Ann. Inst. Henri Poincaré Anal. nonlinear 18, 1 (2001) 1–17 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0294-1449(00)00120-7/FLA
AMERICAN PRICES EMBEDDED IN EUROPEAN PRICES Benjamin JOURDAIN a , Claude MARTINI b a ENPC-CERMICS, 6-8 av. Blaise-Pascal, Cité Descartes, Champs-sur-Marne,
77455 Marne la Vallée cedex 2, France b INRIA Projet Mathfi, Domaine de Voluceau, Rocquencourt, BP105,
78153 Le Chesnay cedex, France Received 12 January 2000
A BSTRACT. – In this paper, we are interested in American option prices in the Black–Scholes model. For a large class of payoffs, we show that in the region where the European price increases with the time to maturity, this price is equal to the American price of another claim. We give examples in which we explicit the corresponding claims. The characterization of the American claims obtained in this way remains an open question. 2001 Éditions scientifiques et médicales Elsevier SAS Keywords: Optimal stopping, Free boundary problems, Martingales, Black–Scholes model, European options, American options AMS classification: 60G40, 60G46, 90A09 R ÉSUMÉ. – Ce travail met en évidence un lien entre prix d’options européennes et prix d’options américaines dans le modèle de Black–Scholes. Nous montrons que pour une large classe de fonctions de payoff ϕ, dans la zone où il augmente avec la maturité, le prix de l’option européenne est égal au prix américain correspondant à une fonction de payoff modifiée ϕb. Nous donnons des exemples où il est possible d’expliciter ϕb. Mais la caractérisation de l’image de ϕ → ϕb reste un problème ouvert. 2001 Éditions scientifiques et médicales Elsevier SAS
Introduction Consider the classical Black–Scholes model: dXtx = ρXtx dt + σ Xtx dBt , X0x = x > 0, ρ ∈ R, σ > 0, E-mail addresses:
[email protected] (B. Jourdain),
[email protected] (C. Martini).
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B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
where B is a standard Brownian motion, ρ the instantaneous interest rate and σ the volatility of X and denote by Af (x) =
σ 2 x 2 00 f (x) + ρxf 0 (x) − ρf (x) 2
(0.1)
the Black–Scholes infinitesimal generator. Given a continuous function ψ : R∗+ → R+ satisfying some growth assumptions, the price of the so-called American option with payoff ψ, maturity t > 0 and spot x is given by the expression
vψam (t, x) = sup E e−ρτ ψ Xτx , τ ∈T (0,t )
(0.2)
where τ runs across the set of stopping times of the Brownian filtration such that τ 6 t almost surely. Except for some very particular class of payoffs ψ (e.g. payoffs satisfying ∀x > 0, Aψ(x) > 0 or ∀x > 0, Aψ(x) 6 0), in general, there is no closedform expressions for vψam(t, x). The computation of vψam(t, x) usually relies either on finite-difference type methods or Markov-chain approximation methods to solve the corresponding optimal stopping problem in a discrete time-space framework. There is also a huge literature on special approximation methods designed for some particular payoffs, among which the case of the Put option, given by ψ(x) = (K − x)+ where K is some positive constant (the strike of the option) has received much attention. The purpose of this paper is to exhibit a new class of payoffs ψ for which a closed-form expression for vψam (t, x) is available. The idea originates from the analytic properties of the function vψam : this function is greater than ψ by (0.2) and typically the space ]0, ∞[ × R∗+ splits into two regions, the so-called Exercise region where by definition vψam = ψ and its complement the Continuation region where vψam > ψ. It is known that vψam solves the evolution equation associated with (0.1) ∂t vψam = Avψam in the Continuation region (at least in the distribution sense). Moreover, as from (0.2) t 7→ vψam (t, x) is non-decreasing, ∂t vψam > 0 holds. In fact, since vψam is a continuous function, it may be remarked that the knowledge of vψam in the Continuation region is enough to get vψam everywhere. This leads to the natural idea to build American prices (i.e. functions vψam for some ψ) by picking up the classical solution vϕ (t, x) of the evolution equation:
∀t, x > 0, ∂t vϕ (t, x) = Avϕ (t, x), ∀x > 0, vϕ (0, x) = ϕ(x),
in the region where it increases with time. From a financial point of view, vϕ (t, x) is the Black–Scholes price of the European option with payoff ϕ and maturity t i.e. vϕ (t, x) = E[e−ρt ϕ(Xtx )]. This embedding idea has been worked out in [2] in case ρ = 0. A similar approach has also been developped in the different context of the free boundary arising in a two-phases problem (see [1]). Trying to generalize things to the
B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
3
case ρ 6= 0, we ran across a probabilistic proof which allows a very compact statement of the embedding result. The first section of the paper is devoted to some basic properties of European and American prices within the Black–Scholes model. Next we state and prove our embedding theorem (Section 2). Then we give some examples (Section 3). Lastly, we discuss some properties of the map which takes a payoff ϕ to the payoff ϕb the American price of which is embedded in its European price (Section 4). The characterization of the payoffs ϕb obtained in this way remains an open question. 1. European and American prices in the Black–Scholes model In this section we recall the very few properties of European and American Black– Scholes prices we shall need in the next section. Let α = 2ρ/σ 2 . The invariant functions of the semigroup associated with (0.1) are easily seen to be the vector space generated by x and x −α . We shall consider payoffs ϕ such that ϕ(x) x ∈ R∗+ 7→ ϕ(x) ∈ R+ is continuous and sup 0 x + x The growth assumption is only there to grant the existence of the various expectations involved. It seems that the continuity assumption could be removed, but the connection with American options would be more intricate, so we keep this hypothesis. P ROPOSITION 1 ([3]). – Under (H0) the price of the European option with maturity t > 0 and payoff ϕ is given by
vϕ (t, x) = E e−ρt ϕ(Xtx ) . In particular vϕ (0, x) = ϕ(x). The function vϕ is continuous from [0, ∞[ × R∗+ into R, and for any t > 0, the process (e−ρu vϕ (t − u, Xux ))06u6t is a continuous square-integrable martingale. Last, for φ(y) = ϕ(ey )/(ey + e−αy ),
σ2 vϕ (t, x) − xE φ σ Bt + ρ + t 2
−x
−α
σ2 E φ σ Bt − ρ + t 2
converges to 0 as t → +∞ locally uniformly in x > 0. Proof. – We only prove the last assertion which is quite unusual. By definition of φ,
vϕ (t, x) = E e−ρt Xtx φ ln(x) + σ Bt + ρ −
σ2 t 2
+ E e−ρt (Xtx )−α φ ln(x) + σ Bt + ρ − Since e−ρt Xtx = xeσ Bt − deduce that
σ2 2 t
.
2
2ρ − 2ρ σ Bt − 2 t
and e−ρt (Xtx )−α = x −α e
σ2 t 2
σ
, by Girsanov theorem, we
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B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
σ2 vϕ (t, x) = xE φ ln(x) + σ Bt + ρ + t 2
+ x −α E φ ln(x) + σ Bt − ρ + Now,
σ2 t 2
.
2 σ2 E φ ln(x) + σ Bt + ρ + σ t − E φ σ Bt + ρ + t 2 2 Z √ 2 2 √ (y−ln(x)/(σ t)) y2 1 σ 2 = √ φ σ ty + ρ + t e− − e− 2 dy 2
2π
R
ϕ(z) 1 6 √ sup 2π z>0 z + z−α
6 e
ln2 (x) 2σ 2 t
− 1 sup z>0
Z √ 2 t)) y2 − (y−ln(x)/(σ 2 − e− 2 dy e R
ϕ(z) z + z−α
converges to 0 as t → +∞ locally uniformly for x > 0. We deal with E(φ(ln(x) + σ Bt − (ρ + σ 2 /2)t)) in the same way to conclude.
2
Let us now turn to American options: P ROPOSITION 2 ([3]). – If ψ satisfies (H0), the price of the American option with maturity t > 0 and payoff ψ is given by
vψam (t, x) = sup E e−ρτ ψ(Xτx ) , τ ∈T (0,t )
where τ runs across the set of stopping times of the Brownian filtration such that τ 6 t almost surely. In particular vψam (0, x) = ψ(x). The function vψam is continuous from [0, ∞[ × R∗+ into R, and for any x ∈ R∗+ the map t 7→ vψam (t, x) is non-decreasing. 2. Embedding American prices in European prices Our main result relates the price vϕ (t, x) of the European option with payoff ϕ to the b price vbϕam (t, x) of the American option with payoff ϕ(x) = inft >0 vϕ (t, x): T HEOREM 3. – Under (H0) let b ϕ(x) = inf vϕ (t, x). t >0
Then ∀(t, x) ∈ [0, +∞) × R∗+ ,
b τx ) 6 vϕ (t, x), sup E e−ρτ ϕ(X
τ ∈T (0,t )
(2.1)
where the supremum is taken over all the stopping times τ of the filtration of the Brownian motion smaller than t.
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Moreover, if there exists a continuous function bt : R∗+ → [0, +∞] such that ∀x > 0,
inf vϕ (t, x) = vϕ bt (x), x
t >0
(where vϕ (∞, x) is defined as lim inft →+∞ vϕ (t, x)) then the converse inequality holds for t > bt (x) and
∀(t, x) ∈ [0, +∞) × R∗+ ,
b τx ) = vϕ t ∨ b sup E e−ρτ ϕ(X t (x), x .
(2.2)
τ ∈T (0,t )
If, lastly, either bt (x0 ) < +∞ for some x0 > 0, or ϕ(x) − ϕ(y) 6 C |x − y| + |x −α − y −α |
∃C > 0, ∀x, y > 0,
ϕ(x) b or the function x+x −α admits limits both for x → 0 and x → +∞, then the function ϕ satisfies (H0) and
∀(t, x) ∈ [0, +∞) × R∗+ ,
vbϕam(t, x) = vϕ t ∨ bt (x), x .
(2.3)
Proof. – Let (t, x) ∈ R+ × R∗+ . According to Proposition 1, the process (e−ρu vϕ (t − u, Xux ))u∈[0,t ] is a martingale. If τ 6 t is a stopping time, by Doob optional sampling theorem
vϕ (t, x) = E e−ρτ vϕ t − τ, Xτx
b τx ) . > E e−ρτ ϕ(X
Since τ is arbitrary, we deduce that (2.1) holds. To prove (2.2), we suppose the existence of bt : R∗+ → [0, +∞] continuous such that ∀x > 0,
b ϕ(x) = vϕ tb(x), x
and we make a distinction between the two following situations: b τx )] is increasing, for u > t • Case bt (x) = +∞: since s → supτ ∈T (0,s) E[e−ρτ ϕ(X
b b τx ) 6 sup E e−ρτ ϕ(X b τx ) 6 vϕ (u, x) ϕ(x) 6 sup E e−ρτ ϕ(X τ ∈T (0,t )
τ ∈T (0,u)
by (2.1).
Letting u → +∞, we deduce that ∀t > 0,
b τx ) = ϕ(x) b sup E e−ρτ ϕ(X = vϕ (∞, x) = vϕ t ∨ bt (x), x .
τ ∈T (0,t )
• Case b t (x) < +∞: let t > bt (x) and τ0 = inf{u: t − u − bt (Xux ) 6 0}. Since b t (Xtx ) > 0, the stopping time τ0 is smaller than t. By continuity of u → t − u − bt (Xux ), t − τ0 = bt (Xτx0 ). Hence
vϕ (t, x) = E e−ρτ0 vϕ t − τ0 , Xτx0
= E e−ρτ0 vϕ bt (Xτx0 ), Xτx0
b τx0 ) 6 sup E e−ρτ ϕ(X b τx ) . = E e−ρτ0 ϕ(X τ ∈T (0,t )
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The converse inequality (2.1) is already proved. Hence ∀t > bt (x), supτ ∈T (0,t ) b τx )] = vϕ (t, x) and τ0 is an optimal stopping time. Since t → vϕam (t, x) E[e−ρτ ϕ(X b is increasing, for t 6 bt (x),
b b τx ) 6 ϕ(x) 6 sup E e−ρτ ϕ(X τ ∈T (0,t )
sup
τ ∈T (0,b t (x))
b τx ) = ϕ(x), b E e−ρτ ϕ(X
which concludes the proof of (2.2). If we check that ϕb is continuous under the various assumptions made in the last assertion of the theorem then (2.3) follows immediately since vbϕam (t, x) = supτ ∈T (0,t ) b τx )]. E[e−ρτ ϕ(X • If bt (x0 ) < +∞ for some x0 > 0 then for x in a neighbourhood of x0 , bt (x) < +∞ b τx )] is increasing, by (2.2) vϕ (t, x) has a limit and since t → supτ ∈T (0,t ) E[e−ρτ ϕ(X for t → +∞. By the last assertion of Proposition 1, we deduce that for φ(y) = ϕ(ey )/(ey + e−αy ), E(φ(σ Bt + (ρ + σ 2 /2)t)) and E(φ(σ Bt − (ρ + σ 2 /2)t)) admit limits as t → +∞. We denote the limits by a and b. Proposition 1 then yields that vϕ (t, x) converges to the invariant function ax + bx −α as t → +∞ locally uniformly for x > 0. The continuity of ϕb follows easily. • If ϕ(x)/(x + x −α ) admits limits for x → 0 and x → +∞ then φ(y) admits limits for y → −∞ and y → +∞. We deduce that E(φ(σ Bt + (ρ + σ 2 /2)t)) and E(φ(σ Bt − (ρ + σ 2 /2)t)) admit limits as t → +∞ and we conclude like in the previous case. • If ∀x, y > 0, |ϕ(x) − ϕ(y)| 6 C(|x − y| + |x −α − y −α |), then ∀t > 0, vϕ (t, x) − vϕ (t, y) 6 E e−ρt ϕ(X x ) − ϕ(X y ) t t 6 C |x − y|E e−ρt Xt1 + x −α − y −α E e−ρt (Xt1 )−α 6 C |x − y| + x −α − y −α .
Hence the functions x → vϕ (t, x) indexed by t > 0 are equicontinuous, which ensures b the continuity of x → inft >0 vϕ (t, x) = ϕ(x). 2 Remark 4. – The continuity of the argument of the infimum is granted in the following b uniqueness situation: suppose that ∀x > 0, ∃!bt (x) 6 T (x), ϕ(x) = vϕ ( bt (x), x) where ∗ T : R+ → R+ is continuous. Then by the continuity of T and vϕ , it is easy to b see that ϕ(x) = inft ∈[0,T (x)] vϕ (t, x) is continuous. Moreover, since b t (x) = inf{t > b b b 0: ϕ(x) = vϕ (t, x)} (respectively t (x) = sup{t 6 T (x): ϕ(x) = vϕ (t, x)}), bt is lower semi-continuous (respectively upper semi-continuous) i.e. bt is continuous and (2.3) holds. ϕ(x) In the above theorem it may happen that the function ϕb is nil: in case limx→0 x+x −α = ϕ(x) limx→+∞ x+x −α = 0, we easily check that
∀x > 0,
lim vϕ (t, x) = 0.
t →+∞
In such a situation, the following localized version of our main result is far more interesting than Theorem 3. It is proved by the same arguments, after noticing that the
B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
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continuity of (t, x) ∈ [0, +∞) × R∗+ → vϕ (t, x) implies the continuity of x → ϕbT (x) = inf06t 6T vϕ (t, x) where T > 0. T HEOREM 5. – Let T > 0. The function ϕbT (x) = inf06t 6T vϕ (t, x) satisfies (H0) and ∀(t, x) ∈ [0, T ] × R∗+ ,
vϕamT (t, x) 6 vϕ (t, x). b
Moreover, if there exists a continuous function bt : R∗+ → [0, T ] such that
inf vϕ (t, x) = vϕ bt (x), x ,
∀x > 0,
06t 6T
then
∀(t, x) ∈ [0, T ] × R∗+ ,
vϕamT (t, x) = vϕ t ∨ bt (x), x . b
Remark 6. – The only feature of the Black–Scholes model which is required in the above results is time-homogeneity. In fact, Propositions 1 and 2 and Theorems 3 and 5 can be adapted to the so-called generalized Black–Scholes model:
σ2 = x exp σ Bt + ρ − δ − t , 2 vϕ (t, x) = E e−ρt ϕ(Xtx ) and Xtx
vψam (t, x) = sup E e−ρτ ψ(Xτx ) , τ ∈T (0,t )
or to the more general time-homogeneous model: X0x = x,
dXtx = Xtx σ (Xtx ) dBt + ρ(Xtx ) − δ(Xtx ) dt h
−
vϕ (t, x) = E e
Rt 0
ρ(Xsx ) ds
h
vψam (t, x) = sup E e− τ ∈T (0,t )
ϕ(Xtx )
Rτ 0
i
ρ(Xsx ) ds
and ψ(Xτx )
i
and also to the multidimensional versions of these models. Of course it would be of great interest to give conditions on ϕ which ensure the existence of a continuous curve in the argument of the infimum. One way is to perform explicit computations, since the Black–Scholes semigroup is explicit. Nevertheless this is not very illuminating. We ran across the following statement, for the local embedding result, which is maybe the simplest in this direction: P ROPOSITION 7. – Let ϕ be a C 4 function which satisfies (H0) and such that for some xc ∈ R∗+ , (i) Aϕ(xc ) = 0 and either ∀x > 0, (x − xc )Aϕ(x) > 0 or ∀x > 0, (x − xc )Aϕ(x) 6 0. (ii) A2 ϕ(xc ) > 0 and ∂x Aϕ(xc ) 6= 0. Then there exists a constant T > 0 such that the assumptions of Theorem 5 are satisfied.
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B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
Proof. – Since ϕ is C 4 , the function vϕ (t, x) belongs to C 2,4 (R+ × R∗+ ) (C 2 in t, C in x) and satisfies the Black–Scholes partial differential equation ∂t vϕ = Avϕ for t > 0 and not only t > 0. Consider the equation ∂t vϕ (t, x) = 0 in a neighbourhood of (0, xc ) in {(t, x), t > 0}. By derivation of the Black–Scholes evolution equation, ∂t2x vϕ (0, xc ) = ∂x Aϕ(xc ) 6= 0. Hence, by the implicit functions theorem, there is for some ε > 0 a curve xb: 4
xb : [0, ε] → R∗+ continuous on [0, ε], with xb(0) = xc , such that ∂t vϕ (t, xb(t)) = 0, and C 1 on ]0, ε[ with
∂t22 vϕ t, xb(t) + ∂t2x vϕ t, xb(t) xb0 (t) = 0. Moreover by taking ε small enough we can assume that xb0 (t) does not vanish and keeps 2 c) the same sign as xb0 (0+ ) = − ∂Ax Aϕ(x . We deduce that there exists a continuous function ϕ(xc ) b b t : [xc , xb(ε)] → [0, ε] such that xb( t (x)) = x. Assume xb0 (0+ ) > 0. Then the function bt is increasing. Moreover, ∂x Aϕ(x) < 0 which ensures ∀x < xc , Aϕ(x) > 0 and ∀x > xc , Aϕ(x) 6 0. We set T = ε and extend bt to R∗+ by setting bt (x) = T for x > xc + ε and bt (x) = 0 for x < xc . The obtained function is continuous and the whole point is to show that for every x, the infimum of t 7→ vϕ (t, x) on [0, T ] is reached at bt (x). This amounts to show that ∂t vϕ (t, x) = Avϕ (t, x) is nonpositive for (t, x) above xb (i.e. for t 6 T and x > xb(t)) and non-negative below. If (Pt )t >0 denotes the semigroup associated with (0.1), Avϕ (t, x) = APt ϕ(x) = Pt Aϕ(x). Let (t, x) belong to the above (respectively below) region. By the optimal stopping theorem, Avϕ (t, x) is equal to the expectation of the value of the martingale (e−ρu Pt −u Aϕ(Xux ))06u6t stopped at the border of the above (respectively below) region {(u, xb(u)), u ∈ [0, ε]} ∪ {(0, x), x > xc } (respectively {(u, xb(u)), u ∈ [0, ε]} ∪ {(0, x), x 6 xc }) which is non-positive (respectively non-negative) since ∀t ∈ ]0, ε], Pu Aϕ(xb(u)) = ∂u vϕ (u, xb(u)) = 0 and ∀x > xc , Aϕ(x) 6 0 (respectively ∀x > xc , Aϕ(x) > 0). The case xb0 (0+ ) < 0 is handled in the same way. 2 Example 8. – As an application, consider the family of payoffs ϕa,b (x) = x −α + x a − x b , where 1 > a > b > −α. Then for x > 1, x a > x b , for x < 1, x −α > x b so that ϕa,b is non-negative. Moreover ϕa,b (x) = 1, x→0 x + x −α lim
ϕa,b (x) =0 x→+∞ x + x −α lim
2
and ϕa,b satisfies (H0). Let λ(y) = ( σ2 y + ρ)(y − 1). Then Aϕa,b (x) = λ(a)x a − λ(b)x b ,
B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
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which gives, since λ(a) < 0 and λ(b) < 0, Aϕa,b (x) < 0 for x > xc and Aϕa,b (x) > 0 for x < xc with λ(a)xca = λ(b)xcb . Moreover
A2 ϕa,b (xc ) = λ(a)2 xca − λ(b)2 xcb = λ(a) − λ(b) λ(b)xcb and A2 ϕa,b (xc ) > 0 as soon as λ(b) > λ(a). Lastly,
∂x Aϕa,b (xc ) = λ(a)axca−1 − λ(b)bxcb−1 =
a b − λ(a)xca 6= 0. xc xc
Of course, in this example, since vϕ (t, x) = x −α + x a et λ(a) − x b et λ(b) , everything can be computed explicitely and it is even possible to check the hypotheses of the global embedding result:
x ∨ xc xc
a−b
= ebt (x)(λ(b)−λ(a))
and ϕba,b (x) = x
−α
+x
a
x ∨ xc xc
λ(a)(a−b) λ(b)−λ(a)
−x
b
x ∨ xc xc
λ(b)(a−b) λ(b)−λ(a)
.
Similarly the hypothesis of Proposition 7 are satisfied by the payoff x + x b − x a where 1 > a > b > −α in case λ(a) > λ(b). In the global case, we could not find any simple condition on ϕ ensuring the existence of a continuous curve in the argument of inft >0 vϕ (t, x). Nevertheless, it is worth mentioning the following interesting class of European payoffs: if ϕ is a non-negative function equal to an invariant function ax + bx −α with a, b > 0, a + b > 0, less a nonφ(x) φ(x) negative function φ satisfying limx→0 x+x −α = limx→+∞ x+x −α = 0, then ∀x > 0, ∀t > 0,
vϕ (t, x) 6 ax + bx −α
lim vϕ (t, x) = ax + bx
t →+∞
−α
and
,
b which implies that ϕ(x) = inft >0 vϕ (t, x) is not trivial and that ∀x ∈ R, ∃bt (x) < b +∞, ϕ(x) = vϕ ( bt (x), x). The only assumption missing to apply Theorem 3 is the continuity of bt . The next section is dedicated to a family of payoffs ϕ included in the above class. In these examples, we explicit some American prices with a non-trivial Exercise region thanks to Theorem 3. We also check that the above mentioned continuity of bt is not always satisfied.
3. Case study: α > −1 and ϕ(x) = x(1{xK2 } ) To be able to compare the invariant functions x and x −α , we need to compare −α and 1. We choose the case α > −1 which is the more interesting from a financial point of view since ρ > 0 ⇔ α > 0. The payoff ϕ is equal to the invariant function x less the
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B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
function φ(x) = x1{K1 6x6K2 } . Since ∀x > 0, ∀t > 0,
0 < vϕ (t, x) < x
lim vϕ (t, x) = x
and
t →+∞
the function t → vϕ (t, x) is likely to be increasing for K1 < x < K2 and decreasing then increasing otherwise. This remark together with the easiness of computations motivate the choice of this example. The function ϕ satisfies the growth assumption in (H0) but is not continuous. Therefore, even if we make the computations for ϕ, we shall after all apply our results to a suitable regularization of ϕ. 3.1. The case K1 = 0 To simplify notations, we replace K2 by K and write ϕ(x) = x1{x>K} . This payoff corresponds to the sum of one Call and K Digit options with common strike K. Its simplicity allows to compute explicitely ϕb and bt . P ROPOSITION 9. – Let ϕ(x) = x1{x>K} . Then
vϕ (t, x) = xN d1 (t, x) , where ln( Kx ) + (ρ + √ d1 (t, x) = σ t and N(d) = Moreover,
Rd
σ2 )t 2
2
− y2 √dy −∞ e 2π
is the cumulative distribution function of the normal law.
2 b ϕ(x) = x1{x>K} N σ
s
x σ2 ρ+ ln 2 K
!
= vϕ bt (x), x
and b t (x) =
ln(x/K)1{x>K} 2
ρ + σ2
,
and ∀x > 0, t → vϕ (t, x) is strictly decreasing on [0, bt (x)] and strictly increasing on [bt (x), +∞). Proof. – Using Girsanov theorem, we get h
vϕ (t, x) = E xeσ Bt −
σ2 2
t
i
1{xeσ Bt +(ρ−σ 2 /2)t >K}
= xP xeσ Bt +(ρ+
σ2 2 )t
> K = xN d1 (t, x) .
By the chain rule, ∂t vϕ (t, x) = xN 0 (d1 (t, x))∂t d1 (t, x). Since ∀x, t > 0, xN 0 (d1 (t, x)) > 0 and ∂t d1 (t, x) =
(ρ +
σ2 )t 2
− ln( Kx )
2σ t 3/2
,
B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
we obtain that
∀x > 0,
11
∀t ∈ ]0, bt (x)[, ∂t vϕ (t, x) < 0, ∀t > bt (x), ∂t vϕ (t, x) > 0.
Hence inft >0 vϕ (t, x) = vϕ ( bt (x), x) and the explicit expression of this function is easily computed. 2 Let us now regularize things in order to apply our theorem. Let u > 0. The function x → vϕ (u, x) is continuous. Let (Pt )t >0 denote the semigroup associated with (0.1). By the semigroup property, the price of the European option with payoff vϕ (u, x) is Pt (Pu ϕ) = Pt +u ϕ. If we set ϕbu = inft >0 Pt (Pu ϕ), then by the previous proposition, ϕbu (x) = vϕ (u ∨ bt (x), x) = P0∨(bt (x)−u)(Pu ϕ)(x). Since bt is a continuous function with values in [0, +∞), so is btu (x) = 0 ∨ ( bt (x) − u). Applying Theorem 3, we obtain the price of the American option with payoff ϕbu : C OROLLARY 10. – Let u > 0. The price of the American option with payoff ϕbu (x) = vϕ (u ∨ bt (x), x) is vbϕam(t, x) = vϕ (t + u) ∨ b t (x), x u
v u
2 2u t ρ+σ =x N σ 2
!
x ln K
2
!
1{t +u6ln(x/K)/(ρ+σ 2/2)}
ln( Kx ) + (ρ + σ2 )(t + u) √ +N 1{t +u>ln(x/K)/(ρ+σ 2/2)} σ t +u
!
and the Exercise region is given by {(t, x): t + u 6 ln(x/K)/(ρ + σ 2 /2)}. Remark 11. – Although the payoff ϕbu has no financial meaning, this example provides a very interesting benchmark for numerical procedures devoted to American options since the price and the Exercise boundary are explicit. Let us also notice that this is a two-parameter (K and u) family of closed-formula. The payoff is of course obtained by setting t to zero in vbϕam (t, x). u
3.2. The case K1 > 0 The main purpose of this subsection is to design an example where there is no continuous curve in the argument of the infimum (Proposition 13). By a slight modification of the computations made in the proof of Proposition 9, we get
vϕ (t, x) = x N(−d1 (t, x) + N d2 t, x) , 2
ln( Kxi ) + (ρ + σ2 )t √ where for i = 1, 2 di (t, x) = . σ t It is not possible to compute ϕb explicitely but using the implicit functions theorem, we can study the sign of ∂t vϕ (t, x) to obtain: L EMMA 12. – There exist two differentiable functions t ∈ R∗+ → ξ1 (t) < ξ2 (t) satisfying
12
B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
(1) limt →0 ξi (t) = Ki (i = 1, 2), √ (2) ∀t > 0, ξ20 (t) > 0 and ∃(β, T ), 0 < β < T 6 (1 + ln K2 /K1 )/(2ρ + σ 2 ), ∀t < β, ξ10 (t) > 0 and ∀t > T , ξ10 (t) < 0, (3) ∀t > 0, ξ2 (t) > K2 e(ρ+
σ2 2 )t
ξ1 (t) < K1 e(ρ+
and
σ2 2 )t
∧
p
K1 K2 e−(ρ+
σ2 2
)t
and such that ∀t > 0, ∀x ∈ ]ξ1 (t), ξ2 (t)[,
∀x ∈ / ξ1 (t), ξ2 (t) ,
∂t vϕ (t, x) > 0
and
∂t vϕ (t, x) < 0.
Proof. – An easy computation yields that ∂t vϕ (t, x) is equal to the product of a strictly positive function with f (t, ln x) where f (t, y) = (y − a1 )eb1 y+c1 + (a2 − y)eb2 y+c2 where for i = 1, 2, σ2 ln Ki ai (t) = ln Ki + ρ + t, bi (t) = 2 , 2 σ t 2 ρ 1 ln Ki ci (t) = + ln K − . i σ2 2 2σ 2 t Since a1 < a2 , f (t, a2 ) = (a2 −a1 )eb1 a2 +c1 > 0. Hence the function y → f (t, y) vanishes at the same points as y → g(t, y) = e(b2 −b1 )y+(c2 −c1 ) −
y − a1 . y − a2
As a1 < a2 and b1 < b2 , the function y → g(t, y) is strictly increasing from −1 to +∞ on ] − ∞, a2 [ and from −∞ to +∞ on ]a2 , +∞[, so it vanishes exactly twice. Let 1 y1 < a2 < y2 denote the corresponding points. Since e(b2 −b1 )y1 +(c2 −c1 ) > 0 and yy11 −a < 1, −a2 we obtain respectively y1 < a1 and (b2 − b1 )y1 < c1 − c2 . We combine these upperbounds to get
y1 (t) < a1 (t) ∧ ln
p
σ2 K1 K2 − ρ + t . 2
(3.1)
We deduce that x → ∂t vϕ (t, x) vanishes exactly twice, at the points ξ1 (t) = ey1 (t ) and ξ2 (t) = ey2 (t ) which satisfy statement (3). As f (t, a2 ) > 0, ∂t vϕ (t, x) is strictly positive for x ∈ (ξ1 (t), ξ2 (t)). Moreover as b1 < b2 , f (t, y) < 0 for |y| large and ∂t vϕ (t, x) is strictly negative for 0 < x < ξ1 (t) and for x > ξ2 (t). Let us study more precisely the functions y1 (t) and y2 (t). Since ∀t > 0, ∀y 6= a2 (t), ∂y g(t, y) > 0, by the implicit function theorem, for i = 1, 2, yi (t) is continuously differentiable and yi0 (t) has the same sign as −∂t g(t, yi (t)). Expliciting the dependence of g on the time variable, we get
g(t, y) = exp
p ln(K2 /K1 ) σ2 y + ρ + t − ln K1 K2 σ 2t 2
−1
B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
+
ln(K1 /K2 ) 2
y − ln(K2 ) − (ρ + σ2 )t
13
,
p ln(K2 /K1 ) ln(K1 /K2 ) σ2 ∂t g(t, y) = y − ln K K exp y + ρ + t 1 2 σ 2t 2 σ 2t 2
− ln
p
K1 K2
+
σ2 ) ln(K1 /K2 ) 2 . 2 − ln K2 − (ρ + σ2 )t)2
(ρ + (y
√
Since y2 (t) > a2 (t) > ln K1 K2 , ∂t g(t, y2 (t)) is strictly negative and ∀t > 0, y20 (t) > 0. Moreover, when t → 0 the first term in g(t, y2 (t)) has a limit equal to +∞ and the equation g(t, y2 (t)) = 0 implies that the second term goes also to ∞ which gives limt →0 y2 (t) = ln K2 . By (3.1), y1 (t) < a1 (t) = ln K1 + (ρ + σ 2 /2)t. Hence when t → 0 the first term in g(t, y1 (t)) has a limit equal to 0. By considering the other terms we deduce that limt →0 y1 (t) = ln K1 . Hence the first term in ∂t g(t, y1 (t)) goes to 0 and limt →0 ∂t g(t, y1 (t)) < 0. Therefore ∃β > 0, ∀t ∈ ]0, β[, y10 (t) > 0. Using the equality g(t, y1 (t)) = 0 to replace the exponential in ∂t g(t, y1 (t)) and multiplying by (y1 (t) − ln K2 − (ρ + σ 2 /2)t)2 / ln(K2 /K1 ), we obtain that ∂t g(t, y1 (t)) has the same sign as
p −1 σ2 y (t) − ln K K y (t) − ln K − ρ + t 1 1 2 1 1 σ 2t 2 2 σ2 σ2 × y1 (t) − ln K2 − ρ + t − ρ+ . 2 2 √ As by √ (3.1), y1 (t) < ln K1 K2 − (ρ + σ 2 /2)t, we conclude that for some T 6 (1 + ln K2 /K1 )/(2ρ + σ 2 ), ∀t > T , ∂t g(t, y1 (t)) > 0 and y10 (t) < 0. 2
So the situation looks like in Fig. 1.
Fig. 1.
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B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
Let u > 0. The payoff vϕ (u, x) satisfies (H0). Let ϕbu (x) = inf vϕ (t + u, x). t >0
Since (t, x) → vϕ (t + u, x) is continuous and t → vϕ (t, x) is increasing for t > t (x) where t (x) is locally bounded (see Lemma 12), the function ϕbu (x) is continuous. According to Lemma 12, there exist 0 < β < T < +∞ such that t → ξ1 (t) is strictly increasing on [0, β] and strictly decreasing on [T , +∞). Concerning the existence of a continuous function btu such that ϕbu (x) = vϕ ( btu (x) + u, x) the situation depends on whether u < β or u > T . P ROPOSITION 13. – • If u > T , then ϕbu (x) = vϕ ( btu (x) + u, x) for the continuous function b tu (x) = 1{x6ξ1 (u)} ξ1−1 (x) − u + 1{x>ξ2 (u)} ξ2−1 (x) − u ,
where ξ1−1 denotes the inverse of the restriction of ξ1 to [T , +∞) and the price of the American option with payoff ϕbu is vbϕam (t, x) = vϕ (t ∨ btu (x)) + u, x
u
= vϕ (t + u) ∨ ξ1−1 (x), x 1{x6ξ1 (u)} + vϕ (t + u, x)1{ξ1 (u) 0, ∀x ∈ ξ1 (t + u), ξ2 (t + u) ,
vϕ (t + u, x) > vbϕam (t, x). u
Proof. – We first suppose that u > T . According to Lemma 12, t ∈ [0, +∞) → ξ1 (t + u) (respectively t ∈ [0, +∞) → ξ2 (t + u)) is decreasing (respectively increasing), and ∀x ∈ ]0, ξ1−1 (u)[ (respectively ∀x ∈ ]ξ2−1 (u), +∞[) t → vϕ (t + u, x) is decreasing on [0, ξ1−1 (x) − u] (respectively [0, ξ2−1 (x) − u]) and increasing on [ξ1−1 (x) − u, +∞[ (respectively [ξ2−1 (x) − u, +∞[). Moreover ∀x ∈ [ξ1 (u), ξ2 (u)], t → vϕ (t + u, x) is increasing. Hence ϕbu (x) = vϕ ( btu (x) + u, x) for the continuous function b tu (x) = 1{x6ξ1 (u)} ξ1−1 (x) − u + 1{x>ξ2 (u)} ξ2−1 (x) − u ,
and we deduce the price of the American option with payoff ϕbu by Theorem 3. We turn to the case u < β. Let F = {(t, x): vϕ (t + u, x) = ϕbu (x)}. According to Lemma 12, t → ξ1 (t) is increasing on [0, β]. We deduce that ∀t ∈ ]u, β[, vϕ (t, ξ1 (t)) > vϕ (u, ξ1 (t)) and (t − u, ξ1 (t)) ∈ / F . Hence F ⊂ F1 ∪ F2 F2 =
where F1 =
t − u, ξ1 (t) , t > β
and
t − u, ξ2 (t) , t > u ∪ (0, x), x ∈ ξ1 (u), ξ2 (u) .
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B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
Let btu be such that ∀x > 0, ϕbu (x) = vϕ ( btu (x) + u, x) i.e. ( btu (x), x) ∈ F . For x small enough ( btu (x), x) ∈ F1 and for x big enough ( btu (x), x) ∈ F2 . Since F1 and F2 are not connected, the function btu is discontinuous. Let t > 0 and x ∈ (ξ1 (t + u), ξ2 (t + u)). The positive continuous function
w ∈ W → Φ(w) = inf e−ρs vϕ t + u − s, xeσ ws +(ρ−
σ2 2 )s
s∈[0,t ]
− ϕbu xeσ ws +(ρ−
σ2 2 )s
,
where W = {w ∈ C([0, t], R), w(0) = 0}, is not constantly equal to 0. Indeed, when ∀s 6 0 ∨ (t + u − β), 2 σ wt +(ρ− σ2 )t
xe
< ξ1 (u),
and ∀s 6 t,
xeσ ws +(ρ−
ξ1 (t + u − s) < xeσ ws +(ρ− σ2 2 )s
σ2 2 )s
,
< ξ2 (t + u − s),
2 σ ws +(ρ− σ2 )s
then ∀s ∈ [0, t], (t − s, xe )∈ / F and Φ(w) > 0. As the support of the Wiener measure is W , E[Φ((Bs )s6t )] > 0. Let τ be a stopping time smaller than t. Then
vϕ (t + u, x) = E e−ρτ vϕ t + u − τ, Xτx
> E e−ρτ ϕbu Xτx
+ E Φ (Bs )s6t .
Since τ is arbitrary, we conclude that vϕ (t + u, x) − vbϕam(t, x) > E[Φ((Bs )s6t )] > 0. u
2
Remark 14. – (1) For any x > 0, t → vϕ (t + u, x) is continuous and increasing for t big enough. Hence t˜u (x) = sup{t: vϕ (t + u, x) = ϕbu (x)} is finite. When u < β, ∀t 6 t˜u (x),
ϕbu (x) 6 vbϕam(t, x) 6 vbϕam t˜u (x), x 6 vϕ t˜u (x) + u, x , u
i.e.
u
vbϕam (t, x) = ϕbu (x), u
but ∃T (x) such that for t > T (x), x ∈ ]ξ1 (t + u), ξ2 (t + u)[ and we cannot deduce vbϕam (t, x) from the price of the European option with payoff ϕ. u (2) Let u < β and x ∗ = sup{x: (t, x) ∈ F1 ∩ F } where F, F1 are defined in the previous proof. Since F1 and F are closed and limt →+∞ ξ1 (t) = 0, ∃t ∗ > β − u such that (t ∗ , x ∗ ) ∈ F1 ∩ F i.e. vϕ (t ∗ + u, x ∗ ) = ϕbu (x ∗ ). Since x ∗ = sup{x: (t, x) ∈ F1 ∩ F }, ∀x ∈ ]x ∗ , ξ2 (u)], vϕ (u, x) = ϕbu (x) and by continuity, vϕ (u, x ∗ ) = ϕbu (x ∗ ). Hence {t > 0, vϕ (t + u, x ∗ ) = ϕbu (x ∗ )} contains at least two elements which is not surprising with regard to Remark 4. 4. Analyticity and some consequences In this section we give some properties of the map ϕ → ϕb which are consequences of the following analyticity of vϕ in the pair (t, x): P ROPOSITION 15. – The function vϕ (t, x) is analytic in ]0, ∞[ × R∗+ . This is a consequence of the same property for the solution of the standard heat equation (which does not seem to be universally known in fact but can be shown by a direct estimation of the derivatives of the solution).
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B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
4.1. One-to-one property b Obviously if ϕ1 is an invariant function Let us study now the injectivity of ϕ → ϕ. (e.g. the nil function), then ϕb1 = ϕ1 and there are plenty of other functions ϕ such that ϕb = ϕb1 , for instance ϕ = ϕ1 + φ where φ is a non-negative continuous function φ(x) φ(x) b satisfying limx→0 x+x −α = limx→+∞ x+x −α = 0. The same phenomenon occurs if t (x) = ∞ everywhere. Therefore the following one-to-one statement is optimal:
P ROPOSITION 16. – Let ϕ1 and ϕ2 satisfy the assumptions of Theorem 3 and assume that ϕ1 is not an invariant function and that there is a value x0 such that b t1 (x0 ) < ∞. Then ϕb1 = ϕb2 ⇒ ϕ1 = ϕ2 . Proof. – For any t0 > b t1 (x0 ) there is an ε > 0 small enough such that the ball B centered at (t0 , x0 ) with radius ε lies in the Continuation region of ϕb1 . In particular vbϕam(t, x) = vϕ1 (t, x) on B and (t, x) 7→ vbϕam (t, x) is analytic on B. Now vbϕam (t, x) = 1
1
1
vbϕam(t, x), therefore vbϕam(t, x) is analytic on B. If bt2 (x0 ) = +∞ then by taking ε small 2 2 enough we can assume vbϕam (t, x) = ϕb2 (x) on B. Therefore ∂t vϕ1 (t, x) = 0 on B and by 2 the analyticity of vϕ1 , ∂t vϕ1 (t, x) = 0 everywhere, which gives that ϕ1 is an invariant function. Since this case is ruled out by assumption, b t2 (x0 ) < ∞. Now on the right of bt1 (x0 ) ∨ bt2 (x0 ), vϕ1 and vϕ2 match on some small enough ball, therefore everywhere, which gives ϕ1 = ϕ2 by the continuity of t 7→ vϕi (t, x) at t = 0. 2 4.2. On the range of ϕ → ϕb
P ROPOSITION 17. – Let ϕ satisfy the assumptions of Theorem 3 and assume that ϕ is not an invariant function and that 0 < bt (x0 ) < ∞ for some point x0 . Then there is an b is analytic. open dense subset of bt −1 (R∗+ ) on which bt , and therefore ϕ, b Proof. – First note that by composition ϕ(x) = vϕ ( bt (x), x) is analytic as soon as bt is. b Pick up some point x1 such that 0 < t (x1 ) < ∞. Since vϕ is analytic, by the implicit function theorem the equation ∂t vϕ (t, x) = 0 on a small enough neighborhood V of ( bt (x1 ), x1 ) defines an analytic curve x 7→ a(x) as soon as ∂t22 vϕ ( bt (x1 ), x1 ) 6= 0. Now by the continuity of bt , ∂t vϕ ( bt (x), x) = 0 in V so that a ≡ bt on a neighborhood of x1 . Unfortunately it is not granted that there exists x1 such that ∂t22 vϕ ( bt (x1 ), x1 ) 6= 0. Let us first remark that there is some point x2 in bt −1 (R∗+ ) such that t 7→ vϕ (t, x2 ) is not constant: otherwise ∂t vϕ (t, x) = 0 would hold on some non-empty open set and therefore everywhere, thus ϕ would be an invariant function. Moreover the set of such points is dense in b t −1 (R∗+ ). Let q = inf{n > 0, ∂tnn vϕ ( bt (x2 ), x2 ) 6= 0}. Then q is finite. If q−1 q = 2 we are over. Otherwise notice first that the equation ∂t q−1 vϕ (t, x) = 0 on a small enough neighborhood V of ( bt (x2 ), x2 ) defines an analytic curve x 7→ b(x). Consider then the quantity q(x) = inf{n > 0, ∂tnn vϕ ( bt (x), x) 6= 0} on a neighborhood of x2 . By the analyticity of vϕ , q(x) 6 q(x2 ) on a sufficiently small neighborhood W of x2 . Either q−1 q(x) ≡ q(x2 ), in which case ∂t q−1 vϕ ( bt (x), x) = 0 on W , therefore bt ≡ b is analytic on W, otherwise there is some point x3 in W such that 0 < bt (x3 ) < ∞ and q(x3 ) < q(x2 ).
B. JOURDAIN, C. MARTINI / Ann. Inst. Henri Poincaré Anal. nonlinear 18 (2001) 1–17
17
By induction we thus either stop and get an analytic curve or reach the level q = 2 at some point. The proof is complete. 2 b This proposition gives a first characterization statement about the functions ϕ:
C OROLLARY 18. – Let ϕ satisfy the assumptions of Theorem 3. Then ϕb is either Asuperharmonic (i.e. Aϕb > 0 in a weak sense) or A-subharmonic (i.e. Aϕb 6 0 in a weak sense) or analytic on a non-empty open subset of R∗+ . In particular ϕ → ϕb is not onto on the space of functions satisfying (H0). Proof. – The first case corresponds to bt = 0 everywhere, the second one to the case b b t = +∞ everywhere (ϕ(x) = lim inft →+∞ vϕ (t, x) is then an A-subharmonic function by Fatou’s lemma) and the last one to the previous proposition.
2
5. Conclusion In this paper, for a fairly general class of payoffs ϕ, we deduce from the European b price vϕ (t, x) the American price of the claim with payoff ϕ(x) = inft >0 vϕ (t, x). We give examples of explicit computations. The characterization of the payoffs ϕb obtained in this way remains an open question. A work devoted to design new approximations of the American Put price relying on our approach is in progress. Acknowledgements We thank Régis Monneau (CERMICS) for numerous stimulating and fruitful discussions. REFERENCES [1] Crank J., Free and Moving Boundary Problems, Oxford University Press, 1984. [2] Martini C., The UVM model and American options, Rapport de Recherche no 3697, INRIA, 1999. [3] Musiela M., Rutkowski M., Martingale Methods In Financial Modelling, Springer, 1998. [4] Revuz D., Yor M., Continuous Martingales and Brownian Motion, Springer, 1991.
