Pricing the Option to Surrender in Incomplete Markets

C© The Journal of Risk and Insurance, 2010, Vol. 77, No. 4, 935-957DOI: 10.1111/j.1539-6975.2010.01358.x

PRICING THE OPTION TO SURRENDERIN INCOMPLETE MARKETSAndrea ConsiglioDomenico De Giovanni

ABSTRACT

New international accounting standards require insurers to reflect the valueof embedded options and guarantees in their products. Pricing techniquesbased on the Black and Scholes paradigm are often used; however, the hy-potheses underneath this model are rarely met. We propose a framework thatencompasses the most known sources of incompleteness. We show that thesurrender option, joined with a wide range of claims embedded in insurancecontracts, can be priced through our tool, and deliver hedging portfolios tomitigate the risk arising from their positions. We provide extensive empiri-cal analysis to highlight the effect of incompleteness on the fair value of theoption.

INTRODUCTION

In a recent paper, the International Association of Insurance Supervisors (Solvencyand Actuarial Issues Subcommittee, 2006) describes best practices for asset-liabilitymanagement (ALM) that a well-managed insurer would be expected to follow. Oneof their statements is addressed to embedded options in insurance contracts:

While the risk of embedded options generally can not be diversified, it isimportant that insurers manage their assets and liabilities in a way thatwould mitigate the potential impact. Management might, for example,take actions such as arranging reinsurance, hedging, offsetting with othertypes of policies, or stopping sales of the product.

The issue of pricing and hedging embedded options in insurance contracts isalso ruled in the International Financial Reporting Standards for insurance contracts(IFRS 4) and Solvency II. The new reporting standard recommends that the costsof options and guarantees embedded in insurance contracts are measured in a

Andrea Consiglio is at the University of Palermo, Palermo, Italy. Domenico De Giovanni is at theUniversity of Calabria, Rende, Italy. The authors can be contacted via e-mail: [email protected] [email protected], respectively.

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936 THE JOURNAL OF RISK AND INSURANCE

market-consistent way, and, to achieve market consistency, it is suggested to splitthe valuation into hedgeable and nonhedgeable components.

If an exposure can be perfectly hedged or replicated on a sufficient liquid and trans-parent market, the hedge or replicating portfolio provides a directly observable price(marked-to-market). For the nonhedgeable liability component and for the remain-ing risk on partial hedges, the valuation process would need to rely on methodologiesto deliver adequate proxies determined on a market consistent basis, that is, arbitrage-free mark-to-model techniques.

Hence, hedging is the “golden rule” suggested by international regulators, and in-surance companies will be judged and ranked according to these aspects.

But, a question arises spontaneously:“Can we always hedge?”

Correctly, regulators subdivide exposures in hedgeable and nonhedgeable compo-nents. However, the boundary between hedgeable and nonhedgeable positions isoften blurred; moreover, this distinction strongly depends on the model adopted.

For example, insurance guarantees, bonus provisions and other related embeddedoptions can be priced under the Black and Scholes framework (see Aase and Persson1997; Bacinello, 2001; Grosen and Jørgensen, 2002; Giraldi et al., 2003). However,their approach implies that the underlying is a tradable asset, and the option can bereplicated by a portfolio of the reference fund and the risk-free rate. This is fairlyimpossible in practice!

This is also unsafe for index-linked policies. For such products, although the under-lying can be easily traded using futures or options written on the index, the liabilitiesgenerated by the insurance contract are long-term ones, and unexpected shortfalls inthe asset returns are not envisaged by the Brownian motion underneath the Black andScholes model. These unpredictable events can generate serious losses. Rolling thehedge forward is not a safe solution, as witnessed by the Metallgesellschaft default(Mello and Parsons, 1995).

To summarize, hedging is suggested as a good practice to mitigate market swings;however, a replicating portfolio is not always available, because of either nontradableassets or too simplistic hypotheses about the market evolution.

In the theory of financial option pricing, such limitations are known as sources ofmarket incompleteness. Recently, scholars have started coping with pricing and hedgingoptions in incomplete markets.

Specifically to the insurance field, Møller (2001) determines risk-minimizing hedgingstrategies for equity-linked pure endowment contracts, where the source of incom-pleteness arises from mortality risk.

Moore and Young (2003) employ a utility method to determine the price of endow-ment contracts linked to risky indexes. In this case, too, the source of incompletenessis the mortality risk.

Coleman et al. (2007) deal with the same problem and solve it in a more generalsetting. In particular, they model the dynamics of the objective price measure bymerging the traditional Black and Scholes price process with Merton’s jump diffusionprocess. They then hedge the insurance claim using the underlying asset and a set of

PRICING THE OPTION TO SURRENDER IN INCOMPLETE MARKETS 937

standard options expiring before the maturity of the claim. The hedging strategy isdetermined by applying the minimum local hedging risk principle by Follmer andSchweizer (1989).

Consiglio and De Giovanni (2008) adopt a super-replication model to determine thehedging portfolio of insurance policies whose final liabilities depend on a minimumguarantee option and a bonus distribution scheme. Their model is general enough todeal with any complex final payoff generated by European path-dependent options.They also account for the bankrupt event by considering the liabilities of the companyas a risky (defaultable) bond. Regulatory restrictions are modeled by assuming thatthe solvency of the company is monitored at discrete points in time.

The aim of the present article is to extend to the surrender option the analysis developedin Consiglio and De Giovanni (2008). Our starting point is the approach proposedby Grosen and Jørgensen (1997) and then extended by Bacinello (2003). They modelthe surrender option as an American option because its maturity is unknown at theinception of the contract. We will also investigate the instance where the lapse rateis represented by a stochastic process, functionally dependent on some economicfactors. In this case, the surrender option turns to a path-dependent European option,and the incompleteness introduced by the additional risk factor is properly treatedthrough our model.

Recently, the interest for lapse modeling is shifted towards a more general surrenderprovision: the Guarantee Minimum Withdrawal Benefits (GMWB). A reference paperfor the valuation of such products is Milevsky and Posner (2001). They propose twovaluation algorithms: one is based on a static approach, viewing the GMWB optionas a Quanto Asian Put; the other is based on a dynamic approach, viewing the GMWBoption as an American put option.

We recall that any exotic payoff can be adapted to be handled with the methodologyproposed in this article. Specifically, the Quanto Asian Put belongs to the class of thepath-dependent options. The valuation of insurance policies with path-dependentpayoff is discussed in Consiglio and De Giovanni (2008), where we determine the priceof the option provision under a regulatory constraints that is modeled as a barrier. Inthe current article, when lapsation rationality is assumed, we use the American optionframework—the dynamic approach, according to Milevsky and Salisbury (2006)—toprice the contingent claim.

The article is organized as follows. In the “Modeling Lapses” section we describe themodels used to deal with lapses; the “Pricing and Hedging in Incomplete Markets”section is devoted to the pricing of embedded options under the incomplete markethypothesis. Implementation notes and discussion of the results are reported in the“Implementation Notes and Results” section. The “Conclusions” section concludesthe article and highlights the major findings.

PRELIMINARIES

We assume that the security prices are stochastic processes that evolve on a discretizedspace and time. The market consists of J tradable securities, whose prices are observedon a finite number of decision stages, t = 0, 1, 2, . . . , T ,

S = (S1

t , . . . , S Jt

)Tt=0. (1)


FIGURE 1A Finite Filtration (left-hand panel) and Its Associated Tree (right-hand panel)

t4t3t2t1t0

At4At3At2At1At0

t4t3t2t1t0

Nt4Nt3Nt2Nt1Nt0

The price process is modeled on the probability space (�,F , P), where the samplespace � is assumed to be finite. Such a formulation allows for a market representationthrough scenario trees (see Pliska, 1997).

We denote by Nt the set of nodes at stage t. Each node n ∈ Nt corresponds, one toone, with an atom of the filtration Ft . In the tree, every node n ∈ Nt , t = 1, . . . , T , hasa unique ancestor node a (n) ∈ Nt−1, and every node n ∈ Nt , t = 0, . . . , T − 1, has anonempty set of child nodes C(n) ⊂ Nt . The collection of all the nodes is denoted byN ≡ ⋃T

t=0 Nt .

In the probabilistic context, if we assume that the sample space |�| is finite, everyalgebra F corresponds to a partitioning of � into mutually disjoint subsets (the F-atoms). In a scenario tree, there is a one-to-one map between the nodes n ∈ Nt and thepartition sets At , for each t = 1, . . . , T . A filtration simply corresponds to a sequenceof algebras generated by successively finer partitions of � (see Figure 1, left-handpanel).

The probabilistic structure implies that the market evolves as a discrete, nonrecombi-nant tree (hence, suitable for incomplete markets). In this context, a European contingentclaim (ECC) is a security associated with the stochastic process S = (St)T

t=0 that givesto its owner a stochastic cashflow F = (Ft)T

t=0. This definition of ECC is an extensionto the case of nonzero payouts at stages other than the expiration. Note that such adefinition also includes securities like futures and coupon bonds.

An American contingent claim (ACC) is a security associated to the real-valued stochas-tic process E = (Et)T

t=0 that gives to its owner the right to withdraw Et euros, afterwhich the security expires.

The value of St , Ft , and Et , in each node n ∈ Nt , is denoted by {Sn}n∈Nt , whereSn = (S1

n, . . . SJn ) ∈ IRJ , {Fn}n∈Nt and {En}n∈Nt , where Fn, En ∈ IR, respectively.


The definitions of ECC and ACC given above are quite general to include put and calloptions with barriers, lookback and Asian payoff, futures, and any derivatives. Notethat such definitions cover claims written on multiple underlyings or nontradableinstruments.

One of the sources of incompleteness is represented by the presence of more riskfactors than available instruments. The main consequence is that not all payoffs arereplicable by trading in marketed securities. For example, mortality risk or lapsedynamics can be modeled as stochastic processes, and liabilities that are triggeredby such events give rise to contingencies that cannot be perfectly hedged or reached.In practice, the process St is augmented by K real-valued variables ξt = (ξ1

t , . . . , ξ Kt )

whose path histories match the nodes n ∈ Nt , for each t = 0, 1, 2, . . . , T .

Particular attention has to be paid in building the scenarios tree. The tree has toincorporate the most relevant features of the multidimensional price process St . Inparticular, the tree has to be able to match expected returns and volatilities, and,above all, the correlations among the set of available assets. Another important fea-ture is the absence of arbitrage. Such a property is crucial for our model since arbi-trage opportunities will lead the optimization model to end up with an unboundedsolution (i.e., a cost of the option equal to minus infinity). In the Appendix we il-lustrate the approach we followed in building the tree used for our experimentalresults.

MODELING LAPSES

The literature on modeling lapses can be subdivided into two strands. One relatesthe lapse behavior to economic and/or social factors, such as the unemployment rate,interest rate, age, education, etc. This approach has been widely used to shape thelapse process for mortgage-backed securities. The other stream describes the lapseprocess as an internal decision process, where the investor rationally exercises hisright to surrender the policy.

Although the former approach allows for some form of irrationality in lapsing apolicy by introducing a minimal lapse rate in each state of the economy, the latterapproach identifies a boundary that separates those states of the economy in whichit is convenient to sell back the policy from those in which it is convenient to keepholding it.

Both approaches have pros and cons. In Albizzati and Geman (1994), the lapse behav-ior is given by a deterministic (piecewise linear) function of a decision criterion that,in its turn, depends on the stochastic interest rate. In this case, the lapse rate does notintroduce a new risk factor, so it can be simply added to the nodes of the tree, {ln}n∈Nt ,for each t = 1, 2, . . . , T − 1, where ln = f (Dn) and Dn depends on the interest rate. Thefunction f (·) is nondecreasing and piecewise linear, and its codomain is [lmin, lmax].Albizzati and Geman assume that the insurance company has at its disposal a largepool of homogenous policies, and by the principle of pooling risk, the optimal stop-ping time transforms into a deterministic aggregate maturity. This implies that theoption is an ECC path-dependent option with expected outflows, due to lapses, atfixed dates.


The Albizzati and Geman (1994) approach has the main advantage of turning acomplex ACC model into a simpler ECC one. However, the lapse mechanism appearsto be too simple to be realistic.

A more robust model is described in Kuo et al. (2003). They estimate an error-correction model with two cointegrating vectors of the lapse rate (lt), interest rate(it), and unemployment rate (ut). By letting ξt = (�lt , �it , �ut)′, and st = (lt , it , ut)′,they obtain the following,

ξt = c + Aξt−1 + α βst−1 + εt , (2)

where c, A, α, and β are parameters to be estimated, together with the variance–covariance matrix of the error, �.

As pointed out in Pennanen (2007), the time series model (2) falls in the general classof the so-called conditional Gaussian processes,

ξt = μ(ξ1, . . . , ξt−1) + σ (ξ1, . . . , ξt−1)εt. (3)

In particular, for model (2), we have that

μ(ξ1, . . . , ξt−1) = c + Aξt−1 + α βst−1, and (4)

σ (ξ1, . . . , ξt−1) = �. (5)

Once the parameters of Model (2) are estimated, a moment matching method (seethe Appendix for details) can be used to build the tree with expected values andvariance–covariance matrix given by (4) and (5), respectively.

Using the pooling-of-risk principle, we can thus price the surrender option as an ECC.However, note that, unlike Albizzati and Geman’s (1994) model, we introduced herean additional independent risk factor that makes the market incomplete. Also notethat the lapse option cannot be hedged as in the Black and Scholes model, since itsunderlying—the lapse rate—is a nontradable risk factor.

If the pooling-of-risk principle cannot be invoked, the surrender option is a typicalACC. Grosen and Jørgensen (1997) first analyze early exercise options embedded inunit-linked insurance contracts and then extend their contribution to policies with asmoothing surplus distribution mechanism (Grosen and Jørgensen, 2000).

As we show in the “Pricing and Hedging in Incomplete Markets” section, the frame-work proposed by Grosen and Jørgensen can be widened to include the hypothesesof market incompleteness discussed in the Introduction.

PRICING AND HEDGING IN INCOMPLETE MARKETS

Our starting point is the policy described in Grosen and Jørgensen (1997). Our choiceis aimed at providing a simple framework to highlight the main findings of our modelwith respect to the complete market paradigm. We stress here that more realistic bonus


schemes and participating formulas can be easily customized (see Consiglio and DeGiovanni, 2008, for details).

We assumed that an insurance company issues contracts that promise to pay somebenefits, at any time t ∈ [0, T], contingent on the value of a reference fund It . Notethat we do not make any assumptions about the dynamics of It .

We denote by I0 the value of the fund at the inception of the contract, and we letL0 ≡ α I0 be the premium paid by the policyholders to enter the contract; the initialinvestments by the equityholders are then given by E0 ≡ (1 − α) I0. To be consistentwith Grosen and Jørgensen (1997), we set α = 1; therefore, the initial investment inthe reference fund fully comes from the premium paid by the policyholders, I0 = L0.

The insurance contract is equipped with a minimum guarantee provision. In particu-lar, at any time period, the policyholder is guaranteed to receive an amount of money,Lt , obtained by compounding the initial premium, L0, at the rate rG

Lt = L0 erG t. (6)

Given an exercise date t ∈ [0, T], the insurance company is liable for either the interestrate guarantee or the claim of the index market value. The value of the claim is thusgiven by

C(t) = max {Lt , It}= max

{L0 erG t − It , 0

} + It. (7)

Relation (7) defines the payoff of a claim on the reference fund, plus an American putoption with a strike increasing at the rate rG .

The model presupposes that in aggregate the whole class of policyholders exercisethe claim and that the option expires. This is a strong assumption, but it could be arealistic approximation in case of high interest rates or unemployment rates. As seenin the “Modeling Lapses” section, a possible alternative is to introduce an additionalrisk factor—the lapse process—and price the option as an ECC in the incompletemarket framework.

Nothing changes from the policyholder’s point of view. If the option is exercised,she will receive the greater of either It or Lt . From the insurer’s point of view, theaggregate liability in the balance sheet is modified to take into account the percentageof policyholders who lapsed the contract,

Lt = (1 − lt)Lt−1erG , (8)

where lt is the lapse rate during the period t − 1 to t. The insurance company will besubjected to an outflow at each time step t given by,

Ot = lt Lt−1erG , (9)

and the claim will expire at the end of the horizon, T.


Super-Replicating an ECCFollowing King (2002) and Consiglio and De Giovanni (2008), we implement astochastic programming model to super-replicate the payout process, {Fn}n∈Nt , t =0, 1, 2, . . . , T , generated by an insurance contract where the lapse dynamics is rep-resented by the process {ln}n∈Nt , for each t = 1, 2, . . . , T − 1. In particular, we havethat

Fn = [Ln − In]+ + In, for all n ∈ NT , (10)

Fn = ln La (n) erG , for all n ∈ Nt ,

t = 1, 2, . . . , T − 1, (11)

Ln = (1 − ln)La (n)erG for all n ∈ Nt ,

t = 1, 2, . . . , T , (12)

with the initial condition at the root node, F0 = 0 and L0 = I0.

In incomplete markets, the price of the option is not unique, and the interval betweenthe buyer’s and seller’s prices describes the possible range of arbitrage-free evalua-tions. Since the main aim of this article is to find an hedging portfolio to mitigate theinsurance company risk, we focus our attention on the seller’s price.

In an ECC, the seller receives an amount, V, and he agrees to pay the buyer thepayoff, F. The seller’s objective is to select a portfolio process of tradable assets thatenables her to meet her obligations without risk (i.e., on all nodes n ∈ N ). The portfolioprocess must be self-financing; that is, the amount of assets bought must be offset bythe amount of assets sold.

Using the tree-notation, we denote by Z jn the number of shares held at each node

n ∈ N , and for each security j = 1, 2, . . . , J .

At the root node, the value (price) of the hedging portfolio plus any payout to becovered is the price of the option, that is,

J∑j=1

S j0 Z j

0 + F0 = V. (13)

At each node n ∈ Nt , t = 1, 2, . . . , T , the self-financing constraints are defined as fol-lows:

J∑j=1

S jn Z j

n + Fn =J∑

j=1

S jn Z j

a (n). (14)


To sum up, the stochastic programming model for the insurance evaluation problemcan be written as follows:

MinimizeZ j

n

V (15)

J∑j=1

S j0 Z j

0 + F0 = V, (16)

J∑j=1

S jn Z j

n + Fn =J∑

j=1

S jn Z j

a (n), for all n ∈ Nt ,

t = 1, 2, . . . , T , (17)

J∑j=1

S jn Z j

n ≥ 0, for all n ∈ NT , (18)

Z jn ∈ IR. for all n ∈ N ,

j = 1, 2, . . . , J . (19)

We highlight here some important points:

1. Problem (15)–(19) is a linear programming model where the objective functionminimizes the value of the hedging portfolio, or rather the price of the option.

2. The payout process {Fn}n∈N is a parameter of problem (15)–(19). This implies thatany complicated structure for Fn does not change the complexity of the model. Inour example, the process {Fn}n∈N is defined by relations (10)–(12).

3. Constraints (18) ensure that at each final node the total position of the hedgingportfolio is not short. In other words, if short positions are allowed, the portfolioprocess must end up with enough long positions so that a positive portfolio valueis delivered.

Super-Replicating an ACCPennanen and King (2005) provide a formal framework to derive the range of pricesfor an ACC. As for the ECC, we need to distinguish between the seller’s and buyer’sprice. For an ACC, the buyer’s price is obtained by hedging against some stoppingtime. This introduces binary variables to keep track of the past history. However,Pennanen and King show that the buyer’s price can be determined as the optimumvalue of a convex (indeed, a linear program) minimization problem. The seller’sprice is technically simpler to solve. In fact, the seller needs to hedge against anystopping time. To this purpose, we modified problem (15)–(19) by introducing a setof constraints such that at each node the portfolio value reaches at least the payoutprocess E.


In the tree notation, for each n ∈ Nt , t = 0, 2, . . . , T , we have that

J∑j=1

S jn Z j

n ≥ En, (20)

where

En = [Ln − In]+ + In, for all n ∈ Nt ,

t = 0, 1, 2, . . . , T , (21)

Ln = L0erG t , for all n ∈ Nt ,

t = 0, 1, 2, . . . , T. (22)

Note that the self-financing constraints are also modified, since the hedging processhas to deliver a value at least equal to En.

In summary, the seller’s price is determined by the following linear program

MinimizeZ j

n

V (23)

J∑j=1

S j0 Z j

0 = V, (24)

J∑j=1

S jn Z j

n =J∑

j=1

S jn Z j

a (n), for all n ∈ Nt ,

t = 1, 2, . . . , T , (25)

J∑j=1

S jn Z j

n ≥ 0, for all n ∈ NT , (26)

J∑j=1

S jn Z j

n ≥ En, for all n ∈ Nt ,

t = 0, 1, . . . , T , (27)

Z jn ∈ IR. for all n ∈ N ,

j = 1, 2, . . . , J (28)


An Alternative Approach: Conditional VaR Super-ReplicationThe1 models proposed in the previous sections prescribe that the value of the self-financing portfolio is greater than the payoff due in each node n ∈ N . Such a re-quirement can be softened by allowing that a certain level of losses (payoff is notmatched) occurs with a given confidence level. In their seminal paper, Follmer andLeukert (1999) introduce the concept of quantile hedging and determine the hedgingstrategy to maximize the probability of success or, alternatively, to bound the shortfallprobability.

In the portfolio optimization community a similar concept was introduced by Rock-afellar and Uryasev (2000). Their model focus on finding an optimal allocation suchthat the conditional value-at-risk (CVaR) measure is minimized, or alternatively,bounded by a given level ω. Note that for practical purposes, the definition of ex-pected shortfall at a specified level α coincides with that of CVaR with a confidencelevel α.

In our exposition we will show that the stochastic programming model used to super-replicate the ACC payoff can be adapted to allow that the expected losses are boundedby a given level ω, with a given confidence level α. We will consider a static approach,in the sense that only the portfolio value at maturity cannot meet the payoff with agiven probability 1 − α. Mathematically, this turns to include constraints (27) only fort = 0, 1, . . . , T − 1, and substitute the Tth constraint with the CVaR equation describedbelow.

As a first step, we define the loss function for each node n ∈ NT ,

Ln = En −J∑

j=1

S jn Z j

n. (29)

Using the tree notation, the CVaR of the final cost distribution is defined by thefollowing set of equations,

ζ + 1|NT |

∑n∈NT

φn

1 − α(30)

φn ≥ Ln − ζ (31)

φn ≥ 0, (32)

where, α is the confidence level. The quantity ζ is an endogenous model variable andit can be proved to be the VaR of the finite distribution of the losses. Note that thesummation in (30) is always positive, since we take only the positive deviations fromthe VaR (φn ≥ 0). This implies that if we minimize (30), or put a bound on it, we will,

1We thank an anonymous referee for suggesting such an approach.


TABLE 1Statistical Properties of the Securities Used in the Experiments

Correlations

Mean St. Dev. Asset-1 Asset-2 Asset-3 Asset-4 Ref. Fund

Asset-1 0.04 0.13 1Asset-2 0.045 0.15 0.6 1Asset-3 0.054 0.2 0.45 0.24 1Asset-4 0.08 0.25 0.1 0.08 0.1 1Ref. fund 0.03 0.1 0.33 0.21 0.12 0.04 1

Note: We built trees by matching the means, variances, and correlations shown in the table. Allthe data are on annual basis.

at the same time, minimize, or bound, the VaR of the distribution (see Rockafellarand Uryasev, 2000, and references therein for details). Also note that the complexityof the problem does not change as we are adding a set of constraints that are linearwith respect to their variables.

IMPLEMENTATION NOTES AND RESULTS

We are now ready for the empirical testing of the model. We first assumed a completemarket, that is, a market where the number of instruments available is equal to thenumber of sources of risk. We calibrated the tree by using model (A1)–(A3) andrestricted the matching problem to constant means, variances, and correlations, withkurtosis β2 = 3 and skewness β1 = 0. If not explicitly stated, we run our experimentsassuming a policy horizon of T = 10 years, and a time interval between two periodsset to 1.67 years, for a total of six time periods. Without loss of generality, we assumedthat the initial prices S j

0 = 100, for all j = 1, 2, . . . , J , and the initial liability L0 = 100.In Table 1 we display the statistical properties for each of the J assets and the referencefund.2

We did not follow a particular criterium in choosing the available set of assets de-scribed in Table 1. We basically tried to provide an ample spectrum of expectedreturns, volatilities, and correlations.

Since the pioneering work of the Nobelist H. M. Markowitz, the choice of the assetuniverse has been considered of greater importance. A similar problem is posed for thesuper-hedging model. In this case, ceteris paribus, we have to look at the correlationsbetween the reference fund and the available assets. The higher is the correlationmagnitude, the smaller will be the price of the option. The sign of the correlationswill determine the direction of the hedge. For example, a shorted option is hedgedby a short portfolio. If we have assets negatively correlated with the reference fund,the shorted option will be hedged by a long portfolio. The worst case scenario forsuper-hedging portfolios is the case of zero correlation.

2To be consistent with Grosen and Jørgensen, we assumed the risk free rate is constant at thelevel of 2.5 percent per year. Stochastic interest rates can be easily modeled (see Consiglio andDe Giovanni, 2008).


Clearly, expected returns and variances also play a role. However, the interplay be-tween correlations, expected returns, and variances is difficult to synthesize in asimple pattern. It is the optimization model to make such a synthesis.

In conclusion, the search for the candidates to enter in the asset universe shouldpoint toward those securities showing a significative degree of comovements with theunderlying asset. The term comovements is emphasized to mean that the correlation isjust a special case. Securities having a nonlinear relationship with the underlying assetcan be included too. The optimization model will select them to be part of the super-hedging portfolios according to their contribution in minimizing the option cost.

Is the Super-Hedging Solution Nontrivial?Before presenting our main results, we provide some evidences about the goodnessof the super-replication approach with respect to the trivial hedging strategy.

In particular, it is argued that the super-replication portfolios, under certain circum-stances, coincide with the trivial buy-and-hold strategy: buy a share and hold untilmaturity.

In our setting the buy-and-hold strategy is given by the value of the underlying assetat inception (I0), plus the present value of the final guarantee value, discounted at therisk-free rate (r f ). Its value is

CT S(0) = I0 + (L0erG T )

e−r f T . (33)

Without loss of generality, we assume that I0 = 100, and we compute the trivialsolution, CT S(0), for rG ranging from 1 to 5 percent. We infuse in the model a certaindegree of incompleteness by removing a subset of assets, such that the number of arcsspringing from each node exceeds the number of independent securities J.

In Table 2 we display the results of such an experiment. Note that we are not in-cluding among the available assets the underlying, because we are assuming that

TABLE 2Price of the Buy-and-Hold Strategy Versus Prices of the Super-Replication Strategy

Incomplete Market

2 Assets 3 Assets 4 Assets Compl. Market

Bid Ask Bid Ask Bid Ask 5 Assets Trivial Sol.

1% 98.41 112.62 99.42 109.65 107.61 108.10 107.32 186.072% 99.94 116.27 100.37 113.68 111.76 112.20 110.83 195.123% 101.00 121.99 113.87 120.00 117.91 118.48 116.30 205.134% 118.26 129.71 123.13 127.88 126.38 126.77 123.68 216.185% 129.88 139.16 133.66 137.16 135.94 136.21 132.04 228.40

Note: The experiment is carried out by letting the minimum guarantee to range between 1 and5 percent. The bid-ask prices are computed for different degree of incompleteness. For example,in the “2 Assets” case, only the risk-free and a risky asset compose the super-hedging portfolio.


the underlying asset could be nontradable. As can be observed, even in case of se-vere incompleteness—the case of a market with only two assets (the risk free andone security)—the price of the super-replication portfolio is much smaller than thetrivial buy-and-hold strategy. Clearly, this result depends on the characteristics of theavailable assets. If we had at our disposal only a zero-coupon bond with maturity at10 years, the super-replication price would be much higher than the trivial buy-and-hold strategy. The same evidence of such a behavior can be found in Edirisinghe etal. (1993) and Broadie et al. (1998).

As expected, a wider bid-ask spread is found when only two assets are available tohedge the option. We remark here that there is no reason to hedge the portfolio withmuch fewer securities than sources of risk. We are reporting such results to show theconsistency of our model. Realistic cases are closer to the “4 Assets” case.

Accuracy and Computational TimeIn principle, the no-arbitrage price of a contingent claim can be computed indepen-dently of the number of time steps. What matters is a significative number of finalnodes to correctly estimate the expected payoff. In the Black and Scholes frameworkthis is not possible because the hedging portfolio is solely based on two assets (the un-derlying and the bond), and therefore, in a single step only two states of the economycan be reached. To keep together completeness and accuracy, the binomial model wasextended to consider intermediate time step, letting the number of possible statesin the next step equal to the number of independent securities. In synthesis, finerdiscretizations are needed to guarantee that with only two independent securities theestimate of the expected payoff is significant. This is true, a fortiori, for recombiningtrees, since they grow linearly, and the number of final nodes is N = T + 1, where Tis the number of time steps.

In an incomplete market setting, such a requirement can be abandoned and we canuse a coarser discretization, provided that the number of final nodes is large enough.However, note that a certain level of discretization is needed when the final payoff ofthe contingent claim depends on the price path before maturity. This is also the caseof American options whose price depends on the optimal exercise boundary.

As pointed out in the Appendix, the major drawback of using trees is that their sizesgrow exponentially with the number of child nodes ν. Hence, the discretization levelhas to be set to a degree such that a reasonable approximation of the option price isreached. Such a tuning is carried out by comparing the price obtained using a binomiallattice tree and the corresponding price obtained through the model (23)–(28) whencompleteness is assumed.

Clearly, nothing ensures that the same degree of discretization adopted in the com-plete market example will deliver the same accuracy in the case of incompleteness.This is an open question.

We took as a benchmark the American put option proposed by Grosen and Jørgensen(1997), and whose payoff is given by Equation (7). We used a recombining tree with1,000 time steps (and therefore 1,000 final nodes), and we found that finer discretiza-tions does not improve the price further. Note that the number of final nodes for the


TABLE 3The Effect of Discretization on the Option Price

Option Price

1% 2% 3% 4% 5%

Recombining tree 107.69 110.49 115.62 123.17 132.773 stages 104.44 106.41 111.26 117.25 124.474 stages 106.18 108.09 111.69 116.07 121.675 stages 107.56 109.78 114.33 121.36 130.886 stages 107.32 110.83 116.30 123.68 132.03

Absolute relative error3 stages 3.016% 3.696% 3.768% 4.807% 6.247%4 stages 1.398% 2.171% 3.398% 5.764% 8.359%5 stages 0.112% 0.641% 1.110% 1.465% 1.421%6 stages 0.339% 0.306% 0.594% 0.414% 0.554%

Note: Assuming completeness, we observe that by partitioning the time horizon in six stagesthe highest absolute relative error is less than 0.6 percent .

six-stages tree is N = 56 = 15,625, much higher than the 1,000s used in the recombin-ing tree.

Of course, the optimal exercise boundary would be better approximated increasingthe number of intermediate time steps, but, as it can be observed from Table 3, sixstages are sufficient to determine a price whose maximum absolute error is less than0.6 percent.

As far as the computational complexity is concerned, the mathematical programmingmodels developed in the “Pricing and Hedging in Incomplete Markets” section are alllinear models. This is a nice feature because it ensures that (1) the algorithm convergesto a global optimum and (2) very large instances can be solved. The latter is a veryimportant property because the dimension of the problem depends on the tree sizes.Fortunately, nowadays the cost of solving large-scale linear programming models isquite cheap. Consider that problems with six stages and N = 56 = 15,625 final nodescorrespond to linear programming models with 23,436 rows, 19,530 columns, and136,705 nonzeros. GAMS/CPLEX is able to find a solution in only 6–13 seconds on alaptop equipped with a Centrino Core 2 Duo, 2.26 GHz. On the same computer, thefree domain solver GLPK finds a solution in about 200 seconds.

Larger instances can be solved with a small additional cost in terms of computationaland memory resources. However, we believe that the methodology proposed hereshould not evaluated according to a “timing” criterion. In fact, the model proposedshould be seen as a planning tool for insurance companies, or a control tool forregulators, instead of a pricing tool for liquid options sold by trader desks.3

3For the interested readers, the GAMS code and sample data can be downloaded fromhttp://www.unipa.it/consiglio/software.html; the MPS sample data to be used in conjunc-tion with the GLPK solver can be downloaded from the same web page.


Conditional VaR FrontierIn the “An Alternative Approach: Conditional VaR Superreplication” section wediscussed a possible extension of the pricing model in incomplete markets by allowingthat a certain level of losses occur with a given confidence level. We show hereaftersome of the results obtained. In particular, we set the confidence level α = 0.99, andwe solve the model (23)–(28) by including the CVaR constraints described in the“An Alternative Approach: Conditional VaR Superreplication” section. The model issolved parametrically by bounding Equation (30) with increasing levels of ω, startingfrom ω = 0 that coincides with the no-arbitrage prices given in Table 3. In Figure 2 weshow the trade-off between ω and the option price. The option price is the value of theminimum guarantee provision, that is, C(0) − L0 (see Grosen and Jørgensen, 1997).As expected, by allowing a greater ω (higher CVaR), the price of the option is lowered.However, this reduction is bounded: a high level of ω, such that the constraint (30)is not binding, does not lead necessarily to a lower price. There exists a minimumprice that cannot be overcome even if an high level of the expected final mismatch isallowed. This is explained by two reasons: (1) we are allowing that only at the finalnodes the payoff is not met, for the rest of the time periods the constraints (27) arestill active, and (2) trivially, the price of the option cannot worth less than zero.

Another important interpretation was pointed out by an anonymous referee:

Basically, the price of the option depends on the level of confidence thatthe company wants to hedge. Then the price directly reflects the level ofrisk the company wants to achieve. This could be useful from the insurer’sperspective and from the regulator’s perspective, and also explains whytwo different companies may propose the same contract at two differentprices: It is a direct consequence of the risk exposure they want to take andare able to take.

Lapse Evaluation Under Different Incompleteness HypothesesIn this section, we release some of the assumptions of the Black and Scholes model. Inparticular, we studied the case of departure from the Gaussian distribution and thecase of additional risk factors, like the lapse rate. In all our examples, we have pricedand shown the values the minimum guarantee option, C(0) − L0.

We built trees with kurtosis β2 = {3, 4, 5, 6} and asymmetry β1 = 0. We then priced theACC originated by the insurance contract given above. We observe that the price ofthe option with rG = 1 percent almost double when moving from β2 = 3—Gaussiandistribution—to β2 = 6—fat-tailed distribution (see Figure 3, top). The price intervalalso depends on the level of the minimum guarantee. In particular, for rG = 4 percent(see Figure 3, bottom), although the magnitude of the option price is higher withrespect to rG = 1 percent, the price interval does not vary across the kurtosis. Thereason is that an insurance policy with a 4 percent minimum guarantee is in the money,and therefore the distribution of the assets does not play any role in determining theprices.

As seen in the “Modeling Lapses” section, the surrender option can be introduced bymodeling the lapse dynamics to take into account of the relationship with economic


FIGURE 2Price of the Minimum Guarantee Option Versus Conditional VaR for Minimum Guaran-tee Rate Equal to 1 Percent (top panel) and 5 Percent (bottom panel)

6.95

7

7.05

7.1

7.15

7.2

7.25

7.3

7.35

0 0.5 1 1.5 2 2.5 3 3.5

Valu

e o

f th

e m

inim

um

gu

aran

tee

op

tio

n

ω

28.5

29

29.5

30

30.5

31

31.5

32

32.5

0 0.5 1 1.5 2 2.5 3 3 .5

Valu

e o

f th

e m

inim

um

gu

aran

tee

op

tio

n

ω

Note: By allowing that the final payoff is not met with a given confidence level, we are able tolower the option price. Such a reduction is more substantial for high values of the minimumguarantee rate and, in any case, is bounded from below.


FIGURE 3The Prices of the ACC Insurance Contract Against Asset Returns With Different Kurtosis

0

2

4

6

8

10

12

14

6543Kurtosis

Pri

ce

0

5

10

15

20

25

30

6543Kurtosis

Pri

ce

Note: The kurtosis has an effect on the magnitude of the option prices. Such an effect alsodepends on the level of the minimum guarantee. For rG = 1 percent, lower than the risk-freerate (top panel), the kurtosis strongly affects the price of the option. If the option is in themoney, rG = 4 percent (bottom panel), the kurtosis of the asset returns is irrelevant.


FIGURE 4Prices of the Surrender Option in the Complete and Incomplete Market Framework

0

5

10

15

20

25

30

35

0% 1% 2% 3% 4% 5%Minimum guarantee rate

Opt

ion

pric

e

ACC Grosen-Jorgensen

ECC Lapse Process

Note: In the complete case, the surrender option is modeled as an ACC; in the incomplete case,as an ECC. The ECC prices are noticeably lower than the ACC ones, and this is due to a finerrepresentation of the policyholder’s lapse behavior.

variables. Using such an approach, the claim embedded in the insurance contractis an ECC, and given the additional risk factor, the pricing has to be carried outin the incomplete market framework. From our point of view, this simply implies amodification of the linear programming model described in the “Pricing and Hedgingin Incomplete Markets” section.

Since the fitting of an error-correction model is outside the scope of this article,we assumed that μ(ξ1, . . . , ξt−1) = μ = 8 percent and σ (ξ1, . . . , ξt−1) = σ = 1 percentper annum. We calibrated a tree by also assuming that the lapse rate is boundedfrom above and from below, respectively, lmax = 10 percent and lmin = 5 percent4

respectively, and set the correlation with the short term interest rate to 0.6.

In Figure 4, we show the prices obtained by modeling the option either as an ACCusing the Grosen and Jørgensen approach, or as an ECC using the incomplete marketapproach described above. The ECC prices are noticeably lower than the ACC ones.We ascribe such a result to a finer representation of the policyholder’s lapse behavior.In particular, the presence of a minimum amount of lapse in each node abates thefinal level of the option payoff and, consequently, the final liability of the insurancecompany.

4These data are taken from the Life Insurance Fact Book 2006 of the American Council of LifeInsurance.


CONCLUSIONS

We have developed a super-replication model to price options embedded in insurancecontracts. Several interesting conclusions can be drawn from the use of the model.First, new regulations press in the direction of mitigating risk expositions. Our modelis able to deliver hedging portfolios even in the worst case situation where the under-lying asset is not tradable. We do not make any assumption about the dynamics of theunderlying assets. Therefore, any distributional hypothesis can be used to calibratethe stochastic processes. We have also showed that even ACC can be priced usingour methodology, thus enlarging the spectrum of possible insurance contracts. Wehave tested the model using the most classic example of ACC in insurance contracts:the surrender option. The same framework can be adopted either by considering thesurrender option as a pure ACC or transforming it in an ECC. We showed that ourmodel improves upon both approaches. In particular, it improves upon Albizzati andGeman’s model, since it allows for richer and more robust modeling of the lapserate dynamics. It improves Grosen and Jørgensen’s model by providing lower optionprices due to a better characterization of the lapse process.

APPENDIX: THE TREE GENERATION MODEL

We generate the tree of the underlying price process S by matching the first M mo-ments of its unknown distribution. Our approach is based on the model by Høylandand Wallace (2001), where the user provides a set of moments, M, of the underlyingdistribution (mean, variance, skewness, correlations, or quantiles), and then pricesand probabilities are jointly determined by solving a nonlinear system of equations,or a nonlinear optimization problem. The method also allows for intertemporal de-pendencies, such as mean reverting or volatility clumping effect.

For a review on alternative scenario generation methods, see Dupacova et al. (2000)and references therein.

We cast the tree generation model as a nonconvex weighted least squares minimiza-tion problem

MinimizeS,p

J∑j=1

⎧⎪⎨⎪⎩

∑i∈M

αi

⎡⎣ ∑

m∈C(n)

pm

(ln

S jm

S jn

)i

− μji

⎤⎦

2⎫⎪⎬⎪⎭

+J∑j

J∑h

h> j

⎧⎪⎨⎪⎩γ jh

⎡⎣ ∑

m∈C(n)

pm

(ln

S jm

S jn

− μj1

)(ln

Shm

Shn

− μh1

)− ρ

⎤⎦

2⎫⎪⎬⎪⎭(A1)

s.t.∑

m∈C(n)

pm = 1 (A2)

pm ≥ 0 m ∈ C(n). (A3)


The goal programming model (A1)–(A3) is much more flexible than solving a sys-tem of nonlinear equations matching the moments of the unknown distribution. Asunderlined in Høyland and Wallace (2001), the solvability of the nonlinear system ofequations increases with the number of arcs. However, to be tractable, the set of arcsthat springs out from each node has to be of limited size. In fact, the size of the treegrows exponentially with the number of branches springing out from each node. Inparticular, if |C(n)| = ν, the total number of nodes is

∑t νt + 1.

Note, however, that infeasibility could also arise from an inconsistency in the spec-ification of the moments (see Høyland and Wallace, 2001, for a discussion on thistopic).

Nonarbitrage ConstraintsTo be consistent with financial asset pricing theory, the tree has to exclude arbitrageopportunities. This is a very important feature, since in the presence of arbitragescenarios, the model (15)–(19) will end up with an unbounded solution.

Arbitrage opportunities can be avoided either by adding nonarbitrage constraints orby increasing the number of arcs. The latter solution is often impracticable due to theexponential growth of the tree size.

Following Klaassen (2002), it is possible to prove that arbitrage opportunities of thefirst type are prevented if and only if

πnS jn −

∑m∈C(n)

πmS jm =

∑m∈C(n)

pmS jm for all j = 1, . . . , J (A4)

πm ≥ 0 m ∈ C(n) (A5)

π0 ∈ IR. (A6)

Likewise, arbitrage opportunities of the second type are excluded if and only if

∑m∈C(n)

νmS jm = S j

n for all j = 1, . . . , J (A7)

νm ≥ 0 m ∈ C(n). (A8)

The set of Equations (A4)–(A6) and (A7)–(A8) can be added as constraints to the goalprogramming model (A1)–(A3) to preclude arbitrage opportunities of both types inthe tree that is generated. Note that Equations (A4) and (A7) are nonlinear constraintsand must be handled with care. In our experiments we added only first type ar-bitrage constraints, and those were sufficient to guarantee trees without arbitrageopportunities.


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Pricing the Option to Surrender in Incomplete Markets

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