1、 Antoon Pelsser Jing Cao Peter den Iseger Pricing Hybrid Options by an Efficient Monte Carlo Approach Discussion Paper 2007 - 019 December 20, 2007 Pricing Hybrid options by an efficient MonteCarlo approachJing Cao, Peter den Iseger and Antoon PelsserDecember 20, 2007Address corresponding to Jing Ca
2、o: University of Amsterdam, Dept. of Quan-titative Economics, Roetersstraat 11, 1018WB Amsterdam, The Netherlands, e-mail:J.Caouva.nl1AbstractIn this paper, inspired by the Least-squares method introduced by Longstaffand Schwartz (2001), we develop an more efficient Monte Carlo techniqueto price the
3、 Hybrid options. Hybrid option, which is by nature with exoticproperties and form, is a combination of two kinds of derivatives, for exam-ple a combination of an interest rate derivative with an equity derivative.Therefore, it is a very useful instrument for pension funds to control theirfunding rat
4、io within required levels. Hybrid payoffs strongly depend both onthe evolution of the yield curve as well as the equity underlying. Unfortu-nately there are no closed-form solutions available for a general (non-zero)correlation between the interest rate and equity process. In the absenceof closed-fo
5、rm formulas, we try to price the hybrid options by an modifiedLeast-squares method in Monte Carlo simulation.1 IntroductionThe purpose of this paper is to develop and analyze pricing methodology forhybrid derivatives used by pension funds to hedge the positions in their ownfunding ratio. These hybri
6、d derivatives are designed specially for pensionfunds which are willing to sacrifice some upward potentials on their fundingratio to obtain downside protections. Since these kind of financial instru-ments blend the properties of debt and equity and their payoff structure isjust like a collar, moreov
7、er, their underlying state variables are often not onlyone or two, but usually multiple. For pension funds, it is critically importantto know the exact value of the derivative at every time point1 so that they arealert to the status of funding ratio. These factors make it quite difficult forus to pr
8、icing these hybrid derivatives accurately and efficiently. Sometimesa closed form formula for a price of such an option is impossible to modeled.Longstaff and Schwartz in 2001 1 proposed a simple but efficient least-squares approach to approximate the value of American options by simula-tion. They m
9、ake use of least squares to estimate the conditional expectedpayoff to the option holder from continuation. This approach is readily ap-plicable in path-dependent and multifactor situations where traditional fi-nite difference techniques cannot be used. Inspired by their method, we tryto develop an
10、more efficient and numerically stable approach which could1Time point2overcome the limits of Longstaff and Schwartzs method in choosing good“basis functions”. We use the properties of Hermite polynomials which areorthogonal with respect to the weight function ez22 , and make the condi-tional expecta
11、tion values we try to calculate expand on Hermite polynomials.Therefore, we can calculate the estimated coefficients of the conditional ex-pectation functions. Moreover, we also make use of an efficient algorithmof evaluation of Kronecker products in multi-dimension developed by DenIseger and Oldenk
12、amp 6 in simulation.We discuss the practical problems that the pension funds are facing, thenecessity for pensions to use hybrid options in dealing with their problemsand how we could pricing the hybrids in section 2. In section 3, we introduce apractically useful method in pricing options in Monte
13、Carlo simulationLeastsquares method. Moreover, we develop a modified Least squares method toimprove the accuracy and efficiency. Section 4 presents our numerical resultsincluding a simple example: Geometric average option on two assets and asimplified hybrid option with two-factor Hull-White model a
14、nd one equityBlack-Scholes model. Section 5 concludes.2 Practical problemTo pension funds, funding ratio, which is calculated as the ratio of assets(at market value) to liabilities2 as reported by each of the funds is a keyfactor to show the solvability of the schemes, as well as the efficiency of t
15、heirinvestment. It is very crucial to keep the funding ratio within certain interval.Because once the funding ratio is smaller than a minimum level3, the pensions2To make the retirement system more secure by further guaranteeing the solvability ofpension schemes, in Netherland, the government requir
16、es the pension funds to value theirliabilities on a marked-to-market basis from January 2007, making the value of liabilitiesmuch more volatile. This will make the matching of assets and liabilities more complicated.However, previously, only the assets of Dutch pension funds were subject to a market
17、valuation. Liabilities were discounted at fixed 4% rate, which made the asset-liabilitymodeling easier.3The minimum level of funding ratio is obviously at least equal to 100% because theliabilities must be fully covered by the assets. For the security reason, some countries havehigher “minimum level
18、”, e.g, in Netherland, the government requires the pension fundsto be supervised by a new framework “Financieel Toetsingskader (FTK)” from January2007. Under the FTK, pension funds that fall below the minimum coverage ratio of 105%of their indexed liabilities will have to correct their situation wit
19、hin one year.3will have deficits to cover their liabilities (which we call “underfunded”) sothat it would not be secure to guarantee the solvability any more. On theother hand, if the funding ratio is getting larger than a maximum level (whichwe call “overfunded”), it will probably require refunds t
20、o the sponsor orimprovements to the benefits. Therefore, the ideal situation of the fundingratio of the pension funds should stay within the range of min,max, where denotes the funding ratio level.Just as we stated above, a stand-alone pension fund is, at time t0 (theinitial time), concerned about t
21、he possibility of its funding ratio at somefuture time t, which we denote as FR(t), falling below a given minimumlevel min. It may wish to sacrifice some upward potential (which the pensionfund does not like too much) to obtain downside protection by entering intoa derivative contract with a third p
22、arty (usually investment bank). Such acontract is called a collar or a hybrid option. As a result, the pension fundcan just keep its funding ratio within the required interval.Now let us state this problem in an analytical model. Let X denote thestate space (state of the economy), such that the liab
23、ilities, the funding ratioare all a function of this state space Lt := L(Xt), FRt = FR(Xt), respec-tively. Let further At := A(Xt) stand for the assets of the pension fund, also(only) dependent on the state of the economy. We consider discrete time(discrete trading) and a horizon T. Lets consider a
24、pension fund wishing tokeep its nominal (unprotected) funding ratio FRN(t) = AN(t)/L(t), accord-ing to the regulation, above a given level min. In return for achieving suchan insurance the pension fund is ready to pay its excess above a level max4.This is in fact an derivative whose payoff is descri
25、bed by a function D(t),defined byD(t) = L(t)maxmin FRN(t),0maxFRN(t)max,0 (2.1)4The upper boundary of the funding ratio is usually chosen such that the initial valueof this derivative contract is zero.4Then the real funding ratio FR(t) with a collar will beFR(t) = A(t)L(t)= AN(t) +D(t)L(t)= FRN(t) +
26、maxmin FRN(t),0maxFRN(t)max,0=min if FRN(t) maxFRN(t), if otherwise.(2.2)Obviously, the payoff structure of the derivative blends characteristics ofdebt and equity. Generally, these kind of financial instruments are called“hybrids”. Since they are mixed by debt and equity, their market price tendto
27、be influenced by both interest rates as well as the equity price. It turns outthat hybrids rely on several risk factors5, in other words, they are generallyhigh-dimension instruments.To make sure their safe position regarding to the funding ratio, pensionsmust have comprehensive information of the h
28、ybrid option and meanwhilechoose certain efficient investment strategies to hedge their position at everytime point. Therefore, the pricing of this derivative becomes an importantproblem for pension funds.As we know, many hybrid options have no analytical (closed-form) solu-tions because they are to
29、o complex (high-dimension and the payoff structureis exotic). Consequently, numerical solutions are necessity. In fact, MonteCarlo simulation becomes an important method when dealing with higherdimensions. The reason is that with a grid-based numerical integration orPDE method, the number of grid po
30、ints grows exponentially with dimen-sions. Monte Carlo on the other hand is a lot less affected by high dimensions,moreover, it can simulate arbitrarily complicated sample paths of state vari-ables. However, a drawback of Monte Carlo is its high computational cost,especially in a high-dimensional se
31、tting. Therefore, we focus on finding anefficient Monte Carlo method to solve the pricing problem of hybrid optionsin a dynamic economy.5In the pension case, there will be inflation risk, interest rate risk, equity risk, contri-bution risk of the sponsors and longevity risk of the members and so on.
32、53 Model and algorithmThere is a fundamental implication of asset pricing theory that under certaincircumstances, the price of a derivative security can be represented as anexpected value, especially the conditional expected value. Valuing derivativesthus reduces to computing (conditional) expectati
33、ons. As we mentionedabove, in many cases, if we were to write the relevant expectation as anintegral, especially when the dimension is large, Monte Carlo method is anattractive approach. Therefore, pricing the hybrid, in principal, becomescomputingconditional expectationsin Monte Carlosimulation. In
34、 our model,we fix a probability space (,Ft,P) with a filtration (Ft)Xtt0. Xt is thestate variable and we assume that Xt+1 = (Xt,Yt+1), where Yt+1 is a randomvariable that is independent of Xt. For example, in the standard Black-Scholes model: dSt = rStdt+StdWt, we could havelog(St+1) = log(St) + (r
35、122)t+tZHere we could understand Xt+1 = log(St+1), Xt = log(St) and Yt+1 = (r 122)t+tZ. In fact, our assumption of economy is general, which couldbe extended to not only the Black-Scholes, but other models.Let V(t,Xt) denote the value of the hybrid option at time t, which couldbe considered as the e
36、xpected value conditional on the filtration Ft.V(t,Xt) = EV(T,XT)|Ft (3.1)It follows from the Markov tower property thatV(t,Xt) = EV(T,XT)|Ft= EE(V(T,XT)|Ft+1)|Ft= EV(t+ 1,Xt+1)|Ft= EV(t+ 1,Xt+1)|Xt = x(3.2)Clearly6, the value V(t,Xt) is the expected value of V(t + 1,Xt+1) condi-tional on the Xt = x
37、. Therefore, our problem now is how to estimate thisconditional expectation value by Monte Carlo. To deal with this problem, wefirst introduce a least-squares algorithm proposed by Longstaff and Schwartz(2001), which at first intended to price American-style options.6For simplicity, we assume the di
38、scount factor here is one.63.1 Least-squares methodLongstaff and Schwartz (2001) 1 introduced a least-squares approach tovalue American options by simulation, which actually focuses on calculatingthe conditional expectation value of the option. They use regressions acrosssimulated sample paths to ev
39、aluate the conditional expectation of the con-tinuation value of the option and compare this expectation to the immediateexercise value at all future dates along each simulated path. Since there isno closed-form expression for the conditional expectation function, LongstaffandSchwartzsuggesttoselect
40、asetofbasis functions suchthattheirweightedcombinations is ”close” to the true function. Still, we generate random vec-tors Xkt for k = 1,.,m and t = 1,.,T, which are independently andidentically distributed samples. (pi(X) denotes basis function and Ai repre-sents the weights on the basis functions
41、. They try to minimize the differencebetween the true value function and the weighted combinations:minAibardblV(t,Xt)summationdisplayi=1Aipi(Xt)bardbl (3.3)so that the weighted combinations can be used to approximate the true valuefunction. With a large number of simulation paths, the Law of Large N
42、um-bers guarantees the weighted combinations to converge to the actual priceof the option. Let C(t,Xt) = summationtexti=1 Aipi(Xt), simply speaking, there exists arelation:V(t,Xt) = EV(t+ 1,Xt+1|Xt = x)summationdisplayi=1Aipi(Xt)= C(t,Xt)(3.4)Thus, in least-squares method, a linear combination of ba
43、sis functions is fit-ted to the data via least squares regression in order to approximate the con-ditional expectation over the entire state space. Specifically, approximatingthe conditional expected function by a linear combination of known functionsof the current state and using least squares regr
44、ession to estimate the bestcoefficients in this approximation, which, then, provides a direct estimateof the conditional expectation function. Therefore, at time t, it is assumedthat V(t,Xt) can be expressed as a linear combination of orthonormal basis7functions (pi(X) such as Laguerre, Hermite, Leg
45、endre, Chebyshev or Jacobipolynomials. In practical implementation, C(t,Xt) is approximated byCI(t,Xt) =Isummationdisplayi=1Aipi(Xt),Ai R (3.5)where I is the number of basis functions we used in estimation. Once thissubset of basis functions has been specified, CI(t,Xt) is estimated by pro-jecting o
46、r regressing the discounted values of V(t + 1,Xt+1), which is givenby: V(T,XT) = D(T,XT)7, onto the basis functions for the paths where theoption is in-the-money8 at time t.Least-squares method has been proved to be a efficient and accurate ap-proach to price derivatives in Monte Carlo. However, its
47、 accuracy dependson the choice of a “good” set of basis functions used in the regression. Thatsthe main drawback of this method. If we did not choose good basis func-tions (regression variables), we would throw away useful information and finda biased price. Moreno but forcomplex options, the choice
48、 of basis functions is not clear, since combiningpolynomials with other functions to represent the information set at eachexercise date is necessary. For these options, their conclusion is that therobustness seems not to be guaranteed and the type and numbers of basisfunctions can slightly affect op
49、tion prices 5.And the choice of basis functions 1 has implications for the statisticalsignificance of individual basis functions in the regression. In particular, somechoices of basis functions are highly correlated with each other, resulting inestimation difficulties for individual regression coefficients akin to the multi-colinearity problem in econometrics. If the choice of basis functions leads toa cross-moment matrix that is nearly singular, then numerical inaccuraciesin some least-squares algori
