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Cheat Sheet: Mathematical Foundations for Finance
Steven Battilana, Autumn Semester 2017
1 Appendix: Some Basic Concepts and Results
1.1 Very Basic Things
Def.
(i) An empty product equals 1 , i.e.
j=1 sj^ = 1. (ii) An empty sum equals 0 , i.e.
j=1 sj^ = 0. Def. (geometric series)
(i) sn = a 0
∑^ n k=
qk
q 6 = = a 0 q
n+1− 1 q− 1 =^ a^0
1 −qn+ 1 −q
(ii) s =
k=
a 0 qk^
|q|< 1 = (^) qa−^01
Def. (conditional probabilities)
Q[C ∩ D] = Q[C]Q[D|C]
Def. (compact) Compactness is a property that generalises the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed di- stance of each other). Examples include a closed interval, a rectangle, or a finite set of points (e.g. [0, 1]). Def. (P -trivial) F 0 is P -trivial iff P [A] ∈ { 0 , 1 }, ∀A ∈ F 0. Useful rules
E[eZ^ ] = eμ+^
1 2 σ
2 , for Z ∼ N (μ, σ^2 ).
Let W be a BM, then 〈W 〉t = t, hence d〈W 〉s = ds.
Def. (Hilbert space) A Hilbert space is a vector space H with an inner product 〈f, g〉such that the norm defined by
|f | =
〈f, g〉
turns H into a complete metric space. If the metric defined by the norm is not complete, then H is instead known as an inner product space. Def. (cauchy-schwarz)
- |〈u, w〉^2 | ≤ 〈u, u〉 · 〈w, w〉
∑N
j=1 uj^ wj^ ≤
N j=1 u
2 j
2
N j=1 w
2 j
2
∫ (^) b a u(τ^ )w(τ^ )dτ^ ≤
b a u
(^2) (τ )dτ
b a w
(^2) (τ )dτ
1 (^2) (a(u, v))
1 2 Def. (Taylor)
j=
(±h)j j!
dj^ f dxj
n=
f (n)(a) n! (x^ −^ a)
n Rem. (uniform convergence on compacts in probability 1) A mode of convergence on the space of processes which occurs often in the study of stochastic calculus, is that of uniform convergence on compacts in probability or ucp convergence for short. First, a sequence of (non-random) functions fn : R+ → R converges uniformly on compacts to a limit f if it converges uniformly on each bounded interval [0, t]. That is,
sup s≤t
|fn(s) − f (s)| → 0
as n → ∞. If stochastic processes are used rather than deterministic functions, then convergence in probability can be used to arrive at the following definition. Def. (uniform convergence on compacts in probability) A sequence of jointly measurable stochastic processes Xn^ converges to the limit X uniformly on compacts in probability if
P
[
sup s≤t
|Xsn − Xs| > K
]
as n → ∞, ∀t, K > 0. Rem. (uniform convergence on compacts in probability 2) The notation Xn^
ucp −−→ X is sometimes used, and Xn^ is said to converge ucp to X. Note that this definition does not make sense for arbitrary stochastic processes, as the supremum is over the uncoun- table index set [0, t] and need not be measurable. However, for right or left continuous processes, the supremum can be restricted to the countable set of rational times, which will be measurable. Def.
M^2 d([0, a]) := {M ∈ M^2 ([0, a]) | t 7 → Mt(ω) RCLL for any ω ∈ Ω}
Def.
(i) We denote by H^2 the vector space of all semimartingales va- nishing at 0 of the form X = M + A with M ∈ M^2 d(0, ∞) and A ∈ F V (finite variation) predictable with total variation V (^) ∞(1) (A) =
0 |dAs| ∈^ L
2 (P ).
(ii)
‖X‖^2 H 2 = ‖M ‖^2 M 2 +‖V
(1) ∞ (A)‖
2 L^2 =^ E
[
[M ]∞ +
0
|dA|
) 2 ]
Thm. (dominated convergence theorem for stochastic integrals) Suppose that X is a semimartingale with decomposition X = M +A as above, and let Gn, n ∈ N, and G be predictable processes. If
Gnt (ω) → Gt(ω) for anyt ≥ 0 , almost surely,
and if there exists a process H that is integrable w.r.t. X such that |Gn| ≤ H for any n ∈ N, then
Gn · X → G·X u.c.p., as n → ∞.
If, in addition to the assumptions above, X is in H^2 and ‖H‖X < ∞ then even
‖Gn · X − G·X‖H 2 → 0 as n → ∞.
”Def.” (Wiki: null set) In set theory, a null set N ⊂ R is a set that can be covered by a countable union of intervals of arbitrarily small total length. The no- tion of null set in set theory anticipates the development of Lebesgue measure since a null set necessarily has measure zero. More generally, on a given measure space M = (X, Σ, μ) a null set is a set S ⊂ X s.t. μ(S) = 0. Def. (power set) We denote by 2 Ω^ the power set of Ω; this is the family of all subsets of Ω. Def. (σ-fiel or σ-algebra) A σ-field or σ-algebra on Ω is a family F of subsets of Ω which con- tains Ω and which is closed under taking complements and countable unions, i.e.
A ∈ F ⇒ Ac^ ∈ F Ai ∈ F, i ∈ N ⇒
i∈N Ai^ is in^ F Rem. F is then also closed under countable intersections. Def. (finite σ-field) A σ-field is called finite if it contains only finitely many sets. Def. (measurable space) A pair (Ω, F) with Ω 6 = ∅ and F a σ-algebra on Ω is called a mea- surable space. Rem. X is measurable (or more precisely Borel-measurable) if for every B ∈ B(R), we have {X ∈ B} ∈ F. Def. (indicator function) For any subset A of Ω, the indicator function IA is the function de-
fined by IA(ω) :=
1 , ω ∈ A 0 , ω /∈ A Def. Let Ω 6 = ∅ and a function X : Ω → R (or more generally to Ω′). Then σ(X) is the smallest σ-field, say, on Ω s.t. X is measurable with respect to G and B(R) (or G and F′, respectively). We call σ(X) the σ-field generated by X.
Rem. We also consider a σ-field generated by a whole familiy of mappings; this is then the smallest σ-field that makes all the mappings in that family measurable. Def. (probability measure, probability space) If (Ω, F) is a masurable space, a probability measure on F is a map- ping P : F → [0, 1] s.t. P [Ω] = 1 and P is σ-additive, i.e.
P
i∈N
Ai
i∈N
P [Ai], Ai ∈ F, i ∈ N, Ai ∩ Aj = ∅, i 6 = j.
The triple (Ω, F, P ) is then called a probability space. Def. (P -almost surely) A statement holds P -almost surely or P -a.s. if the set
A := {ω | the statement does not hold}
is a P -nullset, i.e. has P [A] = 0. We sometimes use instead the for- mulation that a statement holds for P -almost all ω. Notation E.g., X ≥ Y P -a.s. means that P [X < Y ] = 0, or, equivalently P [X ≥ Y ] = 1. Notation:
P [X ≥ Y ] := P [{X ≥ Y }] := P [{ω ∈ Ω | X(ω) ≥ Y (ω)}].
Def. (random variable) Let (Ω, F, P ) be a probability space and X : Ω → R a measurable function. We also say that X is a (real-valued) random variable. If Y is another random variable, we call X and Y equivalent if X = Y P -a.s. Def. (p-integrable) We denote by L^0 or L^0 (F) the family of all equivalence classes of random variables on (Ω, F, P ). For 0 < p < ∞, we denote by Lp(P ) the family of all equivalence classes of random variables X which are p-integrable in the sense that E[|X|p] < ∞, and we write then X ∈ Lp(P ) or X ∈ Lp^ for short. Finally, L∞^ is the family of all equivalence classes of random variables that are bounded by a constant c (where the constant can depend on the random variable). Def. (Atom) If (Ω, F, P ) is a probability space, then an atom of F is a set A ∈ F with the properties that P [A] > 0 and that if B ⊆ A is also in F, then either P [B] = 0 or P [B] = P [A]. Intuitively, atoms are the ’smallest indivisible sets’ in a σ-field. Atoms are pairwise disjoint up to P -nullsets. Def. (atomless) The space (Ω, F, P ) is called atomless if F contains no atoms; this can only happen if F is infinite. Finite σ-fields can be very conve- niently described via their atoms because every set in F is then an union of atoms. Fatou’s Lemma If {fn} is a sequence of nonnegative measurable functions, then ∫ lim infn→∞ fndμ ≤ lim infn→∞
fndμ.
An example of an sequence of functions for which the inequality becomes strict is given by
fn(x) =
0 , x ∈ [−n, n] 1 , otherwise
1.2 Conditional expectations: A survival kit
Rem. Let (Ω, F, P ) be a probability space and U a real-valued random variable, i.e. an F-measurable mapping U : Ω → R. Let G ⊆ F be a fixed sub-σ-field of F; the intuitive interpretation is that G gives us some partial information. The goal is then to find a prediction for U on the basis of the information conveyed by G, or a best estimate for U that uses only information from G. Def. (conditional expectation of U given G) A conditional expectation of U given G is a real-valued random va- riable Y with the following two properties:
(i) Y is G-measurable (ii) E[U IA] = E[Y IA], ∀A ∈ G
Y is then called a version of the conditional expectation and is de- noted by Y = E[U |G]. Thm. 2. Let U be an integrable random variable, i.e. U ∈ L^1 (P ). Then:
(i) There exists a conditional expectation E[U |G], and E[U |G] is again integrable. (ii) E[U |G] is unique up to P -nullsets: If Y, Y ′^ are random variables satisfying the right above definition, then Y ′^ = Y P -a.s.
Lem. (properties and computation rules) Next, we list properties of and computation rules for conditional ex- pectations. Let U, U ′^ be integrable random variables s.t. E[U |G] and E[U ′|G] exist. We denote by bG the set of all bounded G-measurable random variables. Then we have:
(i) E[U Z] = E[E[U |G]Z], ∀Z ∈ b (ii) Linearity: E[aU 1 bU ′|G] = aE[U |G] + bE[U ′|G], P -a.s. ∀a, b, ∈ R (iii) Monotonicity: If U ≥ U ′^ P -a.s., then E[U |G] ≥ E[U ′|G] P -a.s. (iv) Projectivity: E[U |H] = E[E[U |G]|H], P -a.s. ∀σ-field H ⊆ G (v) E[U |G] = U, P -a.s. if U ∈ G-measurable (vi) E[E[U |G]] = E[U ]
(vii) E[ZU |G]
(v) = E[ZE[U |G]|G]
(viii) = ZE[U |G], P -a.s. ∀Z ∈ bG (viii) E[U |G] = E[U ], P -a.s. for U independent of G
Rem.
(i) Instead of integrability of U , one could also assume that U ≥ 0 ; then analogous statements are true. (ii) More generally, (i) and (vii) hold as soon as U and ZU are both integrable or both nonnegative. (iii) If U is Rd-valued, one simply does everything component by com- ponent to obtain analogous results.
Lem. 2. Let U, V be random variables s.t. U is G-measurable and V is inde- pendent of G. For every measurable function F ≥ 0 on R^2 , then
E[F (U, V )|G] = E[F (u, V )]|u=U =: f (U ).
Intuitively, one can compute the conditional expectation E[F (U, V )|G] by ’fixing the known value U and taking the ex- pectation over the independent quantity V. Thm. 2. Suppose (Un)n∈N is a sequence of random variables.
(i) If Un ≥ X P -a.s. for all n and some integrable random variable X, then
E[lim inf n→∞
Un|G] ≤ lim inf n→∞
E[Un|G], P -a.s.
(ii) If (Un) converges to some random variable U P -a.s. and if |Un| ≤ X P -a.s. for all n and some integrable random variable X, then
E[ lim n→∞
Un|G] = E[U |G] = lim n→∞
E[Un|G], P -a.s.
1.3 Stochastic processes and functions
Def. (stochastic process) A (real-valued) stochastic process with index set T is a family of random variables Xt, t ∈ T , which are all defined on the same pro- bability space (Ω, F, P ). We often write X = (Xt)t∈T. Def. (increment of stochastic process) For any stochastic process X = (Xk )k=0, 1 ,...,T , we denote the in- crement from k − 1 to k of X by
∆Xk := Xk − Xk− 1.
Rem. A stochastic process can be viewed as a function depending on two parameters, namely ω ∈ Ω and t ∈ T. Def. (trajectory) If we fix t ∈ T , then ω 7 → Xt(ω) is simply a random variable. If we fix instead ω ∈ Ω, then t 7 → Xω (t) can be viewed as a function T → R, and we often call this the path or the trajectory of the process corresponding to ω. Def. (continuous) A stochastic process is continuous if all or P -almost all its trajecto- ries are continuous functions. Def. (RCLL) A stochastic process is RCLL if all of P -almost all its trajectories are right-continuous (RC) functions admitting left limits (LL). Def. (Wiki: signed measure) Given a measurable space (X, Σ), i.e. a set X with a σ-algebra on it, an signed measure is a function μ : Σ → R, s.t. μ(∅) = 0 and μ is σ-additive, i.e. it satisfies
μ
n=1 An
n=1 μ(An) where the series on the right must converge absolutely, for any se- quence A 1 , A 2 , ..., An, ... of disjoint sets in Σ. Def. (Wiki: total variation in measure theory) Consider a signed measure μ on a measurable space (X, Σ), then it is possible to define two set functions W (μ, ·) and W (μ, ·), respectively
Incremental cost:
∆Ck+1(ϕ) : = (ϕ^0 k+1 − ϕ^0 k ) S^0 k ︸︷︷︸ ︸ ︷︷ =1︸ bank account
∑^ d
i=
ϑik+1 − ϑik
Sik
︸ ︷︷ ︸ portfolio
rewrite the above by adding and subtracting ϕtrk+1Sk+1:
∆Ck+1(ϕ) = ϕ^0 k+1 − ϕ^0 k + (ϑk+1 − ϑk )tr^ Sk
= ϕ^0 k+1 − ϕ^0 k + (ϑk+1 − ϑk )tr^ Sk ± ϕtrk+1Sk+
= ϕ^0 k+1 + ϑtrk+1Sk+1 − ϕ^0 k − ϑtrk Sk − ϑtrk+1∆Sk+ (1) = Vk+1(ϕ) − Vk (ϕ) − ϑtrk+1∆Sk+
= ∆Vk+1(ϕ) − ϑtrk+1∆Sk+
Gains process G
(Gk (ϑ))k=0, 1 ,...,T denotes the discounted gains process associa- ted to ϑ:
Gk (ϑ) :=
∑^ k
j=
ϑtrj ∆Sj , k = 0, 1 , ..., T
= ϑ · S
G is R-valued and adapted.
If we think of a continuous-time model where successive trading dates are infinitely close together, then the increment ∆S beco- mes a differential dS and the sum becomes a stochastic integral
G(ϑ) =
∫ (^) ∑k
i=
ϑidSi, k = 0, 1 , ..., T
Self-financing strategy
A trading strategy ϕ = (ϕ^0 , ϑ) is called self-financing if its cost process C(ϕ) is constant over time.
Proposition 2.1. A self-financing strategy ϕ = (ϕ^0 , ϑ) is uniquely determined by its initial wealth V 0 and its risky asset compo- nent ϑ. In particular, any pair (V 0 , ϑ) specifies in a unique way a self- financing strategy. If ϕ = (ϕ^0 , ϑ) is self-financing, then (ϕ^0 k )k=1,...,T is automati- cally predictable.
It then holds for the corresponding (incremental) cost process:
∆Ck+1(ϕ) = ( ϕ^0 k+1 − ϕ^0 k ) S^0 k + (ϑk+1 − ϑk ) · Sk = 0, P -a.s.
C(ϕ) = C 0 (ϕ) = V 0 (ϕ) = ϕ^00
It then holds for the corresponding value process:
Vk (ϕ) = V 0 (ϕ) + Gk (ϑ) = ϕ^00 + Gk (ϑ)
= ϕ^00 +
∑^ k
j=
ϑtrj ∆Sj
Remarks:
The notion of a strategy being self-financing is a kind of economic budget constraint.
The notion of self-financing is numeraire (discounting) irrelevant, i.e. it does not depend on the units in which the calculations are done.
A first introduction to stopping time Notation:
a ∧ b := min(a, b)
Def. (stopped stochastic process) The stochastic process Sτ^ = (Sτk )k=0, 1 ,...,T defined by
Sτ^ (ω) := Sk∧τ (ω) := Sk∧τ (ω)(ω)
is called the process S stopped at τ. Rem. Sτ^ could fail to be a stochastic process because Skτ = Sk∧τ could fail to be a random variable, i.e. could fail to be measurable. But (in discrete time) this is not a problem if we assume that τ is measura- ble, which is mild and reasonable enough. Def. (stopping time) Note that ϑk = I{k≤τ } is Fk− 1 -measurable for each k iff {τ ≥ k} ∈ Fk− 1 for all k or, equivalently by passing to complements,
{τ ≤ j} ∈ Fj , ∀j.
By definition, this means that τ is a stopping time (w.r.t. F). Example (a doubling strategy, p.16) We denote by the following the (random) time of the first stock price rise:
τ := inf{k | Yk = 1 + u} ∧ T
which is a stopping time because
{τ ≤ k} ︸ ︷︷ ︸ ∈Fk ,∀k
= {Y 1 = 1 + u} ︸ ︷︷ ︸ ∈F 1 ⊆Fk
∪... ∪ {Yk = 1 + u} ︸ ︷︷ ︸ ∈Fk = {max(Y 1 , ..., Yk ) ≥ 1 + u} ∈ Fk , ∀k.
With
ϑk := (^) S^1 k− 1
2 k−^1 I{k≤τ }.
it follows that ϑ is predictable because each ϑk is Fk− 1 -measurable. Note that this uses {k ≤ τ } = {τ < k}c^ = {τ ≤ k − 1 }c.
Admissibility
For a ∈ R, a ≥ 0 , a trading strategy ϕ is called a-admissible if its value process V (ϕ) is uniformly bounded from below by −a, i.e.
Vk (ϕ) ≥ −a, P-a.s., ∀k ≥ 0
A trading strategy is called admissible if it is a-admissible for some a ≥ 0.
Remark:
An admissible strategy can be interpreted as a strategy having some credit line which imposes a lower bound on the associated value process. So one may make debts, but only within clearly defined limits.
2.3 Some important martingale results
Def. (martingale, martingale property) Let (Ω, F, Q) be a probability space with a filtration F = (Fk )k=0, 1 ,...,T. A (real-valued) stochastic process X = (Xk )k=0, 1 ,...,T is called martingale (w.r.t. Q and F) if it is ad- apted to F, is Q-integrable in the sense that Xk ∈ L^1 (Q), ∀k, and satisfies the martingale property
EQ[X|Fk ] = Xk , Q-a.s. for k ≤.
Def. (supermartingale, submartingale)
(i) If we have ” ≤ ” in the above definition (a tendency to go down), X is called a supermartingale.
(ii) If we have ” ≥ ” in the above definition (a tendency to go up), X is called a submartingale.
Def. (local martingale, localising sequence) An adapted process X = (Xk )k=0, 1 ,...,T null at 0 (i.e. X 0 = 0) is called a local martingale (w.r.t. Q and F) if there exists a sequence of stoppint times (τn)n∈N increasing to T s.t. ∀n ∈ N, the stopped process Xτn^ = (Xk∧τn )k=0, 1 ,...,T is a (Q, F)-martingale. We then call (τn)n∈N a localising sequence. Thm. 3. Suppose X = (Xk )k=0, 1 ,...,T is an Rd-valued martingale or local martingale null at 0. For any Rd-valued predictable process ϑ, the stochastic integral process ϑ · X defined by
ϑ · Xk :=
∑k j=
ϑtrj ∆Xj , k = 0, 1 , ..., T
is then a (real-valued) local martingale null at 0. If X is a martingale and ϑ is bounded, then ϑ · X is even a martingale. Rem. In continuous time, the above theorem no longer holds. Rem. If we think of X = S as discounted asset prices, then ϑ · S = G(ϑ)
is the discounted gains process. Cor. 3. For any martingale X and any stopping time τ , the stopped process Xτ^ is again a martingale. In particular, EQ[Xk∧τ ] = EQ[X 0 ], ∀k. Interpretation A martingale describes a fair game in the sense that one cannot predict where it goes next.
(i) Cor. 3.2 says that one cannot change this fundamental character by cleverly stopping the game.
(ii) Thm. 3.1 says that as long as one can only use information from the past, not even complicated clever betting will help.
Thm. 3. Suppose that X is an Rd-valued local Q-martingale null at 0 and ϑ is an Rd-valued predictable process. If the stochastic integral process ϑ · X is uniformly bounded below (i.e. ϑ · X ≥ −b Q-a.s., ∀k, b ≥ 0 ), then ϑ · X is a Q-martingale.
2.4 An example: The multinomial model
Def. (multiplicative model) The multiplicative model with i.i.d. returns is given by
S^ ˜^0 k S^ ˜^0 k− 1
= 1 + r > 0 , ∀k,
S^ ˜^1
k S^ ˜^1 k− 1
= Yk , ∀k,
where S˜ 00 = 1, S˜ 01 = S^10 > 0 is a constant, and Y 1 , ..., YT are i.i.d. and take the finitely many values 1 + y 1 , ..., 1 + ym with respective probabilities p 1 , ..., pm. We assume that all the probabilities pj are
0 and that ym > ym− 1 > ... > y 1 > − 1. This also ensures that S^ ˜^1 remains strictly positive. Rem. Intuition suggests that for a reasonable model, the sure factor 1 + r should lie between the minimal and maximal values 1+y 1 and 1+ym of the (uncertain) random factor. Def. (canonical model) The simplest and in fact canonical model for this setup is a path space. Let
Ω = { 1 , ..., m}T = {ω = (x 1 , ..., xT ) | xk ∈ { 1 , ..., m} for k = 1, ..., T }
be the set of all sequences of length T formed by element of { 1 , ..., m}. Take F = 2Ω, the family of all subsets of Ω, and de- fine P by setting
P [{ω}] = px 1 px 2 · ... · pxT =
∏T
k=
pxk.
Finally, define Y 1 , ..., YT by
Yk (ω) := 1 + yxk (2)
so that Yk (ω) = 1 + yj iff xk = j. We take as filtration the one generated by S˜^1 (or, equivalently, by Y ) s.t.
Fk = σ(Y 1 , ..., Yk ), k = 0, 1 , ..., T.
Def. (atom) A set A ⊆ Ω is an atom of Fk iff there exists a sequence (x 1 , ..., xk ) of length k with elements xi ∈ { 1 , ..., m} s.t. A consists of all those ω ∈ Ω that start with the substring (x 1 , ..., xk ), i.e.
A = Ax 1 ,...,xk := {ω = (x 1 , ..., xT ) ∈ { 1 , ..., m}T^ | xi = xi for i = 1, ..., k}.
Consequences
(i) Each Fk is parametrised by substrings of length k and therefore contains precisely mk^ atoms.
(ii) When going from time k to time k + 1 , each atom A = Ax 1 ,...,xk from Fk splits into precisely m subsets A 1 = Ax 1 ,...,xk , 1 , ..., Am = Ax 1 ,...,xk ,m that are atoms of Fk+1.
(iii) The atoms of Fk are pairwise disjoint and their union is Ω. Fi- nally, each set B ∈ Fk is a union of atoms of Fk ; so the family Fk of events observable up to time k consists of 2 m
k sets.
Def. (one-step transition probabilities) For any atom A = Ax 1 ,...,xk of Fk , we then look at its m suc- cessor atoms A 1 = Ax 1 ,...,xk , 1 , ..., Am = Ax 1 ,...,xk ,m of Fk+1, and we define the one-step transition probabilities for Q at the node corresponding to A by the conditional probabilities
Q[Aj |A] =
Q[Aj ] Q[A] ,^ for^ j^ = 1, ..., m. Because A is the disjoint union of A 1 , ..., Am, we have
0 ≤ Q[Aj |A] ≤ 1 for j = 1, .., m and
∑m j=
Q[Aj |A] = 1.
Rem. The decomposition of factorisation of Q in such a way that for every trajectory ω ∈ Ω, its probability Q[{ω}] is the product of the succes- sive one-step transition probabilities along ω. Rem. We can describe Q equivalently either via its global weights Q[{ω}] or via its local transition behaviour. Def. (independent growth rates) The (coordinate) variables Y 1 , ..., YT from (2) are independent under Q iff for each k, the one-step transition probabilities are the same for each node at time k, but they can still differ across date k. Def. Y 1 , ..., YT are i.i.d. under Q iff at each node throughout the tree, the one-step transition probabilities are the same. Rem. Probability measures with this particular structure can therefore be described by m − 1 parameters; recall that the m one-step transition probabilities at any given node must sum to 1 , which eliminates one degree of freedom.
2.5 Properties of the market
Characterisation of financial markets via EMMs (equivalent mar- tingale measures) The description of a financial market model via EMMs can be summarized as follows:
Existence of an EMM ⇐⇒ the market is arbitrage-free i.e. Pe(S) 6 = ∅ by the 1st^ FTAP
Uniqueness of the EMM ⇐⇒ the market is complete i.e. #(Pe(S)) = 1 by the 2nd^ FTAP
2.5.1 Arbitrage
1 st^ Fundamental Theorem of Asset Pricing (FTAP) Thm. 2.1 (Dalang/Morton/Willinger)
(i) Consider a financial market model in finite discrete time.
(ii) Then S is arbitrage-free iff there exists an EMM for S, i.e.
(NA) ⇐⇒ Pe(S) 6 = ∅
In other words: If there exists a probability measure Q ≈ P on FT s.t. S is a Q-martingale, then S is arbitrage-free.
Limitations: The most important of these assumptions are fric- tionless markets and small investors—and if one tries to relax these to allow for more realism, the theory even in finite discrete time becomes considerably more complicated and partly does not even exist yet.
Cor. 2. The multinomial model with parameters y 1 < ... < ym and r is arbitrage-free iff y 1 < r < ym. Cor. 2. The binomial model with parameters u > d and r is arbitrage-free iff d < r < u. In that case, the EMM Q∗^ for S˜^1 / S˜^0 is unique (on FT ) and is given as in Cor. 1.4.
Arbitrage opportunity Def. (arbitrage opportunity, arbitrage-free)
(i) An arbitrage opportunity is an admissible self-financing strategy ϕ =^ ∧ (0, ϑ) with zero initial wealth, with Vt(ϕ) ≥ 0 , P -a.s. and with P [VT (ϕ) > 0] > 0.
(ii) The financial market (Ω, F, F, P , S^0 , S) or shortly S is called arbitrage-free if there exist no arbitrage opportunities.
(iii) Sometimes one also says that S satisfies (NA).
Def.
(i) (NA+) := forbids to produce something out of nothing with 0 - admissible self-financing strategies.
(i) If we want to bet on a reasonably stable asset price evolution, we might be interested in a payoff of the form H = IB with
B =
a ≤ min i=1,...,d
min k=0, 1 ,...,T
Sik < max i=1,...,d
max k=0, 1 ,...,T
Ski ≤ b
This option pays at time T on unit of money iff all stocks remain between the levels a and b up to time T.
(ii) A payoff of the form
H = IAg
T
∑^ T
k=
Ski
, A ∈ FT , g ≥ 0
gives a payoff which depends on the average price (over time) of asset i, but which is only due in case that a certain event A occurs.
Def. (Attainability)
A payoff H ∈ L^0 +(FT ) is called attainable if there exists an admis- sible self-financing strategy ϕ=^ ∧ (V 0 , ϑ) with VT (ϕ) = H P -a.s.
The stratety ϕ is then said to replicate H and is called a replica- ting strategy for H.
Thm. 1.1 (Arbitrage-free valuation of attainable payoffs)
Consider a financial market in finite discrete time and suppose that S is arbitrage-free and complete and F 0 is trivial.
Then for every payoff H ∈ L^0 +(FT ) has a unique price process V H^ = (V (^) kH )k=0, 1 ,...,T which admits no arbitrage.
V H^ is given by:
V (^) kH = EQ [H|Fk ] = Vk (V 0 , ϑ)
for k = 0, 1 ,... , T , for any EMM Q for S and for any replicating strategy ϕ=^ ∧ (V 0 , ϑ) for H.
Rem.: Because it involves no preferences, but only the assumpti- on of absence of arbitrage, the valuation from this Thm. is often also called risk-neutral valuation, and an EMM Q for S is called a risk-neutral measure
Thm. 1.2 (Characterisation of attainable payoffs)
Consider a financial market in finite discrete time and suppose that S is arbitrage-free and F 0 is trivial. For any payoff H ∈ L^0 +(FT ), the following are equivalent:
(i) H is attainable.
(ii) sup Q∈Pe(S)
EQ[H] < ∞ is attained in some Q∗^ ∈ Pe(S), i.e.
the supremum is finite and a maximum. In other words, we have sup Q∈Pe(S)
EQ[H] = EQ∗ [H] < ∞ for
some Q∗^ ∈ Pe(S).
(iii) The mapping Pe(S) → R, Q 7 → EQ[H] is constant, i.e. H has the same and finite expectation under all EMMs Q for S.
Remark: Note that not all of these relationships necessarily hold for financial markets in infinite discrete time or continuous time. ”2) ⇒ 3)” in general only holds if H is bounded.
Approach to valuing and hedging payoffs For a given payoff H in a financial market in finite discrete time (with F 0 trivial):
(i) Check if S is arbitrage-free by finding at least one EMM Q for S.
(ii) Find all EMMs Q for S.
(iii) Compute EQ[H] for all EMMs Q for S and determine the supre- mum of EQ[H] over Q.
(iv) If the supremum is finite and a maximum, i.e. attained in some Q∗^ ∈ Pe(S), then H is attainble and its price process can be computed as V (^) kH = EQ[H|Fk ], for any Q ∈ P(S). If the supremum is not attained (or, equivalently for finite discrete time, there is a pair of EMMs Q 1 , Q 2 with EQ 1 [H] 6 = EQ 2 [H]), then H is not attainable.
Invariance of the risk-neutral pricing method under a change of num´eraire
The risk-neutral pricing method is invariant under a change of num´eraire, i.e. all assets can be priced under a risk-neutral me- thod independent of the chosen asset used for discounting.
Denote with Q∗∗^ the EMM for Sˆ^0 := S˜
0 S^ ˜^1. Denote with Q∗^ the EMM for S^1 = S˜
1 S^ ˜^0.
Then it holds for a financial market ( S˜^0 , S˜^1 ) and an undiscounted payoff H˜ ∈ L^0 +(FT ) that:
S^ ˜ k^0 EQ∗
[
H˜
S^ ˜^0
T
∣∣ Fk
]
= S˜ k^1 EQ∗∗
[
H˜
S^ ˜^1
T
∣∣ Fk
]
EMMs in submarkets
If a market (S^0 , S^1 ,... , Sk^ ) is (NA), i.e. there exists an EMM Q, then this EMM Q is also an EMM for all submarkets. (e.g. for (Sk^ , Si), k 6 = i, for (Sk^ , Si, Sj^ ), k 6 = i 6 = j etc.)
If there exists a EMM Qj^ for a submarket (S^0 , Sj^ ) which is not also an EMM for another submarket (S^0 , Sk^ ), j 6 = k, then the whole market (S^0 , S^1 ,... , Sk^ ) is not (NA), i.e. it admits arbitra- ge.
2.7 Multiplicative model
Suppose that we start with the RVs r 1 ,... , rT and Y 1 ,... , YT.
Define the bank account/riskless asset by:
S^ ˜ k^0 :=
∏^ k
j=
(1 + rj ),
S^ ˜^0
k S^ ˜^0 k− 1
= 1 + rk , S˜ 00 = 1
Remarks:
- S˜^0 k is Fk− 1 -measurable (i.e. predictable).
- rk denotes the rate for (k − 1 , k].
Define the stock/risky asset by:
S^ ˜ k^1 := S^10
∏^ k
j=
Yj ,
S^ ˜^1
k S^ ˜^1 k− 1
= Yk , S˜ 01 = const., S˜^10 ∈ R
Remarks:
- S˜^1 k is Fk -measurable (i.e. adapted).
- Yk denotes the growth factor for (k − 1 , k].
- The rate of return Rk is given by Yk = 1 + Rk.
2.7.1 Cox-Ross-Rubinstein (CRR) binomial model
Assumptions
Bank account/riskless asset: Suppose all the rk ∈ R are constant with value r > − 1. This means that we have the same nonrandom interest rate over each period. Then the bank account evolves as S˜ k^0 for k = 0, 1 ,... , T.
Stock/risky asset: Suppose that Y 1 ,... YT ∈ R are independent and only take two values, 1 + u with probability p, and 1 + d with probability 1 − p (i.e. all Yk are i.i.d.). Then the stock prices at each step moves either up (by a factor 1 + u) or down (by a factor 1 + d), i.e.
S^ ˜^1
k S^ ˜^1 k− 1
= Yk =
1 + u, with probability p 1 + d, with probability 1 − p
Martingale property The discounted stock price S˜^1 S^ ˜^0 is^ a^ P- martingale iff r = pu + (1 − p)d.
EMM
In the binomial model, there exists a probability measure Q ≈ P s.t. S˜^1 S^ ˜^0 is a^ Q-martingale iff^ u > r > d.
In that case, Q is unique (on FT ) and characterised by the pro- perty that Y 1 ,... , YT are i.i.d. under Q with parameter
q∗^ = Q[Yk = 1 + u] =
r − d u − d
(↗ up)
1 − q∗^ = 1 − Q[Yk = 1 + d] =
u − r u − d
(↘ down)
Arbitrage and completeness The following statements are equiva- lent:
(i) u > r > d
(ii) ∃ a unique EMM Q∗^ for S˜^1 S^ ˜^0 (on^ FT^ )
(iii) The market S is (NA) and complete.
Put-Call parity
Assuming T = 1, it holds that:
V
C(K) 0 −^ V^
P (K) 0 =^ S
1 0 −^
K
1 + r
Pricing binomial contingent claims H
Assume time horizon T = 1, strike K > 0 , and a payoff function H(x, K) : R+ → R+ of a European style contingent claim with strike K.
H may be a European call function C(x, K) = (x − K)+ or a European put function P (x, K) = (K − x)+.
Then H can be replicated using a self-financing strategy ϕH(K)^ = (V 0 H (K), ϑH(K)) s.t.
V 1 (ϕH(K)^ =
H( H˜^11 , K)
1 + r
, P − a.s.
and ϕH(K)^ is given by
V 0 H (K)=
r − d u − d
H(S^10 (1 + u), K) 1 + r
u − r u − d
H(S 01 (1 + d), K) 1 + r
ϑ
H(K) 1 =^
H(1 + u, K/S^10 ) − H(1 + d, K/S^10 ) u − d
Note that this can also be expressed via the martingale pricing approach:
V 0 H (K)= EQ
[
H( S˜^11 , K)
1 + r
]
where
Q
[
S˜ 11 = S 10 (1 + u)
]
= q =
r − d u − d
Q
[
S˜^11 = S^01 (1 + d)
]
= 1 − q =
u − r u − d
Binomial call pricing formula
V^ ˜ (^) kH˜ = S˜^1 k Q∗∗[Wk,T > x] − K˜
S^ ˜^0
k S^ ˜ T^0
Q∗[Wk,T > x]
x =
log K˜ S^ ˜^1 k^ −^ (T^ −^ k) log(1 +^ d) log 1+ 1+ud
Remark:
This is the discrete analogue of the Black-Scholes formula.
2.7.2 Multinomial model
EMM
IOT construct an EMM for S^1 , it needs to hold that:
EQ[S^11 ] = S 01
⇐⇒ EQ[Y 1 ] = 1 + r ⇐⇒
∑^ m
k=
qk (1 + yk ) = 1 + r
with the further conditions:
∑^ m
k=
qi = 1, q 1 ,... , qm ∈ (0, 1)
Arbitrage condition The following statements are equivalent:
(i) y 1 < r < ym
(ii) ∃ an EMM Q ≈ P s.t. S˜^1 S^ ˜^0 is a^ Q-martingale. (iii) The market S is (NA).
Completeness The multinomial model is
complete whenever m ≤ 2 (i.e. there are no nodes that allow for more than two possible stock price evolutions)
incomplete whenever m > 2 (i.e. there is at least one node that allows for more than two possible stock price evolutions)
Inequality of the payoffs of Asian and European call options
Consider a European call option CkE = ( S˜ k^1 )+^ and an Asian call option with
CkA =
k
∑^ k
j=
S^ ˜ j^1 − K
Then it holds for the Q-expectation (risk-neutral) of the two payoffs that:
EQ
[
CkA S^ ˜^0 k
]
≤ EQ
[
CkE S^ ˜^0 k
]
Interpretation: Since the volatility of an Asian style contingent claim is lower than the one of a European style contingent claim, the Asian option bears lower risk and thus yields also lower profit.
American options
Consider an American option with maturity T and nonnegative adapted payoff process U = (Uk )k=0,...,T.
Then the arbitrage-free price process V¯ = ( V¯k )k=0,...,T w.r.t. Q can be expressed as a backward recursive scheme such as:
V^ ¯T = UT V^ ¯k = max(Uk , EQ[ V¯k+1|Fk ]), for k = 0,... , T − 1
Cases of stopping times
Define the stopping time τa for a ∈ R, a > 0 as:
τa := inf{t ≥ 0 |Wt > a}
Then it holds that:
- τa 1 ≤ τa 2 , P -a.s. for a 1 < a 2.
- P [τa < ∞] = 1.
- Wτa = a, P -a.s.
- E
[
Wτa 2
∣∣ F
τa 1
]
6 = Wτa 1 , P-a.s. i.e. the stopping theorem fails for τ = τa 2 and σ = τa 1.
Define the stopping time ρa for a ∈ R, a > 0 as:
ρa := sup{t ≥ 0 |Wt > a}
Then it follows that ρa = ∞ with probability 1 under P.
Def. (stopping time) Again exactly like in discrete time, a stopping time w.r.t. F is a map- ping τ : Ω → [0, ∞] s.t. {τ ≤ t} ∈ Ft for all t ≥ 0. Def. (events observable up to time σ) We define for a stopping time σ, the σ-field of events observable up to time σ as
Fσ := {A ∈ F | A ∩ {σ ≤ t} ∈ Ft, for all t ≥ 0 }.
(One must and can check that Fσ is a σ-field, and one has Fσ ⊆ Fτ for σ ≤ τ .) Def. We also need to define Mτ , the value M at the stopping time τ , by
(Mτ )(ω) := Mτ (ω)(ω).
Note that this implicitly assumes that we have a random variable M∞, because τ can take the value +∞. Def. (hitting times) One useful application of Prop. 2.2 is the computation of the Laplace transforms of certain hitting times. More precisely, let W = (Wt)t≥ 0 be a BM and define for a > 0 , b > 0 the stopping times
τa := inf{t ≥ 0 | Wt > a}, σa,b := inf{t ≥ 0 | Wt > a + bt}.
3.3 Density processes/Girsanov’s theorem
Density in discrete time
Assume (Ω, F) and a filtration F = (Fk )k=0, 1 ,...,T in finite dis- crete time. On (Ω, F), we have two probability measures Q and P , and we assume Q ≈ P.
Radon-Nykodin theorem: There exists a density
dQ dP
:= D
This is a RV D > 0 , P -a.s. s.t. for all A ∈ Fk and for all RVs Y ≥ 0 it holds that:
Q[A] = EP [DIA] ⇐⇒ EQ[Y ] = EP [Y D].
This can also be written as ∫
Ω
Y dQ =
Ω
Y DdP
This formula tells us how to compute Q-expectations in terms of P -expectations and vice-versa.
Density process in discrete time
Assume the same setting as before.
Radon-Nykodin theorem: The density process Z of Q w.r.t. P , or also called the P -martingale Z, is defined as
Zk : = EP [D|Fk ] = EP
[
dQ dP
∣∣ Fk
]
for k = 0, 1 ,... , T
Then for every Fk -measurable RV Y ≥ 0 or Y ∈ L^1 (Q), it holds that
EQ[Y |Fk ] = EP [Y Zk |Fk ]
and for every k ∈ { 0 , 1 ,... , T } and any A ∈ Fk , it holds that
Q[A] = EP [Zk IA]
Properties:
- Zk is a RV and Zk > 0 , P -a.s.
- A process N = (Nk )k=0, 1 ,...,T which is adapted in F is a Q- martingale iff the product ZN is a P -martingale. (This tells us how martingale properties under P and Q are related to each other.)
Bayes formula: If j ≤ k and Uk is Fk -measurable and either ≥ 0 or in L^1 (Q), then
EQ[Uk |Fj ] =
Zj
EP [Zk Uk |Fj ] Q-a.s.
This tells us how conditional expectations under Q and P are related to each other.
Lem. 3.
(i) For every k ∈ { 0 , 1 , ..., T } and any A ∈ Fk or any Fk -measurable random variable Y ≥ 0 or Y ∈ L^1 (Q), we have
Q[A] = EP [Zk IA] ⇔ EQ[Y ] = EP [Zk Y ],
respectively. (This means that Zk is the density of Q w.r.t. P on Fk .)
(ii) If j ≤ k and Uk is Fk -measurable and either ≥ 0 or in L^1 (Q), then we have the Bayes formula
EQ[Uk |Fj ] = (^) Z^1 j
EP [Zk Uk |Fj ] Q-a.s.
(This tells us how conditional expectations under Q and P are related to each other.) Written in terms of Dk , the Bayes formula for j = k − 1 becomes
EQ[Uk |Fk− 1 ] = EP [Dk UK |Fk− 1 ].
This shows that the rations Dk play the role of ”one-step condi- tional densities” of Q with respect to P.
(iii) A process N = (Nk )k=0, 1 ,...,T which is adapted to F is a Q- martingale iff the product ZN is a P -martingale. (This tells us how martingale properties under P and Q are related to each other.)
Proof Lem. 3.
(i)
EQ[Y ] = EP [Y D]
(vi) = EP [EP [Y D|Fk ]] Y Fk -meas. = EP [Y EP [D|Fk ]] = EP [Y Zk ]
(ii) a) LHS:
EQ[Nk |Fj ]
if Q-martingale = Nj ⇒ Nj Zj = EP [Nk Zk |Fj ] ⇒ N Z P -martingale
b) RHS: 1 Zj
EP [Nk Zk |Fj ]
if P -martingale
Zj
Nj Zj
= Nj , i.e. EQ[Nk |Fj ] = Nj ⇒ N is Q-martingale
This concludes the proof for (ii)
Def.
Dk := (^) ZZk−k 1 , for k = 1, ..., T.
The process D is adapted, strictly positive and satisfies by its defini- tion
EP [Dk |Fk− 1 ] = 1,
because Z is a P -martingale. Rem. Again because Z is a martingale and by Lem. 3.1,
EP [Z 0 ] = EP [ZT ] = EP [ZT IΩ]
Lem. 3.1 1) = Q[Ω] = 1,
and we can recover Z from Z 0 and D via
Zk = Z 0
∏k j=1 Dj^ ,^ for^ k^ = 0,^1 , ..., T.
Rem. To construct an equivalent martingale measure for a given process S, all we need are an F 0 -measurable random variable Z 0 > 0 P -a.s. with EP [Z 0 ] = 1 and an adapted strictly positive process D = (Dk )k=1,...,T satisfying EP [Dk |Fk− 1 ] = 1 for all k, and in addition EP [Dk (Sk − Sk− 1 )|Fk− 1 ] = 0 for all k. Def. (i.i.d. returns)
S^ ˜^1
k =^ S
1 0
∏^ k
j=
Yj , S˜ k^0 = (1 + r)k^ ,
where Y 1 , ..., YT are > 0 and i.i.d. under P. Rem. (construction of an EMM Q) Choose Dk like Yk independend of Fk− 1. Then we must have Dk = gk (Yk ) for some measurable function gk , and we have to choose gk in such a way that we get
1 = EP [Dk |Fk− 1 ] = EP [gk (Yk )]
and
1 + r = EP [Dk Yk |Fk− 1 ] = EP [Yk gk (Yk )].
(Note that these calculations both exploit the P independence of Yk from Fk− 1 .) If this choice is possible, we can then choose all the gk ≡ g 1 , because the Yk are (assumed) i.i.d. under P. To ensure that Dk > 0 , we can impose gk > 0.
Density process in continuous time
Suppose we have P and a filtration F = (Ft)t≥ 0. Fix T ∈ (0, ∞) and assume only that Q ≈ P on FT.
If we have this for every T < ∞, we call Q and P locally equiva- lent and write Q
loc ≈ P. For an infinite horizon, this is usually strictly weaker tahtn Q ≈ P.
Def. (density process) Let us for simplicity fix T ∈ (0, ∞) and suppose that Q ≈ P on FT. Denote by
Zt := EP
[
dQ|FT dP |FT
Ft
]
for 0 ≤ t ≤ T
the density process of Q w.r.t. P on [0, T ], choosing an RCLL version of this P -martingale on [0, T ]. Rem. Since Q ≈ P on FT , we have Z > 0 on [0, T ], and because Z is a P -(super)martingale, this implies that also Z− > 0 on [0, T ] by the so-called minimum principle for supermartingales. Lem. 2. Suppose that Q ≈ P on FT. Then
(i) For s ≤ t ≤ T and every Ut which is Ft-measurable and either ≥ 0 or in L^1 (Q), we have the Bayes formula
EQ[Ut|Fs] =
Zs
EP [ZtUt|Fs] Q-a.s.
(ii) An adapted process Y = (Yt) 0 ≤t≤T is a (local) Q-martingale iff the product ZY is a (local) P -martingale.
Rem.
If Q
loc ≈ P , we can use Lem. 2.1 for any T < ∞ and hence obtain a statement for processes Y = (Y )t≥ 0 on [0, ∞).
One consequence of part 2) of Lem. 2.1 is also that (^) Z^1 is a Q - martingale, on [0, ∞] if Q ≈ P on FT , or even on [0, ∞) if Q
loc ≈ P.
Rem. Suppose that Q
loc ≈ P with density process Z, then Q
loc ≈ P implies that Z is a local martingale.
Thm. 2.2 (Girsanov)
(i) Suppose that Q
loc ≈ P with density process Z. If M is a local P-martingale null at 0, then
M^ ˜ := M −
Z
d[Z, M ]
is a local Q-martingale null at 0.
(ii) In particular, every P-semimartingale is also a Q-semimartingale (and vice-versa, by symmetry).
Def. (product rule)
• ZM =
Z−dM +
M−dZ + [Z, M ]
- d(ZM ) = Z−dM + M−dZ + d[Z, M ]
Lem. Let Z, M be of finite variation, then it holds
〈M 〉, 〈Z〉 ⇒ 〈M, Z〉
is also of finite variation. Rem. When [Z, M ] is of finite variation, the following holds [ 1 Z
, [Z, M ]
]
∆[Z, M ] =
Z
d[Z, M ].
Def. (alternative stochastic Itˆo integral definition) ∀V ”that are nice enough”
〈
vdW, V 〉 =
vd〈W, V 〉
Thm. 2.3 (Girsanov (continuous version))
Suppose that Q
loc ≈ P with continuous density process Z. Write Z = Z 0 E(L). If M is a local P-martingale null at 0, then
M^ ˜ := M − [L, M ] = M − 〈L, M 〉
is a local Q-martingale null at 0.
Moreover, if W is a P-BM, then W˜ is a Q-BM.
In particular, if L =
νdW for some ν ∈ L^2 loc(W ), then ˜W = W − 〈
νdW, W 〉 = W −
νsds so that the P-BM W = ˜W +
νsds becomes under Q a BM with (instantaneous) drift ν.
If we have a closer look at W ∗^ defined in the Black-Scholes chap- ter, we see that W ∗^ is a Brownian motion under the probability measure Q∗^ given by
dQ∗ dP
:= E
λdW
T
= exp
−λWT −
λ^2 T
whose density process w.r.t. P is
Z∗ t = E
λdW
t
= exp
−λWt −
λ^2 T
for 0 ≤ t ≤ T.
We express this by saying that along (Πn)n∈N, the BM W has (with probability 1 ) quadratic variation t on [0, t] for every t ≥ 0 , and we write 〈W 〉t = t. (We sometimes also say that P -a.a. trajectories W•(ω) : [0, ∞) → R of BM have quadratic variation t on [0, t], for each t ≥ 0 .) Def. (finite variation) A function g : [0, ∞) → R is of finite variation or has finite 1 - variation if for every T ∈ (0, ∞),
sup Π
ti∈Π
|g(ti ∧ T ) − g(ti− 1 ∧ T )| < ∞,
where the supremum is taken over all partitions Π of [0, ∞). Interpretation The interpretation is that the graph of g has finite length on any time interval. More precisely, if we define the arc length of (the graph of) g on the interval [0, T ] as
sup Π
ti∈Π
(ti ∧ T − ti− 1 ∧ T )^2 + (g(ti ∧ T ) − g(ti− 1 ∧ T ))^2 ,
with the supremum again taken over all partitions Π of [0, ∞), then g has finite variation on [0,T] iff it has finite arc length on [0, T ]. Rem. Any monotonic (increasing or decreasing) function is clearly of finite variation, because the absolute values above disappear and we get a telescoping sum. Moreover, one can show that any function of finite variation can be written as the difference of two increasing functions (and vice versa). Rem. Every continuous function f which has nonzero quadratic variation along a sequence (Πn) as above must have infinite variation, i.e. unbounded oscillations. We also recall that a classical result due to Lebesgue says that any function of finite variation is almost every- where differentiable. So Prop. 1.3 implies that Brownian trajectories must have infinite variation, and Thm 1.4 makes this even quantita- tive. Prop. 2. Suppose W = (Wt)t≥ 0 is a (P, F)-Brownian motion. Then the fol- lowing process are all (P, F)-martingales:
(i) W itself.
(ii) W (^) t^2 − t, t ≥ 0.
(iii) eαWt−^
(^12) α^2 t , t ≥ 0 , for any α ∈ R.
Prop. 2. Let W be a BM an a > 0 , b > 0. Then for any λ > 0 , we have
(i) E[e−λτa^ ] = e−a
√ 2 λ
(ii) E[e−λσa,b^ ] = E[e−λσa,b^ Iσa,b<∞] = e−a(b+
b^2 +2λ).
Rem.
In the proof of the above Prop. 2.3 the following was being used:
Mt := eαWt−^
1 2 α
(^2) t , t ≥ 0.
For a general random variable U ≥ 0 , the function λ 7 → E[e−λU^ ] for λ > 0 is called the Laplace transform of U.
In mathematical finance, both τa and σa,b come up in connection with a number of so-called exotic options. In particular, they are important for barrier options wohse payoff depends on whether or not a (upper or lower) level has been reached by a given time. When computing prices of such options in the Black-Scholes mo- del, one almost immediately encounters the Laplace transforms in Prop. 2.3.
4.2 Markovian properties
Rem. We have already seen in part 2) of Prop. 1.1 that for any fixed time T ∈ (0, ∞), the process
Wt+T − WT , t ≥ 0 , is again a BM
if (Wt)t≥ 0 is a BM. Moreover, one can show that the independence of increments of BM implies that
Wt+T − WT , t ≥ 0 , is independent of F T^0 ,
where F T^0 = σ(Ws; s ≤ T ) is the σ-field generated by BM up to time T. Intuition Intuitively, this means that BM at any fixed time T simply forgets its past up to time T (with the only possible exception that it remem- bers its current position WT at time T ), and starts afresh. Def. (Markov property) One consequence of the above remark is the following. Suppose that at some fixed time T , we are interested in the behaviour of W after time T and try to predict this on the basis of the past of W up to time T , where ”prediction” is done in the sense of a conditional expectation. Then we may as well forget about hte past and look ony at the current value WT at time T. A bit more precisely, we can express this, for functions g ≥ 0 applied to the part of BM after time T , as
E[g(Wu; u ≥ T ) | σ(Ws; s ≤ T )] = E[g(Wu; u ≥ T ) | σ(WT )].
This is called the Markov property of BM. Rem.
BM has even the strong Markov property.
Generalised: If we denote almost as above by FW^ the filtration generated by W (and made right-continuous, to be accurate), and if τ is a stopping time w.r.t. FW^ and s.t. τ < ∞ P -a.s., then
Wt+τ − Wτ , t ≥ 0 , is again a BM and independent of FWτ.
This includes the first remark in this subsection as special case and one can easily believe that it is even more useful than the above definition of Markov property.
4.3 Poisson processes
Poisson processes
A Poisson process N = (Nt)t≥ 0 with parameter λ ∈ R, λ > 0 and w.r.t. (P, F) is a real-valued stochastic process satisfying the following properties:
(PP0) null at zero N is adapted to F and null at 0 (i.e. N 0 ≡ 0 , P-a.s.).
(PP1) independent and stationary increments For 0 ≤ s < t, the increment Nt −Ns is independent (un- der P) of Fs and follows (under P) the Poisson distributi- on with parameter λ(t − s), i.e. Nt − Ns ∼ Poi(λ(t − s)), i.e
P[Nt = k] =
λk^ (t − s)k k!
e−λ(t−s)
(PP2) counting process N is a counting process with jumps of size 1, i.e. for P-a.a. ω, the function t 7 → Nt(ω) is RCLL, piecewise constant and N 0 -valued, and increases by jumps of size 1.
Important properties of Poisson processes: if X ∼ Poi(λ), then:
E[X] = λ, Var[X] = λ
The quadratic variation of a Poisson process equals itself, i.e.
[N ]t = Nt
Examples of Poisson processes: the following Poisson processes are (P, F)-martingales:
- Compensated Poisson process:
N^ ˜t = Nt − λt, t ≥ 0
- Geometric Poisson process:
St = exp (Nt log(1 + σ) − λσt) , t ≥ 0
where σ ∈ R, σ > − 1.
- Two cases of squared compensated Poisson processes: ( N˜t
− Nt,
N˜t
− λt, t ≥ 0
It follows that [ N˜ ]t = Nt.
4.4 Stochastic integration
Def. (recall discrete stochastic integral) The trading gains or losses from a self-financing strategy ϕ =^ ∧ (V 0 , ϑ) are described by the stochastic integral
G(ϑ) = ϑ · S =
ϑdS =
j
ϑtr J ∆Sj =
j
ϑtr j (Sj − Sj− 1 ).
Rem.
Our goal in this section is to construct a stochastic integral pro- cess H · M =
HdM when M is a (real-valued) local martingale null at 0 and H is a (real-valued) predictable process with a sui- table integrability property (relative to M ).
To relate notation-wise to previous chapters we will the following notions:
(i) H =^ ∧ ϑ
(ii) M =^ ∧ S
Rem. Throughout this chapter, we work on a probability space (Ω, F, P ) with a filtration F(Ft)t≥ 0 satisfying the usual conditions of right- continuity and P -completeness. If needed, we define F∞ :=
t≥ 0
Ft.
We also fix a (real-valued) local martingale M = (Mt)t≥ 0 null at 0 and having RCLL trajectories. Since we want to define stochastic integrals
HdM and these are always over half-open intervals of the form (a, b] with 0 ≤ a < b ≤ ∞, the value of M at 0 is irrelevant and it is enough to look at processes H = (Ht) defined for t > 0. Def. (jump)
(i) For any process Y = (Yt)t≥ 0 with RCLL trajectories, we denote by
∆Yt := Yt − Yt− := Yt − lim s↗t
Ys
the jump of Y at time t > 0.
(ii) ∆(
H^2 r d[M ]r )t :=
∫^ t
0
H r^2 d[M ]r − lim s↗t
∫s
0
H r^2 d[M ]r
Optional quadratic variation/square bracket process
For any local martingale M = (Mt)t≥ 0 null at 0, there exists a unique adapted increasing RCLL process [M ] = ([M ]t)t≥ 0 and having the property that M 2 − [M ] is also a local martingale.
This process can be obtained as the quadratic variation of M in the following sense. There exists a sequence (Πn)n∈N of partitions [0, ∞) with |π| →
0 as n → ∞ s.t.
P
[M ]t(ω) = lim n→∞
ti∈Πn
Mti∧t(ω) − Mti− 1 ∧t(ω)
∀t ≥ 0
We call [M ] the optional quadratic variation or square bracket process of M.
If M satisfies sup 0 ≤s≤t |Ms| ∈ L^2 for each t ≥ 0 (and hence is in particular a martingale), then [M ] is integrable (i.e. [M ]t ∈ L^1 for every t ≥ 0 ) and M 2 − [M ] is a martingale.
Thm. 1.
(i) For any local martingale M = (Mt)t≥ 0 null at 0 , there exists a unique adapted increasing RCLL process [M ] = ([M ]t)t≥ 0 null at 0 with ∆[M ] = (∆M )^2 [this is an important property ] and having the property M 2 − [M ] is also a local martingale.
(ii) This process can be obtained as the quadratic variation of M in the following sense: There exists a sequence (Πn)n∈N of partiti- ons of [0, ∞) with |Πn| → 0 as n → ∞ s.t. script:
P
[
[M ]t(ω) = lim n→∞
ti∈Πn
(Mti∧t(ω) − Mti− 1 ∧t(ω))^2 , ∀t ≥ 0
]
lecture:
P
{ω : ∃[M ]t(ω) = lim n→∞
ti∈Πn
(Mti∧t(ω) − Mti− 1 ∧t(ω))^2 , ∀t ≥ 0 }
We call [M ] the optional quadratic variation or square bracket process of M.
(iii) If M satisfies
0 ≤s≤T
|Ms| ∈ L^2 for some T > 0 (and hence is in
particular a martingale on [0, T ]), then [M ] is integrable on [0, T ] (i.e. [M ]t ∈ L^1 ) and M 2 − [M ] is a martingale on [0, T ].
(Optional) covariation process
For two local martingales M, N null at 0, we define the (optional) covariation process [M, N ] by polarisation, i.e.
[M, N ] :=
([M + N ] − [M − N ])
An alternative definition of the covariation process [M, N ]
[M, N ]t := lim n→∞ ∑
ti∈Πn
Mti∧t(ω) − Mti− 1 ∧t(ω)
Nti∧t(ω) − Nti− 1 ∧t(ω)
∀t ≥ 0
The operation [·, ·] is bilinear.
From the characterisation of [M ] in Thm 1.1, it follows that the operation [·, ·] is bilinear, and also that B = [M, N ] is the un- ique adapted RCLL process B null at 0 , of finite variation with ∆B = ∆M ∆N and s.t. M N − B is again a local martingale.
Def. (predictable compensator) There exists a unique increasing predictable process 〈M 〉 = (〈M 〉t)t≥ 0 null at 0 s.t. [M ] − 〈M 〉, and therefore also M 2 − 〈M 〉 [= M 2 − [M ] + [M ] − 〈M 〉], is a local martingale. The process 〈M 〉 is called the shapr bracket (or sometimes the predictable variance) process of M. Note, this might be useful since [M ] is not necessarily predictable. Rem. and Cor.
(i) Any adapted process which is continuous is automatically locally bounded and therefore also locally square-integrable.
(ii) If M is continuous, then so is [M ], because ∆[M ] = (∆M )^2 = 0. This implies then also that [M ] = 〈M 〉. In particular, for a Brow- nian motion W , we have [W ]t = 〈W 〉t = t for all t ≥ 0.
(iii) If bot M and N are locally square-integrable (e.g. if they are continuous), we also get 〈M, N 〉 via polarisation, i.e.
〈M, N 〉 := 14 (〈M + N 〉 − 〈M − N 〉).
(iv) If M is Rd-valued, then [M ] becomes a d × d-matrix-valued pro- cess with entries [M ]ik^ = [M i, M k^ ]. To work with that, one needs to establish more properties. The same applies to 〈M 〉, if it exists. In general the following holds: [M, M ] = [M ].
Def. (Set of all bounded elementary processes)
We denote by bE the set of all bounded elementary process of the form
H =
n∑− 1
i=
hiI(ti,ti+1]
with n ∈ N, 0 ≤ t 0 < t 1 <... < tn < ∞ and each hi a bounded (real-valued) Fti -measurable RV.
Def. (Stochastic integral)
For any stochastic process∫ X = (Xt)t≥ 0 , the stochastic integral HdX of H ∈ bE is defined as
∫ (^) t
0
HsdXs := H · Xt :=
n∑− 1
i=
hi
Xti+1∧t − Xti∧t
for t ≥ 0.
If X and H are both Rd-valued, the integral is still real-valued, and we simply replace products by scalar products everywhere.
Lem 1.2 (Isometry property)
Suppose M is a square-integrable martingale (i.e. Mt ∈ L^2 for all t ≥ 0 ).
a stochastic integral process H · M for every H that can be appro- ximated, w.r.t. the norm ‖ · ‖L (^2) (M ), by processes from bE, and the resulting H · M is again a martingale in M^20 and still satisfies the isometry property described in the equation above. Prop. 1. Suppose that M is in M^20. Then:
(i) bE is dense in L^2 (M ), i.e. the closure of bE in L^2 (M ) is L^2 (M ).
(ii) For every∫ H ∈ L^2 (M ), the stochastic integral process H · M = HdM is well defined, in M^20 and satisfies the above equation.
Rem. Let M ∈ M^20 , we then have E[|Mt|^2 ] < ∞ for every t ≥ 0 s.t. every M ∈ M^20 is also a square-integrable martingale. However, the converse is not true; Brownian motion W for exam- ple is a martingale and has E[W (^) t^2 ] = t. So sup t≥ 0
E[W (^) t^2 ] = ∞ which
means that BM is not in M^20. Def. (locally square-integrable, stochastic interval)
(i) We call a local martingale M null at 0 locally square-integrable and write M ∈ M^20 ,loc if there is a sequence of stopping times τn ↗ ∞ P -a.s. s.t. M τn^ ∈ M^20 for each n.
(ii) We say for a predictable process H that H ∈ L^2 loc(M ) if the- re exists a sequence of stopping times τn ↗ ∞ P -a.s. s.t. HIK 0 ,τnK ∈ L^2 (M ) for each n. Here we use the stochastic in- terval notation K 0 , τnK := {(ω, t) | 0 < t ≤ τn(ω).
Def. (another definition of stochastic integral)
For M ∈ M^20 ,loc and H ∈ L^2 loc(M ), defining the stochastic inte- gral is straightforward, we simply set
H · M := (HIK 0 ,τnK) · M τn^ , on K 0 , τnK
which gives a definition on all of Ω since τn ↗ ∞, s.t. K 0 , τnK increases to Ω.
The only piont we need to check is that this definition is consis- tent, i.e. tht the definition on K 0 , τn+1K ⊇K 0 , τnK does not clash with the defintion on K 0 , τnK. This can be done by using the pro- perties of stachastic integrals.
Of course, H · M is then in M^20 ,loc.
Rem. If M is Rd-valued with components M i^ that are all in M^20 ,loc, one can also define the so-called vector stochastic integral H · M for Rd-valued predictable processes in a suitable space L^2 loc(M ); the re- sult is then a real-valued process. However, one warning is indicated: L^2 loc(M ) is not obtained by just asking that each component Hi should be in L^2 loc(M i) and then setting H · M =
i
Hi^ · M i. In
fact, it can happen that H · M is well defined whereas the individual Hi^ · M i^ are not. So the intuition for the multidimensional case is that ∫ HdM =
i
HidM i^6 =
i
HidM i.
Def. (continuous local martingale, locally bounded)
(i) M is a continuous local martingale null at 0 , briefly written as M ∈ Mc 0 ,loc. This includes in particular the case of a Brownian motion W.
(ii) Then M is in M^20 ,loc because it is even locally bounded: For the stopping times
τn := inf{t ≥ 0 | |Mt| > n} ↗ is∞ P -a.s.,
We have by continuity that |M τn^ | ≤ n for each n, because
|M τ t n| = |Mt∧τn | =
|Mt| ≤ n, t ≤ τn |Mτn | = n, t > τn.
(iii) The set L^2 loc(M ) of nice integrands for M can here be explicitly described as
L^2 loc = {all predictable processes H = (Ht)t> 0 s.t. ∫ (^) t
0
H^2 s d〈M 〉s < ∞, P -a.s. ∀t ≥ 0
Finally, the resulting stochastic integral H · M =
HdM is then, also a continuous local martingale, and of course null at 0.
Properties
(Local) Martingale properties
- If M is a local martingale and H ∈ L^2 loc(M ), then
HdM is a local martingale in M^20 ,loc. If H ∈ L^2 (M ), then
HdM is even a martingale in M^20.
- If M is a local martingale and H is predictable and locally bounded (∗), then
HdM is a local martingale. (∗) : (which means that there are stopping times τn ↗ ∞ P - a.s. s.t. HIK 0 ,τnK is bounded by a constant cn, say, for each n ∈ N)
- If M is a martingale in M^20 and H is predictable and bounded, then
HdM is again a martingale in M^20.
- Warning: If M is a martingale and H is predictable and boun- ded, then
HdM need not be a martingale; this is in striking contrast to the situation in discrete time.
Linearity If M is a local martingale and H, H′^ are in L^2 loc(M ) and a, b ∈ R, then also aH + bH′^ is in L^2 loc(M ) and
(aH + bH′) · M = (aH) · M + (bH′) · M = a(H · M ) + b(H′^ · M ).
Associativity If M is a local martingale and H ∈ L^2 loc(M ), then we already know that H · M is again a local martingale. Then a predictable process K is in L^2 loc(H · M ) iff KH is in L^2 loc(M ), and then
K · (H · M ) = (KH) · M,
i.e.
Kd(
HdM ) =
KHdM.
Behaviour under stopping
- Suppose that M is a local martingale, H ∈ L^2 loc(M ) and τ is a stopping time. Then M τ^ is a local martingale by the stopping theorem, H is in L^2 loc(M τ^ ), HIK 0 ,τ K is in L^2 loc(M ), and we have
(H · M )τ^ = H · (M τ^ ) = (HIK 0 ,τ K) · M = (HIK 0 ,τ K) · (M τ^ ).
- In words: A stopped stochastic integral is computed by either first stopping the integrator and then integrating, or setting the integrand equal to 0 after the stopping time and then in- tegrating, or combining the two.
Quadratic variation and covariation
- Suppose that M, N are local martingales, H ∈ L^2 loc(M ) and K ∈ L^2 loc(N ). Then [∫ HdM, N
]
Hd[M, N ]
and [∫ HdM,
KdN
]
HKd[M, N ].
- The covariation process of two stochastic integrals is obtained by integrating the product of the integrands w.r.t. the cova- riation process of the integrators.
- In particular, [
HdM ] =
H^2 d[M ]. (We have seen this al- ready for H ∈ bE in the remark after Lem. 1.2)
Jumps Suppose M is a local martingale and H ∈ L^2 loc(M ). Then we already know that H · M is in M^20 ,loc and therefore RCLL. Its jumps are given by
HdM
t
= Ht∆Mt, for t > 0 ,
where ∆Yt := Yt − Yt− again denotes the jump at time t of a process Y with trajectories which are RCLL.
4.5 Extension to semimartingales
Def. (semimartingale, special semimartingale)
(i) A semimartingale is a stochastic process X = (Xt)t≥ 0 that can be decomposed as X = X 0 + M + A, where M is a local martin- gale null at 0 and A is an adapted process null at 0 and having RCLL trajectories of finite variation.
(ii) A semimartingale X is called special if there is such a decompo- sition where A is in addition predictable.
Rem. (canonical decomposition, continuous semimartingale, optio- nal quadratic variation)
(i) If X is a special semimartingale, the decomposition with A pre- dictable is unique and called the canonical decomposition. The uniqueness result uses that any local martingale which is predic- table and of finite variation must be constant.
(ii) If X is a continuous semimartingale, both M and A can be cho- sen continuous as well. Therefore X is special because A is then predictable, since A is adapted and continuous.
(iii) If X is a semimartingale, then we define its optional quadratic variation or square bracket process [X] = ([X]t)t≥ 0 via
[X] := [M ] + 2[M, A] + [A] := [M ] + 2
∆M ∆A +
(∆A)^2.
One can show that this is well defined and does not depend on the chosen decomposition of X. Moreover, [X] can also be ob- tained as a quadratic variation similarly as in Thm. 1.1. However, X^2 − [X] is no longer a local martingale, but only a semimartin- gale in general.
Def. (stochastic integral for semimartingale) If∫ X is a semimartingale, we can define a stochastic integral H · X = HdX at least for any process H which is predictable and locally bounded. We simply set
H · X := H · M + H · A,
where H · M is as is the previous section and H · A is defined ω-wise as a Lebesgue-Stieltjes integral.
Properties Rem.
The resulting stochastic integral then has all the properties from the previous section except those that rest in an essential way on the (local) martingale property.
The isometry property for example is of course lost.
We still have, for H predictable and locally bounded:
- H · X is a semimartingale.
- If X is special with canonical decomposition X = X 0 +M +A, then H · X is also special, with canonical decomposition H · X = H · M + H · A. (This uses the non-obvious fact that if A is predictable and of
finite variation and H is predictable and locally bounded, the pathwise defined integral H ·A can be chosen to be predictable again.)
- linearity : same formula as before.
- associativity : same formula as before.
- behaviour under stopping : same formula as before.
- quadratic variation and covariation: same formula as before.
- jumps: same formula as before.
- If X is continuous, then so is H · X; this is clear from ∆(H · X) = H∆X = 0.
Thm. (sort of dominated convergence theorem, continuity proper- ty)
(i) If Hn, n ∈ N , are predictable processes with Hn^ → 0 point- wise on Ω and |Hn| ≤ |H| for some locally bounded H, then Hn^ · X → 0 uniformly on compacts in probability, which means that
sup 0 ≤s≤t
|Hn^ · Xs| → 0 in probability as n → ∞, ∀t ≥ 0.
(ii) This can also be viewed as a continuity property of the stocha- stic integral operator H 7 → H · X, since (pointwise and locally bounded) convergence of (Hn) implies convergence of (Hn^ · X), in the sense of above formula.
Rem. (further properties)
(i) If X is a semimartingale and f is a C^2 -function, then f (X) is again a semimartingale.
(ii) If X is a semimartingale w.r.t. P and R is a probability measure equivalent to P , then X is a semimartingale w.r.t. R. This will follow from Girsanov’s theorem, which even gives a de- composition of X under R.
(iii) If X is any adapted process with RC trajectories, we can always define the (elementary) stochastic integral H · X for processes H ∈ bE. If X is s.t. this mapping on bE also has the continuity property from the above Thm. for any sequence (Hn)n∈N ∈ bE converging pointwise to 0 and with |Hn| ≤ 1 for all n, then X must in fact be a semimartingale.
Rem. The above result implies that if we start with any model where S is not a semimartingale, there will be arbitrage of some kind. Lem. The family of semimartingales is invariant under a transformation by a C^2 -function, i.e. f (X) is a semimartingale whenever X is a semimartingale and f ∈ C^2.
4.6 Stochastic calculus
Good to know
Def. (weak convergence of probability measure) Suppose μj is a sequence of measures on R. By the definition of weak convergence of measures, μj weak converges to μ means that for any bounded continuous function f , there holds that ∫
R
f μj −n−→∞−−→
R
f μ.
Thm. (Wiki: Lebesgue’s Dominated Convergence Theorem) Let {fn} be a sequence of real-valued measurable functions on a mea- sure space (S, Σ, μ). Suppose that the sequence converges pointwise to a function f and is dominated by some integrable function g in the sense that
|fn(x)| ≤ g(x)
∀n in the index set of the sequence and x ∈ S. Then f is integrable and
lim n→∞
S
|fn − f |dμ = 0
which also implies
lim n→∞
S
fndμ =
S
f dμ.
Rem. The statement ”g is integrable” is meant in the sense of Lebesgue, i.e ∫
S
|g|dμ < ∞.
Lem. One can use that any continuous local martingale of finite variation is constant. Throughout this chapter We work on a probability space (Ω, F, P ) with a filtration F = (Ft) satisfying the usual conditions of right-continuity and P - completeness. For all local martingales, we then can and tacitly do choose a version with RCLL trajectories. For the time parameter t, we have either t ∈ [0, T ] with a fixed time horizon T ∈ (0, ∞) or t ≥ 0. In the latter case, we set
F∞ :=
t≥ 0
Ft := σ
t≥ 0
Ft
If f : R → R is in C^2 , α, β ∈ R and the semimartingale X = (Xt)t≥ 0 is given by Xt = αt + βNt, then:
f (Xt) = f (X 0 ) + α
∫ (^) t
0
f ′(Xs−)ds +
0 <s≤t
(f (Xs) − f (Xs−))
One frequently simplification of Thm. 1.2 (i) arises if one or se- veral of the components of X are of finite variation. If Xk^ , say, is of finite variation, then we know from Lem. (a simple result from analysis) that 〈Xk^ 〉 ≡ 0 and hence also 〈Xi, Xk^ 〉 ≡ 0 for all i by Cauchy-Schwarz.
Def. (stochastic differential equation (SDE), geometric Brownian motion (GBM))
Wiki: A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic pro- cess, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as unstable stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion.
A typical SDE is of the form
dXt = μ(Xt, t)dt + σ(Xt, t)dWt.
The solution of this SDE is given by the geometric Brownian mo- tion (GBM)
Xt = X 0 exp
σWt +
μ −
σ^2
t
for t ≥ 0.
Def. (Stochastic exponential) For a general real-valued semimartingale X null at 0, the stochastic exponential of X is defined as the unique solution Z of the SDE
dZt = Zt− dXt, Z 0 = 1
and it follows that the unique solution to this SDE is:
Zt := E(X) = 1 +
∫ (^) t
0
Zs− dXs ∀t ≥ 0
E(X)t = exp
Xt −
[X]t
Yor’s formula:
E(X)E(Y ) = E(X + Y + [X, Y ])
Itˆo process
An Itˆo process is a stochastic process of the form
Xt = X 0 +
∫ (^) t
0
μsds +
∫ (^) t
0
σsdWs, ∀t ≥ 0
where W is some Brownian motion and μ and σ are predictable processes.
More generally, X, μ, W could be vector-valued and σ could be matrix-valued.
For any C^2 function f , the process f (X) is again an Itˆo process, and Itˆo’s formula gives
f (Xt) = f (X 0 ) +
∫ (^) t
0
f ′(Xs)μs +
f ′′(Xs)σ^2 s
ds
∫ (^) t
0
f ′(Xs)σsdWs
Itˆo’s representation theorem
Def. (P -augmented filtration) We start with a BM W = (Wt)t≥ 0 in Rm^ defined on a probability space (Ω, F, P ) without an a priori filtration. We define
F t^0 := σ(Ws, s ≤ t) for t ≥ 0 ,
F ∞^0 := σ(Ws, s ≥ 0),
and construct the filtration FW^ = (FtW ) 0 ≤t≤∞ by adding to each F t^0 all subsets of P -nullsets in F ∞^0 to obtain FtW. This so-called P -augmented filtration FW^ is then P -complete (in (Ω, F ∞^0 , P ), to be accurate) by construction. Rem.
One can show, by using the string Markov property of BM, that FW^ is also automatically right-continuous [RC] (so that it satis- fies the usual conditions).
We usually call FW^ , somewhat misleadingly, the filtration gene- rated by W.
One can show that W is also a BM w.r.t. FW^ ; the key point is to argue that Wt − Ws is still independet of FsW ⊇ F s^0 , even though FsW contains some sets from F ∞^0.
If one works on [0, T ], one replaces ∞ by T ; then F s^0 is not nee- ded separately since we use the P -nullsets from the ”last” σ-field F T^0.
Thm. 3.1 (Itˆo’s representation theorem)
Suppose that W = (Wt)t≥ 0 is a Rm-valued BM.
Then every RV H ∈ L^1 (FW ∞ , P ) has a unique representation as
H = E[H] +
0
ψsdWs, P -a.s.
for an Rm-valued integrand ψ ∈ L^2 loc(W ).
ψ has the additional property that
ψdW is a (P, FW^ )- martingale on [0, ∞] (and is thus uniformly integrable).
Rem. The assumptions on H say that H is integrable and F ∞W -measurable. The latter means intuitively that H(ω) can depend in a measurable way on the entire trajectory W·(ω) of BM, but not on any other source of randomness. Cor. 3. Suppose the filtration F = FW^ is generated by a BM W in Rm. Then:
(i) Every (real-valued) local (P, FW^ )-martingale L is of the form L = L 0 +
γdW for some Rm-valued process γ ∈ L^2 loc(W ).
(ii) Every local (P, FW^ )-martingale is continuous.
Thm. 3.3 (Dudley) Suppose W = (Wt)t≥ 0 is a BM w.r.t. P and F = (Ft≥ 0. As usual, set
F∞ :=
t≥ 0
Ft = σ
t≥ 0
Ft
Then every F∞-measurable random variable H with |H| < ∞ P -a.s. (e.g. every H ∈ L^1 (F∞, P )) can be written as
H =
0
ψsdWs P -a.s.
for some integrand ψ ∈ L^2 loc(W ). Rem. It is almost immediately clear that the integrand ψ in Thm. 3.3 can- not be nice. In fact:
(i) In Thm. 3.3, the stochastic integral process
ψdW is of course a local martingale, but in general not a martingale on [0, ∞]; if it were, it would have constant expectation 0 , which would imply that E[H] = 0.
(ii) In Thm. 3.3, the representation by ψ is not unique.
Itˆo product formula
Define the stochastic process Z = XY , where X and Y are two continuous real-valued semimartingales.
Then Z can be written as the sum of stochastic integrals:
Zt − Z 0 =
∫ (^) t
0
YsdXs +
∫ (^) t
0
XsdYs +
∫ (^) t
0
d[X, Y ]s
General properties/results
Any continuous, adapted process H is also predictable and locally bounded. It furthermore holds for any predictable, locally bounded process H that H ∈ L^2 loc(W ).
Let f : R → R be an arbitrary continuous convex function. Then the process (f (Wt))t≥ 0 is integrable and is a (P , F)- submartingale.
Given a (P , F)-martingale (Mt)t≥ 0 and a measurable function g : R+ → R, the process
(Mt + g(t))t≥ 0
is a:
- (P , F)-supermartingale iff g is decreasing;
- (P , F)-submartingale iff g is increasing.
A continuous local martingale of finite variation is identically con- stant (and hence vanishes if it is null at 0).
For∫ a function f : R → R in C^1 , the stochastic integral · 0 f^
′(Ws)dWs is a continuous local martingale.
Furthermore, for f ∈ C^2 it holds that f (W ) is a continuous local martingale iff
0 f^
′′(Ws)ds = 0.
If a predictable process H = (Ht)t≥ 0 satisfies
E
[
H s^2 ds
]
< ∞, ∀T ≥ 0
then
HdWs is a square-integrable martingale.
If f : R → R is bounded and continuous, then the stochastic integral
f (W )dW is a square-integrable martingale.
If a process H = (Ht)t≥ 0 is predictable and the map s 7 → E[H s^2 ] is continuous, then the stochastic integral
HdW is a square- integrable martingale.
If∫ f : R → R is polynomial, then the stochastic integral f (W )dW is a square-integrable martingale.
5 Black-Scholes Formula
5.1 Black-Scholes (BS) model
Rem. The Black-Scholes model or Samuelson model is the continuous- time analogue os the Cox-Ross-Rubinstein binomial model we have seen at length in earlier chapters. Def. Throughout this we will use the following setting. A fixed time horizon T ∈ (0, ∞) and a probability space (Ω, F, P ) on which there is a Brownian motion W = (Wt) 0 ≤t≤T. We take as filtration F = (F 0 ≤t≤T the one generated by W and augmented by the P -nullsets form F T^0 := σ(Ws; s ≤ T ) s.t. F = FW^ satisfies the usual conditions under P. Def. (undiscounted financial market model) The financial market model has two basic traded assets: a bank ac- count with constant continuously compounded interest rate r ∈ R, and a risky asset (usually called stock) habing two parameters μ ∈ R and σ > 0. Undiscounted prices are given by
S^ ˜^0 t = ert
S^ ˜^1 t = S^10 exp
σWt +
μ −
σ^2
t
with a constant S 01 > 0. Cor. Applying Itˆo’s formula to the above equations yields
d S˜ t^0 = S˜ t^0 rdt,
d S˜ t^1 = S˜ t^1 μdt + S˜^1 t σdWt,
which can be rewritten as
d S˜^0 t S^ ˜^0 t
= rdt,
d S˜^1 t S^ ˜^1 t
= μdt + σdWt,
This means that the bank account has a relative price change ( S˜ t^0 +dt − S˜^0 t )/ S˜^0 t of rdt over a short time period (t, t + dt]; so r is the growth rate of the bank account. In the same way, the relative price change of the stock has a part μdt giving a growth at rate μ, and a second part σdWt ”with mean 0 and variance σ^2 dt” that causes random fluctuations. We call μ the drift (rate) and σ the (instantaneous) volatility of S˜^1. Def. (discounted financial market model) We pass to quantities discounted with S˜^0 ; so we have §^0 = S˜^0 / S˜^0 ≡ 1 , and S^1 = S˜^1 / S˜^0 is by the undiscounted financial market model given by
S^1 t = S^10 exp
σWt +
μ−r −
σ^2
t
We obtain via Itˆo’s formula that S^1 solves the SDE
dS t^1 = S t^1 ((μ − r)dt + σdWt).
For later use, we observe that this gives
d〈S^1 〉t = (S t^1 )^2 σ^2 dt
for the quadratic variation of S^1 , since 〈W 〉t = t. Rem. As in discrete time, we should like to have an equivalent martingale measure for the discounted stock price process S^1. To get an idea how to find this, we rewrite
dS t^1 =S t^1 ((μ − r)dt + σdWt).
⇔ dS t^1 =S t^1 σ
dWt +
μ − r σ
dt
= S^1 t σdW (^) t∗ ,
with W ∗^ = (W (^) t∗ ) 0 ≤t≤T defined by
W (^) t∗ := Wt +
μ − r σ
t = Wt +
∫ (^) t
0
λds for 0 ≤ t ≤ T.
Def. (market price of risk or Sharpe ratio) The quantity
λ := μ−σr
is often called the instantaneous market price of risk or infinitesimal Sharpe ratio of S^1. Rem.
mean portfolio return−risk-free rate standard deviation of portfolio return =^ Sharpe ratio. Rem. By looking at Grisanov’s theorem, we see that W ∗^ is a Brownian motion under the probability measure Q∗^ given by
dQ∗ dP
:= E
λdW
T
= exp
−λWT −
λ^2 T
whose density process w.r.t. P is
Z t∗ = E
λdW
t
= exp
−λWt −
λ^2 T
for 0 ≤ t ≤ T.
Rem. By
dS t^1 = S t^1 σ
dWt +
μ − r σ
dt
= S^1 t σdW (^) t∗ ,
the stochastic integral process
S^1 t = S^10 +
∫ (^) t
0
S u^1 σdW (^) u∗
