Proof. Fatou’s Lemma.
ADVANCED FINANCIAL MODELS SUMMARY
Theorem. If X is a discrete-time local martingale with Xt ≥ 0 for all
t ≥ 0. Then X is a martingale.
ANDREW TULLOCH
Theorem. The probability measure Q is equivalent to the measure P if
and only if there exists a positive random variable ξ such that Q(A) =
EP (ξI(A)).
The random variable ξ is call the density, or Radon-Nikodym derivative
of Q with respect to P.
1. Arbitrage Theory
Definition. An investment/consumption strategy is a predictable process
H satisfying the self-financing condition
Ht−1 · Pt−1 ≥ Ht · Pt−1
Definition. A numeraire is an asset with a strictly positive price at all
times.
(1.1)
Definition. An equivalent martingale measure is any probability measure
St
Q equivalent opt P such that the discounted price process N
is a martint
gale under Q, where Nt is the numeraire price process.
The corresponding consumption process ct is given as
ct = Ht−1 · Pt−1 − Ht · Pt−1
(1.2)
Definition. Xt is predictable if Xt is Ft−1 -measurable for all t ≥ 1.
Definition. Let Y be a state price density, and fix a time horizon T > 0.
Then
YT NT
dQ
=
(1.4)
dP
Y0 N0
is an equivalent martingale measure relative to N for the model.
Definition. A state price density is a strictly positive adapted process Y
such that the process Yt Pt is a martingale.
Definition. An absolute arbitrage is a strategy H such that there exists
a non-random time T > 0 with the properties
(i) X0 (H) = 0 = XT (H) almost surely, and
PT
(ii) P
t=1 ct > 0 > 0.
Definition. Suppose Q is an equivalent martingale measure for the market Pt . Then
N0 P dQ
Yt =
E (
|Ft )
(1.5)
NT
dP
is a state price density.
Definition. An asset is a numeraire if its price is strictly positive for all
time, almost surely.
2. Pricing and Hedging Contingent Claims
Theorem. If a numeraire exists, then we have that if an investment/consumption strategy is an arbitrage for the market model, there exists a pure
investment strategy H 0 and a non-random time horizon T 0 such that
(i) X0 (H 0 ) = 0,
(ii) XT 0 (H 0 ) ≥ 0 almost surely,
(iii) P(XT 0 (H 0 ) > 0) > 0.
Definition. A European claim with payout ξT is replicable (or attainable) if there exists a pure investment strategy H such that XT (H) = ξT
almost surely.
Theorem. Suppose that our market has no arbitrage. Let ξT be the payout of a European option, and let H be the replicating strategy. Suppose
that the option is priced at ξt for 0 ≤ t ≤ T . Then if the augmented
market with the option has no arbitrage,
Theorem. A market model has no arbitrage if and only if there exists a
state price density.
ξt = Xt (H)
Proof. (⇒) H0 = E(Y P1 ), so if 0 ≤ E(Y H · P1 ) = H ·E(Y P1 ) = H ·P0 = 0,
so by pigeonhole H · P1 = 0. (⇐) By separating hyperplane argument, we
have P = {E(Y P1 ) |Y > 0, E(Y kP1 k) < ∞}, so either P0 ∈ P (and so state
price density exists), or there exists H with for all p ∈ P , H · (p − P0 ) ≥ 0
(with p? ∈ P ) with H · (p? − P0 ) > 0.
Then setting Y = Y0 , we have a pigeonhole argument showing that
P (X − H · P0 ) = 0, with X ≥ 0 a.s.
(2.1)
for all 0 ≤ t ≤ T .
Proof. First fundamental theorem applied to (P, ξ).
Theorem. Suppose that the market model has no arbitrage, and let Y
be a state price density process. Let ξT be the payout of an attainable
European contingent claim with maturity date T > 0. Suppose the claim
has price t for 0 ≤ t ≤ T and that the augmented market with the option
has no arbitrage. If either YT ξT is integrable of ξT ≥ 0 almost surely, then
1
ξt =
E(ξT YT |Ft )
(2.2)
Yt
or (if a numeraire exists),
Definition. A supermartingale is an adapted integrable process such that
E(Xt |Fs ) ≤ Xs
(1.3)
for all 0 ≤ s ≤ t.
Definition. A stopping time for a filtration Ft is a random variable τ
such that the event {τ ≤ t} is Ft measurable for all t.
dQ
NT YT
=
dP
N0 Y0
Definition. For an adapted process Xt and a stopping time τ , the stopped
process X τ is given by Xt∧τ .
(2.3)
Definition. A market is complete if every European contingent claim is
attainable, and incomplete otherwise.
Theorem. Let X be a martingale and let τ be a stopping time, then X τ
is a martingale.
Theorem. An arbitrage-free market model is complete if and only if there
exists a unique state price density Y such that Y0 = 1.
Proof. Uniqueness follows from Y = Y 0 , then considering ξ = I YT > YT0
and pigeonhole.
Proof. The process Kt = I(t ≤ τ ) is predictable, bounded, so X τ is a
martingale transform and hence a martingale.
Definition. A local martingale is an adapted process Xt such that there
exists an increasing sequence of stopping times τn with τn ↑ ∞ such that
the stopped process X τn is a martingale for each N .
Theorem. Suppose the adapted process ξt specifies the payout of an
American claim maturing at T > 0. Then there exists a trading strategy
H such that
(i) Xt (H) ≥ ξt for all 0 ≤ qt ≤ T ,
(ii) Xτ ? = ξτ ? for some stopping time τ ? , and
(iii) X0 (H) = supτ ≤T E(Yτ ξτ ).
Theorem. Martingales are local martingales..
Theorem. Let X be a local martingale, with |Xs | < Yt a.s for all 0 ≤
s ≤ t. If E(Yt ) < ∞ for all t ≥ 0, then X is a true martingale.
Theorem. Let U be a discrete-time supermartingale. Then there is a
unique decomposition
Ut = U0 + M t − A t
(2.4)
where M is a martingale and Ais a predictable non-decreasing process with
M0 = A0 = 0.
Proof. Conditional dominated convergence theorem, Xt∧τn is a martingale.
Theorem. Let X be a local martingale, with Xt ≥ 0 for all t ≥ 0. Then
X is a supermartingale.
1
ADVANCED FINANCIAL MODELS SUMMARY
2
for each t ≥ 0, where the limit is in probability. If
Proof. M0 = 0 = A0 ,
Mt+1 = Mt + Ut+1 − E(Ut+1 |Ft )
(2.5)
At+1 = At + Ut − E(Ut+1 |Ft ) .
(2.6)
and telescope.
dXt = αt dWt + βt dt,
(3.6)
dhXit = α2t dt
(3.7)
then
Definition. Let Zt be an integrable adapted discrete-time process. Let Ut
be given by the recursion
UT = ZT
(2.7)
Ut = max(Zt , E(Ut+1 |Ft ))
(2.8)
Ut is called the Snell envelope of Zt . It is the smallest supermartingale
that dominates the process Zt .
Theorem. Let Zt be an integrable adapted process, with Ut its Snell
envelope with Doob decomposition Ut = U0 + Mt − At . Let τ ? = min{t ∈
{0, . . . , T } : At+1 > 0} with the convention τ ? = T on {At = 0∀t}. Then
τ ? is a stopping time, with
Uτ ? = U0 + Mτ ? = Zτ ? .
(2.9)
Theorem. Let Z be an adapted integrable process and let U be its Snell
envelope. Then
U0 = sup E(Zτ ) .
(2.10)
τ ≤T
3. Brownian Motion and Stochastic Calculus
Definition. A Brownian motion is a collection of random variables such
that
(i) W0 (ω) = 0 for all ω ∈ Ω,
(ii) For all 0 ≤ t0 < t1 · · · < tn , Wti+1 − Wti are independent with
distribution N (0, |ti+1 − ti |),
(iii) The sample path t 7→ Wt (ω) is continuous for all ω ∈ Ω.
Rn
Theorem. Let f : R+ ×
→ R where (t, x) 7→ f (t, x) is continuously
differentiable in t and twice-continuously differentiable in x. Then
df (t, Xt ) =
n
n
n
D
X
∂f
∂f
1 X X ∂2f
(i)
(t, Xt )dt +
(t, Xt )dXt +
(t, Xt )d X (i) , X
∂t
∂xi
2 i=1 j=1 ∂xi ∂xj
i=1
(3.8)
Theorem. Let Wt be an m-dimensional Brownian motion, with
Z
Z t
1 t
kαs k2 ds +
αs · dWs )
Zt = exp(−
2 0
−
(3.9)
and Zt be a martingale. Let Q be the equivalent measure with density dQ
=
R dP
ZT . Then the m-dimensional process (Ŵt ) defined by Ŵt = Wt − 0t αs ds
is a Brownian motion on (Ω, FT , Q).
R
Theorem (Novikov’s Condition). If E exp( 12 0T kαs k2 ds) < ∞, then
Z
Z T
1 T
E exp(−
kαs k2 ds +
αs · dWs ) = 1
(3.10)
2 0
0
Theorem. Let (Ω, F , Π) be a probability space with an m-dimensional
Brownian motion W and filtration Ft generated by W . Let X be a continuous local martingale. Then
there exists a unique predictable m-dimensional
R
process αt such that 0t kαs k2 ds < ∞ almost surely for all t ≥ 0 and
R
Xt = X0 + 0t αs · dWs . Furthermore, if Xt > 0 for all t ≥ 0 then there
R
exists a predictable β such that 0t kβs k2 < ∞ and
Z
Z T
1 T
kβs k2 ds +
βs · dWs ).
(3.11)
Xt = X0 exp(−
2 0
0
Definition. A simple predictable process is an adapted process α of the
P
form αt (ω) = N
n=1 I((tn−1 , tn ]) (t)an (ω) where an are bounded and Ftn−1 4. Arbitrage Theory for Continuous-Time Models
measurable for 0 ≤ t0 < · · · < tn < ∞.
Define the stochastic integral by the formula
Definition. An (n + 1)-dimensional predictable process (H, c) such that
H is P -integrable is a self-financing investment/consumption strategy if
Z ∞
N
X
associated with a
αs dWs =
an (Wtn −Wtn−1 )
(3.1) and only if d(Ht · Pt ) = Ht · dPt − ct dt. The wealth
R
R
0
self-financing strategy H is Xt = Ht · Pt = X0 (H) + 0t Hs · dPs − 0t cs ds.
n=1
Theorem (Ito’s Isometry). For a simple predictable integrand α, we have
Z ∞
Z ∞
E (
αs dWs )2 = E
α2s ds
(3.2)
0
0
R
R
Definition. If α is predictable with E 0∞ α2s ds < ∞, then 0∞ αs dWs =
R ∞ (k)dWs
limk 0 αs
where the limit is in L2 (Ω) where α(k) is a sequence of
simple predictable processes converging to α in L2 (R+ × Ω).
R
Theorem. For every predictable α such that E 0t α2s ds < ∞ for all t ≥
R
0, there exists a continuous martingale X such that Xt = 0∞ αs I(s ≤ t) dWs .
R
Theorem. If α is an adapted continuous process then Xt = 0t αs dWs is
R
a continuous local martingale. If we have E 0t α2s ds < ∞ for all t ≥ 0,
then X is a true martingale.
Definition. An Ito process X is an adapted process of the form
Z t
Z t
Xt = X0 +
αs dWs +
βs ds
0
(3.3)
0
where X0 is a fixedR real number andR αt , βt are predictable real-valued
processes such that 0t α2s ds < ∞ and 0t |βs |ds < ∞ almost surely for all
t ≥ 0.
Theorem. Let X be an Ito process and f : R → R twice continuously
differentiable. Then
Z t
Z t
1
f (Xt ) = f (X0 ) +
f 0 (Xs )αs dWs +
[f 0 (Xs )βs + f 00 (Xs )α2s ]ds
2
0
0
(3.4)
Theorem. Let X be an Ito process. There exists a continuous nondecreasing process hXi called the quadratic variation of X, such that
hXit = lim
N
N
X
(X nt − X (n−1)t )2
n=1
N
N
(3.5)
Definition. A trading strategy H is L-admissible if and only if the associated wealth process X(H) is such that Xt (H) ≥ −Lt for all t ≥ 0 a.s.
where L is a given continuous non-negative adapted process.
Definition. An admissible investment/consumption strategy (H, c) is called
an absolute arbitrage if and only if there is a non-random
time T such that
R
X0 (H) = 0 = XT (H) a.s. and P 0T cs ds > 0 > 0.
Definition. A state price density is a positive Ito process Y such that
Y P is an n-dimensional local martingale.
Theorem. If there exists a state price density Y such that Y L is locally
of class D, then there are no L-admissible absolute arbitrages.
Proof. Using the self-financing condition dXt = d(Ht ·dPt ) = Ht ·dPt −ct dt,
we obtain d(Xt Yt ) = Ht · d(Yt Pt ) − Yt ct dt. R
Then if Y L is of class D, we can show E 0T Hs · d(Ys Ps ) + LT YT =
E(LT YT ) be Fatou’s,
and uniform integrability.
R stopped local martingales,
R
Then we show E 0T Ys cs ds = E 0T Hs · (dYs Ps ) ≤ 0 and so ct = 0
a.s.
Definition. A family of random variables Z is called uniformly integrable
if and only if limk→∞ supZ∈Z E(|Z|I(|Z| > k)) = 0.
Theorem. Let Z1 , . . . , Zn be a family of integrable random variables.
The following statements are equivalent:
(i) Zn → Z∞ in L1 , and
(ii) (Zn )is uniformly integrable and Zn → Z∞ in probability.
Definition. A continuous adapted process Z is of class D if the family
of random variables {Zτ } with τ a finite stopping time is uniformly integrable. A process is locally of class D if {Zτ ∧t } for τ a stopping time is
uniformly integrable for
each t ≥ 0.
If E sup0≤s≤t |Zs | < ∞ for each t ≥ 0, then Z is locally of class D.
If Z is a martingale, then Z is locally of class D.
ADVANCED FINANCIAL MODELS SUMMARY
3
Theorem. Suppose (H, c) is a self-financing investment/consumption strat- Proof. Ito’s formula on dV (t, St ), and show V (t, St ) = φt Bt + πt St , and
R
R
egy and let Xt = Ht · Pt = X0 + 0t Hs · dPs − 0t cs ds. Then d(Xt Yt ) = dV (t, St ) = φt dBt = πt · dSt , so H = (φ, π) is a self-financing strategy
that replicates V (t, St ) as required.
Ht · d(Yt Pt ) − Yt ct dt for any Ito process Y .
Definition. A relative arbitrage is a pure investment strategy with wealth
T
process X such that there is a non-random time T > 0 satisfying X
≥
NT
XT
X0
X0
a.s
and
P
>
>
0.
N
N
N
0
0
T
Definition. An equivalent (local) martingale measure relative to the numeraire with price N is a probability measure Q equivalent to P such that
S
is a local martingale.
N
L
Theorem. Let Q be an equivalent local martingale measure. Suppose N
is locally in Q-class D. Then there are no L-admissible relative arbitrages.
Theorem. Suppose that C0 (T, K) = Ee−rT (ST −K)+ (Q). Then
0
(T, K) =
rK ∂C
∂K
2
σ(T,K)
2
∂C0
(T, K)+
∂T
2
K 2 ∂∂KC20 (T, K).
Theorem. Assume that a banker hedges an option assuming constant
volatility, and delta hedges with wealth evolving with dXt = r(Xt −πt St )+
πt St , with πt = VS (t, St , σ̂). If the true dynamics are dSt = St (µdt +
σt dWt ), then using the fact that V solves the BS PDE and that dVt =
rV dt + πt (dSt − rSt dt) + 12 St2 (σt2 − σ̂ 2 )VSS dt, we obtain
Z
1 T r(T −t) 2
XT − g(ST ) =
e
(σ̂ − σt2 )St2 VSS (t, St , σ̂)dt
(5.6)
2 0
Theorem. There exist continuous time markets that have relative arbi6. Interest Rate Models
trage but no absolute arbitrage.
Rt
2
Definition.
A
zero-coupon
bond with maturity T is a European continTheorem. Let λ be a predictable m-dimensional process such that 0 kλs k ds <
Rt
gent
claim
that
pays
one
unit
of currency at time T . P (t, T ) is the price
∞ a.s. for all t ≥ 0 and that σt λt = µt − rt 1. Let Yt = Y0 exp(− 0 (rs +
at time t ∈ [0, T ] of the bond.
Rt
kλs k2
)ds − 0 λs · dWs ) for a constant Y0 > 0 - or equivalently, dYt =
The yield y(t, T ) is defined by y(t, T ) = − T 1−t log P (t, T ).
2
Yt (−rt dt − λt · dWt ). Then Y is a state-price density. Furthermore, if the
∂
log P (t, T ).
The forward rate f (t, T ) is defined by f (t, T ) = − ∂T
filtration is generated by the m-dimensional Brownian motion W , all state
RT
f
(t,s)ds
−
−(T −t)y(t,T ) = e
t
.
Note
that
P
(t,
T
)
=
e
price densities have this form.
Proof. Show Y B and Y S are local martingales.
Martingale
R representation
R theorem shows that all are of the form Mt =
M0 exp(− 21 0t kλs k2 ds − 0t λs · dWs ).
Theorem. Let dBt = Bt rt dt where rt is the sport interest rate. Then
there is no arbitrage relative to the numeraire if there exists an equivalent
P (t,T )
measure Q such that the discounted bond price process B
is
Theorem.R Suppose λR is a predictable process with σt λt = µt − rt 1. If
a local martingale for all T > 0. IN particular, there is no arbitrage if
P (t, T ) = Eexp(− R T r ds)|F (Q) for all 0 ≤ t ≤ T .
−1
t
kλ k2 ds− t λ ·dW
s
0 s
Mt = e 2 − s
is a true martingale, then the measure Q
dQ
defined by dP = MT is an equivalent martingale measure. In particular,
the stock price dynamics are given by
X ij
σt dŴtj )
(4.1)
dSti = Sti (rt dt +
j
where Ŵt = Wt +
Rt
0
λs ds is a Q Brownian motion.
Theorem. Suppose m = d and the d × d matrix σt is invertible for all t, ω
so that in particular, there is a unique (up to scaling) state price density
Y of the form dYt = Yt (−rt dt − λt · dWt ) where λt = σt−1 (µt − rt 1).
Let ξT be non-negative, FT -measurable, and such that ξT YT is integrable. Then there exists a 0-admissible strategy H with initial cost
X0 (H) = EY0 (YT ξT ) which replicates the European claim with payout
ξT .
Furthermore, if LY is locally of class Dand H̃ is an L-admissible strategy replicating the claim, then X0 (H̃) ≥ X0 (H).
Definition. The Black-Scholes model is given by the pair of equations
dBt = Bt rdt
(5.1)
dSt = St (µdt + σdWt )
(5.2)
Consider pricing a European option with payoff ξT = g(ST ). The
2
unique state price density with Y0 = 1 is given by Yt = exp((r− λ2 )t−λWt
with λ = µ−r
.
σ
Thus, the is a trading strategy H which replicates the payout with
1
Xt (H) =
E(YT g(ST )|Ft ) .
(5.3)
Yt
dQ
dP
2
= exp(− λ 2T − λWT ).
Theorem. Suppose that the function V : [0, T ] × Rd → [0, ∞) satisfies
the PDE
d
d
d
X
∂V
∂V
1 XX
∂2V
+
rS i
+
ai,j S i S j
= rV
i
∂t
∂S
2 i=1 j=1
∂S i ∂S j
i=1
(5.4)
and V (T, S) = g(S).
Then there exists a 0-admissible strategy H such that Xt (H) = V (t, St ).
In particular, this strategy replicates the contingent claim with payout
g(ST ).
Furthermore, if H = (φ, π), then the strategy can be calculated as
πt = ∇ V (t, St ) = (
and φt =
V (t,St )−πt ·St
.
Bt
t
s
t∈0,T
t
Definition. The Vasicek model is drt = λ(r − rt )dt + σdŴt .
R
We have E(Q) (rt ) = e−λt r0 +(1−e−λt )r, VQ (rt ) = 0t e−2λ(t−s) σ 2 ds =
σ2
(1
2λ
− e−2λt ).
Indeed, we can deduce that
σ2
(1 − e−λx )2
(6.1)
2λ2
Theorem. Consider now where the short rate is Markovian, and so drt =
α(t, rt )dt + β(t, rt )dŴt for non-random function α, β.
If we fix T > 0 and let V : [0, T ] × R → R satisfy the PDE
f (t, t + x) = rt e−λx + r(1 − e−λx ) −
5. Hedging Contingent Claims in Continuous Time Models
The EMM Q is given by the density
t
∂V
∂V
(t, St ), . . . ,
(t, St ))
∂S1
∂S d
(5.5)
∂V
∂V
1
∂2V
(t, r) + α(t, r)
(t, r) + β(t, r)2
(t, r) = rV (t, r)
(6.2)
∂t
∂r
2
∂r2
with V (T, r) = 1. Assume P (t, T ) = V (t, rt ). Then the discounted price
Rt
process exp(− −
rs ds)P (t, T ) is a Q-local martingale.
References