1. Path integrals for option pricing
Statistical modeling, financial data analysis and applications
Venice, september 2013
Theory of
Quantum and
Complex systems
Path integrals for option pricing
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

2. Part I: path integrals in quantum mechanics

3. Introduction: quantum mechanics with path integrals
Two alternatives: add the amplitudes 1 B A 2

4. Introduction: quantum mechanics with path integralsx
Many alternatives: add the amplitudes B A

5. Introduction: quantum mechanics with path integralsx
Many alternatives: add the amplitudes B A t

6. Introduction: quantum mechanics with path integrals
Many alternatives: add the amplitudes x
is called the path integral propagator
The amplitude corresponding to a given path x(t) is
Here, S is the action functional:
With L the Lagrangian, eg.
x(t) B A t
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed.
(World Scientific, Singapore, 2009)

7. Part II: Path integrals in finance: option pricing

8. 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?
One year from now, you will require 100 tonne of steel; how do you deal with possible price fluctuations?
1. Decide a price now, say 630 EUR/tonne.
sT=730
sT=530
K=630
s0=610 p
1-p
Expected payoff:
p100+(1p)0
Rather than a contract to get steel in one year at 630 EUR/tonne, it is better to obtain the right, not the obligation to buy steel at 630 EUR/tonne one year from now.
This is called an ‘option contract’. All types exist, eg.also the right to sell, with different times, and for different underlying assets.

9. 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 p p
1-p
1-p
Binomial tree method: see eg. Options, Futures, and Other Derivatives by John C. Hull (Prentice Hall publ.)

10. etc. The ‘path-centered’ point of view on Black-Scholes time x
We work with the logreturn and model the changes in the value
of the underlying asset as a Brownian random walk
time x
etc.
Take the sum over all paths: this is the price propagator from t=0 to t=T.
(note: in this example there would already be 212 possible paths…) p
1-p x
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many small ‘binomial’ steps
 according to the central limit theorem, the outcome is Gaussian fluctuations
The probability to end up in xT is also given by the sum over all paths that
end up there, weighed by the probability of these paths

11. The ‘path-centered’ point of view on Black-Scholesx
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many small ‘binomial’ steps
 according to the central limit theorem, the outcome is Gaussian fluctuations
The probability to end up in xT is also given by the sum over all paths that
end up there, weighed by the probability of these paths
We work with the logreturn and model the changes in the value
of the underlying asset as a Brownian random walk
time x
Quantum
Many alternatives: add the amplitudes
this Feynman path integral
determines the propagator
x(t) x0 xT
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:

12. The ‘path-centered’ point of view on Black-Scholesx
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many small ‘binomial’ steps
 according to the central limit theorem, the outcome is Gaussian fluctuations
The probability to end up in xT is also given by the sum over all paths that
end up there, weighed by the probability of these paths
We work with the logreturn and model the changes in the value
of the underlying asset as a Brownian random walk
time x
Stochastic
Many alternatives: add the probabilities
this Wiener path integral
determines the propagator
x(t) x0 xT
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:

13. The ‘path-centered’ point of view on Black-Scholesx
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many small ‘binomial’ steps
 according to the central limit theorem, the outcome is Gaussian fluctuations
The probability to end up in xT is also given by the sum over all paths that
end up there, weighed by the probability of these paths
We work with the logreturn and model the changes in the value
of the underlying asset as a Brownian random walk
time x
Stochastic
Many alternatives: add the probabilities
this Wiener path integral
determines the propagator
x(t) x0 xT
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:

14. The ‘path-centered’ point of view on Black-Scholesx
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many small ‘binomial’ steps
 according to the central limit theorem, the outcome is Gaussian fluctuations
The probability to end up in xT is also given by the sum over all paths that
end up there, weighed by the probability of these paths
We work with the logreturn and model the changes in the value
of the underlying asset as a Brownian random walk
time x
Stochastic
Many alternatives: add the probabilities
this Wiener path integral
determines the propagator
x(t) x0 xT
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:

15. The ‘path-centered’ point of view on Black-Scholesx
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many small ‘binomial’ steps
 according to the central limit theorem, the outcome is Gaussian fluctuations
The probability to end up in xT is also given by the sum over all paths that
end up there, weighed by the probability of these paths
We work with the logreturn and model the changes in the value
of the underlying asset as a Brownian random walk
Black-Scholes: The Galton board
Many alternatives: add the probabilities
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.

16. The ‘path-centered’ point of view on Black-Scholesx
Rather than 2 possible futures, there are many – but they can bee seen as a limit of many small ‘binomial’ steps
 according to the central limit theorem, the outcome is Gaussian fluctuations
The probability to end up in xT is also given by the sum over all paths that
end up there, weighed by the probability of these paths
We work with the logreturn and model the changes in the value
of the underlying asset as a Brownian random walk
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).

17. Part III: Some advantages of
the path integral point of view

18. Two problems with the standard model
Free particle propagator  Black-Scholes option price
Problem 1: The fluctuations are not Gaussian Δx
Black-Scholes-Merton model
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).

19. 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). xA xB x t
Asian option:
payoff is a function of the average of the underlying price?

20. Improving Black-Scholes : stochastic volatilityxt vt 
two particle problem
Black-Scholes-Merton model
Heston model Δx Δx Δx t t
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’

21. Improving Black-Scholes : stochastic volatilityxt vt 
two particle problem Δx
with z = (v/)1/2
From the infinitesimal propagator of the stochastic process we identify the following Lagrangian that corresponds to the same propagator:

22. Improving Black-Scholes : stochastic volatilityxt vt 
two particle problem :
free particle strangely coupled to a radial harmonic oscillator
Black-Scholes-Merton model
Heston model Δx Δx Δx t t
with z = (v/)1/2

23. Improving Black-Scholes : stochastic volatilityxt vt 
two particle problem :
free particle strangely coupled to a radial harmonic oscillator
Black-Scholes-Merton model
Heston model Δx Δx Δx t t
with z = (v/)1/2

24. Improving Black-Scholes : stochastic volatilityxt vt 
two particle problem :
free particle strangely coupled to a radial harmonic oscillator
Black-Scholes-Merton model
Heston model Δx Δx Δx t t
with z = (v/)1/2
Details: D.Lemmens, M. Wouters, JT, S. Foulon, Phys. Rev. E 78, (2008).

25. Other improvements to Black-ScholesBS
Add stoch vol Add jump diff.
Heston
Kou
A) Stochastic Volatility
* Heston model:
* Hull-White model
* 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).

26. Other models and other tricks
Improvements to Black-Scholes translate into to which quantum mechanics
physical actions solving techniques can be applied
A) Stochastic Volatility
1. Heston model free particle coupled to exact solution
radial harmonic oscillator
B) Stochastic volatiltiy + Jump Diffusion
2. Exponential Vasicek model particle in an exponential perturbational
gauge field generated by (Nozieres – Schmitt-Rink
free particle expansion)
3. Kou and Merton’s models particle in complicated variational (Jensen-Feynman
potential (not previously variational principle)
studied)
References
1. D. Lemmens, M. Wouters, J. Tempere, S. Foulon, Phys. Rev. E 78 (2008)
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.

27. 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:

28. 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.

29. The Duru-Kleinert transformation
‘clock’ time is a functional
of the path followed: x t
x(t) x0
with
and F>0 xT
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).

30. 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).

31. 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, (2011).

32. 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, (2011).

33. 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, (2011).

34. Conclusions
Fluctuating paths in finance are described by stochastic models, which can be translated to Lagrangians for path integration.
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 path-
dependent payoffs t St
D. Lemmens, M. Wouters, JT, S. Foulon, Phys. Rev. E 78, (2008); J.P.A. Devreese, D. Lemmens, JT, Physica A 389, (2010);
L. Z. J. Liang, D. Lemmens, JT, European Physical Journal B 75, 335–342 (2010); L. Z. J. Liang, D. Lemmens, JT, Physical Review E 83, (2011).
D. Lemmens, L. Z. J. Liang, JT, A. D. Schepper, Physica A 389, 5193–5207 (2010);