Equity-linked Foreign Exchange Options

S_t^fe^δ(T−t)N

c1(S_t^f, T− t)

− K^fN

c2(S_t^f, T − t) , where δ = r_f− σQ· σS^f and

c1,2(s, t) = ln(s/K^f) + (δ±¹₂|σS^f|²)t

|σS^f|√

t .

4.4.4 Equity-linked Foreign Exchange Options

Finally, assume that an investor desires to hold foreign equity regardless of whether the stock price rises or falls (that is, he is indifferent to the foreign equity exposure), however, wishes to place a floor on the exchange rate risk of his foreign investment. An equity-linked foreign exchange call (an Elf-X call, for short) with payoff at expiry (in units of domestic currency)

C_T^{4 def}= (Q_T − K)⁺S_T^f,

where K is a strike exchange rate expressed in domestic currency per unit of foreign currency, is thus a combination of a currency option with an equity forward. The arbitrage price (in units of domestic currency) of a European Elf-X call with expiry date T equals

C_t⁴= e^−rd(T−t)EP^∗

(QT− K)⁺S_T^fFt

.

We shall first value an equity-linked foreign exchange call option using the domestic martingale measure.

Proposition 4.4.2 The arbitrage price, expressed in domestic currency, of a European equity-linked foreign exchange call option, with strike exchange rate K and expiry date T, is given by the following formula

C_t⁴= S_t^f

 Q_tN

w₁(Q_t, T − t)

− K e^{−γ(T −t)}N

w₂(Q_t, T − t)

, (4.35)

where γ = r_d− rf+ σ_Q· σS^f and

w_1,2(q, t) = ln(q/K) + (γ±¹₂|σQ|²)t

|σQ|√

t .

Chapter 5

American Options

In contrast to the holder of a European option, the holder of an American option is allowed to exercise his right to buy (or sell) the underlying asset at any time before or at the expiry date. This special feature makes the arbitrage pricing of American options much more involved than the valuation of standard European claims. We know already that arbitrage valuation of American claims (for instance, within the framework of the binomial CRR model studied in Chap. 2) is closely related to specific optimal stopping problems. Intuitively, one might expect that the holder of an American option will choose her exercise policy in such a way that the expected payoff from the option will be maximized. Maximization of the expected discounted payoff under subjective probability would lead, of course, to non-uniqueness of the price. It appears, however, that for the purpose of arbitrage valuation, the maximization of the expected discounted payoff should be done under the martingale measure (that is, under risk-neutral probability). Therefore, the uniqueness of the arbitrage price of an American claim holds. One of the earliest works to examine the relationship between the early exercise feature of American options and optimal stopping problems was the paper in McKean (1965).

It should be made clear that the arbitrage valuation of derivative securities was not yet discovered at this time, however. For this reason, the optimal stopping problem associated with the optimal exercise of American put was studied in McKean (1965) under an actual probability P, rather than under the martingale measure P^∗, as is done nowadays. Basic features of American options, within the framework of arbitrage valuation theory, were already examined in van Moerbeke (1976).

However, mathematically rigorous valuation results for American claims were first established by means of arbitrage arguments in Bensoussan (1984) and Karatzas (1988, 1989). For an exhaustive survey of results and techniques related to the arbitrage pricing of American options, we refer the reader to Myneni (1992).

5.1 Valuation of American Claims

We place ourselves within the classic Black-Scholes setup. Hence, the prices of primary securities – that is, the stock price, S, and the savings account, B – are modelled by means of the following differential equations

dSt= µStdt + σStdWt, S0> 0, (5.1) where µ∈ R and σ > 0are real numbers, and

dB_t= rB_tdt, B₀= 1, (5.2)

with r ∈ R, respectively. As usual, we denote by W the standard Brownian motion defined on a filtered probability space (Ω, F, P), where F = F^W. For the sake of notational convenience, we assume here that the underlying Brownian motion W is one-dimensional.

In the context of arbitrage valuation of American contingent claims, it is convenient to assume that an individual may withdraw funds to finance his consumption needs. For any fixed t, we denote

87

by A_tthe cumulative amount of funds that are withdrawn and consumed by an investor up to time t. The term “consumed” refers to the fact that the wealth is dynamically diminished according to the process A. The process A is assumed to be progressively measurable with non-decreasing and RCLL sample paths; also, by convention, A0= A0−= 0. We say that A represents the consumption strategy, as opposed to the trading strategy φ. It is thus natural to call a pair (φ, A) a trading and consumption strategy in (S, B). In the present context, the formal definition of a self-financing strategy reads as follows.

Definition 5.1.1 A trading and consumption strategy (φ, A) in (S, B) is self-financing on [0, T ] if its wealth process V (φ, A), which equals

Vt(φ, A) = φ¹_tSt+ φ²_tBt, ∀ t ∈ [0, T ], (5.3) In view of (5.4), it is clear that A models the flow of funds that are not reinvested in primary securities, but rather are put aside forever. By convention, we say that the amount of funds repre-sented by A_t is consumed by the holder of the dynamic portfolio (φ, A) up to time t. Eliminating the component φ² yields the following equivalent form of (5.4)

dVt= rVtdt + φ¹_tSt

(µ− r) dt + σ dWt

− dAt,

where we write V to denote the wealth process V (φ, A). Equivalently,

dVt= rVtdt + ζt(µ− r) dt + σζtdWt− dAt, (5.5) where ζt= φ¹_tStrepresents the amount of cash invested in shares at time t. The unique solution of the linear SDE (5.5) is given by an explicit formula

V_t= B_t

which holds for every t∈ [0, T ]. We conclude that the wealth process of any self-financing trading and consumption strategy is uniquely determined by the following quantities: the initial endowment V₀, the consumption process A, and the process ζ representing the amount of cash invested in shares. In other words, given an initial endowment V₀, there is one-to-one correspondence between self-financing trading and consumption strategies (φ, A) and two-dimensional processes (ζ, A). We will sometimes find it convenient to identify a self-financing trading and consumption strategy (φ, A) with the corresponding pair (ζ, A), where ζ = φ¹_tSt. Recall that the unique martingale measure P^∗ for the Black-Scholes spot market satisfies

dP^∗

It is easily seen that the dynamics of the wealth process V under the martingale measure P^∗ are given by the following expression

dVt= rVtdt + σζtdW_t^∗− dAt,

where W^∗ follows the standard Brownian motion under P^∗. This yields immediately Vt= Bt

5.1. VALUATION OF AMERICAN CLAIMS 89 Therefore, an auxiliary process Z, which is given by the formula

Zt

def= V_t^∗+

 t 0

B_u⁻¹dAu= V0+

 t 0

σζuB_u⁻¹dW_u^∗,

where V_t^∗ = V_t/B_t, follows a local martingale under P^∗. We say that a self-financing trading and consumption strategy (φ, A) is admissible if the condition

E_P∗

 ^T

0

ζ_u²du

= E_P∗

 ^T

0

(φ¹_uS_u)²du

<∞

is satisfied so that Z is a P^∗-martingale. Similarly to Sect. 3.2.1, this assumption is imposed in order to exclude pathological examples of arbitrage opportunities from the market model. We are now in a position to formally introduce the concept of a contingent claim of American style. To this end, we take an arbitrary continuous reward function g : R+× [0, T ] → R satisfying the linear growth condition. An American claim with the reward function g and expiry date T is a financial security which pays to its holder the amount g(S_t, t) when exercised at time t.

The writer of an American claim with the reward function g accepts the obligation to pay the amount g(St, t) at any time t. It should be emphasized that the choice of the exercise time is at discretion of the holder of an American claim (that is, of a party assuming a long position). In order to formalize the concept of an American claim, we need to introduce first a suitable class of admissible exercise times. Since we exclude clairvoyance, the admissible exercise time τ is assumed to be a stopping time of filtration F. Let us recall that a random variable τ : (Ω,FT, P)→ [0, T ] is a stopping time of filtration F if, for every t∈ [0, T ], the event {τ ≤ t} belongs to the σ-field Ft. Since in the Black-Scholes model we have F = F^W = F^W∗ = F^S, any stopping time of the filtration F is also a stopping time of the filtration F^S generated by the stock price process S. In intuitive terms, it is assumed throughout that the decision to exercise an American claim at time t is based on the observations of stock price fluctuations up to time t, but not after this date. This interpretation is consistent with our general assumption that the σ-fieldFt represents the information available to all investors at time t. Let us denote byT[t,T ] the set of all stopping times of the filtration F which satisfy t≤ τ ≤ T (with probability 1).

Definition 5.1.2 An American contingent claim X^a with the reward function g : R+× [0, T ] → R is a financial instrument consisting of: (a) an expiry date T ; (b) the selection of a stopping time τ∈ T[0,T ]; and (c) a payoff X_τ^a = g(Sτ, τ ) on exercise.

Typical examples of American claims are American options with constant strike price K and expiry date T. The payoffs of American call and put options, when exercised at the random time τ, are equal to Xτ = (Sτ − K)⁺ and Yτ = (K − Sτ)⁺ respectively. Our aim is to derive the

“rational” price and to determine the “rational” exercise time of an American contingent claim by means of purely arbitrage arguments. To this end, we shall first introduce a specific class of trading strategies. For expositional simplicity, we shall search for the price of an American claim X^aat time 0; the general case can be treated along the same lines, but is more cumbersome from the notational viewpoint. It will be sufficient to consider a very special class of trading strategies associated with the American contingent claim X^a, namely the buy-and-hold strategies. By a buy-and-hold strategy associated with an American claim X^a, we mean a pair (c, τ ), where τ ∈ T[0,T ] and c is a real number. In financial interpretation, a buy-and-hold strategy (c, τ ) assumes that c > 0units of the American security X^a are acquired (or shorted, if c < 0) at time 0, and then held in the portfolio up to the exercise time τ. Observe that such a strategy excludes trading in the American claim after the initial date. In other words, dynamic trading in the American claim is not considered at this stage.

Let us assume that there exists a “market” price, say U0, at which the American claim X^a trades in the market at time 0. Our first task is to find the right value of U0 by means of no-arbitrage arguments (as mentioned above, the arguments which lead to the arbitrage valuation of the claim

X^a at time t > 0are much the same as in the case of t = 0, therefore the general case is left to the reader).

Definition 5.1.3 By a self-financing trading strategy in (S, B, X^a), we mean a collection (φ, A, c, τ ), where (φ, A) is a trading and consumption strategy in (S, B) and (c, τ ) is a buy-and-hold strategy associated with X^a. In addition, we assume that on the random interval (τ, T ] we have

φ¹_t = 0, φ²_t = φ¹_τSτB_τ⁻¹+ φ²_τ+ cg(Sτ, τ )B_τ⁻¹. (5.6) It will soon become apparent that it is enough to consider the cases of c = 1 and c =−1; that is, the long and short positions in the American claim X^a. An analysis of condition (5.6) shows that the definition of a self-financing strategy (φ, A, c, τ ) implicitly assumes that the American claim is exercised at a random time τ, existing positions in shares are closed at time τ, and all the proceeds are invested in risk-free bonds. For brevity, we shall sometimes write ˜ψ to denote the dynamic portfolio (φ, A, c, τ ) in what follows. Note that the wealth process V ( ˜ψ) of any self-financing strategy in (S, B, X^a) satisfies the following initial and terminal conditions

V0( ˜ψ) = φ¹₀S0+ φ²₀+ cU0 (5.7) and

VT( ˜ψ) = e^r(T−τ)(φ¹_τSτ+ cg(Sτ, τ )) + e^rTφ²_τ. (5.8) In what follows, we shall restrict our attention to the class of admissible trading strategies ˜ψ = (φ, A, c, τ ) in (S, B, X^a), which are defined in the following way.

Definition 5.1.4 A self-financing trading strategy (φ, A, c, τ ) in (S, B, X^a) is said to be admissible if a trading and consumption strategy (φ, A) is admissible and A_T = A_τ. The class of all admissible strategies (φ, A, c, τ ) is denoted by ˜Ψ.

Let us introduce the class ˜Ψ⁰of those admissible trading strategies ˜ψ for which the initial wealth satisfies V0( ˜ψ) < 0, and the terminal wealth has the non-negative value; that is¹ VT( ˜ψ) = φ²_TBt≥ 0.

In order to precisely define an arbitrage opportunity, we have to take into account the early exercise feature of American claims. It is intuitively clear that it is enough to consider two cases – a long and a short position in one unit of an American claim. This is due to the fact that we need to exclude the existence of arbitrage opportunities for both the seller and the buyer of an American claim.

Indeed, the position of both parties involved in a contract of American style is no longer symmetric, as it was in the case of European claims. The holder of an American claim can actively choose his exercise policy. The seller of an American claim, on the contrary, should be prepared to meet his obligations at any (random) time. We therefore set down the following definition of arbitrage and an arbitrage-free market model.

Definition 5.1.5 There is arbitrage in the market model with trading in the American claim X^a with initial price U₀ if either (a) there is long arbitrage, i.e., there exists a stopping time τ such that for some trading and consumption strategy (φ, A) the strategy (φ, A, 1, τ ) belongs to the class Ψ˜⁰, or (b) there is short arbitrage, i.e., there exists a trading and consumption strategy (φ, A) such that for any stopping time τ the strategy (φ, A,−1, τ) belongs to the class ˜Ψ⁰. In the absence of arbitrage in the market model, we say that the model is arbitrage-free.

Definition 5.1.5 can be reformulated in the following way: there is absence of arbitrage in the market if the following conditions are satisfied: (a) for any stopping time τ and any trading and consumption strategy (φ, A), the strategy (φ, A, 1, τ ) is not in ˜Ψ⁰; and (b) for any trading and consumption strategy (φ, A), there exists a stopping time τ such that the strategy (φ, A,−1, τ) is

1Since the existence of a strictly positive savings account is assumed,one can alternatively define the class Ψ⁰ as the set of those strategies ˜ψ from ˜Ψ for which V₀( ˜ψ) = 0, V_T( ˜ψ) = φ²_TB_T≥ 0, and the latter inequality is strict with positive probability.

5.1. VALUATION OF AMERICAN CLAIMS 91 not in ˜Ψ⁰. Intuitively, under the absence of arbitrage in the market, the holder of an American claim is unable to find an exercise policy τ and a trading and consumption strategy (φ, A) that would yield a risk-free profit. Also, under the absence of arbitrage, it is not possible to make risk-free profit by selling the American claim at time 0, provided that the buyer makes a clever choice of the exercise date. More precisely, there exists an exercise policy for the long party which prevents the short party from locking in a risk-free profit.

By definition, the arbitrage price at time 0of the American claim X^a, denoted by π0(X^a), is that level of the price U₀which makes the model arbitrage-free. Our aim is now to show that the assumed absence of arbitrage in the sense of Definition 5.1.5 leads to a unique value for the arbitrage price π₀(X^a) of X^a (as already mentioned, it is not hard to extend this reasoning in order to determine the arbitrage price π_t(X^a) of the American claim X^a at any date t ∈ [0, T ]). Also, we shall find the rational exercise policy of the holder – that is, the stopping time that excludes the possibility of short arbitrage.

The following auxiliary result relates the value process associated with the specific optimal stop-ping problem to the wealth process of a certain admissible trading strategy. For any reward function g, we define an adapted process V by setting

V_t= ess sup_{τ∈T[t,T ]}E_P∗

e^−r(τ−t)g(S_τ, τ )|Ft

 (5.9)

for every t∈ [0, T ], provided that the right-hand side in (5.9) is well-defined.

Proposition 5.1.1 Let V be an adapted process defined by formula (5.9) for some reward function g. Then there exists an admissible trading and consumption strategy (φ, A) such that V_t= V_t(φ, A) for every t∈ [0, T ].

Proof. We shall give the outline of the proof (for technical details, we refer to Karatzas (1988) and Myneni (1992)). Let us introduce the Snell envelope J of the discounted reward process Z_t^∗= e^−rtg(St, t). By definition, the process J is the smallest supermartingale majorant to the process Z^∗. From the general theory of optimal stopping, we know that

J_t= ess sup_{τ∈T[t,T ]}E_P∗

e^−rτg(S_τ, τ )|Ft

= ess sup_{τ∈T[t,T ]}E_P∗ Z_τ^∗|Ft



for every t∈ [0, T ], so that Vt= e^rtJt. Since J is a RCLL regular supermartingale of class DL,² it follows from general results that J admits the unique Doob-Meyer decomposition J = M−H, where M is a (square-integrable) martingale and H is a continuous non-decreasing process with H0= 0.

Consequently,

d e^rtJt

= re^rtJtdt + e^rtdMt− e^rtdHt.

By virtue of the predictable representation property (see Theorem 3.1.2) we have

Mt= M0+

 t 0

ξudW_u^∗, ∀ t ∈ [0, T ],

for some progressively measurable process ξ with EP^∗

_T

0 ξ_u²du

<∞. Hence, upon setting

φ¹_t = e^rtξ_tσ⁻¹S_t⁻¹, φ²_t = J_t− ξtσ⁻¹, A_t=

 _t

0

e^rudH_u, (5.10)

we conclude that the process V represents the wealth process of some (admissible) trading and

consumption strategy. ✷

2Basically,one needs to check that the family{Jτ| τ ∈ T[0,T ]} of random variables is uniformly integrable under P^∗. We refer the reader to Sect. 1.4 in Karatzas and Shreve (1998) for the definition of a regular process and for the concept of the Doob-Meyer decomposition of a semimartingale.

By the general theory of optimal stopping, we know also that the random time τ_tthat maximizes the expected discounted reward after the date t is the first instant at which the process J drops to the level of the discounted reward, that is

τt= inf{u ∈ [t, T ] | Ju= Z_u^∗}, ∀ t ∈ [0, T ]. (5.11) In other words, the optimal (under P^∗) exercise policy of the American claim with reward function g is given by the equality

τ0= inf{u ∈ [0, T ] | Ju= e^−rug(Su, u)}. (5.12) Observe that the stopping time τ0 is well-defined (i.e., the set on the right-hand side is non-empty with probability 1), and necessarily

V_τ0 = g(S_τ0, τ₀). (5.13)

In addition, the stopped process Jt∧τ0 is a martingale, so that the process H is constant on the interval [0, τ0]. This means also that At= 0on the random interval [0, τ0], so that no consumption is present before time τ0. We find it convenient to introduce the following definition.

Definition 5.1.6 An admissible trading and consumption strategy (φ, A) is said to be a perfect hedging against the American contingent claim X^a with reward function g if, with probability 1,

V_t(φ)≥ g(St, t), ∀ t ∈ [0, T ]. (5.14) We write Φ(X^a) to denote the class of all perfect hedging strategies against the American contingent claim X^a.

From the majorizing property of the Snell envelope, we infer that the trading and consumption strategy (φ, A) introduced in the proof of Proposition 5.1.1 is a perfect hedging against the American claim with reward function g. Moreover, this strategy has the special property of minimal initial endowment amongst all admissible perfect hedging strategies against the American claim. We shall now explicitly determine π0(X^a) by assuming that trading in the American claim X^a would not destroy the arbitrage-free features of the Black-Scholes model.

Theorem 5.1.1 There is absence of arbitrage (in the sense of Definition 5.1.5) in the market model with trading in an American claim if and only if the price π0(X^a) is given by the formula

π0(X^a) = sup_{τ∈T[0,T ]}EP^∗

e^−rτg(Sτ, τ )

. (5.15)

More generally, the arbitrage price at time t of an American claim with reward function g equals π_t(X^a) = ess sup_{τ∈T[t,T ]}E_P∗

e^−r(τ−t)g(S_τ, τ )|Ft

.

Proof. We shall follow Myneni (1992). Let us assume that the “market” price of the option is U0 > V0. We shall show that in this case, the American claim is overpriced – that is, a short arbitrage is possible. Let (φ, A) be the trading and consumption strategy considered in the proof of Proposition 5.1.1 (see formula (5.10)). Suppose that the option’s buyer selects an arbitrary stopping time τ ∈ T[0,T ] as his exercise policy. Let us consider the following strategy ( ˆφ, ˆA,−1, τ) (observe that in implementing this strategy, we do not need to assume that the exercise time τ is known in advance)

φˆ¹_t = φ¹_tI_{[0,τ ]}(t), φˆ²_t = φ²_tI_{[0,τ ]}(t) +

φ²_τ+ φ¹_τSτB⁻¹_τ − g(Sτ, τ )B⁻¹_τ 

I_{(τ,T ]}(t),

and ˆA_t= A_t∧τ. Since (φ, A) is assumed to be a perfect hedging, we have ˆφ¹_τS_τ+ ˆφ²_τB_τ≥ g(Sτ, τ ), so that ˆφ²_TBT ≥ 0, P^∗-a.s. On the other hand, by construction, the initial wealth of ( ˆφ, ˆA,−1, τ)

5.2. AMERICAN CALL AND PUT OPTIONS 93

In document Introduction to Arbitrage Pricing (Page 87-94)