Notes on Stochastic Finance - NTU

1 Chapter 12. Change of Num raire and Forward Measures In this chapter we introduce the notion of num raire. This allows us to con- sider pricing under random discount rates using forward measures, with the pricing of exchange options (Margrabe formula) and foreign exchange op- tions (Garman-Kohlagen formula) as main applications. A short introduc- tion to the computation of self-financing hedging strategies under change of num raire is also given in Section The change of num raire technique and associated forward measures will also be applied to the pricing of bonds and interest rate derivatives such as bond options in Chapter 14. Notion of Num raire .. 431. Change of Num raire .. 434. Foreign Exchange .. 443. Pricing Exchange Options .. 450. Hedging by Change of Num raire .. 452. Exercises.

2 456. Notion of Num raire A num raire is any strictly positive (Ft )t R+ -adapted Stochastic process (Nt )t R+ that can be taken as a unit of reference when pricing an asset or a claim. In general, the price St of an asset, when quoted in terms of the num raire Nt , is given by St S t := , t R+ . Nt Deterministic num raires transformations are easy to handle as a change of num raire by a deterministic factor is a formal algebraic transformation that does not involve any risk. This can be the case for example when a currency is pegged to another currency, the exchange rate from Euro to 431. N. Privault French Franc has been fixed on January 1st, 1999. On the other hand, a random num raire may involve risk and allow for arbitrage opportunities. Examples of num raire processes (Nt )t R+ include: - Money market account.

3 Given (rt )t R+ a possibly random, time-dependent and (Ft )t R+ -adapted risk-free interest rate process, let . w . t Nt := exp rs ds . 0. In this case, St rt S t = = e 0 rs ds St , t R+ , Nt represents the discounted price of the asset at time 0. - Currenty exchange rates. In this case, Nt := Rt denotes the SGD/EUR (SGDEUR=X) exchange rate between a domestic currency (SGD) and a foreign currency (EUR), one unit of local currency (SGD) corresponds to Rt units in foreign currency (EUR). Let St S t := , t R+ , Rt denote the price of a foreign (EUR) asset quoted in units of the lo- cal currency (SGD). For example, if Rt = and St = e1, then S t = St /Rt = St ' S$ , and 1/Rt is the foreign EUR/SGD. exchange rate. - Forward num raire. The price P (t, T ) of a bond paying P (T , T ) = $1 at maturity T can be taken as num raire.

4 In this case we have h rT i Nt := P (t, T ) = IE e t rs ds Ft , 0 6 t 6 T. Recall that . Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist , Kenneth E. Boulding, in: Energy Reorganization Act of 1973: Hearings, Ninety-third Congress, First Session, on 11510, page 248, United States Congress, Government Printing Office, 1973. 432 ". This version: January 16, 2019. Change of Num raire and Forward Measures My foreign currency account St grew by 5%. My foreign currency account St grew by 5%. this year. this year. The foreign exchange rate dropped by 10%. Q: Did I achieve a positive return? Q: Did I achieve a positive return? A: A: (a) Scenario A. (b) Scenario B. Fig. : Why change of num raire? rt h rT i t 7 e 0. rs ds P (t, T ) = IE e 0 rs ds Ft , 0 6 t 6 T, is an Ft - martingale.

5 - Annuity num raires. Processes of the form n X. Nt := (Tk Tk 1 )P (t, Tk ), 0 6 t 6 T1 , k =1. where P (t, T1 ), P (t, T2 ), .. , P (t, Tn ) are bond prices with maturities T1 <. T2 < < Tn arranged according to a tenor structure. - Combinations rt f of the above: for example a foreign money market ac- count e 0 rs ds Rt , expressed in local (or domestic) units of currency, where (rtf )t R+ represents a short term interest rate on the foreign market. When the num raire is a random process, the pricing of a claim whose value has been transformed under change of num raire, under a change of cur- rency, has to take into account the risks existing on the foreign market. In particular, in order to perform a fair pricing, one has to determine a probability measure (for example on the foreign market), under which the transformed (or forward, or deflated) process S t = St /Nt will be a martin- gale.

6 Rt For example in case Nt := e 0 rs ds is the money market account, the risk- neutral probability measure P is a measure under which the discounted price process St rt S t = = e 0 rs ds St , t R+ , Nt " 433. This version: January 16, 2019. N. Privault is a martingale. In the next section we will see that this property can be extended to any kind of num raire. Change of Num raire In this section we review the pricing of options by a change of measure asso- ciated to a num raire Nt , cf. [GKR95] and references therein. Most of the results of this chapter rely on the following assumption, which expresses absence of arbitrage. In the foreign exchange setting where Nt =. Rt , this condition states that the price of one unit of foreign currenty is a martingale when quoted and discounted in the domestic currency.

7 Assumption (A) Under the risk-neutral probability measure P , the discounted num raire rt t 7 Mt := e 0. rs ds Nt is an Ft -martingale. ((A)). Definition Given (Nt )t [0,T ] a num raire process, the associated for- ward measure P. is defined by dP MT rT N. := = e 0 rs ds T . ( ). dP M0 N0. Recall that from Section the above Relation ( ) rewrites as MT rT NT. dP. = dP = e 0 rs ds dP , M0 N0. which is equivalent to stating that w w rT. rs ds NT. X ( ) dP. ( ) = e 0 XdP . N0. for any (bounded) random variable S or, under a different notation, r . [X ] = IE e 0 rs ds NT X , T. IE. N0. 434 ". This version: January 16, 2019. Change of Num raire and Forward Measures for all integrable FT -measurable random variables X. More generally, by ( ) and the fact that the process rt t 7 Mt := e 0. rs ds Nt is an Ft -martingale under P under Assumption (A), we find that.

8 DP rT Nt r t rs ds NT 0 rs ds Mt IE Ft = I. E e Ft = e 0 = , ( ). dP N0 N0 M0. 0 6 t 6 T . In Proposition we will show, as a consequence of next Lemma below, that for any integrable random claim C we have h rT i IE C e t rs ds NT Ft = Nt IE. [C | Ft ], 0 6 t 6 T. Note that ( ), which is Ft -measurable, should not be confused with ( ), which is FT -measurable. In the next Lemma we compute the probability density dP |F /dP of P |F with respect to P . t |Ft t |Ft Lemma We have dP. |F M rT N.. t = T = e t rs ds T , 0 6 t 6 T. ( ). dP|Ft Mt Nt Proof. The proof of ( ) relies on the abstract version of the Bayes formula. We start by noting that for all integrable Ft -measurable random variable G, by ( ) and the tower property ( ) we have rT . GX = IE GX e 0 rs ds NT. IE.. N0. Nt r t rs ds r T rs ds NT.

9 = IE G e 0 IE X e t Ft N0 Nt rT. dP . t rs ds NT = IE G IE F t IE X e Ft dP Nt . dP . rT N . = IE G IE X e t rs ds T Ft dP Nt rT N . T. = IE. G IE X e t rs ds Ft , Nt for all integrable random variable X, which shows that rT . X | Ft = IE X e t rs ds NT Ft , IE.. Nt " 435. This version: January 16, 2019. N. Privault ( ) holds.. rt We note that in case the num raire Nt = e 0. rs ds is equal to the money market account we simply have P. = P . Pricing using Change of Num raire The change of num raire technique is specially useful for pricing under ran- dom interest rates, in which case an expectation of the form h rT i IE e t rs ds C Ft becomes a path integral, see [Das04] for a recent account of path integral methods in quantitative Finance . The next proposition is the basic result of this section, it provides a way to price r an option with arbitrary payoff C.

10 T. under a random discount factor e t rs ds by use of the forward measure. It will be applied in Chapter 14 to the pricing of bond options and caplets, cf. Propositions , and below. Proposition An option with integrable claim payoff C L1 (P , FT ). is priced at time t as h rT . C Ft , i IE e t rs ds C Ft = Nt IE 0 6 t 6 T, ( ). NT. provided that C/NT L1 P, FT .. Proof. By Relation ( ) in Lemma we have h rT dP. " #. i |F N t rs ds t IE e C Ft = IE t C Ft dP |Ft NT. dP. " #. |F C = Nt IE t F t dP |Ft NT.. C Ft , = Nt IE 0 6 t 6 T. NT. Equivalently we can write C dP. " #.. C |F . Nt IE. Ft = Nt IE. t Ft NT NT dP |Ft h rT i = IE e t rs ds C Ft , 0 6 t 6 T.. 436 ". This version: January 16, 2019. Change of Num raire and Forward Measures Each application of the formula ( ) will require to a) identify a suitable num raire (Nt )t R+ , and to b) make sure that the ratio C/NT takes a sufficiently simple form, in order to allow for the computation of the expectation in the right-hand side of ( ).