How to Model a Financial Bubble Mathematically Lecture 1 April 12, 2013
How to Model a Financial Bubble Mathematically
Lecture 1
Philip Protter, Columbia University
Statistical Methods and Applications in Insurance and
Finance
CIMPA School
April 12, 2013
A Little History
• Stock markets date back to at least 1531 in Antwerp, Belgium
• There are over 150 stock market exchanges world wide, of
which the most significant count 103:
• There are 34 in Europe, including 5 in the U.K. and 4 in
France
• There are 20 in North America (7 in Canada, 12 in the U.S.,
and 1 in Mexico)
• There are 3 principal stock exchanges in Africa:
Johannesburg, Lagos, and Casablanca
Why do Stock Markets Exist?
• In the U.S., for example, the railroads needed vast amounts of
capital to build their tracks, and created the need for a stock
exchange
• The Dow Jones Industrial Average officially began in 1896
• In 1884, 12 years earlier, its predecessor began: Customer’s
Afternoon Letter which contained 11 stocks, 9 of which
were railroads
• In 1885, there were 12 railroads and 2 industrials in the Dow
Jones letter
• In 1886, there were 10 railroads and 2 industrials in the Dow
Jones letter
The Dow Jones Industrial Average in January, 1896
*American Sugar
Chicago, Milwaukee & St. Paul
Chicago, Rock Island & Pacific
Delaware, Lackawanna & Western
Missouri Pacific
Union Pacific
Chicago, Burlington & Quincy
Chicago & North Western
Delaware & Hudson Canal
Louisville & Nashville
Northern Pacific preferred
*Western Union
*Indicates an industrial (not a railroad)
The initial Dow Jones Industrial Average without
Railroads (May 26, 1896)
American Cotton Oil
American Tobacco
Distilling & Cattle Feeding
Laclede Gas
North American
U.S. Leather preferred
American Sugar
Chicago Gas
General Electric
National Lead
Tennessee Coal & Iron
U.S. Rubber
North American was replaced by US. Cordage Preferred, and
Distilling & Cattle Feeding became American Spirits, in August,
1896
Basic Mathematical Models for Asset Pricing
Finance
• Let S = (St )0≤t≤T represent the (nonnegative) price process
of a risky asset (e.g., the price of a stock, a commodity such
as “pork bellies,” a currency exchange rate, etc.)
• The present is often thought of as time t = 0. One is
interested in the unknown price at some future time T , and
thus ST constitutes a “risk.”
• Example: An American company contracts at time t = 0 to
deliver machine parts to Germany at time T . Then the
unknown price of Euros at time T (in dollars) constitutes a
risk for that company.
• In order to reduce this risk, one may use “derivatives”: one
can purchase — at time t = 0 — the right to buy Euros at
time T at a price that is fixed at time 0, and which is called
the “strike price.”
• If the price of Euros is higher at time T , then one exercises
this right to buy the Euros, and the risk is removed. This is
one example of a derivative, called a call option.
• More generally, a derivative is any financial security whose
value is derived from the price of another asset, financial
security, or commodity.
Call and Put Options
• A call option with strike price K , and payoff at time T can
be represented mathematically as
C = (ST − K )+
where x + = max(x, 0).
• Analogously, the payoff to a put option with strike price K at
time T is
P = (K − ST )+
and this corresponds to the right to sell the security at price
K at time T .
• Calls and Puts are related, and we have
ST − K = (ST − K )+ − (K − ST )+ .
This relation is known as put – call parity.
More complicated simple options
• We can calls and puts as building blocks for more complicated
derivatives.
• For example, if
V = max(K , ST )
then
V = ST + (K − ST )+ = K + (ST − K )+ .
• More generally, if f : R+ → R+ is convex then
f (x) = f (0) + f+0 (0)x +
Z
∞
(x − y )+ µ(dy )
(1)
0
where f+0 (x) is the right continuous version of the
(mathematical) derivative of f , and µ is a positive measure on
R with µ = f 00 , where the mathematical derivative is in the
generalized function sense.
Thus if f is convex, and if
V = f (ST )
is our financial derivative, then V is a portfolio consisting of a
continuum of European call options:
Z ∞
V = f (0) + f+0 (0)ST +
(ST − K )+ µ(dK ).
0
Other kinds of derivatives
• We can also have path dependent derivatives.
V
= F (S)T
= F (St ; 0 ≤ t ≤ T )
which are functionals of the paths of S.
• For example if S has c`
adl`ag paths (c`adl`ag is a French
acronym for “right continuous with left limits”) then
F : D → R+ , where D is the space of functions
f : [0, T ] → R+ which are right continuous with left limits.
The time value of money
• Inflation makes money worth less as time goes on
• Deflation makes it worth more
• Evaluating a claim that pays off $D at time T , when current
time is zero, can be done in time T dollars, or in time 0
dollars; if we use time T dollars for the payoff, but time 0
dollars for the evaluation, we must discount the payoff by the
rate of inflation (deflation)
• Suppose we have $D at time 0, and invest it in a bank which
pays interest rate r for one time unit (eg, one year). After one
year, we have $(D + rD).
• If we are paid interest every 3 months , or 1/4 year, and leave
the interest in the bank, we have $D + Dr /4 after the first
quarter, $D(1 + r /4)2 after the second, and $D(1 + r /4)4
after one year.
• If we compound n times in one year and leave the money in
the bank, we have $D(1 + r /n)n
• Taking limits limn→∞ $D(1 + r /n)n = $De r ; for t time units
analogously the limit = $De rt , which solves the ODE
dRt
= r;
Rt
R0 = D
• In general if r is a stochastic process (rt )t≥0 , then
Z
Rt = D +
t
rs Rs ds
0
⇒
Rt = De
Rt
0
rs ds
.
A simple Portfolio
• A simple portfolio has a varying quantity of shares of a stock,
plus a varying amount of money in a liquid, risk-free money
account.
• The value of a portfolio, V , depends on the trading strategy a
for stocks, and b for the money account
• A trading strategy is a vector of stochastic processes (a, b)
• Following a strategy (a, b) gives a dynamic portfolio value
process:
Vt (a, b) = at St + bt Rt .
• A trading strategy (a, b) is called self-financing if
Z
at St + bt Rt = a0 S0 + b0 R0 +
t
Z
as dSs +
0
t
bs dRs
0
Comments on self-financing
Z
at St + bt Rt = a0 S0 + b0 R0 +
t
Z
as dSs +
0
t
bs dRs
(2)
0
• Intuitively Self-financing means that we do not consume
money for other purposes, or add new money; we will soon
give a heuristic justification of equation (2)
• S is taken, by assumption, to have sample paths which are
right continuous and have left limits (c`adl`ag), and R is
continuous; hence the right side of (2) is at least c`adl`ag
• This creates implicit restrictions on the illusory arbitrariness of
the choice of a (predictable) and b (right continuous)
• If r ≡ 0 then (Rt )t≥0 ≡ 1, hence dRt = 0 and (2) becomes
Z t
at St + bt = a0 S0 + b0 +
as dSs
(3)
0
• This means once we have chosen strategy a, then b is
determined
Heuristic justification of self-financing
• Suppose a, b, S are all three semimartingales, and that R ≡ 1.
Then we have:
(at+dt − at )St+dt = −(bt+dt − bt )
(4)
which says that the change in stock holdings creates a
corresponding change in the money account.
• Equation (4) becomes
(at+dt − at )(St+dt − St ) + (at+dt − at )St
= −(bt+dt − bt )
≈ d[a, S]t + St− dat
= −dbt
(5)
• By Integration by parts, (5) becomes
Z
at St
=
a0 S0 + b0 +
t
t
Z
as dSs +
0
Ss− das + [a, S]t
0
⇒ d(at St ) − at− dSt = −dbt
Z t
≡ at St + bt = a0 S0 + b0 +
as dSs .
0
(6)
What is Arbitrage?
• In language: Arbitrage is the chance, no matter how small, to
make a profit without taking any risk
•
Definition
A model is arbitrage free on [0, 1] if there does not exist a
self-financing strategy (a, b) such that
V0 (a, b) = 0,
VT (a, b) ≥ 0,
P(VT (a, b) > 0) > 0.
(7)
• We want to convert this idea into useful mathematics
• Folk Theorem: There is no arbitrage if and only if there
exist a new probability Q, equivalent to P (ie, same sets of
probability zero, written Q ∼ P), such that S is a martingale.
• The above folk theorem is based on it being true in simple
cases (eg, finite probability space Ω [J.M. Harrison & S.R.
Pliska])
Martingales, local martingales, and sigma
martingales
• We assume given a complete, filtered probability space
(Ω, F, F, P), where F = (Ft )t≥0
• A stochastic process M is a martingale if E (|Mt |) < ∞, and
for s ≤ t, E (Mt |Fs ) = Ms a.s.
• Martingales are insufficient; for example:
• If X is a submartingale, we want a decomposition of
X = M + A, where M is a martingale and A is an increasing,
predictably measurable process. This is not true in general,
instead we need the concept of local martingale.
• If
R tN is a martingale, we would like the stochastic integral
Hs dNs to be a martingale, too. This is not true in general,
0
but instead (if X has continuous paths) it is a local martingale.
• In
R t general, if N is a martingale, then the stochastic integral
Hs dNs is a σ martingale
0
Definitions
• A stochastic process X with X0 = 0 is a local martingale if
there exists a sequence of stopping times (Tn )n≥1 with
limn→∞ Tn = ∞ a.s., such that Xt∧Tn is a martingale for
every n ≥ 1
• A stochastic process X with X0 = 0 is a σ martingale if there
exists Ra martingale M and a predictable process H such that
t
Xt = 0 Hs dMs for all t ≥ 0
• Note: Martingales ⊂ Local Martingales ⊂ σ Martingales
• If X is a nonnegative (or just bounded from below) σ
martingale, then it is a local martingale. So X ≥ 0 ⇒ Local
Martingales = σ Martingales
• Stochastic integration is closed for σ martingales
• For continuous processes, stochastic integration is closed for
local martingales
One way local martingales can arise
• Let X be the unique weak solution of the stochastic
differential equation
dXt = σ(Xt )dBt ,
X0 = 1
where B is a standard Brownian motion
• (Blei-Engelbert) If there exists an α ∈ (0, 1) such that
Z
∞
α
1
dy < ∞
σ(y )2
then X is a local martingale, and not a martingale. We call
such a process a strict local martingale
The Canonical Example of a Local Martingale
• Let Bt = (Bt1 , Bt2 , Bt3 ) be standard 3 dimensional Brownian
motion, with B0 = (1, 0, 0).
• Let u : R3 \{(0, 0, 0)} → R+ be given by
u(x) =
1
kx k
• Xt = u(Bt ) is a positive real valued local martingale, with
E (X0 ) = 1; X is called the inverse Bessel process
• X is not a martingale, because one can show that
lim E (Xt ) = 0
t→∞
and therefore it is not constant
• The inverse Bessel process satisfies the SDE
dXt = −(Xt )2 dBt ,
X0 = x0 > 0
The Canonical Example of a σ martingale
• τ is an exponential r.v. with parameter λ = 1
• U is independent of τ and P(U = 1) = P(U = −1) = 12
• Xt = U1{t≥τ } ; then X is a martingale
Rt
• Let Hs = 1s for s > 0, and let Mt = 0 Hs dXs
• Note that M has unbounded positive and negative jumps
• E (|Mν |) = ∞ for every stopping time ν with P(ν > 0) > 0,
so M is not a martingale, and not a local martingale, but M
is in fact a σ martingale.
Semimartingales and arbitrage
• Suppose S has continuous paths and is a semimartingale with
decomposition St = S0 + Mt + At , with M0 = A0 = 0, and
Q ∼ P; take
dQ
|Ft )
Z t = EP (
dP
which is a martingale
• By Girsanov’s theorem the decomposition of S under Q is
given by
Z
St = (Mt −
0
t
1
d[Z , M]s ) + (At +
Zs−
Z
0
t
1
d[Z , M]s );
Zs−
• Therefore if Z can be chosen so that
Z
t
At = −
0
1
d[Z , M]s ,
Zs−
we have that S is a Q-local martingale.
• By the Kunita-Watanabe inequality, from (8) we have
d[Z , M]t d[Z , Z ]t
d[M, M]t
(8)
• Recall (8):
Z
At = −
0
t
1
d[Z , M]s ,
Zs−
therefore we must have that
dAt d[M, M]t
in order for M to be martingale or local martingale under Q.
• This is not always the case; for example by Tanaka’s formula,
if
Z
St
= 1 + |Bt | = 1 +
t
sign(Bs )dBs + L0t
0
= 1 + βt + L0 (B)t ,
then d[β, β]t = dt, but dL0t 6 dt.
• Therefore 6 ∃ Q ∼ P for (9) such that S is a Q (local)
martingale
(9)
What if S is continuous and not a semimartingale?
• If there exists a Q ∼ P such that S is a local martingale under
dP
|Ft )
Q, then let Yt = EQ ( dQ
• By Girsanov,
Z
St = (St +
0
t
1
d[Y , S]s ) −
Ys
Z
0
t
1
d[Y , S]s
Ys
is a P decomposition of S. Therefore S is a P
semimartingale, a contradiction
• Thus a necessary condition for Q ∼ P is that S be a P
semimartingale
Why are (local) martingales so important?
• Martingales model fair gambling games
• A price process which is a model under the risk neutral
measure should have constant expectation
• Martingales have the property that t 7→ E (Mt ) is constant
• Theorem A stochastic process X is a martingale if and only if
E (Mτ ) = E (M0 ) for every bounded stopping time τ .
• Thus, M has constant expectation not just for fixed times,
but for stopping times as well.
The First Fundamental Theorem of Asset Pricing
• First Version: J. M. Harrison and S. R. Pliska, circa 1979
showed that a finite probability space (Ω, F, (Sn )n=0,1,2,... , P)
has No Arbitrage if and only if there exists another
probability measure Q ∼ P such that S is a martingale
• Second Version: David Kreps, circa 1981 realized that No
Arbitrage was not a strong enough condition to guarantee
such a result in a more general case. He created a new
condition and called it No Free Lunch
• Ignoring admissibility conditions for now, Kreps said that S
admits a Free Lunch on [0, T ] if there exists a function
f ∈ L∞
+ (Ω, F, P) such that P(f > 0) > 0, and a net
RT
(fα )α∈I = (gα − hα )α∈I , with hα ≥ 0 and gα = 0 Hsα dSs , for
admissible H α . And also fα → f in the Mackey topology on
L∞ induced by L1
• The Mackey topology is often written as σ(L∞ , L1 ), which
means that for a sequence (Xn )n≥1 ∈ L∞ , then Xn → X , if for
any Y ∈ L1 , E (Xn Y ) → E (XY ).
Economic intuition of No Free Lunch
RT
• Often we think of f as being of the form f = 0 Hs dSs
• Kreps saw that f could not in general be restricted to this
form for an admissible process H. (If it were, one could follow
this trading strategy H and replicated f , and have classical
arbitrage [starting with 0 and ending with f ≥ 0])
• But suppose f can be approximated by (fα )α∈I in a suitable
topology
• Let (hα )α∈I be the “errors” in the approximation, representing
“money thrown away.”
• No Free Lunch does not allow arbitrage, but it does allow
arbitrage to exist in the limit
Kreps’ Theorem
Theorem (Kreps, 1981) A bounded process S = (St )0≤t≤T
admits NFL if and only if there exists Q ∼ P such that S is a
martingale under Q.
This creates three immediate questions:
1. Can we replace [0, T ] with [0, ∞)?
2. What if S is not bounded?
3. What does convergence in nets mean vis `a vis an economics
interpretation?
The Four Fundamental Papers that Clarified the
Issues Surrounding the First Fundamental Theorem
1. Harrison, J.M, Kreps, D.M. (1979) Martingales and Arbitrage
in Multiperiod Securities Markets, Journal of Economic
Theory 20, 381-408
2. Harrison, J.M, Pliska, S.R. (1981) Martingales and Stochastic
Integrals in the Theory of Continuous Trading, Stochastic
Processes and their Applications 11, 215-260
3. Kreps, D.M. (1981) Arbitrage and Equilibrium in Economics
with infinitely many Commodities, Journal of Mathematical
Economics 8, 15-35
4. Harrison, J.M, Pliska, S.R. (1983) A stochastic calculus model
of continuous trading: Complete markets, Stochastic
Processes and their Application 11, 313-316
13 years later: Delbaen and Schachermayer
• Delbaen and Schachermayer, 1994: Convergence with nets
is replaced with convergence of sequences; S bounded is
replaced with S locally bounded, and M a martingale is
replaced with M a local martingale
• Delbaen and Schachermayer, 1998: The general case is
treated, where S can be c`adl`ag, and does not have to be
locally bounded, and M is replaced with a σ martingale.
• Before we discuss these results, we need the concept of an
admissible trading strategy
The Doubling Strategy
• Bet $1 at even money
• Stop betting if you win and collect $1 net winnings; otherwise
bet again, waging $2
• Stop if you win; you have now lost $1 and won $2, for a profit
of $1; otherwise bet again, waging $4
• In general: stop whenever you win, otherwise bet again,
doubling your last bet; your net winnings will be $1
• The probability is 1 that you will eventually win $1, so this is
an arbitrage strategy, known as the doubling strategy
Problems with the Doubling Strategy
• Need to make an unlimited number of bets (time constraints)
• Need “no fees” to make such bets (transaction costs)
• Need to have a counterparty (liquidity)
• But the above are practical problems; a theoretical problem is
the need for infinite resources
• We can eliminate the doubling strategy with an admissibility
condition
Admissibility
Definition: Let S be a semimartingale, α R> 0. A predictable
t
process H is α-admissible if H0 = 0, and 0 Hs dSs ≥ −α, for all
t ≥ 0.
H is admissible if there exists an α > 0 such that H is
α-admissible.
Note:
• We are implicity assuming That if H is admissible it is
predictably measurable and is in the space of S-integrable
processes
• This condition of admissibility is intrinsically asymmetric: H
can increase without bound, but is strictly limited in how
much it can be negative
The Kreps-Delbaen-Schachermayer Theory
• We work on the semi-infinite time interval [0, ∞], on a filtered
complete probability space (Ω, F, F, P), where F = (Ft )t≥0 .
• We further assume we have a risky asset price process
S = (St )t≥0 and that the spot interest rate r = 0
• A Contingent Claim is simply an FT measurable random
variable; examples are C = (ST − K )+ , which is a call at
strike price K and maturity time T ; another example is
P = (K − ST )+ which is a Rput We let H · S denote the
t
stochastic integral process ( 0 Hs dSs )t≥0
• Note that a call C is unbounded if S is unbounded, but a put
P is always bounded, irrespective of the behavior of S
Definitions
• We let L0+ denote finite-valued, nonnegative random variables
(a.s.). We define
K = {(H · S)∞ |H is admissible}
Kα = {H · S)∞ |H is α − admissible}
• No Arbitrage (NA): K ∩ L0+ = {0}
• Intuition: Starting with nothing, the only nonnegative result
we can end up with is identically 0; i.e., nothing
• Next we define
A0 = K − L0+ = {X = H · S)∞ |H is admissible, f ≥ 0, finite}
A = A0 ∩ L∞ = {|X | ≤ k, some k : X = (H · S)∞ − f }
• No Free Lunch (NFL) [Kreps]: A¯M ∩ L∞
+ = {0}, where the
¯ M denotes closure in the Mackey topology σ(L1 , L∞ )
(·)
• No Free Lunch with Vanishing Risk (NFLVR)
[Delbaen-Schachermayer]: A¯ ∩ L∞
+ = {0}, where the
∞
closure of A is in L , that is, the a.s. sup norm, as opposed
to the Mackey closure of Kreps and NFL
• Theorem: NFLVR is invariant under a change to an
equivalent probability measure
• NFLVR has become the accepted definition of no arbitrage; it
is considered to be the “gold standard.”
• However, we will see when we consider bubbles, that NFLVR
is just a bit too weak.
• The idea of No Dominance was introduced by Robert Merton
in 1973, but largely forgotten
No Dominance
• Let P(S) be all probabilities equivalent to the underlying
probability P such that if Q ∈ P(S) then S is a Qσ
martingale. Let
J
= {J ∈ FT |J is bounded from below and
sup EQ (S) < ∞}
Q∈P(S)
Λ(J)t
=
{the market price at time t of the contingent claim J}
• Definition: An element D of J Q-dominates another
element C of J if there exists a time t < T such that
C − Λ(C )t
Q{C − Λ(C )t
≤ D − Λ(D)t , for all t ≥ 0, Q a.s., and
< D − Λ(D)t } > 0 for some t ≥ 0
• We say that the model has No Dominance (ND) under P if
for any contingent claim C ∈ P(S), there does not exist
another claim D in P(S) which dominates C
• Theorem: If No Dominance holds for one Q ∈ P(S), then it
holds for Q ∈ P(S)
• Theorem: If for any H ∈ A we have Λ((H · S)T )0 = 0, then
No Dominance implies (NA).
• Theorem: If for any H ∈ A we have Λ((H · S)T )0 = 0 and Λ
is lower semicontinuous on L∞ with the k · k norm, then No
Dominance implies (NFLVR)
End of Lecture 1
How to Model a Financial Bubble Mathematically
Lecture 2
Philip Protter, Columbia University
Statistical Methods and Applications in Insurance and
Finance
CIMPA School
April 12, 2013
• We have a stochastic process (St )t≥0 modeling the price of a
risky asset; S exists on a complete, filtered probability space
(Ω, F, P, F), where F = (Ft )t≥0
• We discussed that S needs to be a semimartingale: a
semimartingale is a process S that has a decomposition
St = S0 + Mt + At , where M0 = A0 = 0 a.s., and M is a local
martingale and A is a c`adl`
gprocess with paths of finite
variation on compact time intervals; a local martingale M is a
process such that there exists an increasing sequence of
stopping times (Tn )n≥0 with T0 = 0 a.s. and limn→∞ Tn = ∞
a.s.
• Folk Theorem: There are no arbitrage opportunities if and
only if there is an equivalent probability measure Q such that
under Q the risky asset price process S is a martingale.
• We let L0+ denote finite-valued, nonnegative random variables
(a.s.). We define
K = {(H · S)∞ |H is admissible}
Kα = {H · S)∞ |H is α − admissible}
• No Arbitrage (NA): K ∩ L0+ = {0}
• Intuition: Starting with nothing, the only nonnegative result
we can end up with is identically 0; i.e., nothing
• Next we define
A0 = K − L0+ = {X = H · S)∞ |H is admissible, f ≥ 0, finite}
A = A0 ∩ L∞ = {|X | ≤ k, some k : X = (H · S)∞ − f }
• No Free Lunch (NFL) [Kreps]: A¯ ∩ L∞
+ = {0}, where the
¯ denotes closure in the Mackey topology, that is σ(L∞ , L1 )
(·)
• No Free Lunch with Vanishing Risk (NFLVR)
[Delbaen-Schachermayer]: A¯ ∩ L∞
+ = {0}, where the
∞
closure of A is in L , that is, the a.s. sup norm, as opposed
to the Mackey closure of Kreps and NFL
• Theorem: NFLVR is invariant under a change to an
equivalent probability measure
• NFLVR has become the accepted definition of no arbitrage; it
is considered to be the “gold standard.”
• However, we will see when we consider bubbles, that NFLVR
is just a bit too weak.
• The idea of No Dominance was introduced by Robert Merton
in 1973, but largely forgotten
No Dominance
• Let P(S) be all probabilities equivalent to the underlying
probability P such that if Q ∈ P(S) then S is a Qσ
martingale. Let
J
= {J ∈ FT |J is bounded from below and
sup EQ (S) < ∞}
Q∈P(S)
Λ(J)t
=
is the market price at time t of the contingent claim J
• Definition: An element D of J Q-dominates another
element C of J if there exists a time t < T such that
C − Λ(C )t
Q{C − Λ(C )t
≤ D − Λ(D)t , for all t ≥ 0, Q a.s., and
< D − Λ(D)t } > 0 for somet ≥ 0
We say that the model has No Dominance (ND) under P if
for any contingent claim C ∈ P(S), there does not exist
another claim D in P(S) which dominates C
• Theorem: If No Dominance holds for one Q ∈ P(S), then it
holds for Q ∈ P(S)
• Theorem: If for any H ∈ A we have Λ((H · S)T )0 = 0, then
No Dominance implies (NA).
• Theorem: If for any H ∈ A we have Λ((H · S)T )0 = 0 and Λ
is lower semicontinuous on L∞ with the k · k norm, then No
Dominance implies (NFLVR)
A General Framework
• Let S be a semimartingale modeling a risky asset price process
(so S ≥ 0)
• Assume that NFLVR holds
• Let D = (Dt )(t ≥ 0 be its cumulative cash flow of
dividends
• Assume the spot interest rate r ≡ 0
• We assume that the risky asset has a (finite valued) maturity,
or lifetime, of the risky asset
• Let Xτ = the terminal payoff, or liquidation value at time τ
• Assume that Xτ ≥ 0 and Dt ≥ 0
• ∆Dt = Dt − Dt− , and if ∆Dt0 > 0, for some t0 , then St0
denotes the price ex dividend
• Ex dividend refers to the trading of shares when a declared
dividend belongs to the seller, rather than to the buyer.
The Wealth Process
• The wealth of the investor at time t is given by
Z
τ ∧t
Wt = St +
dDu + Xτ 1{t≥τ }
0
• We assume there exists a probability measure Q ∼ P such
that W is a Q-local martingale
• Note that since W ≥ 0 always, we do not have need of σ
martingales
Trading Strategies
• A trading strategy is a vector process (πt , ηt )t≥0
• (πt )t≥0 is the trading strategy for the risky asset
• (ηt )t≥0 is the (risk-free) money market trading strategy
• Wtπ = πt St + ηt is the Wealth process corresponding to
the strategy (π, η)
Self-Financing Trading Strategies
• W0π = 0
• The Value Process V corresponding to the strategy (π, η) is
given by
Vtπ,η
Z
=
t
πu dWu
0
• Let α > 0. A self-financing strategy π is α-admissible if
Vtπ,η ≥ −α. The strategy π is admissible if it is α-admissible
for some α > 0
The Case of Complete Markets
• A market is complete for a class of contingent claims H if
every claim X ∈ H can be perfectly hedged
• How to interpret this statement in mathematics?
• First we must discount for the time value of money; usually
we work in a finite horizon, ie, a time interval [0, T ]
• We can take, for example, H to be all bounded X ∈ FT ; or
all random variables X in FT such that X /RT is bounded,
RT
where RT = 1 + 0 Rs rs ds, and r is the spot interest rate
process
• The Second Fundamental Theorem of Finance: A market
under H is complete if and only if every X ∈ H can be
perfectly hedged. That is, for any X ∈ H there exists a
hedging strategy π such that
Z
X =α+
T
πs dWs
0
This is equivalent to there being only one equivalent
probability measure Q such that W is a Q local
martingale.
• The unique equivalent measure Q that turns W into a local
martingale is called the risk neutral measure
• Recall that we are operating under the assumption of NFLVR,
so we know that at least one such Q exists
• The price of such a claim X is now intuitively clear: if
RT
1
X ∈ LQ (FT ) and 0 πs dWs is a martingale, then the price
should be EQ (X ) = α
The Problem of Unique Prices
• A suicide strategy is a strategy σ such that W0σ = 1 but
WTσ = 0
• Suicide strategies can lead to the non-uniqueness of prices
• For example let Y be a contingent claim that suppose θ is a
strategy such that
Y = c + WTθ
• The fair price of Y should be EQ (Y ) = c. But if one adds the
suicide strategy σ, one gets
Y = c + 1 + WTθ+σ
which by the same reasoning should have price c + 1.
Eliminating suicide strategies
• The problem arises because a local martingale Z need not
have constant expectation, and can even have ZT = 0 at a
finite time T
• Harrison and Pliska eliminate this possibility be restricting
themselves to martingales, at the cost of generality
• One can also eliminate suicide strategies with Merton’s No
Dominance assumption, and then allow local martingales
(recall that No Dominance implies NFLVR)
NFLVR does not imply ND
• Consider two risky assets maturing at time τ with payoff Xτ
and Yτ , respectively. Suppose Xτ ≥ Yτ a.s. Then:
Xt? = EQ {Xτ |Ft }1{t<τ }
≥ EQ {Yτ |Ft }1{t<τ } = Yt?
• Let β be a nonnegative local martingale such that βτ = 0 a.s.,
and βt0 > Xt?0 − Yt?0 for some t0
• Such a non-trivial β exists, and it is unbounded, of necessity
(if it were bounded, it would be a nonnegative [true]
martingale with terminal value 0, and hence identically 0)
• Suppose next the two risky asset prices are
Xt
= Xt?
Yt
= βt + Yt?
• No Dominance is violated, because
Yt0
> Xt0 for some t0 ,
Xτ
≥ Yτ
• But NFLVR is not violated. Why?
• Suppose we have a strategy designed to take advantage of
this mis-pricing: Sell Y and buy X , say at time t0 , and then
hold it to maturity. This gives a gain of βt0 > 0 at time t0 ,
and since Xτ − Yτ ≥ 0, with no other cash flows, we are safe
to keep our gain βt0 ; this creates an apparent arbitrage
• However, to do this, if t0 < u ≤ τ , then the value of the
trading strategy is
−Yu + Xu = βu + (Xu? − Yu? )
• Since β is of necessity unbounded, the admissibility condition
eliminates this strategy of apparent arbitrage; doubling all
over again!
What is the difference between S being a martingale
under the risk neutral measure, and a local
martingale?
• It turns out that this nuance can be used to give an
explanation of financial bubbles
• But first, let us describe the popular conception of a financial
bubble
US Stock Prices 1929 (Donaldson & Kamstra [1996])
Important Recent Bubbles
• Minor crashes in the 1960s and 1980s
• Junk bond financing led to the major crash of 1987
• Japanese housing bubble circa 1970 to 1989
• The “dot com” crash, from March 11th, 2000 to October 9th,
2002. Led by speculation due to the promise of the internet;
The Nasdaq Composite lost 78% of its value as it fell from
5046.86 to 1114.11.
• The 2008 US housing bubble and subprime mortgages
NASDAQ Index 1998-2000 (Brunnermeir & Nagel)
Current US Housing Price Trend (Center for Responsible Lending)
Oil Futures (WTRG Economics)
Oil Futures (WTRG Economics)
Oil Futures (WTRG Economics)
Oil Futures (WTRG Economics)
Oil Futures (WTRG Economics)
Oil Futures (WTRG Economics)
We can also see the bubble in the entire world’s stock markets
bursting. The next slide gives the results from the Morgan Stanley
Capital International All Country World Index.
Global Stock Markets (Greed and Fear, Oct 9, 2008)
Our Framework for Modeling Bubbles
• We assume NFLVR
• We have a risky asset price process S with S ≥ 0, and the
spot rate rt ≡ 0, so Rt ≡ 1
• We have a dividend process D, a wealth process W , and a
liquidation vale Xτ
• We are in a complete market framework, with a unique risk
neutral measure Q
• a fair price to pay for the risky asset at time t is the
conditional expectation of the future cash flows (taken under
the risk neutral measure)
The Fundamental Price
In complete markets with a finite horizon T , we use the risk
neutral measure Q, and for t < T the fundamental price of the
risky asset is defined to be:
St?
Z
= EQ {
T
dDu + XT |Ft }
t
Definition (Bubble)
A bubble in a static market for an asset with price process S is
defined to be:
β = S − S?
Static Markets
Theorem (Three types of bubbles)
1. β is a local martingale (which could be a uniformly
martingale) if P(τ = ∞) > 0;
2. β is a local martingale but not a uniformly integrable
martingale, if it is unbounded, but with P(τ < ∞) = 1;
3. β is a strict Q local martingale, if τ is a bounded stopping
time.
• Type 1 is akin to fiat money
• Type 2 is tested in the empirical literature
• Type 3 is essentially “new.” Type 3 are the most interesting!
• Fiat money (Type 1 bubbles) has no intrinsic value, and
become almost worthless
• German hyper inflation of the 1920s
• Zimbabwean hyperinflation of the current day
100 million German marks, 1923
100 trillion Zimbabwe dollars, 2009
Hyperinflation in Zimbabwe
In 1980, 1ZWR=US $1
Month
ZWR per USD
Sept 2008
1, 000
Oct 2008
90, 000
Nov 2008
1, 200, 000
Mid Dec 2008
60, 000, 000
End Dec 2008
2, 000, 000, 000
Mid Jan 2009
1, 000, 000, 000, 000
2 Feb 2009
300, 000, 000, 000, 000
Zimbabwean Currency Follow-up
• The Zimbabwe dollar was abandoned officially on April 12,
2009
• Since then it has been legal to use other currencies
• Most common are reported to be the South African Rand, the
Botswana Pula, The British pound sterling, and the US dollar
Theorem (Bubble Decomposition)
The risky asset price admits a unique decomposition
S = S ? + (β 1 + β 2 + β 3 )
where
1. β 1 is a c`adl`ag nonnegative uniformly integrable martingale
with limt→∞ βt1 = X∞ a.s.
2. β 2 is a c`adl`ag nonnegative NON uniformly integrable
martingale with limt→∞ βt2 = 0 a.s.
3. β 3 is a c`adl`ag non-negative supermartingale (and strict local
martingale) such that limt→∞ E {βt3 } = 0 and limt→∞ βt3 = 0
a.s.
Ben Bernanke and the Federal Reserve
End of Lecture 2
How to Model a Financial Bubble Mathematically
Lecture 3
Philip Protter, Columbia University
Statistical Methods and Applications in Insurance and
Finance
CIMPA School
April 13, 2013
Review from Lecture 2
• Let S be a semimartingale modeling a risky asset price process
(so S ≥ 0)
• Assume that NFLVR holds
• Let D = (Dt )t≥0 be its cumulative cash flow of dividends
• Assume the spot interest rate r ≡ 0
• Let Xτ = the terminal payoff, or liquidation value at time τ
• The wealth of the investor at time t is given by
Z
τ ∧t
Wt = St +
dDu + Xτ 1{t≥τ }
0
• We assume there exists a probability measure Q ∼ P such
that W is a Q-local martingale
• A trading strategy is a vector process (πt , ηt )t≥0
• W0π = 0
• The Value Process V corresponding to the strategy (π, η) is
given by
Vtπ,η =
t
Z
πu dWu
0
• Let α > 0. A strategy π is α-admissible if Vtπ,η ≥ −α. The
strategy π is admissible if it is α-admissible for some α > 0
• The Second Fundamental Theorem of Finance: A market
under H is complete if and only if for every X ∈ H there
exists a hedging strategy π such that
Z
X =α+
T
πs dWs
0
This is equivalent to there being only one equivalent
probability measure Q such that W is a Q local
martingale.
The Fundamental Price in a Complete Market
Setting
• Since markets are assumed complete, let Q ∼ P be the
unique risk neutral measure
• We define the fundamental price of the risky asset S,
denoted S ? , to be the future discounted future cash flow one
expects to get, conditional on current information
• In mathematics, S ? is given by
St?
Z
= EQ {
T
dDs + Xτ 1{τ <∞} |Ft }1{t<τ }
t
• The payoff at time t = ∞ does not contribute to S ?
Theorem:
St? is well defined. Moreover,
lim S ?
t→∞ t
= 0 a.s.
• Observe that W is a nonnegative Q supermartingale, so
St? ∈ L1 (dQ), and the result follows the supermartingale
convergence theorem, and the facts that (Dt )t≥0 and Xτ are
nonnegative.
• Note that in contrast, we cannot assume that St? is in L1 (dP)
• Corollary:
Z t∧τ
?
?
Wt = St +
dDu + Xτ 1{τ ≤t}
0
is a uniformly integrable martingale under Q, and
Z τ
?
W∞
=
dDu + Xτ 1{τ <∞}
0
Bubbles
• A bubble is defined to be a process β = (βt )t≥0 given by
βt = St − St?
• Note that βt ≥ 0 for all t, a.s.
• Theorem: If there exists a non trivial bubble (ie, βt 6= 0 for
some t > 0) for the risky asset price process S, then
1. If P(τ = ∞) > 0 then β is a local martingale without
restrictions (it can even be a uniformly integrable martingale)
2. If P(τ < ∞) = 1, and β is unbounded, then β is a local
martingale, and it cannot be a uniformly integrable martingale
3. If τ is bounded, and then β must be a strict local martingale
Theorem (Bubble Decomposition)
The risky asset price admits a unique decomposition
S = S ? + (β 1 + β 2 + β 3 )
where
1. β 1 is a c`adl`ag nonnegative uniformly integrable martingale
with limt→∞ βt1 = X∞ a.s.
2. β 2 is a c`adl`ag nonnegative NON uniformly integrable
martingale with limt→∞ βt2 = 0 a.s.
3. β 3 is a c`adl`ag non-negative supermartingale (and strict local
martingale) such that limt→∞ E {βt3 } = 0 and limt→∞ βt3 = 0
a.s.
Examples
• We call the three types of bubbles in the decomposition
bubbles of Type 1, Type 2 and Type 3
• Example of a Type 1 bubble: Let St = 1, all t, 0 < t < ∞,
and no dividends. This is an example of fiat money
• In this case τ = ∞ a.s., and X∞ = 1, and Dt ≡ 0 all t ≥ 0.
• Therefore
?
S∞
Z
t∧τ
dDu + Xτ 1{t≥τ } |Ft
= EQ
0
• Hence
βt = St − St? = 1
=0
A second example of a Type 1 bubble
• Let Bt = (Bt1 , Bt2 , Bt3 ), the three dimensional standard
Brownian motion
• k Bt k is called the Bessel process;
Xt =
1
k Bt k
is known as the inverse Bessel process
• Assume there are no dividends, only the asset price. One can
show that
lim Xt = 0
t→∞
and that X is a local martingale; indeed, X satisfies the SDE
dXt = −Xt2 dBt ;
X0 = 1
where B is a Brownian motion
• Also E (X0 ) = 1 and limt→∞ E (Xt ) = 0
Example of a Type 2 bubble
• Let τ be a stopping time with P(τ > t) > 0, for all t > 0,
and P(τ < ∞) = 1
• Let
St? = 1{t<τ } , payoff 1 at time τ
1 − 1{τ ≤t}
βt =
P(τ > t)
St = St? + βt
• One can show that β is a martingale which is not uniformly
integrable, and β∞ = 0
• So β is a bubble which is not uniformly integrable
Example of a Type 3 bubble
[A. Cox and D. Hobson, 2005]
• Let T be a fixed (non random) time, and define
St? = 1{[0,T )} (t),
XT = 1
• Let the bubble be given by
Z
βt =
0
t
√
βu
dBu
T −u
• Then β is a strict local martingale, with Bt = 0; define
St = St? + βt
Historical example of an option
• Aristotle, in his treatise Politics; Book 1, Part XI, writes of
Thales of Miletus, a pre-Socratic Greek philosopher and one
of the Seven Sages of Greece
• Thales wanted to justify his beliefs in astronomy, which
allowed him to predict (correctly, as it turned out) that there
would be a bumper olive crop harvest (Source: Walter
Schachermayer)
• According to Aristotle, Thales “gave deposits for the use of all
the olive-presses in Chios and Miletus, which he hired at a low
price because no one bid against him. When the harvest-time
came, and many were wanted all at once and of a sudden, he
let them out at any rate which he pleased, and made a
quantity of money.”
An ancient olive press used to make oil
Put-Call Parity in the Presence of Bubbles
• A Call Option has the payoff structure at the maturity time T
of (ST − K )+ and a put (K − ST )+ and a forward contract
at strike price K and maturity time T has a payoff at time T
of ST − K
• Recall that trivially
(ST − K )+ − (K − ST )+ = ST − K
• Let Ct (K ), Pt (K ), and Vt (K ) be the market prices at time t
and strike price K with common maturity time T of a call, a
put, and a forward
• Let Ct (K )? , Pt (K )? , and Vt (K )? be the fundamental prices
at time t and strike price K with common maturity time T of
a call, a put, and a forward
• The traditional approach for a complete market for put call
parity is to define the time t price of (for example) a
European call to be
EQ {(ST − K )+ |Ft }
and then put-call parity follows from the linearity of
conditional expectation
• The issue of whether or not market prices agree with the
conditional expectation prices is assumed to be true
• With bubbles, the market prices of calls, puts, and forwards
need not satisfy put-call parity
Example of Put-Call Parity Failing
• Let B i , 1 ≤ i ≤ 5 be five iid standard Brownian motions
• Define
Mt1 = exp(Bt1 − t/2)
Z t
Mi
√ s dBsi ,
Mti = 1 +
T −s
0
2≤i ≤5
• Consider a market with finite time horizon T
• It is complete, given M i , 1 ≤ i ≤ 5
• M 1 is a uniformly integrable martingale, and the rest are strict
local martingales on [0, T ]
• Let
St? = sup Mt1 ;
P(K )t
s≤t
?
St = St? + Mt2 ;
= P (K )t + Mt4 ;
C (K )t = C ? (K )t + Mt3
V (K )t = V ? (K )t + Mt5
• All the traded securities in the this example have bubbles
• Let δtC , δtP , and δtF be the bubbles parts of the market prices
for the Call, Put, and Forward.
• Under special conditions only (the absence of bubbles) do we
have market price put-call parity:
Ct (K ) − Pt (K ) = Ft (K ) if and only if δtF = δtC − δtP
Ct (K ) − Pt (K ) = St − K if and only if δtS = δtC − δtP
Implications for Models in the Black-Scholes
Paradigm
• To take advantage of these bubbles based on the convergence
at time T , one needs only to short sell at least one asset
• Such a strategy, however, is not admissible due to possible
unbounded losses
• By the Black-Scholes paradigm we mean a continuous risky
asset price process under the now standard NFLVR structure
• The important consequence is that in the presence of bubbles,
the Black-Scholes formula need not hold
• This is because the time t market price of a call option,
Ct (K ), can differ from the price EQ {(ST − K )+ |Ft }
Consequence for Black-Scholes Paradigm Models
• Implied volatility from the B-S formula need not equal
historical volatility; indeed, if there is a bubble, implied
volatility should exceed historical volatility
• However, if one assumes No Dominance, then the usual
understanding of the Black-Scholes model applies
• Another issue is Merton’s No Early Exercise Theorem
• This theorem states that while an American call option with
strike price K and maturity time T has the a priori impression
of presenting more flexibility in the exercise of the option, in
reality the optimal strategy is to exercise it at maturity T .
Therefore the fair prices of an American call option and that
of a European call option are the same
• The proof of Merton’s theorem uses Jensen’s inequality and
assumes the risky asset risk neutral price process is a
martingale
Under NFLVR and continuous complete markets,
Merton’s No Early Exercise Theorem need not hold
• We give an example where No Early Exercise fails to hold
• Let Bt = (Bt1 , Bt2 , Bt3 ) be a standard Brownian motion with
B0 = (1, 0, 0)
• Recall that the inverse Bessel process is
Xt =
1
k Bt k
which is a strict local martingale
• If X models a risky asset price process, then the price process
is a bubble
• (Xt )t≥0 is a uniformly integrable collection, E (X0 ) = 1, and
limt→∞ Xt = 0 a.s. and in L1
• If X is a risk neutral (Q) martingale, then by Jensen’s
inequality, t 7→ EQ {(Xt − K )+ } is monotone increasing
• For the inverse Bessel process, with Soumik Pal, we have
shown that the prices of European calls decrease as a function
of time to expiration
• That is, for S the inverse Bessel process, the function
T 7→ E {(ST − K )+ }
is monotone decreasing if K ≤ 12 , and otherwise it is initially
increasing and then strictly decreasing for
2K + 1 −1
T ≥ K log
.
2K − 1
• A similar results holds for all continuous strict local
martingales with asymptotic behavior similar to that of the
inverse Bessel process
• This result is intuitive in the presence of bubbles, since in a
bubble, the best strategy is to get in and out early, and not to
wait a long time to liquidate your positions
Bubble Decomposition
Theorem [Bubble Decomposition]:
St = St? + βt = St? + βt1 + βt2 + βt3 ,
is a unique decomposition such that
• β 1 ≥ 0 is a uniformly integrable martingale with
limt→∞ βt1 = X∞ a.s.
• β 2 ≥ 0, is not a uniformly integrable martingale, but of
course is a local martingale and is possibly a martingale, and
limt→∞ βt2 = 0 a.s.
• β 3 ≥ 0 is a strict local martingale such that limt→∞ βt3 = 0
a.s. and in L1
Why Does Short Selling Not Correct for Bubbles?
• Two reasons are proposed in the literature:
• The first is structural limitations: This is the limited ability
and/or expensive cost to borrow an asset for short sales (eg,
Duffie, Gˆarleanu, and Pederson [2002])
• As regards the first, in markets where short selling does not
exist (especially the third world), there do not seem to be
more bubbles
• The second is the risk the short seller takes that the price will
continue to go up (the danger of trying to predict a bubble)
• In mathematics this translates into admissibility violations
Two Problems with Complete Markets and Bubbles
• What is nice is that the risk neutral measure Q is unique, and
we therefore have a unique fundamental price
• An undesirable property is the impossibility of bubble birth:
A nonnegative local martingale cannot spring up after being
zero; once a nonnegative local martingale reaches zero, it
sticks at zero forever after
• The biggest problem is that while bubbles make sense in
complete markets under NFLVR, bubbles do not exist under
No Dominance. This is serious, because we will see later we
need No Dominance to establish fundamental put-call parity
• Theorem: Under No Dominance, Type 2 and Type 3 bubbles
do not exist in a complete market (with NFLVR)
Proof that Bubbles Do Not Exist in Complete
Markets under ND
• Theorem: Under No Dominance, Type 2 and Type 3 bubbles
do not exist in a complete market (with NFLVR)
• Proof: For Type 2 and Type 3 bubbles, β∞ = 0. Let W be
the wealth process corresponding to the risky asset price
process S
• There exist hedging processes π 1 and π 2 such that
Wt?
βt
Z
t
+
πu1 dWu
Z t0
= β0 +
πu2 dWu
=
W0?
0
• Let η 1 and η 2 make π 1 and π 2 self-financing, so that both π 1
and π 2 are admissible
• We have two ways to generate W
• The first way is buy and hold
• The second way is to follow π 1 , obtaining W ?
• The cost of the first position is W0 ≥ W0? , with W0 > W0? if
there is a non-trivial bubble
• That means that π 1 dominates the buy and hold strategy,
which violates No Dominance; so β cannot exist.
• We conclude: bubbles exist in a complete market under
NFLVR [Lowenstein and Willard, Cox and Hobson], but
cannot be born after time t = 0, create a Black-Scholes
paradox, and violate put-call parity.
• Bubbles do not exist in a complete market under No
Dominance, which is stronger than NFLVR
• The non existence of bubbles under ND solves the
Black-Scholes paradigm paradox, for example
• What happens in incomplete markets? In incomplete
markets under No Dominance the argument showing
bubbles do not exist, no longer applies
Do Bubbles Exist in Incomplete Markets?
• To discuss bubbles in incomplete markets, we need to decide
what we mean by a fundamental price, since there is an
infinite choice of risk neutral measures
• There are five basic methods to choose such a measure
• The first is Utility Indifference Pricing: Risk Neutral prices
span an interval on the real line, and choosing the right price
depends on the utility function of preferences of the agent
selling the contingent claim
• The Egocentric Method: Simply choose one arbitrarily
• The Convenience Method: Choose a risk neutral measure
•
•
•
•
that gives the price process mathematically nice properties: for
example, makes it a Markov process, or even a L´evy process
The Canonical Method: Find a reasonable criterion (eg,
minimal variance of the error, minimal distance to the
historical measure in a distance one chooses, minimal entropy)
and let it determine the risk neutral measure
The Ostrich Method: Prove results under a risk neutral
measure already chosen; that is, you do not specify it, but
pretend someone else has done so already
All of these methods assume that, once chosen, the risk
neutral measure is fixed and and never changes
The current methods share two properties:
1. In one sense or another they can be considered to be ad hoc
2. Once a risk neutral measure is chosen, it remains fixed in the
model for all time (which is often a finite horizon model, and
time is modeled by an interval [0, T ])
• Instead of these approaches, one can let the market choose
the risk neutral measure
• En passant, the market is often wrong (otherwise we would
not have bubbles!), but the issue is not “truth,” as it is, for
example, in physics
• The usual approach is as follows:
• Begin with the risky asset price process and a (possibly
random) savings rate;
• By a change of num´
eraire argument, assume interest rates are
zero;
• Find and choose a risk neutral measure Q;
• If g (XT ) is a financial (European style) derivative at time T ,
the price process is declared to be EQ {g (Xt )|Ft } for
0 ≤ t ≤ T.
The approach of Jacod and myself:
• Begin with the risky asset price process and a (possibly
random) savings rate;
• Next assume there are market given price processes for a large
number of (European style) derivatives of the form g (XT ),
where T varies;
• Find the collection Q of risk neutral measures that make both
the price process and all of the derivative price processes local
martingales;
• If there are enough derivative prices, the cardinality of Q
might be one;
• Alternatively the cardinality of Q might be zero or ∞
• compatibility issues are serious here.
Other Approaches; Some key ones follow:
• H. Dengler and R. Jarrow (1997): A simple jump model
where the price process is pure jump, with two distinct jump
sizes; two call options are used to complete the market.
• B. Dupire (1997): A simple stochastic volatility model, where
the stock price varies but the expiration time T is fixed. A
PDE is obtained, and when (and if) solved, it gives a unique
martingale measure;
• E. Derman and I. Kani (1997): the stochastic volatility
case, where the strike price K varies and T is fixed.
Smoothness in K is assumed, and by twice differentiating
obtain a density for the call option, leading to a PDE, which
when (and if) solved leads to a martingale measure choice.
• M. Schweizer and J. Wissel (2006), to appear in Math
Finance: improved on previous ideas by using different
maturities, creating a term structure of volatilities, within a
Brownian based stochastic volatility framework. They find
conditions for equivalent martingale measures to exist, and
also a condition for it to be unique.
How Do We Let the Market Choose the Risk
Neutral Pricing Measure?
• Begin with the risky asset price process S and a (possibly
random) savings rate;
• Next assume there are market-given price processes for a large
number of (European style) derivatives of the form g (ST ),
where T varies;
• Find the collection Q of risk neutral measures that make both
the price process and all of the derivative price processes local
martingales
• If there are enough derivative prices, the cardinality of Q
might be one;
• Alternatively the cardinality of Q might be zero or ∞
(compatibility issues are serious here).
Regime Change
• Recall that we have an infinite number of possible risk neutral
measures
• Let us assume that market has chosen a unique one
determined by (an infinite number of) option prices
• A fortiori the option prices must be internally consistent as
well; else we would have no risk neutral measure matching,
due to arbitrage opportunities (Delbaen-Schachermayer)
• Given the nature of the market over time, it is unreasonable to
assume that it stays with that risk neutral measure for all time
• When the market changes from one risk neutral choice to a
different one, we call it a regime change
• One can construct regime changes via stopping times
• Very recently Biagini, F¨
ollmer, and Nedelcu have proposed
a regime change via a continuous flow of measures
• This gives an elegant way for a bubble to be born gradually
• Finally, one can also simply change the coefficients of the
SDE governing the price process, either at a random time, or
via a continuous flow
The Fundamental Price for Incomplete Markets
• What follows is based on work with Robert Jarrow and
Kazuhiro Shimbo
• Recall that for the case of complete markets with a finite
horizon T , with risk neutral measure Q, and for t < T the
fundamental price of the risky asset is defined to be:
Z
St? = EQ {
t
T
dDu + XT |Ft }
• In incomplete markets, if one Q is chosen by the market for all
time (a “static market”), the definition is analogous.
• In an incomplete market with an infinite horizon, we assume
there exists a countable sequence of stopping times
0 = T0 < T1 < T2 < . . . increasing to ∞ a.s. which
represents change times from one risk neutral probability to
another
• The stochastic interval [Ti , Ti+1 ) consists of the i th regime
• In an incomplete market with an infinite horizon and regime
change, the fundamental price of the risky asset with end
time τ for the asset, t < τ , and for regime i at time t, is
defined to be:
Z τ
St? = EQ i {
dDu + Xτ 1{τ <∞} |Ft }
t
where Q i is the risk neutral measure chosen by the market
• Note that Xτ 1{τ =∞ } is not included
The Evaluation Measure
• We can piece all of these measures Q i together to get one
measure Q ?
• Q ? need not be a risk neutral measure; (if it were, then in
effect we would not have regime change)
• Some people find Q ? not being a risk neutral measure,
although it is equivalent, to be troubling
• We call Q ? the evaluation measure, and write it Q t? to
denote that it changes with the time t
• The Fundamental Price can be written compactly:
St? =
X
Z
1[Ti ,Ti+1 ) (t)EQ i {
τ
dDu + Xτ 1{τ <∞} |Ft }
t
i
or using Q ? = Q t?
Z
St? = EQ t? {
t
τ
dDu + Xτ 1{τ <∞} |Ft }
End of Lecture 3
Thank you for your
attention
How to Model a Financial Bubble Mathematically
Lecture 4
Philip Protter, Columbia University
Statistical Methods and Applications in Insurance and
Finance
CIMPA School
April 13, 2013
• At the end of Lecture 3 we explained how the market chooses
one of the infinite number of risk neutral measures
• The observer can tell this by observing an infinite number of
option prices
• The Fundamental Price can be written compactly:
St? =
X
Z
1[Ti ,Ti+1 ) (t)EQ i {
τ
dDu + Xτ 1{τ <∞} |Ft }
t
i
or using Q ? = Q t?
St?
= E
Q t?
Z
{
t
τ
dDu + Xτ 1{τ <∞} |Ft }
• In incomplete markets, if one Q is chosen by the market for all
time (a “static market”), the definition is analogous.
• In an incomplete market with an infinite horizon, we assume
there exists a countable sequence of stopping times
0 = T0 < T1 < T2 < . . . increasing to ∞ a.s. which
represents change times from one risk neutral probability to
another
• The stochastic interval [Ti , Ti+1 ) consists of the i th regime
• In an incomplete market with an infinite horizon and regime
change, the fundamental price of the risky asset with end
time τ for the asset, t < τ , and for regime i at time t, is
defined to be:
Z τ
St? = EQ i {
dDu + Xτ 1{τ <∞} |Ft }
t
where Q i is the risk neutral measure chosen by the market
• Note that Xτ 1{τ =∞ } is not included
The Evaluation Measure
• We can piece all of these measures Q i together to get one
measure Q ?
• Q ? need not be a risk neutral measure; (if it were, then in
effect we would not have regime change)
• Some people find Q ? not being a risk neutral measure,
although it is equivalent, to be troubling
• We call Q ? the evaluation measure, and write it Q t? to
denote that it changes with the time t
• The Fundamental Price can be written compactly:
St? =
X
Z
1[Ti ,Ti+1 ) (t)EQ i {
τ
dDu + Xτ 1{τ <∞} |Ft }
t
i
or using Q ? = Q t?
Z
St? = EQ t? {
t
τ
dDu + Xτ 1{τ <∞} |Ft }
Bubble Birth
• Recall the definition of a bubble:
Defintion[Bubble]: A bubble in a static market for an asset
with price process S is defined to be:
βt = St − St? ≥ 0,
t≥0
• A bubble in a dynamic market for t < τ in regime i is:
Z
βt = St − EQ t? {
τ
dDu + Xτ 1{τ <∞} |Ft } ≥ 0
t
• Since we are in regime i, we have in this case Q t? = Q i .
• If there are no bubbles for regime i − 1, a change to a new
risk neutral measure at change time Ti might in effect create
a bubble; if it does, we call this bubble birth
Derivatives
• Recall we touched on European Calls and Puts in Lecture 3
• For a European call with payoff (ST − K )+ at time T , we
denote its market price at time t < T as Ct (K )
• Analogously for a put with payoff (K − ST )+ at time T , its
time t market price is Pt (K )
• A forward with payoff ST − K has market price at time t
denoted Vt (K )
• Their fundamental prices are given by
Ct? (K ) = EQ t? {(ST − K )+ }
Pt? (K ) = EQ t? {(K − ST )+ }
Vt? (K ) = EQ t? {(ST − K )}
• Theorem [Put-Call Parity]: We have put-call parity for
fundamental prices
Ct? (K ) − Pt? (K ) = Vt? (K )
• The proof follows from the linearity of the conditional
expectation with respect to Q t?
• Theorem: Pt (K ) = Pt? (K )
• The proof follows from the fact that it is bounded, plus the
hypothesis of No Dominance
• In contrast, if there is a bubble in the risky asset price S, then
it is captured by the call option market price process Ct (K )
• Theorem: for K > 0 we have
Ct (K ) − Ct? (K ) = St − EQ t? {ST |Ft }
• The proof of this theorem follows from the observation
regarding bubbles of forwards:
Vt? (K ) = EQ t? {ST − K |Ft }
= EQ t? {ST |Ft } − K
= St? − K ≤ St − K = Vtf (K )
which means that a forward has a type 3 bubble of size
δt3 = Vtf (K ) − Vtf ? (K ) = (St − K ) − (St? − K )
= St − St?
• Put-call parity and the fact that Pt (K ) = Pt? (K ) imply
Ct (K ) − Ct? (K ) = St − EQ t? {ST |Ft }
• Note that with calls we are dealing with Type 3 bubbles, since
they exist on the compact time interval [0, T ]
Testing for Bubbles
• Recall that we have put-call parity even in the presence of
bubbles
• This is in contrast to the views expressed in the literature
• For example, Battalio and Schultz find no violations of put
call parity during the internet stock price bubble, and argue
that this evidence is inconsistent with an internet stock price
bubble
• However a “correct” model for the price operator EQ ( ·| Ft ),
when applied to both call and put options on the same spot
commodity, would give differential results in the presence of
price bubbles
• Puts would be priced correctly, but calls would not. If type 3
bubbles exist, this difference would be observable
• Also, the mispricings would be independent of the moneyness
of the options, but dependent on the time to maturity
Existence
•
dXt
= σ(Xt )dWt + b(t, Xt , Yt )dt
dYt
= g (Yt )dBt + f (Yt )dt
• W and B are iid Brownian motions
• The process X is a strict local martingale under a risk
neutral measure Q if and only if
Z ∞
x
dx < ∞, any > 0
2
σ (x)
• This model is incomplete and satisfies No Dominance
Quotes
• Federal Reserve Chairman Ben Bernanke:
“It is extraordinarily difficult in real time to know if an asset
price is appropriate or not.” Congressional Testimony,
December, 2009
• New York Fed President William Dudley: “What I am
proposing is that we try to identify bubbles in real time, try to
develop tools to address those bubbles, try to use those tools
when appropriate to limit the size of those bubbles and,
therefore, try to limit the damage when those bubbles burst.”
Interview with Planet Money, April 9, 2010
Detecting Bubbles - Joint Hypo with {S}
• Specify a stochastic process
dSs = σ(Ss )dZs + µ(Ss , Ys )ds;
S0 = x
where Z a standard BM, and Y is a diffusion driven by an
independent Brownian motion
• Under a risk neutral measure Q, this simplifies to
Z t
St
S0
=
+
σ(Ss )dZs .
Bt
B0
0
• The process S is a strict local martingale if and only if
Z
∞
x
dx < ∞.
σ(x)2
• Empirically test this integral, which we can do under the
original probability measure P
• Can independently test for fit of asset price evolution.
• Results: Asset price bubbles existed during the Tech bubble.
Bubble Detection Procedure
• We assume the price process follows an SDE of the form
dXt = σ(Xt )dWt + b(t, Xt , Yt )dt
(1)
• First we estimate σ from available historical data, using
estimators from the literature, slightly adapted for our
purposes
• If the function σ exhibits some regularity of behavior, we
attempt to extrapolate an extension of its behavior beyond
the range of values assumed by X
• If we assume σ is a parameterized family of functions, then we
need only estimate the parameter, and this is routine
• Otherwise, if we can show it is contained by a parameterized
family via comparison theorems, we can extend it as well
A simple model
• Let S satisfy a stochastic differential equation of the form:
dSt
= σ ? (St )dBt + µ(St , Yt )dt;
St
S0 > 0
where Y can be (for example) an independent diffusion.
• Our goal is to determine the function σ ? (x) from frequent
observations of St as time evolves
Natural Limitations of the Problem
• By the assumption of the structure of S, we have St > 0 for
all t ≥ 0
• We assume that σ ? and µ are such that a unique solution
exists, and without explosions; for example, xσ ? (x) and
xµ(x, y ) are Lipshitz continuous
• Due to the lack of explosions, on a finite time interval [0, T ]
the process S is bounded, ω by ω, so we can know xσ ? (x)
only on a compact interval of R+
• We consider only the one dimensional problem; multiple
dimensions are much harder, as is the case of stochastic
volatility
• The definitive paper on this subject is doe to Jean Jacod:
Non-parametric kernel estimation of the coefficient of a
diffusion, Scandinavian J. Statist., 27 (2000), pp. 83-96.
A Key Tool: The Space Time Formula
• Let X be a semimartingale with a local time process Lat for
the level a; then
[X , X ]ct
∞
Z
=
−∞
Lat da
• More generally,
Z
t
g (Xs− )d[X , X ]cs
Z
∞
=
0
−∞
Lat g (a)da
for any bounded, Borel function g
• In the case of our SDE, S = X and d[S, S]ct = St2 σ ? (St )2 dt
• Let us simplify notation by assuming σ(x) = xσ ? (x); then
d[S, S]ct = σ 2 (St )dt
The Idea of Florens-Zmirou
• In this Brownian paradigm,
LxT
1
= lim
→0 2
T
Z
0
1{|Ss −x|<} ds
• The actual local time of interest is given by
`xT
1
= lim
→0 2
Z
0
T
1{|Ss −x|<} d[S, S]s
• Since [S, S]s = σ 2 (Ss )ds, we have `xT = σ 2 (x)LxT , whence
`xT
= σ 2 (x)
LxT
and we can create an estimator for σ 2 (x)
The Estimator of Florens-Zmirou
• We consider two sets of sums:
LnT (x) =
n
T X
1{|Sti −x|<hn }
2nhn
i=1
n
T X
n
1{|Sti −x|<hn } n(Sti+1 − Sti )2
`T (x) =
2nhn
i=1
where hn is a sequence of positive real numbers converging to
0 and satisfying some constraints.
• This allows us to construct an estimator of σ(x) given by:
Pn
Sn (x) =
i=1 1{|Sti −x|<hn } n(Sti+1 −
Pn
i=1 1{|Sti −x|<hn }
Sti )2
(2)
Consistency of the Estimator of Florens-Zmirou
• Florens-Zmriou proved the following SLLN type theorem:
Theorem
If σ is bounded above and below from zero, has three continuous
and bounded derivatives, and if (hn )n≥1 satisfies nhn → ∞ and
nhn4 → 0 then Sn (x) is a consistent estimator of σ 2 (x).
• The proof of this theorem is based on the expansion of the
transition density. The choice of a sequence hn converging to
0 and satisfying nhn → ∞ and nhn4 → 0 allows one to show
that LnT (x) and `nT (x) converge in L2 (dQ) to LT (x) and
σ 2 (x)LT (x), respectively. Hence Sn (x) is a consistent
estimator of σ 2 (x), for any x that has been visited by the
diffusion.
Confidence Intervals of the Estimator of
Florens-Zmirou
• Florens-Zmirou also proved another CLT type theorem, useful
to obtain confidence intervals for the estimator Sn (x) of σ(x).
Theorem
p
If moreover nhn3 → 0 then Nxn ( σSn2 (x)
− 1) converges in
(x)
√
distributionPto 2Z where Z is a standard normal random variable
and Nxn = ni=1 1{|Sti −x|<hn } .
• The requirement that nhn4 → 0 requires impractical amounts
of data
• By introducing a smoothing kernel, we can relax the condition
to
nhn2 → ∞
The Smooth Kernel Approach
• The idea underlying the smooth kernel estimator is to replace
the kernel K (x) = 12 1{|x|≤1} by a smooth kernel φ, which is a
6
C
R positive function with compact support and such that
R+ φ = 1
• We are interested in some kind of convergence of the
following quantities:
Vnx
=
n−1
1 X S ni − x
)n(S i+1 − S i )2
φ(
n
n
nhn
hn
(3)
n−1
1 X S ni − x
φ(
)
nhn
hn
(4)
i=0
Lxn =
i=0
to σ 2 (x)Lx and Lx respectively, where hn satisfies nhn2 → ∞.
The Theorem
• After massive calculations and estimates, we get:
Theorem
x
If nhn2 → ∞ then Snx = VLnx converges in probability to σ 2 (x) and
n
provides a consistent estimator of σ 2 (x).
• In the work of Jacod of 2000, he is able to take hn = √1n and
moreover obtains a rate of convergence and an associated
Central Limit Theorem, as well as treating the
multidimensional case.
Relaxing the restriction that σ > 0
• Let F = (Ft )t≥0 be the underlying Brownian filtration.
• Let τp be the first exit of S from the interval ( p1 , p]. We use
the theory of the enlargement of filtrations
• Let Gt = σ(Ss ; s ≤ t) ∨ σ(τp )
• Or more generally, Gt? = σ(Ss ; s ≤ t) ∨ U, where U = 1{τp >T }
• If we work on the space (Ω, G ? , Q, G? ), where Q is the
conditional probability that σ(S) stays in the interval ( p1 , p],
then we can obtain our previous results for Q; all works
because this special event is in the filtration.
• We piece the convergence in probability together. This works
if, in practice, S never attains the value 0
Going Beyond the Visible Range
• Suppose we can estimate σ(x) with reasonable accuracy for
all x in some compact interval [a, b] of R+ , as we now know
that we can
• What does σ(x) look like outside of [a, b]?
• This is, of course, intrinsically unknowable
• But we need to know it for (for example) a theory of bubble
detection
• In particular, we would like to know the asymptotic properties
of σ(x) as x % ∞
• What to do?
First Method: Parametric Estimation
• This approach uses the classic paper of V. Genon-Catalot and
J. Jacod, AIHP, 1993
• The idea is to consider a parameterized family of coefficients
σ(α, x) and then once knows α, one knows the function
σ(α, x) for all x ∈ R+
• It is too much to expect σ to be in such a family, but it might
be “squeezed” in between two members of the parameterized
family
• A measure of asymptotic growth of interest in the theory of
bubbles is whether or not
Z ∞
x
dx < ∞
2
σ(x)
• If the above integral is finite, then there is a bubble
• Basically, σ(x) has to increase asymptotically faster than
f (x) = x for the integral to be finite
The Comparison Theorem
•
Theorem (Comparison Theorem)
Assume that dSt = σ(t, St )dWt and that there exist two functions
Σ and σ such that : for all t and x, σ(x) ≤ σ(t, x) ≤ Σ(x) and
such that σ, Σ and σ are continuous, locally H¨
older continuous
with exponent 12 , then:
R∞
R∞ x
1. if for all c > 0, c Σ2x(x) dx = ∞ then σ(x)
2 dx = ∞
R∞ x
2. if there exists c > 0 such that c σ2 (x) dx < ∞ then
R∞ x
dx < ∞
σ(x)2
Bubble Birth and Regime Change
• One way we can have bubble birth is to incorporate the ideas
of regime change
• At a sequence of stopping times, the function σ in the
volatility of equation (1) changes to a different σ
• Another possibility for regime change is for the risk neutral
measure Q to change to another one
• Biagini, F¨
ollmer, and Nedelcu have a new result that allows
the risk neutral measures to change via a continuous flow of
equivalent measures
• This gives a continuous change to risk neutral measure, giving
an elegant method for bubble birth
A Comparison Theorem
Comparison Theorem:
Assume that dSt = σ(t, St )dWt and that there exist two functions
Σ and σ such that : ∀t, ∀x, σ(x) ≤ σ(t, x) ≤ Σ(x) and such that
σ, Σ and σ are continuous, locally Holder continuous with
exponent 12 , then:
R∞
1. If ∀c > 0, c Σ2x(x) dx = ∞ then S is a martingale.
R∞
2. If ∃c > 0, c σ2x(x) dx < ∞ then S is a strict local martingale.
Another approach is that of Reproducing Kernel Hilbert Spaces
(RKHS)
Estimation of σ
Method 1: Parametric Estimation of σ
• We choose a class of volatility functions large enough to
include many of the forms used in practice
• An example: power functions σ(x) = σ
¯ x α , where σ
¯ and α are
the unknown parameters to be estimated
• Next we use non-parametric estimators for the function σ
• If the parametric estimators and the non-parametric
estimators are similar, then we know the behavior for large x
through the parametric continuation, and we can test for
convergence or divergence
• If they are not comparable, then this method fails
Estimation of σ, Continued
Method 2: Reproducing Kernel Hilbert Spaces [RKHS]
• Step 1: We first interpolate an estimate of σ within the
bounded interval where we have observations
• Consequence: We lose the irregularities of non parametric
estimators
• We then choose an RKHS family corresponding to this
interpolated σ so as to optimize how close we are to σ on the
bounded interval. We give a criterion for such an optimization.
• We do this by choosing a certain extrapolating RKHS, which
once again determines the tail behavior of σ
• More precisely: Let (Hm )m∈N denote our family of RKHS
spaces; Given any choice of m, call it m0 , we can interpolate
perfectly
• If we stop here, this method is just as arbitrary as parametric
estimation and continuation
Estimation of σ, Continued
• Instead we use the data a second time to choose m in a
reasonable way
• Thus this method is sort of a two step bootstrap method
• Define the Sobolev space:
H n (D) = u ∈ L2 (D)st.∀k ∈ [1, n], u (k) ∈ L2 (D) where u (k)
is the weak derivative of u.
• The norm that is usually chosen is
R
||u||2 =
Pn
k=0 D (u
(k) )2 (x)dx.
• Due to Sobolev inequalities, an equivalent and more
R
R
appropriate norm is ||u|| = D u 2 (x)dx +
where τ keeps things away from 0
1
τ 2n
D (u
(n) )2 (x)dx,
• We denote by K the kernel function of H n (]a, b[), where in
this case D =]a, b[.
• We then can establish the form of K , which is simple but
messy to state
Estimation of σ, Continued
• We next must decide if an extrapolation is required: If the
natural extension of the estimated volatility σ shows the
integral in question does not diverge as x → ∞ and remains
bounded on R+ , no extrapolation is required, the process is a
martingale
• If however it appears that the σ(x) → ∞ as x → ∞, then
more work is required
• Extrapolate σ using our RKHS spaces and choose the one
that minimizes the distance, using a weighted norm, away
from σ
ˆ on the bounded interval where observations exist.
• In this way we obtain a function that interpolates all the
available data points, but also remains as close as possible to
the interpolated function on the observation interval
Does this work with data?
We test this method with data from the Dot Com Bubble
Era (2000-2002): Recall we need to estimate σ and determine
the convergence or divergence of the integral
Z ∞
x
dx, any > 0
σ(x)2
LastMinute.com Historical open/close data
LastMinute.com Estimates of σ using parametric methods
There is a bubble
Infospace.com Historical open/close data
Infospace.com Estimates of σ
Infospace.com Estimates of σ using RKHS
There is a bubble
TheGlobe.com Historical open/close data
TheGlobe.com Estimates of σ
There is a bubble
Geocities Historical open/close data
Geocities Estimates of σ
There is NOT a bubble
eToys.com Historical open/close data
eToys.com Estimates of σ using parametric methods
The Test is inconclusive
The Recent Case of LinkedIn Stock
• LinkedIn shares jumped 109.4 percent on their first day of
trade
• Chicago Fed President Charles Evans said he was withholding
judgment over whether a new dot-com bubble was under way
• “I have no way of knowing that those aren’t just exactly the
right valuations,” Evans told reporters
• We used data from the IPO on May 19 until May 24 to show
there is a bubble
Tick Data of LinkedIn
RKHS best estimate of σ
The Recent Case of Gold Prices
• There was a lot of speculation recently that gold prices were
in a bubble
• An example was the Dealbook article of August 30, 2011, by
S.M. Davidoff, “How to Deflate a Gold Bubble (That Might
Not Even Exist).”
Gold Spot Prices
RKHS best estimate of σ
There is Not a Bubble
Implications for Risk Management
• Bubbles distort economic decisions of financial institutions,
regulators, investment firms.
• The tools for risk management - financial derivatives - are not
priced nor hedged in the standard text book fashion.
• Organizations that use the standard textbook formulations
will mismanage risk.
• Regulatory perspective - the determination of economic
capital will be too low, because in the presence of bubbles, the
drift on the relevant asset price evolutions are too large (too
optimistic).
• Investment decisions of pension funds and institutions will
also be distorted because risky asset prices will look more
attractive than they should.
Thank you for your
Attention
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