ON THE USE OF NUMERAIRES IN OPTION PRICING by Simon ...

ON THE USE OF NUMERAIRES IN OPTIONPRICINGbySimon Benninga*Tomas Bjork**Zvi Wiener***Working Paper No 13/2002 June 2002* Faculty of Management, Tel Aviv University, ISRAEL.e-mail: benniga@post.tau.ac.il** Department of Finance, Stockholm School of Economics, SWEDEN.e-mail: fintb@hhs.se*** School of Business, Hebrew University of Jerusalem, ISRAEL.e-mail: mswiener@mscc.huji.ac.ilBenninga acknowledges a grant from the Israel Institute of Business Research atTel Aviv University.Wiener’s research was funded by the Krueger Center at the Hebrew University.IIBR working papers are intended for preliminary circulation of tentative researchresults. Comments are welcome and should be addressed directly to the authors.

On the Use of Numeraires in Option PricingSimon Benninga ∗Faculty of ManagementTel-Aviv University ISRAELe-mail: benninga@post.tau.ac.ilTomas BjörkDepartment of FinanceStockholm School of EconomicsSWEDENe-mail: fintb@hhs.seZvi Wiener †School of BusinessHebrew University of JerusalemISRAELe-mail: mswiener@mscc.huji.ac.ilDecember 1, 2001AbstractIn this paper we discuss the significant computational simplificationthat occurs when option pricing is approached through the change ofnumeraire technique. The original impetus was a paper (Hoang, Powell,Shi 1999) on endowment options recently published in this journal; in thepresent paper we extend these results to the case of stochastic interestrates. We also discuss four additional option pricing problems within theframework of a change of numeraire:• Pricing savings plans which incorporate a choice of linkage.• Pricing convertible bonds.• Pricing employee stock ownership plans• Pricing options where the strike price is in a currency different fromthe stock price.∗ Benninga acknowledges a grant from the Israel Institute of Business Research at Tel-AvivUniversity.† Wiener’s research was funded by the Krueger Center at the Hebrew University.1

Contents1 Introduction 32 The change of numeraire approach 43 Endowment warrants 73.1 Institutionalsetup .......................... 73.2 Mathematicalmodel ......................... 74 Pricing savings plans with choice of linkage 104.1 Institutionalsetup .......................... 104.2 Mathematicalmodel ......................... 115 Pricing convertible bonds 135.1 Institutionalsetup .......................... 135.2 Mathematicalmodel ......................... 146 Employee stock ownership plans 166.1 Institutionalsetup .......................... 166.2 Mathematicalmodel ......................... 167 Options with a foreign-currency strike price 197.1 Institutionalsetup .......................... 197.2 Mathematicalmodel ......................... 207.3 Pricingtheoptionindollars..................... 217.4 Pricingtheoptiondirectlyinpounds................ 228 Conclusion 242

1 IntroductionIn this paper we explore five possible applications of the numeraire method inoption pricing. While the numeraire methodiswell-knowninthetheoreticalliterature, it appears to be infrequently used in more applied papers, and manypractitioners seem to be unaware of how to use it as well as when it is profitable(or not) to use it. In order to illustrate the uses (and possible misuses) ofthe method we discuss in some detail five concrete applied problems in optionpricing:² Pricing endowment warrants.² Pricing savings plans which incorporate a choice of linkage.² Pricing convertible bonds within a stochastic interest rate franework.² Pricing employee stock ownership plans (ESOPs)² Pricing options where the strike price is in a currency different from thestock price.The standard Black-Scholes (BS) formula prices a European option on an assetthat follows a geometric Brownian motion. The asset’s uncertainty is the onlyrisk factor in the model. A more general approach developed by Black-Merton-Scholes leads to a partial differential equation. The most general method developedso far for the pricing of contingent claims is the martingale approachto arbitrage thory developed by Harrison-Kreps (1981), Harrison-Pliska (1981)and others. However, whether one uses the PDE or the standard “risk neutralvaluation” formulas of the martingale method, it is in most cases very hard toobtain analytic pricing formulas. Thus, for many important cases, special formulas(typically modifications of the original BS formula), were developed. SeeHaug (1997) for an extensive set of examples.One of the most typical cases of several risk factors occurs when an option isto choose among two assets with stochastic prices. In such a case it is oftenof considerable advantage to use a change of numeraire in the pricing of theoption. In what follows we demonstrate examples where the numeraire approachleads to significant simplifications but, in order not to oversell the method, alsoexamples where the numeraire change is trivial or where an obvious numerairechange really does not simplify the computations. The main message is still thatin many cases the change of numeraire approach leads to a drastic simplificationof the computational work.In section 2 we start with a brief introductory review of the numeraire method,followed by a mathematical summary (which can be skipped on first readingof this paper). In sections 3-7 we then present five different option pricingproblems. For each problem we present the possible choices of numeraire, discussthe pros and cons of the various numeraires, and compute the option prices.3

2 The change of numeraire approachThe basic idea of the numeraire approach can be described as follows: Supposethat an option’s price depends on several (say n) sources of risk. We may thencompute the price of the option according to the following scheme:² Fix a security which embodies one of the sources of risk, and choose thissecurity as the numeraire.² Express all prices on the market, including that of the option, in terms ofthechosennumeraire. Inotherwords, we perform all our computationsin a relative price system.² Sincethenumeraireassetinthenewpricesystemisriskless(bydefinition)we have decreased the number of risk factors by one from n to n¡1. If, forexample, we started out with two sources of risk, we can now often applystandard one-risk-factor option pricing formulas (such as Black-Scholes).² We thus derive the option price in terms of the numeraire. A simpletranslation from the numeraire back to the local currency will then givethe price of the option in monetary terms.These ideas were developed independently by Geman (1989) and Jamshidian(1989). 1 The standard reference in an abstract setting is Geman, et.al. (1995).In the remainder of this section, we consider a Markovian framework which issimpler than that of the last paper, but which is still reasonably general. Alldetails and proofs can be found in Björk (1999). 2Assumption 2.1 The following objects are given a priori.² An empirically observable (k +1)-dimensional stochastic processwith the notational conventionX =(X 1 ,...,X k+1 ),X k+1 (t) =r(t).² We assume that under a fixed risk neutral martingale measure Q, thefactor dynamics have the formdX i (t) =µ i (t, X(t)) dt + δ i (t, X(t)) dW (t),i =1,...,k+1,where W =(W 1 ,...,W d ) ? is a standard d-dimensional Q-Wiener process.The superscript ? denotes transpose.1 The earliest incarnation of a similar idea is to be found in papers by Fischer (1978) andGalai (1983).2 The remainder of this section can be skipped by readers interested only in the implementationof the numeraire method.4

² A risk free asset (money account) with the dynamicsdB(t) =r(t)B(t)dt.The interpetation of this is that the components of the vector process X are theunderlying factors in the economy. We make no a priori market assumptions,so whether or not a particular component is the price process of a traded assetin the market will depend on the particular application. We now also introduceasset prices, driven by the underlying factors, in the economy.Assumption 2.2² We consider a fixed set of price processes S 0 (t),...,S n (t), each of which isassumed to be the arbitrage free price process for some traded asset withoutdividends.² Under the risk neutral measure Q, theS-dynamics have the formfor i =0,...,n¡ 1.dS i (t) =r(t)S i (t)dt + S i (t)² The nth asset price is always given bydXσ ij (t, X(t))dW j (t), (1)j=1S n (t) =B(t),and thus (1) also holds for i = n with σ nj =0for j =1,...,d.We now fix an arbitary asset as the numeraire, and for notational conveninenceweassumethatitisS0 . We may then express all other asset pricesin terms of the numeraire S 0 , thus obtaining the normalized price vectorZ =(Z 0 ,Z 1 ,...,Z n ), defined byZ i (t) = S i(t)S 0 (t) .The main result is the following theorem, which shows how to price an arbitrarycontingent claim in terms of the chosen numeraire. For brevity, a contingentclaim with exercise date T will henceforth be referred to as a “T -claim”.Theorem 2.1 (Main theorem) Let the numeraire S 0 be the price process foratradedasssetwithS 0 (t) > 0 for all t. Then there exists a probability measure,denoted by Q 0 , with the following properties.² For every T -claim Y, the corresonding arbitrage free price process Π (t; Y)is given by∙ ¸ YΠ (t; Y) =S 0 (t)Et,X(t)0 , (2)S 0 (T )where E 0 denotes expectation w.r.t. Q 0 .5

² The Q 0 -dynamics of the Z-processes are given bydZ i = Z i [σ i ¡ σ 0 ] dW 0 , i =0,...,n. (3)² The Q 0 -dynamics of the price processes are given bywhere W 0 is a Q 0 -Wiener process.dS i = S i (r + σ i σ ? 0) dt + S i σ i dW 0 , (4)² The Q 0 -dynamics of the X-processes are given bydX i (t) = ¡ µ i + δ i σ ? 0¢dt + δ i dW 0 (t). (5)² The measure Q 0 depends upon the choice of numeraire asset S 0 ,butthesame measure is used for all claims, regardless of their exercise dates.In passing we note that if we use the money account B as the numeraire, then thepricing formula above reduces to the well known standard risk neutral valuationformulaΠ (t; Y) =B(t)E 0 t,X(t)∙YB(T )¸R= Et,X(t)∙e 0 − Tt¸r(s)ds YIn more pedestrian terms, the main points oftheTheoremaboveareasfollows.² The pricing formula (2) shows that the measure Q 0 “takes care of thestochasticity” related to the numeraire S 0 . Note that we do not have tocompute the price S 0 (t)—we simply use the observed market price. Wealso see that if the claim Y is of the form Y = X¢S 0 (T ) then the changeof numeraire is a huge simplification of the standard risk neutral formula(6): Instead of computing the joint distribution of R Tr(s)ds and Y (undertQ) we only have to compute the distribution of X (under Q 0 ).² Formula (3) says that the normalized price processes are martingales (i.e.zero drift) under Q 0 ,andidentifies the relevant volatility.² Formulas (4)-(5) shows how the dynamics of the asset prices and the underlyinga factors change when we move from Q to Q 0 . Note that thecrucial object is the volatility σ 0 if the numeraire asset.In the following sections we show examples of the use of the numeraire methodwhich illustrate the considerable conceptual and implementational simplificationto which this method leads.(6)6

3 Endowment warrantsEndowment options, which are primarily traded in Australia and New Zealand,were recently discussed in Hoang-Powell-Shi (1999, henceforth HPS), where thearbitrage free price of the warrant was derived in the case of a deterministic shortrate. The authors in HPS also provide an approximation of the option price forthe stochastic interest rate case. Our contribution in this section is to prove thatthe exact option pricing formula of HPS, for the case of a deterministic shortrate, is in fact invariant under the introduction of stochastic interest rates, thusmaking an approximation procedure unnecessary.3.1 Institutional setupAn endowment option is a very long term call option. Typically we have thefollowing setup:² At issue, the initial strike price K 0 is set to approximatively 50% of thecurrent stock price, so the option is initially deep in the money.² The endowment options are European.² The time to exercise is typically 10+ years.² The options are interest rate and dividend protected. The protection isperformed by the following two adjustments:— Thestrikepriceisnotfixedovertime.Insteaditisincreasedbytheshort-term interest rate.— The strike price is decreased by the size of the dividend each time adividend is paid.² The payoff attheexercisedateT is that of a standard call option, butwith the adjusted (as above) strike price K T .3.2 Mathematical modelWe model the underlying stock price process S t in a standard Black-Scholessetting. In other words, under the objective probability measure P , the priceprocess S t follows Geometrical Brownian Motion (between dividends) as:dS t = αS t dt + S t σW t ,where α and σ are deterministic constants. We allow the short rate r t to bean arbitrary random process, thus giving us the following P -dynamics of themoney-market account:dB t = r t B t dt, (7)B 0 = 1. (8)7

In order to analyze this option we have to formalize the protection features ofthe option. This is done in the following way.² We assume that the strike price process K t is changed at the continuouslycompounded instantaneous interest rate. The formal model is thus asfollowsdK t = r t K t dt. (9)² For simplicity we assume (see Remark 3.1 below) that the dividend protectionis perfect. More precisely we assume that the dividend protectionis done by reinvesting the dividends into the stock itself. Under this assumptionwe can view the stock price as the theoretical price of a mutualfund which includes all dividends invested in the stock. Formally thisimplies that we can treat the stock price process S t defined above as theprice process of a stock without dividends.The value of the option at the exercise date T is given by the contingent claimX ,defined byX =max[S T ¡ K T , 0]Clearly there are two sources of risk in endowment options: The stock price riskand the risk of the short-term interest rate. In order to analyze this option, weobserve that from (7)-(9) it follows thatThus we can express the claim X asK T = K 0 B T .X =max[S T ¡ K 0 B T , 0]and from this expression we see that the natural numeraire process is nowobviously the money account B t . The martingale measure for this numeraire isthe standard risk neutral martingale measure Q under which we have the stockprice dynamicsdS t = r t S t dt + S t σdWt B , (10)where W B is a Q-Wiener process.A direct application of Theorem 2.1 gives us the pricing formula∙ ¸1Π (0; X )=B 0 E Q max [S T ¡ K 0 B T , 0] .B TAfter a simple algebraic manipulation, and using the fact that B 0 =1,wethusobtainΠ (0; X )=E Q [max [Z T ¡ K 0 , 0]] (11)where Z t = S t /B t is the normalized stock price process. It follows immediatelyfrom (7), (10), and the Itô formula that under Q we have Z-dynamics given bydZ t = Z t σdW B t . (12)8

and from (11)-(12) we now see that our original pricing problem has been reducedto that of computing the price of a standard European call, with strikeprice K 0 , on an underlying stock with volatility σ in a world where the shortrate is zero. Thus the Black-Scholes formula gives the endowment warrant priceat t =0directlyaswhereC EW = Π (0; X )=S 0 N(d 1 ) ¡ KN(d 2 ) (13)d 1 = ln (S 0/K 0 )+ 1 2 σ2 Tσ p Td 2 = d 1 ¡ σ p T.Using the numeraire approach price of the endowment option in (13) is givenby a standard Black-Scholes formula for the case where r =0. Theresultdoesnot in any way depend upon assumptions made about the stochastic short rateprocess r t .The pricing formula (13) was in fact earlier derived in HPS, but only for thecase of a deterministic short rate. The case of a stochastic short rate is nottreatedindetailinHPS.InsteadtheauthorsofHPSattempttoincludetheeffect of a stochastic interest rate by introducing the following scheme:² They assume that the short rate r is deterministic and constant.² The strike price process is assumed to have dynamics of the form,dK t = rK t dt + γdV twhere V is a new Wiener process (possibly correlated with W ).² They then go on to value the claim X =max[S T ¡ K T , 0] by using theMargrabe (1978) result about exchange options.The claim made in HPS is that this setup is an approximation to the case ofa stochastic interest rate. Whether it is a good approximation or not is neverclarified in HPS, and from our analysis above we see that the entire schemeis in fact unnecessary, since the pricing formula (13) is invariant under theintroduction of a stochastic short rate.Remark 3.1 We note that the result above relies upon our simplifying assumptionabout perfect dividend protection. 3 A more realistic modeling of thedividend protection would lead to severe computational problems. To see thisassume that the stock pays a constant dividend yield rate δ. This would change3 In reality is performed by reducing the strike price by the dividend amount with therestriction that the resulting strike is never negative.9

our model in two ways: The Q-dynamics of the stock price would be different,and the dynamics of the strike process K t would have to be changed.As for the Q-dynamics of the stock price, standard theory immediately gives usdS t =(r t ¡ δ)S t dt + S t σdW B t .Furthermore, from the institutional description above we see that in real life(as opposed to in our simplified model), the dividend protection is done bydecreasing the strike price process with the dividend amount at every dividendpayment. In terms of our model this means that over an infinitesimal interval[t, t + dt],thestrikepriceshoulddecreasewiththeamountδS t dt. Thus theK t -dynamics are given bydK t =(r t K t ¡ δS t ) dt.This equation can be solved asR TZ T RK T = ertdt Tr0 K 0 ¡ δ eu du t S t dt0The moral is that in the expression of the contingent claimX =max[S T ¡ K T , 0]we now have the unpleasant integral expressionZ T R Treu du t S t dt.0Even in the simple case of a deterministic short rate this integral is quite problematic.It is then basically a sum of lognormally distributed random variables,and thus we have the same hard computational problems as in the case of anAsian option.4 Pricing savings plans with choice of linkageThese plans are common. Typically they give savers an ex-post choice of interestrates to be paid on their account. With the inception of capital requirements,many financial institutions have to recognize these options and price them.4.1 Institutional setupWe use the example of a common bank account from the Israeli context; thisaccount gives savers the ex-post choice of indexing their savings to an Israelishekelinterest rate or a US dollar rate.10

² The saver deposits NIS 100 (“NIS”= Israeli shekels) today in a shekel/dollarsavings account with a maturity of 1 year.² In one year, the account pays the maximum of:— NIS 100, indexed to the inflation rate + a shekel interest rate.— Today’s dollar equivalent of NIS 100 + dollar interest rate.The savings plan is thus an option to exchange the Israeli interest rate for theUS interest rate, while at the same time taking on the exchange rate risk. Sincethe choice is made ex-post, it is clear that both the shekel and the dollar interestrates offered on such an account must be below their respective market rates.4.2 Mathematical modelIn this section we derive the value of the exchange option described above; theresult is given in equation (18) below.We consider two economies, one domestic and one foreign, and we introduce thefollowing notation.r d = domestic short rater f = foreign short rateI t = domestic inflation processX t = the exchange rate in terms of domestic currency/foreign currency.Y t = Xt−1 = the exchange rate in terms of foreign currency/ domestic currency.T = the maturity of the savings plan.The value of the option is linear in the initial shekel amount invested in thesavings plan; without loss in generality, we assume that this amount is 1 shekel.In the domestic currency the contingent T -claim Ξ d to be priced, is thus givenbyΞ d =max £ e rdT I T ,X0 −1 er f T ¤X TIn the foreign currency the claim Ξ f is given byΞ f =max £ e rdT I T Y T ,Y 0 e r f T ¤It turns out that it is easier to work with Ξ f than with Ξ d , and we haveΞ f =max £ e r dT I T Y T ¡ Y 0 e r f T , 0 ¤ + Y 0 e r f T .The price (in the foreign currency) at t = 0 of this claim is now given byΠ (0; Ξ f ) = e −r f T E Q £ fmax © e rdT I T Y T ¡ e r f T Y 0 , 0 ª ¡ e r f T ¤Y 0h noi= E Q fmax e (r d−r f )T I T Y T ¡ Y 0 , 0 + Y 0 , (14)11

where Q f denotes the risk neutral martingale measure for the foreign market.At this point we have to make some probabilistic assumptions, and in fact weassume that we have a Garman-Kohlhagen model for Y . Standard theory thengives us the Q f dynamics of Y asdY t = Y t (r f ¡ r d )dt + Y t σ Y dW t . (15)For simplicity we assume that also the inflation follows a geometric Brownianmotion, with Q f -dynamics given bydI t = I t α I dt + I t σ I dW t . (16)Note that W is assumed to be two-dimensional, thus allowing for correlationbetween Y and I. Also note that economic theory does not say anything aboutthe mean inflation rate α I under Q f .When computing the expectation in (14) we cannot use a standard change ofnumeraire technique, the reason being that none of the processes Y , I or Y ¢I areprice processes of traded assets without dividends. Instead we have to attackthe expectation directly.To that end we define the process Z as Z t = Y t ¢ I t and obtain the followingQ f -dynamics.dZ t = Z t (r f ¡ r d + α I + σ Y σ ? I ) dt + Z t (σ Y + σ I ) dW t .¿Fromthisitiseasytoseethatifwedefine S t bythen we will have the Q f -dynamicsS t = e −(r f −r d +α I +σ Y σ ? I )t Z t ,dS t = S t (σ Y + σ I ) dW t ,thepointbeingthatwecaninterperetS t as a stock price in a Black-Scholesworld with zero short rate and Q f as the risk neutral measure. With thisnotation we obtain easilyΠ (0; Ξ f )=e cT E Q £ £f max ST ¡ e −cT Y 0 , 0 ¤¤ + Y 0 ,wherec = α I + σ Y σ I ? .The expectation above can now be expressed by the Black-Scholes formula fora call option with strike price e −cT Y 0 , zero short rate and a volatility given byqσ = kσ Y k 2 + kσ I k 2 +2σ Y σI?12

The price, at t = 0 of the claim, expressed in the foreign currency is thus givenby the formulaΠ (0; Ξ f ) = e cT I 0 Y 0 N[d 1 ] ¡ Y 0 N[d 2 ]+Y 0 , (17)d 1 = ln (I 0)+ ¡ c + 1 2 σ2¢ Tσ p Td 2 = d 1 ¡ σ p T.Finally, the price at t = 0 in domestic terms is given byΠ (0; Ξ d )=X 0 Π (0; Ξ f )=e cT I 0 N[d 1 ] ¡ N[d 2 ]+1. (18)Remark 4.1 For practical purposes it may be more convenient to model Y andI asdY t = Y t (r f ¡ r d )dt + Y t σ Y dWt Y ,dI t = I t α I dt + I t σ I dWt I ,where now σ Y and σ Y are constant scalars, whereas W Y and W I are scalarWiener processes with local correlation given by dWt Y dWtI = ρdt.In this model (which of course is logically equivalent to the one above) we havethe pricing formulas (17)-(18), but now with the notationc = α I + ρσ Y σ I ,qσ = σY 2 + σ2 I +2ρσ Y σ I,5 Pricing convertible bondsStandard pricing models of convertible bonds concentrate on pricing the bondand its conversion option at date t = 0 (see, for example, Brennan-Schwartz1977, Bardhan et.al. 1993). A somewhat less-standard problem is the pricing ofthe bond at some date 0

² The bond matures at a fixed date T 1 .² The underlying stock pays no dividends² At a fixed date T 0 ,withT 0

Assumption 5.1 Define, as usual, the forward rates by f(t, T )=¡ ∂∂Tln p(t, T ),We now make the following assumptions, all under the risk neutral martingalemeasure Q.² The bond market can be described by an HJM model for the forward ratesof the formà Z !Tdf (t, T )= σ f (t, T ) σ f(t, ? u)du dt + σ f (t, T )dW ttwhere the volatility structure σ f (t, T ) is assumed to be deterministic. Wis a (possibly multidimensional) Q-Wiener process.² The stock price follows a geometric Brownian motion, i.e.dS t = r t S t dt + S t σ S dW t ,where r t = f(t, t) is the short rate. The row vector σ S is assumed to beconstant and deterministic.In essence we have thus assumed a standard Black-Scholes model for the stockprice S, and a Gaussian forward rate model. The point of this is that it willlead to (see below) a lognormal distribution for Z, thus allowing us to use astandard Black-Scholes formula. From the forward rate dynamics above if nowfollows (Björk (1999), prop. 15.5) that we have bond price dynamics given bydp(t, T )=r t p(t, T )dt ¡ p(t, T )Σ p (t, T )dW t ,where the bond price volatility is given byΣ p (t, T )=Z Ttσ f (t, u)du.We may now attack the expectation in (19), and to this end we compute theZ-dynamics under Q T1 . It follows directly from the Itô formula that the Q-dynamics of Z are givendZ t = Z(t)α Z (t)dt + Z t fσ S + Σ p (t, T 1 )g dW twhere for the moment we do not bother about the drift process α Z .Furthermorewe know from the general theory (see Theorem 2.1) that the following hold² The Z processisaQ T 1martingale (i.e. zero drift term).² The volatility does not change when we change measure from Q to Q T 1.The Q T 1dynamics of Z are thus given bydZ t = Z t σ Z (t)dW 1 t (20)15

whereσ Z (t) =σ S + Σ p (t, T 1 ), (21)and where W 1 is a Q T 1Wiener process.Under the assumptions above the volatility σ Z is deterministic, thus guaranteeingthat Z has a lognormal distribution. We can in fact writedZ t = Z t kσ Z (t)k dV 1t ,where V 1 is a scalar Q T 1Wiener process. We may thus use a small variation ofthe Black-Scholes formula to obtain the final pricing resultProposition 5.1 The price, at t, of the convertible bond is given by the formulaΠ (t; X )=S t N[d 1 ] ¡ p(t, T 1 )N[d 2 ]+p(t, T 1 ),whered 1 =µ µ 1Stp ln+ 1 σ2 (t, T 0 ) p(t, T 1 ) 2 σ2 (t, T 0 ) ,d 2 = d 1 ¡ p σ 2 (t, T 0 ),σ 2 (t, T 0 ) =Z T0σ Z (t) = σ S +tkσ Z (u)k 2 du,Z T1tσ f (t, s)ds6 Employee stock ownership plans6.1 Institutional setupIn employee stock ownership plans (ESOP) it is common to include an optionof essentially the following form: The holder has the right to buy a stock at theminimum between its price in 6 months and in 1 year minus a rebate (say 15%).The exercise is one year.6.2 Mathematical modelIn a more general setting the ESOP is a contingent claim X ,tobepayedoutat time T 1 ,oftheformX = S T ¡ β min [S T1 ,S T0 ] , (22)sointheconcretecaseabovewewouldhaveβ =0.85, T 0 =1/2 andT 1 =1.The problem is to price X at some time t · T 0 , and to this end we assume a16

standard Black-Scholes model where, under the usual risk neutral measure Qwe have the dynamicsdS t = rS t dt + σS t dW t , (23)dB t = rB t dt, (24)with a deterministic and constant short rate r. The price Π (t; X ) of the optioncan obviously be writtenwhere the T 1 -claim Y is defined byΠ (t; X )=S t ¡ βΠ (t; Y)Y =min[S T1 ,S T0 ] .In order to compute the price of Y we now basically want to do as follows.² Perform a suitable change of numeraire.² Use a standard version of some well known option pricing formula.The problem with carrying out this small program is that, at the exercise timeT 1 ,thetermS T0 does not have a natural interpretation as a spot price of atraded asset. In order to overcome this difficulty we therefore introduce a newasset S 0 defined by½St 0 S=t , 0 · t · T 0 ,S T0 e r(t−T0) , T 0 · t · T 1 .In other words, S 0 can be thought of as the value of a self financing portfoliowhere you at t = 0 buy one share of the underlying stock and keep it untilt = T 0 .Att = T 0 you then sell the share and put all the money into the bankaccount.We then have ST 0 1= S T0 e r(T1−T0) so we can now express Y in terms of ST 0 1asY =min £ S T1 ,K¢ ST 0 ¤1(25)whereK = e −r(T1−T0) (26)The point of this is that ST 0 1in (25) can formally be treated as the price at T 1 ofatradedasset. Infact,fromthedefinition above we have the following trivialQ-dynamics for S 0dSt 0 = rSt 0 dt + σt 0 St 0 dW twhere the deterministic volatility is defined byσ 0 t =½ σ, 0 · t · T0 ,0, T 0 · t · T 1 .(27)17

It is now time to perform a change of numeraire, and we can choose either Sor S 0 as the numeraire. From a logical point of view the choice is irrelevant,but the computations become somewhat easier if we choose S 0 .WithS 0 as thenumeraire we obtain (always with t

whereΠ (t; ESOP) =S t ¡ βS t N[d 1 ]+βS t KN[d 2 ], (30)and where K is given by (26).d 1 = ln (1/K)+ 1 2 σ2 (T 1 ¡ T 0 )σ p ,T 1 ¡ T 0d 2 = d 1 ¡ σ p T 1 ¡ T 0 ,7 Options with a foreign-currency strike priceIn this section we discuss options whose strike price is linked to a non-domesticcurrency. We illustrate with the example of an option with a US dollar strikeprice on a stock denominated in UK pounds. Such options might be part ofan executive compensation program; such options might be given to motivatemanagers to maximize the dollar price of their stock. Another example is anoption where the strike price is CPI-indexed.7.1 Institutional setupFor purposes of illustration we assume that the underlying security is traded inthe UK in pound sterling and that the option exercise price is in dollars. Theinstitutional setup is as follows.² The option is initially (i.e. at t = 0) an at-the-money option, when thestrike price is expressed in pounds. 4² This pound strike price is, at t = 0, converted into dollars.² The dollar strike price thus computed is kept constant during the life ofthe option.² Attheexercisedatet = T the holder can pay the fixed dollar strike pricein order to obtain the underlying stock.² The option is fully dividend protected.Since the stock is traded in pounds, the fixed dollar strike corresponds to arandomly changing strike price when expressed in pounds; thus we have a nontrivialvaluation problem. The numeraire approach can be used to simplify thevaluation of such an option. The resulting valuation is given in (36).4 For tax reasons most executive stock options are initially at-the-money.19

7.2 Mathematical modelWe model the stock price S (in pounds) as a standard geometric Brownianmotion under the objective probability measure P , and we assume deterministicshort rates r p and r d in the UK and the US market respectively. Since we haveassumed complete dividend protection we may as well assume (from a formalpoint of view) that S is without dividends. We thus have the following P -dynamics for the stock price.dS t = αS t dt + S t δ S W S t ,We denote the dollar/pound exchange rate by X, and assume a standard Garman-Kohlhagen (1983) model for X. WethushaveP -dynamics given bydX t = α X Xdt + X t δ X dW X t ,Denoting the pound/dollar exchange rate by Y ,whereY =1/X, we immediatelyhave the dynamicsdY t = α Y Ydt+ Y t δ Y dW Y twhere α Y is of no interest for pricing purposes. Here W S , W X and W Y arescalar Wiener processes and we have the relationsδ Y = δ X (31)W Y = ¡W X , (32)dWt S ¢ dWt X = ρdt, (33)dWt S ¢ dWt Y = ¡ρdt. (34)For computational purposes it is sometimes convenient to express the dynamicsin terms of a two dimensional Wiener process W with independent componentsintead of using the two correlated processes W X and W S . Logically the twoapproaches are equivalent, and in the new W -formalismwethenhavetheP -dynamicsdS t = αS t dt + S t σ S W t ,dX t = α X Xdt + X t σ X dW t ,dY t = α Y Y t dt + Y t σ Y dW T .The volatilities σ S , σ X and σ Y are two-dimensional row vectors with the propertiesthatσ Y = ¡σ Xkσ X k 2 = δX,2kσ Y k 2 = δY 2 ,kσ S k 2 = δS,2σ X σ S ? = ρδ X δ Sσ Y σ ? S = ¡ρδ Y δ S20

where ? denotes transpose.The initial strike price expressed in pounds is by definition given byK p 0 = S 0,and the corresponding dollar strike price is thusK d = K p 0 ¢ X 0 = S 0 X 0 .The dollar strike price is kept constant until the exercise date. However, expressedin pounds the strike price evolves dynamically as a result of the varyingexchange rate, so the pound strike at maturity is given byK p T = Kd ¢ X −1T= S 0 ¢ X 0 ¢ X −1T . (35)There are now two natural ways to value this option: we can work in dollars orin pounds, and initially it is not obvious which way is the easier. We will in factperform the calculations in both alternatives and compare the computationaleffort. As will be seen below it turms out to be slightly easier to work in dollarsthan in pounds.7.3 Pricing the option in dollarsIn this approach we transfer all data into dollars. The stock price, expressed indollars, is given bySt d = S t ¢ X t ,so in dollar terms the payout Ξ of the option at maturity is given by the expressionΞ d =max £ S T X T ¡ K d , 0 ¤Since the dollar strike K d is constant we can use the Black-Scholes formulaapplied to the dollar price process St d . The Itô formula applied to Std = S t X timmediately gives us the P -dynamics of Std asWe can write this asdS d t = S d t (α + α X + σ S σ ? X) dt + S d t (σ S + σ X ) dW tdS d t = S d t (α + α X + σ S σ ? X) dt + S d t δ S,d dV twhere V is a scalar Wiener process and whereqδ S,d = kσ S + σ X k = δS 2 + δ2 X +2ρδ Sδ Xis the dollar volatility of the stock price.21

The dollar price (expressed in dollar data) at t of the option is now obtaineddirectly from the Black-Scholes formula asCt d = St d N[d 1 ] ¡ e −r d(T −t) K d N[d 2 ], (36)ln ¡ St d /K d¢ ³ ´+ r d + 1 2 δ2 S,d(T ¡ t)d 1 =p ,δ S,d T ¡ tpd 2 = d 1 ¡ δ S,d T ¡ t.The corresponding price in pound terms is finally obtained asso we have the final pricing formulaC p t = C d t ¢1X t,C p t = S t N[d 1 ] ¡ e −r d(T −t) S 0X 0N[d 2 ], (37)X³ ´ ³t´lnSt X tS 0 X 0+ r d + 1 2 δ2 S,d(T ¡ t)d 1 =p ,δ S,d T ¡ tpd 2 = d 1 ¡ δ S,d T ¡ t,qδ S,d = δS 2 + δ2 X +2ρδ Sδ X7.4 Pricing the option directly in poundsAlthough this is not immediately obvious, pricing the option directly in poundsis a bit more complicated than pricing the option in dollars. The pricing problem,expressed in pound terms, is that of pricing the T -claim Ξ p defined byΞ p =max[S T ¡ K p T , 0] .Using (35) and denoting the pound/dollar exchange rate by Y (where of courseY =1/X) weobtain∙¸Ξ P Y T=max S T ¡ S 0 , 0 .Y 0It is now tempting to use the punds/dollar exchange rate Y as the numeraire butthis is not allowed. The reason is that although Y is the price of a traded asset(dollar bills) it is not the price of a traded asset without dividends, the obviousreason being that dollars are put into an American (or perhaps Eurodollar)account where they will command the interest rate r d . Thus the role of Y israther that of the price of an asset with the continuous dividend yield r d . Inorder to convert the present situation into the standard case covered by Theorem2.1 we therefore do as follows.22

² We denote the dollar bank account by B d with dynamicsdB d t = r d B d t dt.² The value in pounds, of the dollar bank account, is then given by theprocess Ŷt, defined byŶ dt = B d t ¢ Y t = Y t e r dt .² The process Ŷt can now be interpreted as the price process (denoted inpounds) of a traded asset without dividends.² We may thus use Ŷt as a numeraire.SincewehaveY T = ŶT e −rdT we can write∙Ξ P =max S T ¡ ỸT e −r dT S ¸0, 0 .Y 0Using Ŷ as the numeraire we immediately obtain from Theorem 2.1Π ¡ t; Ξ P ¢ = ŶtE Q Ŷt [max [Z T ¡ K, 0]] , (38)where QŶ denotes the martingale measure with Ŷ as the numeraire, where Zis defined byZ t = S t,Ŷ tand where K is given byK = e −r dT S 0. (39)Y 0¿From Theorem 2.1 we know that Z has zero drift under QŶ , and a simplecalculation shows that the QŶ dynamics of Z are given bydZ t = Z t (σ S ¡ σ Y ) dW t .Thus the expectation in (38) is given by the Black-Scholes formula for a call,with strike price K, written on an asset with (scalar) volatilityqδ Z = kσ S ¡ σ Y k =qkσ S k 2 + kσ Y k 2 ¡ 2σ S σY ? = δS 2 + δ2 Y +2ρδ Sδ Yin a world with zero interst rate. We thus obtain the pricing formulaΠ ¡ t; Ξ P ¢ = Ŷt fZ t N[d 1 ] ¡ KN[d 2 ]g½ µ 1 Ztd 1 = p ln + 1 ¾δ Z T ¡ t K 2 δ2 Z(T ¡ t) ,pd 2 = d 1 ¡ δ Z T ¡ t23

After simplification this reduces to the pricing formula, which of course coincideswith (37).Ct P = Π ¡ t; Ξ P ¢ = S t N[d 1 ] ¡ Y t e −r d(T −t) S 0N[d 2 ], (40)Y 0where½ µ ½1 St Yd 1 = p 0ln + r d + 1 ¾ ¾δ Z T ¡ t Y t S 0 2 δ2 Z (T ¡ t) ,pd 2 = d 1 ¡ δ Z T ¡ t,qδ Z = δS 2 + δ2 Y +2ρδ Sδ Y .In this case we have thus seen that there were two distinct (but logically equivalent)ways of pricing the option. From the computations above it is also clear(ex post) that the easiest way was by using the dollar bank account as thenumeraire, rather than using the pound value of the same account.8 ConclusionNumeraire methods have been in the derivative pricing literature since papersby Geman (1989) and Jamshidian (1989). These methods afford a considerablesimplification in the pricing of many complex options; however, they appearnottobewell-known. Inthispaperwehaveconsideredfive problems whosecomputation is vastly aided by the use of numeraire methods. The first of theseis the pricing of endowment options (discussed in the Journal of Derivatives ina recent paper by Hoang, Powell, and Shi (1999). We also discuss the pricing ofoptions where the strike price is denominated in a currency different from thatof the underlying stock, the pricing of savings plans where the choice of interestpaid is ex-post chosen by the saver, the pricing of convertible bonds and thepricing of employee stock ownership plans.Numeraire methods are not a panacea for complex option pricing. However,when there are several risk factors which impact an option’s price, choosing oneof the factors as a numeraire reduces the dimensionality of the computationalproblem by one. A clever choice of the numeraire can, in addition, lead to asignificant computational simplification in the option’s pricing.References[1] Bardhan I., A. Bergier, E. Derman, C. Dosemblet, I Kani, P. Karasinski(1993). Valuing convertible bonds as derivatives. Goldman Sachs QuantitativeStrategies Research Notes.[2] Björk T. (1999). Arbitrage Theory in Continuous Time, OxfordUniversityPress.24

[3] Brennan M.J., E.S. Schwartz (1977). Convertible bonds: Valuation andoptimal strategies for call and conversion. The Journal of Finance, 32,1699-1715.[4] Core J., W. Guay (2000). Stock option plans for non-executive employees,Working paper, Wharton School.[5] Fischer, S. (1978). Call option pricing when the exercise price is uncertain,and the valuation of index bonds. Journal of Finance, 33, 169-176.[6] Galai, D. (1983). Pricing of Optional Bonds. Journal of Banking and Finance,7, 323-337.[7] Geman, H. (1989). The Importance of the Forward Neutral Probability ina Stochastic Approach of Interest Rates. Working paper, ESSEC.[8] Geman, H., N. El Karoui and J. C. Rochet (1995). ”Change of Numeraire,Changes of Probability Measures and Pricing of Options.” Journal of AppliedProbability, 32, 443-458.[9] Harrison, J. & Kreps, J. (1981). Martingales and Arbitrage in MultiperiodSecurities Markets. Journal of Economic Theory 11, 418—443.[10] Harrison, J. & Pliska, S. (1981). Martingales and Stochastic Integrals inthe Theory of Continuous Trading. Stochastic Processes & Applications11, 215—260.[11] Haug E. G. (1997), The complete guide to option pricing formulas, McGrawHill.[12] Heath C., S. Huddart, and M. Lang (1999). Psychological factors and stockoption exercise, Quarterly Journal of Economics, 114, 601-628.[13] Hoang, Philip, John G. Powell, and Jing Shi (1999). Endowment WarrantValuation, Journal of Derivatives, 91-103.[14] Huddart S. (1994). Employee stock options, Journal Of Accounting AndEconomics 18, 2, 207-231.[15] Jamshidian, F. (1989). An Exact Bond Option Formula. Journal of Finance44, 205—209.[16] Margrabe W. (1978). The value of an option to exchange one asset foranother, Journal of Finance, 33, 1, 177-186.[17] Reiner E. (1992). Quanto mechanics, Risk Magazine, 5, 59-63.25