The aim of this section is to describe the specific features that distinguish the arbitrage valuation of contingent claims within the classic Black-Scholes framework from the pricing of options on stocks and bonds under stochastic interest rates. We assume throughout that the price B(t, T ) of a zero-coupon bond of maturity T ≤ T∗ (T∗ > 0is a fixed horizon date) follows an Itˆo process under the martingale measure P∗2
dB(t, T ) = B(t, T )
rtdt + b(t, T )· dWt∗
, (9.17)
with B(T, T ) = 1, where W∗ denotes a d-dimensional standard Brownian motion defined on a filtered probability space (Ω, F, P∗), and rtstands for the instantaneous, continuously compounded rate of interest. In other words, we take for granted the existence of an arbitrage-free family B(t, T ) of bond prices associated with a certain process r which models the short-term interest rate. Moreover, it is implicitly assumed that we have already constructed an arbitrage-free model of a market in which all bonds of different maturities, as well as a certain number of other assets (called stocks in what follows), are primary traded securities. It should be stressed that the way in which such a construction is achieved is not relevant for the results presented in what follows. In particular, the concept of the instantaneous forward interest rate, which is known to play an essential role in the HJM methodology, is not employed. As already mentioned, in addition to zero-coupon bonds, we shall also consider other primary assets, referred to as stocks in what follows. The dynamics of a stock price Si, i = 1, . . . , M, under the martingale measure P∗are given by the following expression
dSti= Sti
rtdt + σit· dWt∗
, S0i > 0, (9.18)
where σirepresents the volatility of the stock price Si. Unless explicitly stated otherwise, for every T and i, the bond price volatility b(t, T ) and the stock price volatility σitare assumed to be Rd-valued, bounded, adapted processes. Generally speaking, we assume that the prices of all primary securities follow strictly positive processes with continuous sample paths. It should be observed, however, that certain results presented in this section are independent of the particular form of bond and stock prices introduced above. We denote by πt(X) the arbitrage price at time t of an attainable contingent claim X which settles at time T. Therefore
πt(X) = BtEP∗(XBT−1| Ft), ∀ t ∈ [0, T ], (9.19) by virtue of the standard risk-neutral valuation formula. In (9.19), B represents the savings account given by (7.8). Recall that the price B(t, T ) of a zero-coupon bond which matures at time T admits the following representation (cf. (8.2))
B(t, T ) = BtEP∗(BT−1| Ft), ∀ t ∈ [0, T ], (9.20) for any maturity 0 ≤ T ≤ T∗. Suppose now that we wish to price a European call option, with expiry date T, which is written on a zero-coupon bond of maturity U > T. The option’s payoff at expiry equals
CT = (B(T, U )− K)+, so that the option price Ctat any date t≤ T is
Ct= BtEP∗
BT−1(B(T, U )− K)+Ft
.
To find the option’s price using the last equality, we need to know the joint (conditional) probability law of FT-measurable random variables BT and B(T, U ). The technique which was developed to circumvent this step is based on an equivalent change of probability measure. It appears that it is possible to find a probability measure PT such that the following holds
Ct= B(t, T )EPT
(B(T, U )− K)+Ft
.
2The reader may find it convenient to assume that the probability measure P∗is the unique martingale measure for the family B(t, T ), T≤ T∗; this is not essential,however.
9.2. FORWARD MEASURE APPROACH 143 Consequently,
Ct= B(t, T )EPT
(FB(T, U, T )− K)+Ft
,
where FB(t, U, T ) is the forward price at time t, for settlement at the date T, of the U -maturity zero-coupon bond (see formula (9.22)). If b(t, U )− b(t, T ) is a deterministic function, then the forward price FB(t, U, T ) can be shown to follow a lognormal martingale under PT; therefore, a Black-Scholes-like expression for the option’s price is available.
9.2.1 Forward Price
Recall that a forward contract is an agreement, established at the date t < T, to pay or receive on settlement date T a preassigned payoff, say X, at an agreed forward price. It should be emphasized that there is no cash flow at the contract’s initiation and the contract is not marked to market.
We may and do assume, without loss of generality, that a forward contract is settled by cash on date T. Therefore, a forward contract written at time t with the underlying contingent claim X and prescribed settlement date T > t may be summarized by the following two basic rules: (a) a cash amount X will be received at time T, and a preassigned amount FX(t, T ) of cash will be paid at time T ; (b) the amount FX(t, T ) should be predetermined at time t (according to the information available at this time) in such a way that the arbitrage price of the forward contract at time t is zero.
In fact, since nothing is paid up front, it is natural to admit that a forward contract is worthless at its initiation. We adopt the following formal definition of a forward contract.
Definition 9.2.1 Let us fix 0 ≤ t ≤ T ≤ T∗. A forward contract written at time t on a time T contingent claim X is represented by the time T contingent claim GT = X− FX(t, T ) that satisfies the following conditions: (a) FX(t, T ) is a Ft-measurable random variable; (b) the arbitrage price at time t of a contingent claim GT equals zero, i.e., πt(GT) = 0.
The random variable FX(t, T ) is referred to as the forward price of a contingent claim X at time t for the settlement date T. The contingent claim X may be defined in particular as a preassigned amount of the underlying financial asset to be delivered at the settlement date. For instance, if the underlying asset of a forward contract is one share of a stock S, then clearly X = ST. Similarly, if the asset to be delivered at time T is a zero-coupon bond of maturity U≥ T, we have X = B(T, U).
Note that both ST and B(T, U ) are attainable contingent claims in our market model. The following well-known result expresses the forward price of a claim X in terms of its arbitrage price πt(X) and the price B(t, T ) of a zero-coupon bond which matures at time T.
Lemma 9.2.1 The forward price FX(t, T ) at time t≤ T, for the settlement date T, of an attainable contingent claim X equals
FX(t, T ) = EP∗(XBT−1| Ft)
EP∗(B−1T | Ft) = πt(X)
B(t, T ). (9.21)
Proof. It is sufficient to observe that
πt(GT) = BtEP∗(GTBT−1| Ft)
= Bt
EP∗(XB−1T | Ft)− FX(t, T )EP∗(B−1T | Ft)
= 0,
where the last equality follows by condition (b) of Definition 9.2.1. This proves the first equality;
the second follows immediately from (9.18)–(9.20). ✷
Let us examine the two typical cases of forward contracts mentioned above. If the underlying asset for delivery at time T is a zero-coupon bond of maturity U≥ T, then (9.21) becomes
FB(T,U )(t, T ) = B(t, U )
B(t, T ), ∀ t ∈ [0, T ]. (9.22)
On the other hand, the forward price of a stock S (S stands hereafter for Si for some i) equals FST(t, T ) = St
B(t, T ), ∀ t ∈ [0, T ]. (9.23)
For the sake of brevity, we shall write FB(t, U, T ) and FS(t, T ) instead of FB(T,U )(t, T ) and FST(t, T ), respectively. More generally, for any tradable asset Z, we write FZ(t, T ) to denote the forward price of the asset – that is, FZ(t, T ) = Zt/B(t, T ) for t∈ [0, T ].
9.2.2 Forward Martingale Measure
To the best of our knowledge, within the framework of arbitrage valuation of interest rate derivatives, the method of a forward risk adjustment was pioneered under the name of a forward risk-adjusted process in Jamshidian (1987) (the corresponding equivalent change of probability measure was then used by Jamshidian (1989a) in the Gaussian framework). The formal definition of a forward bility measure was explicitly introduced in Geman (1989) under the name of forward neutral proba-bility. In particular, Geman observed that the forward price of any financial asset follows a (local) martingale under the forward neutral probability associated with the settlement date of a forward contract. Most results in this section do not rely on specific assumptions imposed on the dynamics of bond and stock prices. We assume that we are given an arbitrage-free family B(t, T ) of bond prices and the related savings account B. Note that by assumption, 0 < B(0, T ) = EP∗(B−1T ) <∞.
Definition 9.2.2 A probability measure PT on (Ω,FT) equivalent to P∗, with the Radon-Nikod´ym derivative given by the formula
dPT
dP∗ = BT−1
EP∗(B−1T ) = 1
BTB(0, T ), P∗-a.s., (9.24) is called the forward martingale measure (or briefly, the forward measure) for the settlement date T.
Notice that the above Radon-Nikod´ym derivative, when restricted to the σ-fieldFt, satisfies for every t∈ [0, T ]
ηt def
= dPT
dP∗Ft = EP∗
1
BTB(0, T )
Ft
= B(t, T ) BtB(0, T ).
When the bond price is governed by (9.17), an explicit representation for the density process ηtis available. Namely, we have
ηt= exp
t
0
b(u, T )· dWu∗−1 2
t 0
|b(u, T )|2du
. (9.25)
In other words, ηt=Et(UT), where UtT =t
0b(u, T )· dWu∗. Furthermore, the process WT given by the formula
WtT = Wt∗−
t 0
b(u, T ) du, ∀ t ∈ [0, T ], (9.26) follows a standard Brownian motion under the forward measure PT. We shall sometimes refer to WT as the forward Brownian motion for the date T. The next result shows that the forward price of a European contingent claim X which settles at time T can be easily expressed in terms of the conditional expectation under the forward measure PT.
Lemma 9.2.2 The forward price at t for the date T of an attainable contingent claim X which settles at time T equals
FX(t, T ) = EPT(X| Ft), ∀ t ∈ [0, T ], (9.27) provided that X is PT-integrable. In particular, the forward price process FX(t, T ), t∈ [0, T ], follows a martingale under the forward measure PT.
9.2. FORWARD MEASURE APPROACH 145 Proof. The Bayes rule yields
EPT(X| Ft) = EP∗(ηTX| Ft)
EP∗(ηT| Ft) = EP∗(ηTη−1t X| Ft), (9.28) where
ηT = dPT
dP∗ = 1
BTB(0, T )
and ηt= EP∗(ηT| Ft). Combining (9.21) with (9.28), we obtain the desired result. ✷ Under (9.17), (9.28) can be given a more explicit form, namely
EPT(X| Ft) = EP∗
X exp
T
t
b(u, T )· dWu∗−1 2
T t
|b(u, T )|2du Ft
. The following equalities:
FB(t, T, U ) = EPT(B(T, U )| Ft), ∀ 0 < t ≤ T ≤ U ≤ T∗, and
FS(t, T ) = EPT(ST| Ft) ∀t ∈ [0, T ],
are immediate consequences of the last lemma. More generally, the relative price of any traded secu-rity (which pays no coupons or dividends) follows a local martingale under the forward probability measure PT, provided that the price of a bond which matures at time T is taken as a numeraire.
The next lemma establishes a version of the risk-neutral valuation formula that is tailored to the stochastic interest rate framework.
Lemma 9.2.3 The arbitrage price of an attainable contingent claim X which settles at time T is given by the formula
πt(X) = B(t, T ) EPT(X| Ft), ∀ t ∈ [0, T ]. (9.29) Proof. Equality (9.29) is an immediate consequence of (9.21) combined with (9.27). For a more direct proof, note that the price πt(X) can be re-expressed as follows
πt(X) = BtEP∗(XB−1T | Ft) = BtB(0, T ) EP∗(ηTX| Ft).
An application of the Bayes rule yields
πt(X) = BtB(0, T ) EPT(X| Ft) EP∗(ηT| Ft)
= BtB(0, T ) EPT(X| Ft) EP∗
1
BTB(0, T )
Ft
= B(t, T ) EPT(X| Ft),
as expected. ✷
The following corollary deals with a contingent claim which settles at time U = T. Our aim is to express the value of this claim in terms of the forward measure for the date T.
Corollary 9.2.1 Let X be an arbitrary attainable contingent claim which settles at time U. (i) If U ≤ T, then the price of X at time t ≤ U equals
πt(X) = B(t, T ) EPT(XB−1(U, T )| Ft). (9.30) (ii) If U≥ T and X is FT-measurable, then for any t≤ U we have
πt(X) = B(t, T ) EPT(XB(T, U )| Ft). (9.31)
Proof. Both equalities are intuitively clear. In case (i), we invest at time U a FU-measurable payoff X in zero-coupon bonds which mature at time T. For the second case, observe that in order to replicate aFT-measurable claim X at time U, it is enough to purchase, at time T, X units of a zero-coupon bond maturing at time U. Both strategies are manifestly self-financing, and thus the result follows.
An alternative way of deriving (9.30) is to observe that since X is FU-measurable, we have for every t∈ [0, U]
This means that the claim X that settles at time U has, at any date t∈ [0, U], an identical arbitrage price to the claim Y = XB−1(U, T ) that settles at time T. Formula (9.30) now follows from relation (9.29) applied to the claim Y. Similarly, to prove the second formula, we observe that since X is FT-measurable, we have for t∈ [0, T ]
We conclude once again that a FT-measurable claim X which settles at time U ≥ T is essentially
equivalent to a claim Y = XB(T, U ) which settles at time T. ✷