On Option Pricing by Quantum Mechanics Approach Hiroshi Inoue School of Management, Tokyo University of Science Kuki, Saitama 346-851 Japan e-mail:
[email protected]
Abstract
a deterministic function of t , which is given by Newton’s law of motion. In We discuss the path integral contrast, in quantum mechanics the methodology of quantum mechanics particle’s evolution is random, analogous for option pricing and the relevant to the case of the evolution of a stock other materials. price having no-zero volatility. When we come to analyze option that depend on a path-dependent 2. Quantum Mechanics quantity, the straightforward BS approach is inadequate. We discuss Quantum mechanics was discovered some options that are only weakly in two different forms: wave mechanics path-dependent in that they can be and matrix mechanics. In 1900 Planck valued using only the current values introduced the quantum of action h of the underlying asset and time, and in 1905 Einstein postulated without the need for any variable to particles of light with energy represent the path-dependent E = hν ( ν :frequency). In 1924 de quantity so that they satisfy the BS Broglie put the two formulas 2 equation. E = mc (m : mass ) , E = hν together and invented matter waves. Then, the wave 1. Introduction nature of matter was experimentally confirmed by the Davisson-Germer The breakthrough in option pricing electron diffraction experiment in 1927 theory came with the famous Black and and theoretically supported by the work Scholes paper in 1973.They were the first to show that options could be priced of Schrӧdinger in 1926. Suppose we have a particle (say an by constructing a risk-free hedge by electron) of mass m in a potential dynamically managing a simple portfolio 3 consisting of the underlying asset and V where V is a real function on R cash. After that many directions of representing the potential energy. research relevant to financial Schrӧdinger attempted to describe the derivatives have been produced and motion of the electron by means of a quantity φ subject to a wave equation. contributed to different areas. On the other hand, quantum theory as His hypothesis is that a stationary state a part of physics has been developed vibrates according to the equation during various historical findings, and h2 Δφ + ( E − V )φ = 0 , option pricing has a mathematical 2m description, which corresponds to a where h is Planck’s constant h quantum system. In classical mechanics divided by 2 π , and E (with the the position of a particle at time t , xt is dimensions of energy) plays the role of
an eigen value.
be suitable to be used for option pricing.
3. Hamiltonian
4. Path-dependent Option
Hamiltonians are applied to the study of stock options and stochastic interest rate models, which are characterized by having finite numbers of degrees of freedom. The problem of the pricing of derivative securities is recast as a problem of quantum mechanics, and the Hamiltonians driving the prices of options are considered for stock prices with constant and stochastic volatility. Martingale condition required for risk-neutral evolution is re-expressed in terms of the Hamiltonian. Potential terms in the Hamiltonian are shown to represent a class of path-dependent options. By assuming of several conditions such as no-arbitrage, constant spot rate r, continuous rebalancing of the portfolio, no transaction costs and infinite divisibility of the stock, the Black-Scholes equation is expressed as
Use of a path integral formulation has some advantages since it is related to Lagrangian description of diffusion process so that it enables the use of quantum mechanical methods. In classical physics, time evolution of dynamical systems is governed by Least Action Principle. Equations of motion such as Newton’s equations can be viewed as Euler-Lagrange equations for a minimum of a certain action functional, a time integral of the Lagrangian function defining the dynamical system. Thus, their deterministic solutions, which are trajectories of the classical dynamical system, minimize the action functional that is the least action principle. In quantum physics, we can think about probabilities of different paths a quantum dynamical system can take. It is possible to define a measure on the set of all possible paths from the initial state x 0 to the final state xT of
∂f 1 2 2 ∂ 2 f ∂f + σ S + rS = rf 2 ∂t 2 ∂S ∂S
the quantum dynamical system so that The equation above can be transformed expected values of different quantities into a quantum mechanical version by a dependent on paths are given by path change of variable, S = e x with x a real integrals over all possible path from x 0 to xT . variable[2]. This yields
∂f = H BS f ∂t
where H BS is an Hamiltonian given by
1 ∂2 1 ∂ H BS = − σ 2 2 + ( σ 2 − r ) + r . 2 2 ∂x ∂x
Then,the BS Hamiltonian can be generalized to satisfy martingale condition,
1 ∂2 1 ∂ H V = − σ 2 2 + ( σ 2 − V ( x)) 2 2 ∂x ∂x (3.1) + V (x) where the potential V (x) is an arbitrary function of x . Note that a derivative evolving with H BS will yield a risk-free measure and
4-1. Classical Mechanics and Action Functional In classical physics, it is known that the product of acceleration of motion and mass of a particle equals the force the particle receives. The motion following Newton’s mechanics can be described in another way. By the principle of least action a particular path x (t ) out of all the possible paths can be determined. That is, there exists a certain quantity A(x ) which can be computed for each path. The path x (t ) is that for which A(x ) becomes a minimum. The quantity A(x ) is defined by
s
A( x) = ∫ L( x& , x, t )dt s
time s and reaches to x at time t . Then, the integral kernel is defined as
(4.1)
where L(⋅) is the Lagrangian for system. For a particle of mass moving in a potential V ( x, t ) , which function of position and time, Lagrangian is
the
m
K ( x , t y , s ) = C ∫ e ih
−1
A( r )
Ω
Dr
where Dr is the volume element is a integrating with respect to r over Ω the and C is a constant weight. This can be expressed by using Lagrange function m 2 instead of A( r ) L( x& , x, t ) = x& − V ( x, t ) （4.2)
2
t
The phase of the contribution from a K ( x, t y, s ) = e ∫s Dr (4.4) Ω given path is expressed as the action A(x ) for that path in units of the which is called Feynman path integral. quantum of action h . In other words, 4-2. Schrӧdinger Equation the probability P (b, a ) to go from a point
∫
ih −1 L ( u , r , r& ) du
x a at the time t a to the point x b at
Let L ( x, t , v ) be a Lagrange function t b is P (b, a ) = K (b, a ) of an amplitude of time t , position x and velocity v . Define 3 dimensional vector K ( a, b) to go from a to b . This p = ( p1 , p 2 , p3 ) so that amplitude is the sum of ∂ contribution φ [ x (t )] from each path. p = L( x, t , v), j = 1,2,3 2
j
K (b, a) = ∑ φ[ x(t )]
∂v j
which express the momentum of a The contribution of a path has a phase particle. With Lagrange function (4.1) proportional to the action A(x ) . the expression above is i / hA[ x ( t )]
φ [ x(t )] = const.e
1
v j = p j , j = 1,2,3 Note that in quantum mechanics the m dynamics of a particle follows a wave Then, Hamilton function H ( x, t , p ) can function φ ( x), x ∈ R 3 and, assuming be defined
∫
R3
2
φ ( x) dx = 1
(4.3)
H ( x, t , p ) =< v ⋅ p > − L( x, t , v )
which expresses mechanical energy of a
φ (x ) is interpreted as the density of particle. In particular, in our case it 2
probability for which the particle is observed at x. time s If a wave function φ ( x, s ) at becomes φ ( x, t ) at every different time t , as basic law of quantum mechanics, the correspondence is linear and expressed φ ( x, t ) = U (t , s )φ ( x, s ) , where the linear transformation U (t , s ) is the motion law of quantum mechanics describing dynamics of a particle. Thus, it is written as integral transformation
φ ( x, t ) = ∫ K ( x, t y, s)φ ( y, s)dy R3
becomes
H ( x, t , p ) =
1 2 p + eV ( x, t ) 2m
where p =
p12 + p 22 + p32
Now,
replace
we
each
p j ( j = 1,2,3) of momentum by
element
partial differential action − ih
∂ ∂x j
to
obtain Hamiltonian, H ( x, t ,−ih ∂∂x ) . Thus,
in this case
H ( x, t ,−ih
∂ − h2 )= ∂x 2m
∂2 ∑ j =1 ∂x 2 + eV ( x, t ) function φ (t , x ) explaining 3
Let Ω be the space consisting of whole A wave curves for which a particle starts y at particle’s state of quantum mechanics is
the solution to the partial differential equation
∞
xT
−∞
x
= e − rτ ∫ ( ∫ F (e xT )e − ABS [ x (t )] Dx(t ' ))dxT , '
∂ ∂ ih φ ( x, t ) = H ( x, t ,−ih )φ ( x, t ) ∂t ∂x
where the average is over the risk-neutral process. Note that the ' The equation is called Schrӧdinger action functional ABS [ x (t )] defined on equation. With the function paths x(t ' ), t ≤ t ' ≤ T defines the path K ( x, t y, s) of (4.4), integration measure with BS Lagrangian ∂ ∂
{
)) K ( x, t y , s ) = 0 ∂x and ∫ 3 K ( x, t y, s ) f ( y )dy = f ( x) hold for (ih
∂t
− H (t , x,−ih
T
ABS [ x(t ' )] = ∫ L BS dt ' ,
K ( x, t y, s) is the basic solution to Schrӧdinger equation.
.
t
L BS =
R
every function f ( x ). In other words,
}
1 ( x& (t ' ) − μ ) 2 2σ 2
If F has the following simple form[8]
F = f ( S T )e − f1[ S (t )] , where f 1 [ S (t ' )] can be represented as '
5. Option Pricing
T
f1 [ S (t ' )] = ∫ V ( x(t ' ), t ' )dt ' t
Consider a complete probability of some potential V ( x, t ' ) ,then it space with a becomes (Ω, ℑ, P ) filtration {ℑt }0≤t ≤∞ ,which is right G ( S , t ) = e − rτ ∞ f (e xT )e ( μ / σ 2 )( xT − x ) −( μ 2τ / 2σ 2 )
ℑ0 contains all the P –null sets of ℑ . According to the
continuous and
papers ([10],[11]) we can assume the existence of a risk-neutral probability measure Q equivalent to P such that the discounted prices of the securities are martingales. Under Q ,the dynamics of the underlying stock price S (t ) follows the stochastic differential equation dS (t ) = rS (t )dt + σS (t )dWt .
∫
F
−∞
(5.1)
KV ( xT , T x, t )dxT , where
K V ( xT , T x, t ) is
the
Green
function for zero-drift Brownian motion with killing at rate V ( x, t ' ) such that
K V ( xT , T x, t ) =
∫
xT
x
T
exp(− ∫ ( L0 + V )dt ' ) Dx(t ' ), (5.2) t
where L0 is the Lagrangian for a zero drift process. The representation of (5.2) satisfy
We partially follow the notation and 1 2 ∂ 2 K V ∂K V description in [6] to explain path σ + = V ( x, t ) K V 2 ∂t ∂x 2 dependent situation relevant to a with initial condition potential. K V ( xT , T x, t ) = δ ( xT − x) . Suppose we consider a path dependent option defined by its payoff at expiration where δ ( x ) is the Dirac delta function. G F (T ) = F [ S (t ' )], Also, the option price (5.1) satisfies the where F [ S (t ' )] is a given functional on BS partial differential equation with price paths S (t ' ), t ≤ t ' ≤ T .Then the potential
{
}
∂ 2G
∂K
∂G
1 2 F F present value of this path dependent σ +μ V + = 2 2 ∂x ∂t ∂x option at the contract t is given by (r + V ( x, t ))G F Feynman-Kac formula − rτ ' G F ( S , T ) = e E[ F ( S (t ))] , τ = T − t and the terminal condition
GF (ST , T ) = f (ST )
Thus,it is showed that the BSequation path-dependent options. includes risk-free rate, r + V ( x, t ) . (4)From the point of view that Lagrange 6. Some Remarks and Hamilton physical systems are Some relationships between path characterized as only one function, then integral of quantum mechanics and motion equation, definition of physical option pricing were discussed along with quantity and conservation law are the risk-neutral valuation, including determined if either Lagrange or some correspondence of quantum Hamilton is given. mechanics and financial derivatives. equation is time The definition of path integrals is (5)Schrӧdinger originally used to describe quantum reversible since the Hamiltonian is phenomena, and this definition is Hermitian though Black-Scholes process time-irreversible due to the completely rigorous and the limit does is Hamiltonian being non-Hermitian. converge [2]. (1)In finance, the fundamental principle comes from the no-arbitrage ([3],[7]etc). Since it plays a role similar to the least action principle and the law of energy conservation in physics, similar to physical dynamical systems Lagrangian functions and action functionals can be introduced for financial models. Also, financial models are stochastic ones so that expected values of different quantities dependent on price paths are given by path integrals, where the action functional for the underlying risk-neutral price process defines a risk-neutral measure on the set of all paths.
(6)In quantum mechanics a particle in mechanical system is considered to be not isolated but a member of the whole mechanical system. In other words, the value of some physical quantity is not determined in a moment at just neighborhood but the entire system is involved for that. However, Newton’s standpoint shows that the position of a particle is determined by knowing force which acts on the particle (initial values and velocity need to be provided), not taking into account of other particles. (7)There are several options whose payoff functions depend on the path.In particular, the barrier option and Asian option, which are recently popular for investors, will be considered to use the methodology above under various conditions to find the solutions.
(2)Averages satisfy the BS partial differential equation, which is a finance counterpart of the Schrӧdinger equation of quantum mechanics, and the risk-neutral valuation formula is interpreted as the Feynman-Kac References representation of the PDE solution. Thus, the path-integral formalism [1]Black,F.,and M.Scholes(1973) The Pricing of Options and Corporate provides a natural bridge between the Liabilities J.Polt.Econ.,81,637-654 risk-neutral martingale pricing and the [2]Baaquie,B.E.,C.Coriano,M.Srikant arbitrage-free PDE based pricing[6]. (2004) Hamiltonian and Potentials in Derivative Pricing Models: Exact (3)By martingale condition required for Results and Lattice Simulations risk-neutral evolution, risk-free rate r Physica A 4 531-557 in Hamiltonian is replaced by potential [3]Duffie,D (1996) Dynamic Asset V ( x ) in (3.1). Potential terms in the nd Pricing 2 ed. Princeton Univ. Press, Hamiltonian show to represent a class of
New Jersey [4]Nelson,E. (2001) Dynamic Theories of Brownian Motion, Princeton University Press [5]Harrison,J.M.,and S.Pliska (1981) Martingales and Stochastic Integrals in the Theory of Continuous Trading Stochastic Process and Their Applications,11,215 [6]Linetsky,V. (1998) The Path Integral Approach to Financial Modeling an Option Pricing, Computational Economics 11,129-163 [7]Merton,R.C (1990) Continuous –Time Finance,Blacwell, Cambridge,MA [8]Prigogine,I (1989) Exploring Complexity [9]Feynman,R.P. and A.R.Hibbs(1965) Quantum Mechanics and Path Integrals, McGraw-Hill, New York [10]Harrison,J.M. and D. Kreps,(1979) Martingale and Arbitrage in Multiperiods Securities Markets, J.Econom Theory,20,381-408 [11]Kac,M.(1951) On some connections between probability and differential and integral equations, Proc. of 2nd Berkeley symp. Math. Stat. and Prob.,Univ. of Calif. Press, 189-215
