Geometrc Brownan moton wth ntal value Ŝ, constant drft r and volatlty σ (Black-Scholes model). It satsfes the followng stochastc dfferental equaton: dsˆ = rsˆ dt+ σ Sˆ d W, (3.1) t t t t de n the nterval [ + ] ( ( ) where W t s a Brownan moton. Let the mortalty functon M ( t ) be defned as the fracton of the orgnal owners of the Guaranteed Lfelong Wthdrawal Benefts (GLWB) contract who tt, dt M t s assumed to be constant n ths fxed nterval of sze dt ). Ths number can be computed from a mortalty table, gven the probablty ( x) that an owner whch s x years old wll de the next year. The fracton of the orgnal owners stll alve at tme t, denoted by R( t ) s t R( t) 1 M ( s) d. s = Tme t s measured n years from the contract ncepton date, so the maturty corresponds to the moment when everyone has ded (and of course the end of the mortalty table). Let S be the amount n the nvestment account (.e. mutual fund) of any holder of the GLWB contract stll alve at tme t. We suppose that ths nvestment account tracks an ndex Ŝ prevously descrbed. Let A be the guarantee account balance. We suppose that percentage fees based on the value of the nvestment account S are charged to the polcy holder at the annual rate α tot and wthdrawn contnuously from that account. These fees nclude mutual fund management fees α m and a fee charged to fund the guarantee (also known as the rder) α, so that αtot = αg + α. Let V m ( S, At, ) be the value of the entre g DOI: /tel Theoretcal Economcs Letters

5 contract (express n backward tme) as sum of the no-arbtrage value of the guarantee only porton of the contract (the GLWB rder) and the amount n the nvestment accounts of those remanng alve. We have the followng dynamcs for V( S, At, ) : 2 2 σ S Vt = VSS + ( r αtot ) SVS rv + αmr( t) S + M ( t) S, (3.2) 2 wth ntal value V ( S, A,) = (whch s n fact the termnal condton n real lfe tme). Between two annual dates t and t + 1, the contract follows ths partal dfferental equaton. At date t + 1 there are jumps dependng on the specfcaton of the contract Jumps Events At each montorng date t there are jumps events to take nto account the product features, e.g. wthdrawal, bonus, ratchet, penaltes. We assume that the order of event occurrng at an event tme t s frst the ratchet event then the wthdrawal events contanng possbly bonus or penaltes. Ratchet Event. If the contract specfes a ratchet (step-up) feature, then the value of the guarantee account A s ncreased f the nvestment account has ncreased. The guarantee account A can never decrease, unless the contract s partally or fully surrendered. At a ratchet event tme t we have the followng relaton + ( ) = ( ( ) ) V S, A, t V S, max S, A, t. General Wthdrawal Event. The contract wll typcally specfy a wthdrawal rate G r. Gven a tme nterval of t t 1 between wthdrawals, the contract wthdrawal amount at t = t s Gr( t t 1) A. At ths pont we do not make any partcular assumptons about the wthdrawal strategy of the polcy holder. In general terms, the polcy holder s actons at t can be represented by a polcy parameter γ, where γ 2. Wthdrawals of amounts less than or equal to the contract wthdrawal amount Gr( t t 1) A are represented by γ [,1]. Wthdrawals n excess of the contract amount are ndcated by γ ( 1,2], wth γ = 2 correspondng to full surrender. Wthdrawal events can be wrtten n the general form + γ γ ( ) = ( ) + ( γ ) V S, At, V S, A, t Cash Flow S, At,,, where S γ and A γ are partcular values dependng on the wthdrawal event, and where Cash Flow s the cash flow from the event dependng on the wthdrawal event represented by the value γ. Bonus Event ( γ = ). If the contract holder chooses not to wthdraw at t = t, ths s ndcated by γ =. Let the bonus fracton be denoted by B( t ) (f no bonus s possble at t = t, then B( t ) = ). Thus we defne ( ( )) γ γ S : = S, A : = A 1+ B t and Cash Flow =. DOI: /tel Theoretcal Economcs Letters

6 Wthdrawal Not Exceedng Contract Amount ( γ (,1] ). Note that wthdrawals at the contract rate (or less) are allowed even f the amount n the nvestment account s S =. The wthdrawal amount s γ Gr( t t 1) A and we obtan the followng defntons ). Fnally consder the case of a wth- G t t A,.e. the γ S : = max ( S γ Gr ( t t 1 ) A, ), γ A : = A and Cash Flow = R( t) γ Gr ( t t 1 ) A. Partal or Full Surrender ( γ ( 1,2] drawal of an amount greater than the contract amount ( ) wthdrawal amount s gven by r 1 Gr ( t t 1) A+ ( γ 1) max ( S Gr ( t t 1) A,) ( 1 κ( t) ) where ( ) [,1] κ t s a penalty for wthdrawal above the contract amount. In ths case we have to defne the followng quanttes and γ ( γ) ( γ r ( 1 ) ) ( γ) γ S : = 2 max S G t t A,, A : = A 2 ( ) r ( 1) ( γ ) ( r ( 1) )( κ( ) ) Cash Flow = R t G t t A+ 1 max S G t t A, 1 t. Note that t s assumed that the guarantee account value A s reduced proportonately for any wthdrawal above the contract rate. Now we consder the wthdrawal strategy. The rsk neutral prce s the cost of hedgng. If we consder that the nsurer should charge a prce whch ensures that no losses can occur (assumng that the clam s hedged), then the wthdrawal strategy s assumed to verfy γ ( ( ) ( )) γ γ = arg max V S, A, t + Cash Flow S, At,, γ. γ [,2] Assumng such a strategy by polcy holders, and hedgng aganst t, s obvously very conservatve from the standpont of the nsurer, snce t seeks to provde complete protecton aganst polcy holder wthdrawal behavor (gven assumptons about parameter values such as volatltes). In other words, f nvestors follow ths strategy, and f the nsurer hedges contnuously, the balance n the nsurer s overall hedged portfolo wll be zero. On the other hand, f the nvestor devates from ths strategy, then the nsurer s portfolo wll have a postve balance. 4. Numercal Results At each date t, we need to compute the wthdrawal strategy, so we look for the maxma by a lnear search on values γ [,2]. From the prevous experments on ths model, the authors can say that only values, 1 and 2 are actually selected by ths lnear search. That s to say: when wthdrawal s chosen, only full wthdrawal s selected, and n case of surrender event, only a full surrender event s selected. Wth ths optmal dynamcal approach, or n a statc approach where all γ DOI: /tel Theoretcal Economcs Letters

7 values are fxed at the begnnng of the contract, t s possble to compute (backward n real lfe tme, but forward here) the value V( S, At, ) for all varables S, A and t [, T]. The prcng problem now reduces to fnd the rder fee * α g (drvng the dynamc between tme events) such that ( ) V α S, A, T = S, * g (4.1) * when V α g s the soluton of the partal dfferental Equaton (3.2) when * αg = αg. Vewng V α g as beng parametrzed by the rder fee α, we solve g the Equaton (4.1) usng a classcal secant method. Typcally, only 5 or 7 teratons are necessary to obtan convergence of the algorthm under a fxed tolerance of Numercal Parameters In all our numercal experments we have used the followng parameters whch are compatble wth actual lterature on GLWB. Table 1 refers to the Black-Scholes parameters, Table 2 refers to the product features and Table 3 refers to the polcy holder parameters descrbng ts behavor. We consder that there are no ntal fees and that all the fees and the death benefts are pad contnuously. There s no ratchet feature. The bonus s of 5% f there s no wthdrawal. If there s a partal surrender then there are penaltes gven n Table 4. Moreover we used the DAV 24R mortalty Table, 65 year old German male (see [9] for the table). We have used the followng parameters for the Fourer-Cosne method. The range of ntegraton s gven by a = ( r αg ) Lσ + log ( S ) and b= r α + Lσ + log S wth L = We have remarked that a number ( g ) ( ) Table 1. The Black-Scholes parameters. nvestment volatlty nterest rate dvdend guarantee S σ r q A 1.1/ Table 2. The product features. maturty wthdrawal rate management fees T Gr α m 57.5 Table 3. The polcy holder parameters. polcy holder behavor age of polcyholder wthdrawal frequency frst wthdrawal wthdraw every tme 65 years old 1 per year 1 st year DOI: /tel Theoretcal Economcs Letters

8 Table 4. The surrender penaltes. κ t.5 t 1.4 1< t 2.3 2< t 3.2 3< t 4.1 4< t 5. 5 < t N = 2 of Fourer-Cosne coeffcents provdes a fast method, but the choce of N = 9 seems better and more stable. For the partal dfferental equaton method (PDE), we use an mplct fnte dfference scheme wth a tme step δ t =.1 and a non-unform mesh of 75 ponts n the space varable Prcng and Hedgng the Insurance Product We compute the fee α such that (4.1) s verfed wth a tolerance of 1 5.e. g ( ) * 5 V α g S, A, T S < 1. We perform all these computatons wth parameters detaled n Tables1-4. We only change the value of the volatlty parameter, respectvely σ =.1 and σ =.15.The numercal results are presented n Table 5. Wth ths method t s possble to numercally compute the classcal greeks. But n the nsurance market, and for ths knd of product, an nterestng value could be gven by the sensblty wth respect to the mortalty table. The authors have mplemented a ±1% shock on the probablty ( x) gven by the mortalty table. Table 6 gves the sensblty wth respect to ths shock when σ =.15. The numercal results provde that Fourer-Cosne method may be a relable method n order to prce varables annutes. The parameters defnng the Fourer-Cosne method have been chosen n order to obtan fast and accurate results. 5. Conclusons In ths artcle we have ntroduced the Fourer-Cosne method for prcng and hedgng nsurance dervatves. We have nvestgated the effcency of such a method on a partcular nsurance product whch s the Guaranteed Long lfe Wthdrawal Benefts model of varable annutes. We have assumed that the ndex tracked by the nvestment account follows a Geometrc Brownan Moton, but ths method can be extended to varous models as soon as we have an explct formula for the densty functon for the underlyng process. Frst numercal results say us that Fourer-Cosne method seems to be a promsng technque n order to prce and hedge varable annutes. Future research can be extended to take nto account a model for the ndex DOI: /tel Theoretcal Economcs Letters

9 Table 5. Values α g of the fee. Fourer-Cosne PDE σ = σ = Table 6. Sensblty wth respect to ±1% shocks on mortalty probabltes. σ =.15 Fourer-Cosne PDE shock +1% no shock shock 1% wth stochastc volatlty and/or stochastc nterest rate. For nstance n the Heston model we have an explct formula for the densty functon, then the extenson to ths case could be straghtforward. References [1] Hardy, M. (23) Investment Guarantees: Modelng and Rsk Management for Equty Lnked Lfe Insurance. Wley, New York. [2] Boyle, P.P. and Schwartz, E. (1977) Equlbrum Prces of Guarantees under Equty-Lnked Contracts. Journal of Rsk and Insurance, 44, [3] Brennan, M.J. and Schwartz, E. (1976) The Prcng of Equty-Lnked Lfe Insurance Polces wth an Asset Value Guarantee. Journal of Fnancal Economcs, 3, [4] Bauer, D., Klng, A. and Russ, J. (28) A Unversal Prcng Framework for Guaranteed Mnmum Benefts n Varable Annutes. Astn Bulletn, 38, [5] Bacnello, A.R., Mllossovch, P., Olver, A. and Ptacco, E. (211) Varable Annutes: A Unfyng 953 Valuaton Approach. Insurance: Mathematcs and Economcs, 49, [6] Holz, D., Klng, A. and Ruß, J. (27) GMWB for Lfe: An Analyss of Lfelong Wthdrawal Guarantees. Zetschrft für de gesamte Verscherungswssenschaft, 11, [7] Klng, A., Ruez, F. and Ruß, J. (214) The Impact of Stochastc Volatlty on Prcng, Hedgng, and Hedge Effcency of Varable Annuty Guarantees. European Actuaral Journal, 4, [8] Chen, Z., Vetzal, K. and Forsyth P. (28) The Effect of Modellng Parameters on the Value of GMWB Guarantees. Insurance: Mathematcs and Economcs, 43, [9] Forsyth, P. and Vetzal, K. (214) An Optmal Stochastc Control Framework for Determnng the Cost of Hedgng of Varable Annutes. Journal of Economc Dynamcs and Control, 44, [1] Belanger, A., Forsyth, P. and Labahn, G. (29) Valung the Guaranteed Mnmum Death Benet 957 Clause wth Partal Wthdrawals. Appled Mathematcal Fnance, 16, DOI: /tel Theoretcal Economcs Letters

10 [11] Fang, F. and Oosterlee, C.W. (28) A Novel Prcng Method for European Optons Based on Fourer-Cosne Seres Expansons. SIAM Journal on Scentfc Computng, 32, [12] Fang, F. and Oosterlee, C.W. (21) A Fourer-Based Valuaton Method for Bermudan and Barrer Optons under Heston Model. SIAM Journal on Fnancal Mathematcs, 2, [13] Goudenège, L., Molent, A. and Zanette, A. (216) Prcng and Hedgng GLWB n the Heston and n the Black-Scholes wth Stochastc Interest Rate Models. Insurance: Mathematcs and Economcs, 7, DOI: /tel Theoretcal Economcs Letters