Using the Donsker Delta Function to Compute Hedging Strategies

Transcription

1 Using the Donsker Delta Function Using the Donsker Delta Function to Compute Hedging Strategies Knut Aase ),), Bernt Øksendal ),) and Jan Ubøe 3) ) Norwegian School of Economics and Business Administration Helleveien 3, N-535 Bergen - Sandviken, Norway ) Department of Mathematics, University of Oslo, Box 53 Blindern, N-36 Oslo, Norway 3) Stord/Haugesund College, Skåregaten 3, N-5 5, Haugesund, Norway ABSAC. We use white noise calculus and the Donsker Delta Function to find explicit formulas for the replicating portfolios in a Black-Scholes market for a class of contingent -claims.. Introduction As a motivation for this paper we start by considering the following problem from mathematical economics: Fix > and consider the following simple market model, with two securities: ) A risk-free asset (e.g., a bank account), where the price A t per unit at time t is given by the differential equation da t = ρ(t)a t dt, A = (.) ) A risky asset (e.g., a stock), where the price S t per unit at time t is given by the stochastic differential equation ds t = µ(t)s t dt + σ(t)s t db t,s > constant (.) Here ρ(t),µ(t) and σ(t) are given deterministic functions with the property that ( ρ(s) + µ(s) + σ (s) ) ds < (.3) For simplicity we assume that σ is bounded away from zero. B t = B t (ω); t,ω Ω denotes - dimensional Brownian motion starting at zero. he probability law of B t is denoted by P and the σ-algebra generated by {B s ( )} s t is denoted by F t. We refer to, e.g., Ø for more information about stochastic differential equations. Now let φ :, be another deterministic function such that φ (t)dt <, and define Z(t) =Z(t, ω) = Let F (ω) be a contingent -claim of Markovian type, i.e., given by t φ(s)db s (ω); t (.4) F (ω) =h(z( )) (.5) where h : is a bounded measurable function. Since this simple extension of the Black and Scholes market is complete, it is well known that the claim F can be hedged, i.e., there exist a replicating (self-financing) portfolio for F (see details below). he problem we study in this paper is:

2 Knut Aase et al. POBLEM. How do we find explicitly such a replicating portfolio for F? We now want to describe this in more detail: Let (ξ t,η t )beaportfolio, i.e., ξ t = ξ t (ω),η t = η t (ω) are F t -adapted stochastic processes interpreted as the number of units held by a person at time t of assets # and #, respectively. hen the value V t = V t (ω) of this portfolio at time t is defined by he portfolio is called self-financing if V t = ξ t A t + η t S t (.6) dv t = ξ t da t + η t ds t (.7) his means that no external funds are added to the portfolio and that no funds are extracted from the portfolio as time evolves. From now on we consider only self-financing portfolios. From (.6) we get Substituting this in (.7) and using (.) (.), we get ξ t = V t η t S t A t (.8) dv t =(V t η t S t ) da t + η t ds t A t ( ) = ρ(t)v t dt + η t S t (µ(t) ρ(t))dt + σ(t)dbt Since σ(t) for a.a. t, this can be written where dv t = ρ(t)v t dt + σ(t)η t S t ( α(t)dt + dbt ) α(t) = Multiplying (.9) by the integrating factor e t µ(t) ρ(t) σ(t) ρ(s)ds,weget d (e t ) ρ(s)ds V t = e t ρ(s)ds ( ) σ(t)η t S t α(t)dt + dbt (.9) (.) Hence e ρ(s)ds V (ω) =V + e t ρ(s)ds ( ) σ(t)η t S t α(t)dt + dbt (.) Now suppose that F (ω) is a given (European) contingent -claim, i.e., F (ω) is a given F -measurable, lower bounded random variable. o hedge such a claim means to find a constant V and a self-financing portfolio (ξ t,η t ) such that the corresponding value process V t starts up with value V for t = and ends up with the value V (ω) =F (ω) a.s. (.) at time. V is then the market value of F at time. We also require that the process {V t } t, is (t, ω)-a.s. lower bounded. By (.) combined with (.8) we see that it suffices to find V and a process u(t, ω) such that e ρ(s)ds F (ω) =V + u(t, ω) ( ) α(t)dt + db t (.3) and P u (s, ω)ds < = P u(s, ω)α(s) ds < = (.4)

3 and such that put Using the Donsker Delta Function { t u(t, ω)( ) } α(t)dt + db t is lower bounded. If such a process u(t, ω) is found, we t, t η t = e ρ(s)ds σ(t) St u(t, ω) (.5) and solve for ξ t using (.8). It is well known and easy to see by the Girsanov theorem that if α (s)ds < (.6) then V is unique and given by V = E Q e ρ(s)ds F (.7) (provided this quantity is finite), where E Q denotes expectation with respect to the measure Q defined on F by dq(ω) = exp α(s)db s α (s)ds dp (ω) (.8) so that B t := t α(s)ds + B t (.9) is a Brownian motion with respect to Q. o find u(t, ω), there are several known methods: a) If the claim F (ω) is of Markovian type, i.e., F (ω) =h(s (ω)) for some (deterministic) function h :, then u(t, ω) can (in principle) be found by solving a (deterministic) boundary value problem for a parabolic partial differential equation. See BS, M, and D, Section 5D for details. b) For some not necessarily Markovian type claims F (ω) one can (in principle) apply the Clark-Ocone theorem (as extended by Karatzas and Ocone KO) to express u(t, ω) as follows: u(t, ω) =E Q D t F F t (.) where D t F is the Malliavin derivative of F at t. he problems with this formula are: i) It is in general difficult to compute conditional expectations and ii) the Malliavin derivative D t F only exists under additional restrictions on F. For example, it is not sufficient that F L (F,P) and it does not exist for the F given by (.5) if h is not differentiable. he purpose of this paper is to give an alternative approach based on white noise calculus and the Donsker delta function. We will show how this approach gives explicit formulas quickly and with easy, intuitive proofs, once the basic white noise calculus has been established. We illustrate this by using the method to solve Problem.. See heorem 3.9. and Corollary 3. together with the remarks following the corollary. Although Problem. could also be solved by Method a) and - with some additional work - by Method b), it is conceivable that the white noise approach can cover some cases which are not well adapted to Methods a) and b). Moreover, it may give new insights. See (3.3) and the corresponding remark. In Section we briefly recall some of the basic white noise theory. hen in Section 3 we give a special representation of the Donsker delta function and combine it with white noise calculus to compute explicitly the hedging strategies requested in Problem.. In Section 4 we prove a similar formula for the n- dimensional case.. White noise, Hida distributions and the Wick product Here we briefly recall some of the main concepts and results from white noise theory. For more information we refer the reader to HKPS and HØUZ. Our notation will follow that from HØUZ. 3

4 Knut Aase et al. From now on we will assume that our Brownian motion is constructed on a white noise probability space (Ω, F,P) and we let (S) and (S) denote the space of stochastic test functions and the space of stochastic distributions (Hida distributions), respectively. Using the Hermite functions e (x),e (x),... (which form an orthonormal basis for L ()) and the Hermite polynomials h n (x); n =,,,..., one constructs an orthogonal L (P ) basis {H α (ω)} α I where I denotes the set of all multi-indices α =(α,α,...) of arbitrary but finite length, where α,α,... are non-negative integers. hus every X L (P ) has a unique representation X(ω) = α c α H α (ω); c α where X L (P ) = E P X = α α!c α (.) and where α! =α!α! when α =(α,α,...) I. he space (S) of stochastic test functions can be described as the set of all X(ω) = α c αh α (ω) L (P ) such that X,k := α!c α(n) qα < for all q (.) α where (N) β = β ( ) β (k) βk if β =(β,β,...) I Similarly, the space (S) of Hida distributions can be described as the set of formal series X(ω) = α c αh α (ω) such that there exists q such that X, q := α α!c α(n) qα < (.3) hus we have (S) L (P ) (S) he family of seminorms,k ; k gives a natural projective topology on (S) and an inductive topology on (S). With these topologies (S) becomes the dual of (S). he action of F (ω) = α a αh α (S) on f(ω) = α b αh α (S) is given by <F,f>= α α!a α b α (.4) One of the important features about the Hida space (S) is that it contains the singular white noise W t (ω) for all t. More precisely, if we define W t (ω) = e i (t)h ɛi (ω) (.5) i= where ɛ i =(,,...,,...) with on the ith place, then W t (ω) (S) for each t and we have the crucial identities d dt B t(ω) =W t (in (S) ) (.6) and t B t (ω) = W s ds (integration in (S) ) (.7) he last identity can be generalized considerably by means of the Wick product: 4

5 Using the Donsker Delta Function DEFINIION. he Wick product X Y of X(ω) = α a αh α (ω) (S) and Y (ω) = α b βh β (ω) (S) is defined by For example we have and more generally ( ) φ(s)db s ( ) X Y (ω) = a α b β H α+β (ω) = γ ( α,β ) ψ(s)db s = α+β=γ a α b β H γ (ω) (B t B t )(ω) =B t (ω) t (.8) ( ) φ(s)db s ( ) ψ(s)db s φ(s)ψ(s)ds (.9) for all φ, ψ L (). Some important properties of the Wick product are listed below: X (S),Y (S) X Y (S) (.) X (S),Y (S) X Y (S) (.) X Y = Y X (commutative law) (.) X (Y Z) =(X Y ) Z (associative law) (.3) X (Y + Z) =X Y + X Z (distributive law) (.4) Using the associative law we can define Wick powers X Y = X Y if X or Y is deterministic (.5) EX Y =EX EY (when defined) (.6) X n = X X X (n times) More generally, if f(z) = a k z k k= is entire, i.e., an analytic function of the complex variable z in the complex plane C, we can - for some X (S) - define the Wick version f (X) = a k X k (S) (.7) k= For example, if φ L () is deterministic, then exp φ(s)db s = exp φ(s)db s φ (s)ds (.8) We also mention the chain rule in (S) : Suppose t X t : (S) is continuously differentiable and let f : C C be entire such that f() and f (X t ) (S) for all t, then d dt f (X t ) = f (X t ) d dt X t in (S) (.9) Finally we recall the following important connection between Ito integration and the Wick product: Let u(t, ω) beanf t -adapted process such that E b a u (t, ω)dt <. hen u(t, ω) W t is integrable in (S) and b b u(t, ω)db t (ω) = u(t, ω) W t (ω)dt (.) a 5 a

6 Knut Aase et al. (See LØU, B and HØUZ, heorem.5.9 and the references therein). As a simple example to illustrate the above, first note that by the chain rule we have and hence d dt exp B t = exp B t db t = exp B t W t dt exp B t = exp B + =+ t t exp B s db s exp B s W s ds which is a direct proof (without using the Ito formula) that exp B t is a martingale. he Hermite transform In white noise analysis one makes use of several different transforms, the most popular being the S- transform and the Hermite transform, H. he construction of these transforms depends on the particular choice of Hermite functions as a basis for L (). When expanded along this basis, the H-transform can be defined as follows DEFINIION. Let X(ω) = α a αh α (ω) (S), then the Hermite transform of X (with respect to the basis {e k } k ), denoted by HX or X, is defined by HX(z) = X(z) = α a α z α C (when convergent) (.) where z =(z,z,...) C N (the set of all sequences of complex numbers), and if α =(α,α,...) I, where z j =. z α = z α zα zαn n (.) One can verify that the sum in (.) converges for all z C N c (the set of all finite length sequences of complex numbers), and that any element in (S) is uniquely characterized through its H-transform. We recall the important relation HX Y (z) =HX(z) HY(z) (.3) (.3) can be extended to cover Wick-versions, so in general if f : C C is entire, f() and f (X) (S). Hf (X)(z) =f(hx(z)) (when convergent) (.4) While the basis of Hermite functions is necessary for the definition of the topological structure in the Hida distribution space, it turns out that other bases may be convenient for computational purposes. If we remain within L (P ), the Wick product can be expanded along any orthonormal basis for L (), see HØUZ, Appendix D: Base invariance of the Wick product. In what follows we will sometimes specialize the theory to Wick powers of smoothed white noise. Within this context a different version of the H- transform can be considered. By abuse of notation, we do not distinguish between the (strictly speaking, different) versions of the transform. Given any φ L (), we define the smoothed white noise, w(φ) =w(φ, ω), by w(φ, ω) := φ(s)db s (ω) (.5) If we consider the context of random variables on the form X(ω) = k= a kw(φ) k, then it is convenient to make use of the following formulation, see GHLØUZ, 4.: 6

7 Using the Donsker Delta Function POPOSIION.3 Let φ L () with φ L () =. Suppose X(ω) = k= a kw(φ) k (S), and define f(z) = k= a kz k for z C. Suppose y f(x + iy) is integrable with respect to the measure e y / dy on for all x and put F (x) = f(x + iy)e y / dy π Suppose V (ω) :=F (w(φ, ω)) L (P ). hen X(ω) =V (ω) a.s., i.e., X(ω) = f(x + iy)e y / dy x=w(φ,ω) (.6) π 3. he Donsker delta function and the first main theorem Donskers δ-function is a generalized white noise functional which have been treated in several papers within white noise analysis, see, e.g., H, K and also HKPS and the references therein. For completeness we give an independent presentation here. DEFINIION 3. Let Y :Ω be a random variable which also belongs to (S). hen a continuous function δ Y ( ) : (S) is called a Donsker delta function of Y if it has the property that g(y)δ Y (y)dy = g(y ) a.s. (3.) for all (measurable) g : such that the integral converges. POPOSIION 3. Suppose Y is a normally distributed random variable with mean m and variance v>. hen δ Y is unique and is given by the expression δ Y (y) = exp (y Y ) (S) (3.) πv v POOF Let G Y (y) denote the right hand side of (3.). It follows from the characterization theorem for (S) (see PS) that G Y (y) (S) for all y and that y G Y (y) is continuous for y. We verify that G Y satisfies (3.), i.e., that g(y)g Y (y)dy = g(y ) a.s. (3.3) First let us assume that g has the form g(y) =e λy for some λ C (3.4) hen by taking the Hermite transform of the left hand side of (3.3), we get H g(y)g Y (y)dy = = e λy H G Y (y) dy e λy (y Ỹ ) exp dy πv v (3.5) 7

8 Knut Aase et al. where Ỹ = Ỹ (z) is the Hermite transform of Y at z =(z,z,...) C N. he expression (3.5) may be regarded as the expectation of e λz where Z is a normally distributed random variable with mean Ỹ and variance v. NowZ := Y m+ỹ is such a random variable. Hence (3.5) can be written as Eeλ(Y m+ỹ ), which by the well known formula for the characteristic function of a normal random variable is equal to expλỹ + λ v. We conclude that H g(y)g Y (y)dy = expλỹ + λ v=h exp λy + λ v = HexpλY = Hg(Y ) his proves that (3.3) holds for functions g given by (3.4). herefore (3.3) also holds for linear combinations of such functions. By a well known density argument, (3.3) holds for all g such that the integral in (3.) converges. It remains to prove uniqueness: If H : (S) and H : (S) are two continuous functions such that g(y)h i (y)dy = g(y ); i =, (3.6) for all g such that the integral converges, then in particular (3.6) must hold for all continuous functions with compact support. But then clearly we must have H (y) =H (y) for a.a. y and hence for all y by continuity. LEMMA 3.3 Let ψ :,,φ :, be deterministic functions and such that φ, := φ (s)ds <. Define Y (t) = t ψ(s)ds + t ψ(s) ds < and φ(s)db s, t (3.7) hen exp + exp (y Y ( )) φ = exp, (y Y (t)) φ, y φ, y Y (t) φ, (ψ(t)+φ(t)w t )dt (3.8) POOF his is just an application of the fundamental theorem of calculus plus the chain rule in (S) : Define H :, (S) by H(t) = exp (y Y (t)) φ ; t (3.9), 8

9 Using the Donsker Delta Function hen dh H( )=H() + dt dt = exp y φ, + exp (y Y (t)) φ d (y Y (t)), dt φ dt, = exp y φ, + exp (y Y (t)) φ, y Y (t) φ, (ψ(t)+φ(t)w t )dt We are now ready for the first main result in this section: HEOEM 3.4 Let φ :,,α:, be deterministic functions such that < φ, := φ (s)ds < and α (s)ds < (3.) Define Y (t) =Y (t, ω) = Let f : be bounded. hen where and t f(y ( )) = V + V = u(t, ω) =φ(t) φ(s)db s + t f(y) exp π φ, f(y) exp π φ, φ(s)α(s)ds; t (3.) u(t, ω) (α(t)+w t )dt (3.) y φ, (y Y (t)) φ, dy (3.3) y Y (t) φ dy (3.4), POOF We now combine Proposition 3. and Lemma 3.3 to get, with ψ(s) = φ(s)α(s) f(y) f(y ( )) = f(y)δ Y ( ) (y)dy = exp π φ, f(y) y f(y) = exp π φ, φ dy +, π φ, ( ) exp (y Y (t)) φ y Y (t), φ (φ(t)α(t)+φ(t)w t )dt dy, ( f(y) = V + φ(t) π φ, exp (y Y (t)) φ, = V + u(t, ω) (α(t)+w t )dt ) y Y (t) φ dy (α(t)+w t )dt, (y Y ( )) φ, dy (3.5) 9

10 Knut Aase et al. In the application of heorem 3.4 the following result will be useful: POPOSIION 3.5 Let p(x) =ax +bx+c, where a, b, c are real constants. Let ψ be as before and suppose that a ψ <, where ψ = ψ (s)ds. Define Y (ω) = ψ(s)db s hen exp ay + by + c =K ψ exp K ψ where the constant K ψ is defined by (ay + by + c + (4ac b ) ψ ) (3.6) K ψ := +a ψ (3.7) POOF ψ ψ We expand the Wick product along a base with φ := Y (ω) = ψ w(φ). With reference to Proposition.3, consider as its first base element, and note that Fix x. hen f(z) :=e a ψ z +b ψ z+c F (x) := f(x + iy)e y / dy π = e a ψ (x y +ixy)+b ψ (x+iy)+c e y / dy π = e a ψ x +b ψ x+c e i(xa ψ +b) ψ y ( +a ψ )y dy π = e a ψ x +b ψ x+c +a ψ (ax ψ +b) ψ e +4a ψ (3.8) (3.9) In this calculation we made use of the familiar formula e iαt β t dt = e α 4β (3.) π β Hence V (ω) :=F (w(ψ)) = e aw(ψ) +bw(ψ)+c +a ψ (aw(ψ)+b) ψ e +4a ψ = = +bw(ψ)+c +a ψ eaw(ψ) e a ψ w(ψ) +ab ψ w(ψ)+ b ψ +a ψ aw(ψ) +a ψ w(ψ) +bw(ψ)+ab ψ w(ψ)+c+ac ψ a ψ w(ψ) ab ψ w(ψ) b ψ e +a ψ +a ψ (3.) = +a ψ e +a ψ (aw(ψ) +bw(ψ)+c+ (4ac b ) ψ ) L (P ) herefore the result (3.6), (3.7) follows from Proposition.3.

11 Using the Donsker Delta Function COOLLAY 3.6 Under the same conditions as in the previous proposition, we have exp a(y Y ) =K ψ expak ψ (y Y ) (3.) POOF Just note that b 4ac = in this case. COOLLAY 3.7 Let φ(t),y(t) be as in heorem 3.4. Let t< and assume that hen exp φ, φ t, := φ (s)ds > (y Y (t)) φ =, t exp φ t, (y Y (t)) φ t, (3.3) POOF Put ψ(s) =φ(s)x,t in Corollary 3.6. hen a =, and we see that φ, a ψ = φ,t φ = φ, φ t,, φ, < (3.4) by our assumptions. Moreover K ψ = +a ψ = φ,t φ, = φ t, φ, (3.5) Hence K ψ = φ, and ak ψ φ = t, φ t, Corollary 3.7 then follows directly from Corollary 3.6. (3.6) LEMMA 3.8 Let φ(t),y(t) and φ t, be as in Corollary 3.7. hen exp (y Y (t)) φ, = exp φ t, φ, (y Y (t)) φ t, y Y (t) φ, y Y (t) φ t, (3.7) POOF If we differentiate both sides of (3.3) w.r.t. y, the result follows. We can now give a more explicit (and familiar) representation than the one given in heorem 3.4:

12 Knut Aase et al. HEOEM 3.9 Let φ(t),y(t) be as in heorem 3.4 and assume that Let f : be bounded. hen φ t, := φ (s)ds > for all t< (3.8) t f(y ( )) = V + g(t, ω) ( ) α(t)dt + db t where and V = g(t, ω) =φ(t) f(y) exp π φ, f(y) exp π φ t, y φ, (y Y (t)) φ t, dy (3.9) y Y (t) φ dy (3.3) t, POOF We will apply heorem 3.4, and therefore we consider u(t, ω) :=φ(t) By Lemma 3.8, u(t, ω) =g(t, ω). Hence f(y) exp π φ, (y Y (t)) φ, y Y (t) φ dy (3.3), E u (t, ω)dt =E g (t, ω)dt < and heorem 3.4 gives with V as in (3.9) (or (3.3)), that f(y ( )) = V + = V + = V + u(t, ω) (α(t)+w t )dt g(t, ω) (α(t)+w t )dt g(t, ω)(α(t)dt + db t ) as claimed. EMAK he conclusion of heorem 3.9 remains true without the assumption (3.8) if we interprete g(t, ω) as when φ t, =.

13 Using the Donsker Delta Function EMAK Although the expression (3.3) clearly has a computational advantage to the Wick version (3.4), it should be noted that (3.4) may give some insight which is not evident from (3.3). For example, we might ask for the limiting behaviour as t of the replicating portfolio g(t, ω) in (3.3). If φ(t) is continuous at t =, then by (3.4) we see that lim g(t, ω) = lim t t =φ( ) u(t, ω) f(y) exp π φ, (y Y ( )) φ, y Y ( ) φ dy, (3.3) his limit clearly exists in (S). COOLLAY 3. For the digital payoff F (ω) =X K, ) Y ( ) we have the representation X K, ) (Y ( )) = V + u(t, ω)(α(t)dt + db t ) where V = K exp π φ, y φ, dy (3.33) and u(t, ω) = φ(t) (K Y (t)) exp π φ t, φ t, (3.34) POOF Here f(y) =X K, ) y, so we see that (3.33) follows from (3.3) by performing the integration with respect to y. EMAK o be precise, the hedging procedure w.r.t. the contingent -claim, F (ω) = h(z( )), in (.5), is carried out as follows: Put Y (t) =Z(t)+ t α(s)φ(s)ds and let f(x) :=e ρ(s)ds h(x α(s)φ(s)ds) With these definitions e ρ(s)ds F (ω) =f(y ( )) and V and u(t, ω) in (.3) are then provided by the explicit expressions in heorem he multi-dimensional case In this section we generalize the results of the previous section to arbitrary dimension n. We let B(t) = (B (t),...,b n (t)) denote n-dimensional Brownian motion (where in general M denotes the transpose of the matrix M). Similarly W (t) =(W (t),...,w n (t)) is n-dimensional white noise. 3

14 Knut Aase et al. DEFINIION 4. Let Y =(Y,...,Y n ):Ω n be a random variable, each component of which belongs to (S). hen a continuous function δ Y ( ) : n (S) is called a Donsker delta function of Y if it has the property that n g(y)δ Y (y)dy = g(y ) a.s. (4.) for all (measurable) g : n such that the integral converges. Here - and in the following - dy = dy dy n denotes n-dimensional Lebesgue measure. POPOSIION 4. Suppose Y :Ω n is a normally distributed random variable with mean m = EY and covariance matrix C =c ij i,j n. Suppose C is invertible with inverse A =a ij i,j n. hen δ Y (y) is unique and is given by the expression where A is the determinant of A. δ Y (y) =(π) n/ A exp n a ij (y i Y i ) (y j Y j ) (4.) POOF Let G Y (y) denote the right hand side of (4.). We verify that G Y satisfies (4.), i.e., that g(y)δ Y (y)dy = g(y ) a.s. (4.3) n o this end let us first assume that g has the form i,j= g(y) =e λ y = e λy+ +λnyn (4.4) for some λ =(λ,...,λ n ) C n. hen taking the H-transform of the left hand side of (4.3), we get H = g(y)g Y (y)dy = e λ y HG Y (y)dy n n e λ y (π) n/ A exp n a ij (y i Ỹi) (y j Ỹj) n where Ỹ = Ỹ (z) =(Ỹ(z),...,Ỹn(z)) is the Hermite transform of Y =(Y,...,Y n )atz =(z,z,...) C N. he expression (4.5) may be regarded as the expectation of e λ Z where Z is a normally distributed random variable with mean Ỹ and covariance matrix C = A. Now Z := Y m + Ỹ is such a random variable. Hence (4.5) can be written as Ee λ (Y m+ỹ ), which by the well known formula for the characteristic function of a normal random variable is equal to expλ Ỹ + n i,j= c ijλ i λ j. We conclude that i,j= H g(y)g Y (y)dy = exp λ Ỹ + n c ij λ i λ j n i,j= =H exp λ Y + n c ij λ i λ j = Hexpλ Y = Hg(Y ) i,j= (4.5) 4

15 Using the Donsker Delta Function his proves that (4.3) holds for all functions g given by (4.4). Hence using, e.g., the Fourier transform, we see that (4.3) holds in general. It remains to prove uniqueness: If H : n (S) and H : n (S) are two continuous functions such that n g(y)h i (y)dy = g(y ) for i =, (4.6) for all g such that the integral converges, then in particular (4.6) must hold for all continuous functions with compact support. But then we clearly must have H (y) =H (y) for a.a. y n and hence for all y n by continuity. In the following we let ψ :, n,φ:, n n be deterministic functions such that ψ(s) ds < and φ := n i,j= φ ij(s)ds < (4.7) Define Y (t) = = t t φ(s)db(s)+ t ψ(s)ds (φ(s)w (s)+ψ(s))ds ; t (4.8) m = EY ( ) = ψ(s)ds n (4.9) and, for i, j n c ij = E(Y i ( ) m i )(Y j ( ) m j ) = Assume that the matrix C =c ij i,j n is invertible and put (φφ ) ij (s)ds (4.) A =a ij i,j n = C (4.) Define H(t) =H(t, y) = exp n a ij (y i Y i (t)) (y j Y j (t)) ; t i,j= = exp (y Y (t)) (y Y (t)) ; t (4.) 5

16 Knut Aase et al. LEMMA 4.3 ( H( )=H() + H(t) n a ij ((y i Y i (t)) (φ j (t)w (t)+ψ j (t)) i,j= +(y j Y j (t)) (φ i (t)w (t)+ψ i (t)) ) dt (4.3) where φ j is row number j of the matrix φ. POOF By the chain rule H( ) = H() + = H() + = H() + H(t) dh dt = H() + H(t) d dt dt H(t) ( n a ij (y i Y i (t)) (y j Y j (t)) dt i,j= n a ij ((y i(t)) i Y i (t)) d dt Y j(t)+(y j Y j (t)) d dt Y dt i,j= n a ij ((y i Y i (t)) (φ j (t)w (t)+ψ j (t)) i,j= +(y j Y j (t)) (φ i (t)w (t)+ψ i (t))) ) dt We can now prove the main result of this section: HEOEM 4.4 Let α :, n be a deterministic function such that α = Let φ :, n n be as in (4.7) and define Y (t) = Let f : n be bounded. hen t φ(s)db(s)+ α (s)ds < (4.4) t φ(s)α(s)ds ; t (4.5) f(y ( )) = V + u(t, ω) (α(t) +W (t))dt (4.6) where and V =(π) n/ A f(y) exp y Aydy (4.7) n u(t, ω) =(π) n/ A f(y) exp ( n ) (y Y (t)) Aφ(t) dy (y Y (t)) A(y Y (t)) (4.8) 6

17 Using the Donsker Delta Function POOF We apply Proposition 4. and Lemma 4.3 with ψ(t) =φ(t)α(t) and a = φ(t)α(t)dt to get f(y ( )) = f(y)δ Y ( ) (y)dy n =(π) n/ A f(y) exp n a ij (y i Y i ( )) (y j Y j ( )) dy n i,j= =(π) n/ A f(y)h(,y)dy =(π) n/ A f(y)h(,y)dy n n +(π) n/ ( n A f(y) H(t, y) a ij ((y i Y i (t)) (φ j (t)w (t)+ψ j (t)) n +(y j Y j (t)) (φ i (t)w (t)+ψ i (t)) ) dt i,j= =(π) n/ A f(y)h(,y)dy n +(π) n/ ( ) A f(y)h(t, y) (y Y (t)) Aφ(t)dy (α(t)+w(t))dt n which by (4.) is the same as (4.6)-(4.8). LEMMA 4.5 Let φ(t),y(t) be as in (4.7),(4.5). Let t< and define the n n matrix C t, = t φ(s)φ (s)ds (4.9) Assume that and put hen C t, > (4.) A t, = C t, (4.) A, exp (y Y (t)) A, (y Y (t)) = A t, exp (4.) (y Y (t)) A t, (y Y (t)) POOF he proof uses the same method as the proof of Corollary 3.7. We omit the details. 7

18 Knut Aase et al. COOLLAY 4.6 Let φ(t),y(t) and A t, be as in Lemma 4.5. hen A, exp (y Y (t)) A, (y Y (t)) (y Y (t)) A, φ(t) = A t, exp (y Y (t)) A t, (y Y (t)) (y Y (t)) A t, φ(t) (4.3) POOF Differentiate (4.) with respect to y,y,...,y n. We can now give a more explicit formulation than the one given in heorem 4.4: HEOEM 4.7 Let φ(t),y(t) and A t, be as in Lemma 4.5. Assume that Let f : n be bounded. hen C t, > for all t, (4.4) f(y ( )) = V + u(t, ω)(α(t)dt + db(t)) (4.5) where V is as in (4.7) and u(t, ω) =(π) A n/ t, f(y) exp (y Y (t)) A t, (y Y (t)) (y Y (t)) A t, φ(t)dy (4.6) EXAMPLE 4.8 he general results in heorem 4.4 and heorem 4.7 can be used to study the replicating portfolios for more exotic options than those of the type (.5). For example, one can study the portfolios of pathdependent options like a knock-out option of the form F (ω) =X K, ) max Z(t, ω) (4.7) t where Z(t, ω) is the (-dimensional) process in (.4). he idea is the following: Let = t <t < <t n = be an equidistant partition of,, and define X,t(t) X,t(t) φ(t) = n n (4.8) X,tn(t) hen X,t X,t X,t φφ X,t X,t X,t = X,t X,t X,tn (4.9) 8

19 and hence C = Since C = t (t t )(t 3 t ) (t n t n )=( t) n, where Using the Donsker Delta Function t t t t φφ (t)dt = t t (4.3) t t t n t = t i t i = n ; i n the matrix C is invertible. Hence heorem 4.4 applies to t t B (t t )+ t t α (s)ds Y (t) := φ(s)db(s)+ φ(s)α(s)ds =. B (t t n )+ (4.3) t t n α (s)ds where α =(α,...,α n ):, n is as in (4.4). In this case we see that A = C = n Now let f : n be bounded. hen t tn f(y (,ω)) = f(b (t )+ α (s)ds,...,b (t n )+ α (s)ds) In particular, if α =, we get the following representation by heorem 4.4: f(b (t ),...,B (t n )) = V + u (t, ω)db (t) where (4.3) u (t, ω) ( n ) n/ = f(y,...,y n ) exp n ( n (y i B (t t i )) +(y n B (t)) π n i= n ) (y i B (t t i )) (y i+ B (t t i+ )) n i= ( n n (y i B (t t i ))X,ti(t)+(y n B (t)) i= n ) (y i B (t t i ))X,ti+(t) (y i+ B (t t i+ ))X,ti(t) dy i= i= (4.33) hus we see that if F is the knock out option F (ω) =X K, ) max B (t, ω) t then we obtain an approximation of the corresponding replicating portfolio u(t, ω) by choosing n large and f(y,...,y n ) = max{y i i n} in (4.33). With some extra work one could obtain a similar representation without Wick products using heorem

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