### x(t)

```Statistical modeling, financial data analysis and applications
Venice, 11-14 september 2013
Path integrals for option pricing
Theory of
Quantum and
Complex systems
J. Tempere, S.N. Klimin, J.T. Devreese, TQC, Universiteit Antwerpen
January 2013
From left to right: dr. Kai Ji, Maarten Baeten, dr. Serghei Klimin, Stijn Ceuppens, Dries Sels, prof. Jacques Tempere, Ben Anthonis,
prof. Michiel Wouters, dr. Jeroen Devreese, Enya Vermeyen, Giovanni Lombardi, Selma Koghee, dr. Onur Umucalilar, Nick Van den Broeck.
Not shown: prof.em. Jozef Devreese, prof.em. Fons Brosens, dr. Vladimir Gladilin, dr. Wim Casteels
Financial support by the Fund for Scientific Research-Flanders
Part I: path integrals in quantum mechanics
Introduction: quantum mechanics with path integrals
1
B
A
2
Introduction: quantum mechanics with path integrals
x
B
A
Introduction: quantum mechanics with path integrals
x
B
A
t
Introduction: quantum mechanics with path integrals
x
is called the path integral propagator
The amplitude corresponding to
a given path x(t) is
x(t)
B
A
Here, S is the action functional:
With L the Lagrangian, eg.
t
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed.
(World Scientific, Singapore, 2009)
Part II: Path integrals in finance: option pricing
Options: the right to buy/sell in the future at a fixed price
Main question in option pricing: how much is this ‘right to buy in the future’ worth?
sT=730
K=630
s0=610
sT=530
Expected payoff:
p100+(1p)0
One year
from
now, you
will steel
require
100year
tonne
how do ityou
deal with
Rather
than
a contract
to get
in one
at of
630steel;
EUR/tonne,
is better
to obtain
the
right, price
not the
obligation to buy steel at 630 EUR/tonne one year from now.
possible
fluctuations?
This
is called
an ‘option
contract’.
All types exist, eg.also the right to sell, with different
1. Decide
a price
now, say
630 EUR/tonne.
times, and for different underlying assets.
The ‘standard model’ of option pricing: Black-Scholes
We work with the logreturn
and model the changes in the value
of the underlying asset as a Brownian random walk
time
x
From the payoff at the final
position, work a step back to
obtain a differential equation
Binomial tree method: see eg. Options, Futures, and Other Derivatives by John C. Hull (Prentice Hall publ.)
The ‘path-centered’ point of view on Black-Scholes
We work with the logreturn
and model the changes in the value
of the underlying asset as a Brownian random walk
time
x
Take the sum over all paths: this
is the price propagator from t=0
to t=T.
(note: in this example there
paths…)
etc.
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many
The probability to end up in xT is also given by the sum over all paths that
small
x ‘binomial’ steps
endup
there, weighed
by the
of outcome
these paths
according
to the central
limitprobability
theorem, the
is Gaussian fluctuations
The ‘path-centered’ point of view on Black-Scholes
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many
Thework
probability
to
end up in xT is also givenand
by model
the sum
all paths
that
We
with the
logreturn
theover
changes
in the value
small
steps
x ‘binomial’
end
there, weighed
the
of outcome
these paths
of
the
underlying
as aby
Brownian
randomthe
walk
up
according
toasset
the central
limitprobability
theorem,
is Gaussian fluctuations
time
x
Quantum
x(t)
x0
xT
this Feynman path integral
determines the propagator
The amplitude for a given path is a phase factor :
where S is the action functional
,
fixed by integrating the Lagrangian along the path, eg. for a free particle:
The ‘path-centered’ point of view on Black-Scholes
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many
Thework
probability
to
end up in xT is also givenand
by model
the sum
all paths
that
We
with the
logreturn
theover
changes
in the value
small
steps
x ‘binomial’
end
there, weighed
the
of outcome
these paths
of
the
underlying
as aby
Brownian
randomthe
walk
up
according
toasset
the central
limitprobability
theorem,
is Gaussian fluctuations
time
x
Stochastic
x(t)
x0
xT
this Wiener path integral
determines the propagator
The amplitude for a given path is a phase factor :
where S is the action functional
,
fixed by integrating the Lagrangian along the path, eg. for a free particle:
The ‘path-centered’ point of view on Black-Scholes
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many
Thework
probability
to
end up in xT is also givenand
by model
the sum
all paths
that
We
with the
logreturn
theover
changes
in the value
small
steps
x ‘binomial’
end
there, weighed
the
of outcome
these paths
of
the
underlying
as aby
Brownian
randomthe
walk
up
according
toasset
the central
limitprobability
theorem,
is Gaussian fluctuations
time
x
Stochastic
x(t)
x0
xT
this Wiener path integral
determines the propagator
The probability for a given path is a real number:
where S is the action functional
,
fixed by integrating the Lagrangian along the path, eg. for a free particle:
The ‘path-centered’ point of view on Black-Scholes
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many
Thework
probability
to
end up in xT is also givenand
by model
the sum
all paths
that
We
with the
logreturn
theover
changes
in the value
small
steps
x ‘binomial’
end
there, weighed
the
of outcome
these paths
of
the
underlying
as aby
Brownian
randomthe
walk
up
according
toasset
the central
limitprobability
theorem,
is Gaussian fluctuations
time
x
Stochastic
x(t)
x0
xT
this Wiener path integral
determines the propagator
The probability for a given path is a real number:
where S is the action functional
,
fixed by integrating the Lagrangian along the path, eg. for the BS model:
The ‘path-centered’ point of view on Black-Scholes
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many
Thework
probability
to
end up in xT is also givenand
by model
the sum
all paths
that
We
with the
logreturn
theover
changes
in the value
small
steps
x ‘binomial’
end
there, weighed
the
of outcome
these paths
of
the
underlying
as aby
Brownian
randomthe
walk
up
according
toasset
the central
limitprobability
theorem,
is Gaussian fluctuations
Black-Scholes: The Galton board
this Wiener path integral
determines the propagator
The probability for a given path is a real number:
where S is the action functional
,
fixed by integrating the Lagrangian along the path, eg. for the BS model:
The application of path integrals to option prices in BS has been pioneered by various authors: Dash, Linetsky, Rosa-Clot, Kleinert.
The ‘path-centered’ point of view on Black-Scholes
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many
Thework
probability
to
end up in xT is also givenand
by model
the sum
all paths
that
We
with the
logreturn
theover
changes
in the value
small
steps
x ‘binomial’
end
there, weighed
the
of outcome
these paths
of
the
underlying
as aby
Brownian
randomthe
walk
up
according
toasset
the central
limitprobability
theorem,
is Gaussian fluctuations
Fisher Black & Myron Scholes, "The Pricing of Options and Corporate Liabilities". Journal of Political Economy
81 (3): 637–654 (1973).
Robert C. Merton, "Theory of Rational Option Pricing“, Bell Journal of Economics and Management Science
(The RAND Corporation) 4 (1): 141–183 (1973).
the path integral point of view
Two problems with the standard model
Free particle propagator  Black-Scholes option price
Problem 1: The fluctuations are not Gaussian
Black-Scholes-Merton model
Δx
Δx
Problem 2: Not all options have a payoff that depends only on x(t=T), many options have a path-dependent
payoff, i.e. payoff is a functional of x(t).
Two problems with the standard model
Free particle propagator  Black-Scholes option price
Problem 1: The fluctuations are not Gaussian
Problem 2: Not all options have a payoff that depends only on x(t=T), many options have a path-dependent
payoff, i.e. payoff is a functional of x(t).
x
xA
Asian option:
payoff is a function of the average of
the underlying price?
xB
t
Improving Black-Scholes : stochastic volatility
Black-Scholes-Merton
model
vt
xt
Δx
Δx

Heston model
Δx
t
t
t
two particle problem
with z = (v/)1/2
The Heston model treats the variance as a second stochastic variable, satisfying its own
stochastic differential equation:
mean reversion rate
mean reversion
level
the ‘volatility of the volatility’
t
Improving Black-Scholes : stochastic volatility
vt
xt
Δx

t
t
two particle problem
with z = (v/)1/2
From the infinitesimal propagator of the stochastic process we identify the following
Lagrangian that corresponds to the same propagator:
Improving Black-Scholes : stochastic volatility
Black-Scholes-Merton
model
vt
xt
Δx
Δx

Heston model
Δx
t
t
two particle problem :
free particle strangely coupled to a radial harmonic oscillator
with z = (v/)1/2
t
t
Improving Black-Scholes : stochastic volatility
Black-Scholes-Merton
model
vt
xt
Δx
Δx

Heston model
Δx
t
t
two particle problem :
free particle strangely coupled to a radial harmonic oscillator
with z = (v/)1/2
t
t
Improving Black-Scholes : stochastic volatility
Black-Scholes-Merton
model
vt
xt
Δx
Δx
Heston model
Δx

t
t
two particle problem :
free particle strangely coupled to a radial harmonic oscillator
with z = (v/)1/2
Details: D.Lemmens, M. Wouters, JT, S. Foulon, Phys. Rev. E 78, 016101 (2008).
t
t
Other improvements to Black-Scholes
BS
A) Stochastic Volatility
* Heston model:
* Hull-White model
Heston
* Exponential Vasicek model
B) Jump Diffusion (and Levy models)
again a zoo of proposals
poisson process
H. Kleinert , Option Pricing from Path Integral for Non-Gaussian Fluctuations.
Natural Martingale and Application to Truncated Lévy Distributions ,
Physica A 312, 217 (2002).
Kou
Other models and other tricks
Improvements to Black-Scholes
...translate into
physical actions
...to which quantum mechanics
solving techniques can be applied
A) Stochastic Volatility
1. Heston model
free particle coupled to
exact solution
2. Exponential Vasicek model
particle in an exponential
gauge field generated by
free particle
perturbational
(Nozieres – Schmitt-Rink
expansion)
3. Kou and Merton’s models
particle in complicated
potential (not previously
studied)
variational (Jensen-Feynman
variational principle)
B) Stochastic volatiltiy + Jump Diffusion
References
1. D. Lemmens, M. Wouters, J. Tempere, S. Foulon, Phys. Rev. E 78 (2008) 016101.
2. L. Z. Liang, D. Lemmens, J. Tempere, European Physical Journal B 75 (2010) 335–342.
3. D. Lemmens, L. Z. J. Liang, J. Tempere, A. D. Schepper, Physica A 389 (2010) 5193 – 5207.
More complicated payoffs
‘Plain vanilla’ or simple options have a payoff that only depends on the value of the underlying
at expiration, x(t=T). For such options we have:
Many other option contracts have a payoff that depends on the entire path, such as:
 Asian options: payoff depends on the average price during the option lifetime
 Timer options: contract duration depends on a volatility budget
 Barrier options: contract becomes void if price goes above/below some value
For such options, the price is given by
Feynman-Kac ‘interpretation’  include payoff in the path weight:
Part IV: A concrete and recent example:
Timer options
Timer options have an uncertain expiry time,
equal to the time at which a certain “variance
budget” has been used up.
The Duru-Kleinert transformation
‘clock’ time is a functional
of the path followed:
x
t
x(t)
x0
xT
with
and F>0
final time depends on path
q
q()

qB
qA
H. Duru and H. Kleinert, Solution of the Path Integral for the H-Atom, Phys. Letters B 84, 185 (1979).
H. Duru and N. Unal, Phys. Rev. D 34, 959 (1986).
The Duru-Kleinert transformation
This transformation results in the equivalency between the following
two path integrals:
F(q) can be chosen to regularize a singular potential.
This technique was used to solve the propagator of
the electron in the hydrogen atom, transforming
the a 3D singular potential into a 4D harmonic
oscillator problem.
H. Duru and H. Kleinert, Solution of the Path Integral for the H-Atom, Phys. Letters B 84, 185 (1979).
H. Duru and N. Unal, Phys. Rev. D 34, 959 (1986).
More complicated payoffs: timer option
Timer options have an uncertain expiry time, equal to the time at which a certain pre-specified
“variance budget” has been used up. Their description requires a stochastic volatility model:
The expiry time is determined by the variance budget B :
This now defines a
Duru-Kleinert
pseudotime !
L. Z. J. Liang, D. Lemmens, and J. Tempere, Physical Review E 83, 056112 (2011).
More complicated payoffs: timer option
Timer options have an uncertain expiry time, equal to the time at which a certain pre-specified
“variance budget” has been used up. Their description requires a stochastic volatility model:
The Duru-Kleinert transformation
has a well defined inverse
.
Denoting
and
we find that these obey new SDE’s:
now X and V evolve up to a fixed time, 
L. Z. J. Liang, D. Lemmens, and J. Tempere, Physical Review E 83, 056112 (2011).
More complicated payoffs: timer option
The 3/2 model:
results in a particle in a
Morse potential.
The Heston model
results here in particle in a
Kratzer potential
L. Z. J. Liang, D. Lemmens, and J. Tempere, Physical Review E 83, 056112 (2011).
Conclusions
Fluctuating paths in finance are described
by stochastic models, which can be
translated to Lagrangians for path
integration.
St
t
Path integrals can solve in a unifying
framework the two problems of the
‘standard model of option pricing’:
1/ The real fluctuations are not gaussian
2/ New types of option contracts have pathdependent payoffs
D. Lemmens, M. Wouters, JT, S. Foulon, Phys. Rev. E 78, 016101 (2008);
L. Z. J. Liang, D. Lemmens, JT, European Physical Journal B 75, 335–342 (2010);
D. Lemmens, L. Z. J. Liang, JT, A. D. Schepper, Physica A 389, 5193–5207 (2010);
J.P.A. Devreese, D. Lemmens, JT, Physica A 389, 780-788 (2010);
L. Z. J. Liang, D. Lemmens, JT, Physical Review E 83, 056112 (2011).
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