# Path integrals for option pricing

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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

Part I: path integrals in quantum mechanics

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

Introduction: quantum mechanics with path integrals
x Many alternatives: add the amplitudes B A

Introduction: quantum mechanics with path integrals
x Many alternatives: add the amplitudes B A t

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)

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? 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.

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.)

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

The ‘path-centered’ point of view on Black-Scholes
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 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:

The ‘path-centered’ point of view on Black-Scholes
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 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:

The ‘path-centered’ point of view on Black-Scholes
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 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:

The ‘path-centered’ point of view on Black-Scholes
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 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:

The ‘path-centered’ point of view on Black-Scholes
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 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.

The ‘path-centered’ point of view on Black-Scholes
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 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).

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 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).

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?

Improving Black-Scholes : stochastic volatility
xt 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’

Improving Black-Scholes : stochastic volatility
xt 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:

Improving Black-Scholes : stochastic volatility
xt 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

Improving Black-Scholes : stochastic volatility
xt 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

Improving Black-Scholes : stochastic volatility
xt 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).

Other improvements to Black-Scholes
BS 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).

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.

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 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).

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, (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, (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, (2011).

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);