Background on quantum finance Quantum finance

1 background on quantum finance

1.1 quantum continuous model
1.2 quantum binomial model
1.3 multi-step quantum binomial model

1.3.1 bose–einstein assumption

background on quantum finance

one of these alternatives quantum computing. physics models have evolved classical quantum, has computing. quantum computers have been shown outperform classical computers when comes simulating quantum mechanics several other algorithms such shor s algorithm factorization , grover s algorithm quantum search, making them attractive area research solving computational finance problems.

quantum continuous model

most quantum option pricing research typically focuses on quantization of classical black–scholes–merton equation perspective of continuous equations schrödinger equation. haven builds on work of chen , others, considers market perspective of schrödinger equation. key message in haven s work black–scholes–merton equation special case of schrödinger equation markets assumed efficient. schrödinger-based equation haven derives has parameter ħ (not confused complex conjugate of h) represents amount of arbitrage present in market resulting variety of sources including non-infinitely fast price changes, non-infinitely fast information dissemination , unequal wealth among traders. haven argues setting value appropriately, more accurate option price can derived, because in reality, markets not efficient.

this 1 of reasons why possible quantum option pricing model more accurate classical one. baaquie has published many papers on quantum finance , written book brings many of them together. core baaquie s research , others matacz feynman s path integrals.

baaquie applies path integrals several exotic options , presents analytical results comparing results results of black–scholes–merton equation showing similar. piotrowski et al. take different approach changing black–scholes–merton assumption regarding behavior of stock underlying option. instead of assuming follows wiener-bachelier process, assume follows ornstein-uhlenbeck process. new assumption in place, derive quantum finance model european call option formula.

other models such hull-white , cox-ingersoll-ross have used same approach in classical setting interest rate derivatives. khrennikov builds on work of haven , others , further bolsters idea market efficiency assumption made black–scholes–merton equation may not appropriate. support idea, khrennikov builds on framework of contextual probabilities using agents way of overcoming criticism of applying quantum theory finance. accardi , boukas again quantize black–scholes–merton equation, in case, consider underlying stock have both brownian , poisson processes.

quantum binomial model

chen published paper in 2001, presents quantum binomial options pricing model or abbreviated quantum binomial model. metaphorically speaking, chen s quantum binomial options pricing model (referred hereafter quantum binomial model) existing quantum finance models cox-ross-rubinstein classical binomial options pricing model black–scholes–merton model: discretized , simpler version of same result. these simplifications make respective theories not easier analyze easier implement on computer.

multi-step quantum binomial model

in multi-step model quantum pricing formula is:

c

0


n


=

t
r

[
(

⨂

j
=
1


n



ρ

j


)


[

s

n


−
k
]


+


]


{\displaystyle c_{0}^{n}=\mathrm {tr} [(\bigotimes _{j=1}^{n}\rho _{j}){[s_{n}-k]}^{+}]}

which equivalent of cox-ross-rubinstein binomial options pricing model formula follows:

c

0


n


=
(
1
+
r

)

−
n



∑

n
=
0


n





n
!


n
!
(
n
−
n
)
!




q

n




(
1
−
q
)


n
−
n




[

s

0




(
1
+
b
)


n




(
1
+
a
)


n
−
n


−
k
]


+




{\displaystyle c_{0}^{n}=(1+r)^{-n}\sum _{n=0}^{n}{\frac {n!}{n!(n-n)!}}q^{n}{(1-q)}^{n-n}{[s_{0}{(1+b)}^{n}{(1+a)}^{n-n}-k]}^{+}}

this shows assuming stocks behave according maxwell-boltzmann classical statistics, quantum binomial model indeed collapse classical binomial model.

quantum volatility follows per meyer:

σ
=



ln
⁡

(
1
+

x

0


+



x

1


2


+

x

2


2


+

x

3


2




)



1

/

t





{\displaystyle \sigma ={\frac {\ln {(1+x_{0}+{\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}})}}{\sqrt {1/t}}}}



bose–einstein assumption

maxwell–boltzmann statistics can replaced quantum bose–einstein statistics resulting in following option price formula:

c

0


n


=
(
1
+
r

)

−
n



∑

n
=
0


n



(




q

n




(
1
−
q
)


n
−
n





∑

k
=
0


n



q

k




(
1
−
q
)


n
−
k





)



[

s

0




(
1
+
b
)


n




(
1
+
a
)


n
−
n


−
k
]


+




{\displaystyle c_{0}^{n}=(1+r)^{-n}\sum _{n=0}^{n}\left({\frac {q^{n}{(1-q)}^{n-n}}{\sum _{k=0}^{n}q^{k}{(1-q)}^{n-k}}}\right){[s_{0}{(1+b)}^{n}{(1+a)}^{n-n}-k]}^{+}}

the bose-einstein equation produce option prices differ produced cox-ross-rubinstein option pricing formula in circumstances. because stock being treated quantum boson particle instead of a

classical particle.