Feynman Path Integral and Financial Market

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Feynman Path Integral and Financial Market
Most delicate but richest ﬁeld for theorist
Vivek Kumar Bhartiya
Skolkovo Institute of Science and Technology and
Moscow Institute of Physics and Technology, Moscow Branch
MA06043, Spring Term
Instructor- Prof. Leonid Levitov(MIT)
March 26, 2015
Vivek Kumar Bhartiya (Skolkovo Institute of Science and Technology and Moscow Institute of Physics and Technology, Moscow BranchMFeynman Path Integral and Financial Market March 26, 2015 1 / 26

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1 Introduction
2 Black-Scholes Equation
3 Feynman Path Integral Formulation
4 Path Integral in Finance
5 Challenges

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Introduction
1 Introduction
5 Challenges

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History
Figure : Louis Bachelier, French
mathematician, 1876-1946
”Bachelier’s work on random walks was more mathematical and predated
Einstein’s celebrated study of Brownian motion by ﬁve years”

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History
Figure : Louis Bachelier, French
mathematician, 1876-1946
Figure : Fischer Black and Myron Scholes, 1973

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Present and Future
R. P. Feynman, 1948; discovery of space-time approach to quantum
mechanics- path integral”
John W. Dash, 1980, ﬁrst person to use quantum physics in mathematical
ﬁnance.

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Black-Scholes Equation
1 Introduction
5 Challenges

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”European call option” f (t) is given by
f (T, S(T)) =
S(T) − K S(T) > K,
0 S(T) < K.
where K = strike price.
The problem of option is the following: given the price of the security S(t)
at time t, what should be the price of option f at time t < T?

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”European call option” f (t) is given by
f (T, S(T)) =
S(T) − K S(T) > K,
0 S(T) < K.
where K = strike price.
The problem of option is the following: given the price of the security S(t)
at time t, what should be the price of option f at time t < T?
S(t) as a stochastic variable through Langevin Equation as
dS(t)
dt
= φS(t) + σSR(t),

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S(t) as a stochastic variable through Langevin Equation as
dS(t)
dt
= φS(t) + σSR(t),
where,
R is drwan from a δ-correlated random distribution with zero mean and
correlation
R(t)R(t ) = δ(t − t ).
φ is expected return on the security S,
σ is its volatility.

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For the Gaussian process, in an inﬁnitesimal time interval dt,
(Rdt)2 = dt + O(dt3/2). Rewriting dS = φS(t) + σSR(t)dt, we get
(dS)2
= σ2
S2
dt + O(dt3/2
).
From these equations we obtains
df =
∂f
∂t
dt +
∂f
∂S
dS +
1
2
∂2f
∂S2
(dS)2
+ O(dt3/2
)
∂f
∂t
=
∂f
∂t
+
1
2
σ2
S2 ∂2f
∂S2
+ σRS
∂f
∂S
+ φS
∂f
∂S

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The change of the portfolio is independent of the Gaussian white noise R.
π = f −
∂f
∂S
S
π is portfolio in which an investor sells an option f and buys ∂f
∂S amount of
security S.
dπ
dt
=
df
dt
−
∂f
∂S
dS
dt
=
df
dt
+
1
2
σ2
S2 ∂2f
∂S2
The portfolio free from stochastic nature of security. - hedging.
dπ
dt
= rπ
Black-Scholes diﬀrential equation
∂f
∂t
+ rS
∂f
∂S
+
1
2
σ2
S2 ∂2f
∂S2
= rf

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1 Introduction
5 Challenges

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Feynman Path Integral Formulation
|Ψ(t ) = Û(t , t)|Ψ(t) , Û(t , t) = e− i ˆH(t −t)
Θ(t − t). (1)
In real space representation
Ψ(q , t ) = q |Û(t , t)Ψ(t) = dq U(q , t ; q, t)Ψ(q, t),
where U(q , t ; q, t) = q |e− i ˆH(t −t)
|q Θ(t − t) defines the (q , q) -
known as the propagator of the theory.
e−i ˆHt/
= e−i ˆH t/
N
(2)
factors e−i ˆH t/ are small. The first simplification- exponential can be
factorized into two pieces, each of which can be readily diagonalized.

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e−i ˆHt/
= e−i ˆH t/
N
(3)
each of which can be readily diagonalized.
e−i ˆH t/
= e−i ˆT t/
e−i ˆV t/
+ O( t2
)
The advantage of this factorization is that the eigenstates of each factor
e−i ˆT t/ and e−i ˆV t/ are known independently.
qf | e−i ˆH t/
N
|qi qf |∧e−i ˆT t/
e−i ˆV t/
∧.....∧e−i ˆT t/
e−i ˆV t/
|qi ,
(4)
and inserted at each positions indicated by the symbol ” ∧ ” the resolution
of identity
id = dqn dpn|qn qn|pn pn|. (5)

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The unit operator is arranged in such a way that both ˆT and ˆV act on
corresponding eigenstates. Substituting Eq.(4) into (3) by making use of
identity q|p = p|q ∗ = eiqp/ /(2π )2, one obtains
qf |e−i ˆHt/
|qi
N−1
n=1,qN =qf ,q0=qi
dqn
N
n=1
dpn
2π
e
−i t
ΣN−1
n=0 V (qn)+T(pn+1)−pn+1
q
(6)
At each ”time step” tn = n t, n = 1, ......, N, we are integrating over a
pair of coordinates xn ≡ (qn, pn) parametrizing the classical phase space
(Fig :2).

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N → ∞ t = N t ﬁxed, set of the phase space points xn becomes a
continuous curve x(t).
t
N−1
n=0
→
t
0
dt ,
qn+1 − qn
t
→ ∂t q|t =tn ≡ ˙q|t =tn
while
[V (qn) + T (pn+1)] → [T(p|t =tn + V (q|t =tn ) ≡ H (x|t =tn )
denotes the classical Hamiltonian. In the limit N → ∞, the fact that
Kinetic and potential energies are evaluated at neighboring slices, n and
n+1, becomes irrelevant. Finally,
lim
N→∞
N−1
n=1,qN =qf ,q0=qi
dqn
N
n=1
dpn
2π
≡
q(t)=qf ,q(0)=qi
Dx,

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qf |e−i ˆHt/
|qi =
q(t)=qf ,q(0)=qi
Dx exp
i t
0
dt (p ˙q − H(p, q)) .
(7)
Integration extends over all possible paths through the classical
phase space of the system which begin and end at the same
conﬁguration points qi and qf respectively. The contribution of
each path is weighted by its Hamiltonian action.
qf |e−i ˆHt/
|qi =
q(t)=qf ,q(0)=qi
Dq e− i t
0 dt V (q)
Dp e
− i t
0 dt p2
2m
−p ˙q
(8)
qf |e−i ˆHt/
|qi =
q(t)=qf ,q(0)=qi
Dq exp
i t
0
dt L(q, ˙q) (9)

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1 Introduction
5 Challenges

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Path Integral in Finance
Change of variable S = ex
∂f
∂t
= ˆH + r f
ˆH = −
σ2
2
∂2
∂x2
+
σ2
2
− r
∂
∂x
.
To evaluate the price of the European call option with constant volatility
Feynman-Kac formula
f (t, x) = e−r(T−t)
∞
−∞
dx x|e−(T−t)ˆH
|x f (T, x ).
The problem of determining the price has been reduced to the evaluation
of the propagator of some eﬀective quantum Hamiltonian.
In the simplest case, x|e−τ ˆH
|x

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Path Integral in Finance
In this case,
x|e−τ ˆH|
x =
1
√
2πτσ2
exp −
1
2τσ2
x − x + τ(r − σ2
/2)
2
.
Black-Scholes distribution
where,
ln S(t) has a normal distribution with
mean = ln S(t) + (r − σ2
2 )(T − τ) and
the variance = σ2(T − t).
But for
non-uniform volatility-path integral becomes essential.

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1 Introduction
5 Challenges

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Challenges
brainchild of economists
Fischer Black and Myron
Scholes, provided a rational
way to price a ﬁnancial
contract when it still had time
to run. It was like buying or
selling a bet on a horse,
halfway through the race.

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Challenges- Most Important slide
The Black-Scholes equation relates price of the option to four other
quantities:
1 time
2 the price of the asset upon which the option is secured
3 the risk-free interest rate
4 volatility of the asset. This is a measure of how erratically its market
value changes.
asset’s volatility remains the same for the lifetime of the option

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Reality:
Figure : (a) Index S&P 500 for 13-year period Jan. 1, 1984 — Dec. 14, 1996,
recorded every minute, and (b) volatility in time intervals 30 min

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Feynman Path Integral and Financial Market

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