Option pricing under quantum theory of securities price formation

Jack Sarkissian

Managing Member, Algostox Trading LLC

641 Lexington Avenue, 15th floor, New York, NY 10022

email: jack@algostox.com

Abstract

Recent work showed that securities prices behave as quantum chaotic quantities that described

by quantum equations. We study pricing of European style options under that framework. The

resulting volatility surface exhibits the smile and other characteristics of equity options.

Additionally, we propose a modification to pricing process that allows to compute bid and ask

prices marked to the general price level.

1. Quantum probability distribution of returns

Recent work showed that securities prices behave as quantum chaotic quantities [1-3]. This

description allowed to understand the behavior of bid-ask spread and model price dynamics of

securities with limited liquidity. According to such framework, price dynamics can be described

with a wave of probability amplitude propagating in time through random trading environment

and obeying quantum equations. For highly liquid securities it turns into the classical diffusion

theory, in which prices just follow a Brownian motion in price space.

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Please cite as: J. Sarkissian, “Option Pricing Under Quantum Theory of Securities Price Formation” (October 4, 2016). Available at SSRN: http://ssrn.com/abstract=2848014

© 2016 Algostox Trading. The enclosed materials are copyrighted materials. Federal law prohibits the unauthorized reproduction, distribution or exhibition of the materials. Violations of copyright law will be prosecuted.

One of the main results of this theory is that probability distribution of returns, usually thought to

be a smooth bell shaped curve, is not actually bell shaped at all. Even though still localized, it is

erratic across price space, forming the so-called “speckle pattern” shown in Fig. 1. This erratic

curve occurs due to randomness of trading environment in every possible scenario. The well-

known smooth bell curve forms only after including all possible scenarios, and averaging over

their ensemble. In reality, however, we must always face a single scenario with its erratic

behavior, not an ensemble average.

0

0.05

0.1

0.15

0.2

-1 -0.5 0 0.5 1

Prob

abili

ty

Return

Fig. 1. Typical probability distribution in quantum

price formation framework is erratic across price space.

Evolution of return probability distribution is erratic as well. It is usually thought that larger price

shifts have a lower probability of occurring, which is also an artefact of the smooth bell curve. In

quantum description any price shift, even big, can at times build up a good chance of occurring,

leading to “Black Swans”, see Fig. 2.

2

-100 -50 0 50 1000

0.1

0.2

0.3

0.4

x, (%)

Prob

abili

ty

(a)

InitialFinal

0 20 40 600

0.1

0.2

0.3

0.4

Time horizon

Prob

abili

ty

(b)

return = 0%return = -10%return = -20%

0 20 40 600

20

40

60

80

Time horizon

%

(c)

Volatility95% VaR95% ES

0 20 40 600

1

2

3

4

Time horizon

(d)

95% VaR/Volatility95% ES/Volatility

Fig. 2. Dynamics of probabilities for different return levels. Despite being generally low for

distant levels, probability may sometimes build up to substantial values – effect representing the

“Black swans”.

It is important to see how these findings reflect on option pricing, whether or not they lead to any

new effects or explain the already known ones. For example, the non-Gaussian nature of

probability distribution must result in some kind of distorted volatility surface even for vanilla

options. The ever-changing, so-to-speak “breathing” of the speckle pattern of that probability

distribution could result in higher premiums for certain path-dependent options and lower

premiums for others. Options with early exercise features, may also be affected, since in addition

to regular uncertainty they would now have to cover the risk of “breathing”.

In this paper we study the effects of these findings on pricing of European style options. For

simplicity, but without loss of generality, we will take interest rates and dividends equal to zero.

Their absence will not have qualitative effect on the features discussed here.

2. Overview of quantum theory of price formation

Here we give a brief overview of quantum theory of price formation. The basic element of this

approach is the probability amplitude ψ, which describes the state of the security, and whose

absolute value squared represents the probability of finding the security in the given state:

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p=|ψ|2 (1)

Security prices are governed by the price operator S. Price operator’s eigenfunctions represent

the pure states in which the security can be found, and eigenvalues represent the spectrum of

prices that the security can attain in each corresponding state with quantum number n:

S ψn=sn ψn (2)

To guarantee that prices are real numbers the price operator S must be Hermitian. Price spectrum

is determined by the following equation

det ‖smn−sn δmn‖=0 (3)

Matrix elements of price operator fluctuate in time, which introduces randomness to eigenvalues

and eigenfunctions:

S (t+δt )= S (t )+δ S (t) (4)

and therefore:

sn ( t+δt )=sn (t )+δ sn(t) (5)

Dynamics of probability amplitude is described with the following equation.

iτs ∂ ψ∂ t

=S ψ (6)

It is natural to use the nearest-neighbor approximation, in which only the adjacent price levels

interact and the others are neglected. Under this approximation and in the so-called coordinate

representation with x= ln ( s ) for a coordinate, Eq. (6) becomes:

iτ ∂ f∂ t

=i ∂ ( μf )∂ x

+ ∂2 (γf )∂ x2 (7)

where f is connected to ψ with a phase multiplier, that has no effect on probability itself. Here

μ=μ (x , t ) and γ=γ ( x , t) are allowed to fluctuate both in coordinate and time. This equation

combined with initial condition f ( x , t=0 )= f 0(x ) formulates an initial value problem that

describes evolution of probability amplitude of security’s logarithmic price (or effectively

probability amplitude of returns) with time. Each realization r of μ or γ in time and coordinate

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results in a particular f (r ). In order to obtain a complete probability distribution ensemble

averaging should be performed over the ensemble of all realizations:

P ( x , t )=⟨|f ( r ) ( x ,t )|2 ⟩r (8)

where ⟨ …⟩ r stands for ensemble averaging. Ensemble averaging should also be performed to

calculate the expected values of operators:

A (t )=⟨ ⟨ f (r )|A|f (r ) ⟩ ⟩r (9)

For a Gaussian input probability distribution with width w0, solutions resulting from Eq. (7) lead

to expanding probability distribution, erratic in profile, but localized with its width increasing

over time as

w ( t )=wΔ t √1+( ϵwΔ t )

2

( tΔ t

−1) (10)

where

w Δt=w0 √1+( ϵw0 )

2

(11)

Here ϵ is the price granularity size and Δt is the time step. A particular realization |f (r )|2 may be

seen in Fig. 3, and the ensemble-averaged probability distribution is shown in Fig. 4.

time

x (%

)

10 20 30 40 50 60

-100

-50

0

50

0

0.1

0.2

0.3

0.4

0.5

Fig. 3. Dynamics of probability distribution in a single realization

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time

x (%

)

10 20 30 40 50 60

-100

-50

0

50 0.05

0.1

0.15

0.2

Fig. 4. Ensemble average dynamics of probability distribution.

3. Option valuation and volatility surface

With this framework the European option values can be calculated as the ensemble average of

expected payout over all possible realizations of f . For example, for a call option in a particular

realization, we have:

C(r )(K ,T )=∫0

∞

|f (r ) [ ln ( S ) , T ]|2max (S−K ,0)dS (12)

Averaging this over all r’s gives us the call price:

C ( K ,T )=⟨ C(r) ( K ,T ) ⟩r (13)

The resulting implied volatility surface is shown in Fig. 5. This surface has all the typical

characteristics observed in equity options. It does exhibit the smile curvature along the strike

axis. The curve is steeper in in-the-money options than in out-of-the-money options. The smile is

more pronounced in short-term options and it decreases in longer term options. All these features

arise from the quantum model naturally, without a need for external adjustments.

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15

400%

20%40%60%80%

100%

70 80 90 100 110 120 130 140 150

Tim

e to

exp

iratio

n

impl

ied

vola

tility

Strike/price

Volatility surface

0%-20% 20%-40% 40%-60% 60%-80% 80%-100%

Fig. 5. Volatility surface of a European style call option priced under quantum framework. In this

framework typical features of equity volatility surface appear naturally.

Pricing formula Eq. (13) follows the traditional paradigm of arbitrage-free pricing, in which a

security price is equal to the total discounted cash flows expected from it. Trading performed by

these prices will lead to exchange of securities of equal economic value. In reality, no trader

wants to exchange his securities for other securities of equal value. The point of trading is

precisely in exchanging securities of lesser value for securities of higher value. This is why we

should consider different pricing mechanisms, not necessarily based on expectation value.

Prices C(r ) form a probability distribution p(C(r )), as shown in Fig. 6. In classical Black-Scholes

theory that distribution is the portion of the log-normal distribution corresponding to values

above the strike price. In quantum framework they are distributed and scattered around their

expectation value.

7

0

20

40

60

80

1.5 2.4 3.2 4.1 5.0 5.8 6.7 7.5 8.4 9.2p(

C)

C

Fig. 6. Probability distribution for an ATM call with S=100, T=0.06 and volatility σ=40 %.

Using this probability distribution, we can define option price as its α percentile:

Cα :∑C (r )

Cα

p (C (r ) )=α (14)

where C(r ) have to be arranged in an increasing order before summation. Such approach allows

more flexibility in pricing, depending on whether option is being bought or sold, whether the

quote is intended to provide liquidity or take it, etc. Call prices corresponding to 10 th, and 90th

percentiles along with their mean values are shown in Fig. 7

0

10

20

30

40

70 90 110 130 150

C

K

Mean 90th percentile 10th percentile

Fig. 7. Call prices corresponding to different percentiles of p (C(r))

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4. Discussion

Quantum theory of price formation appears to capture important elements of financial

instruments not usually covered in textbooks. Among them are: (a) statistical behavior of spread,

(b) spread curve scaling, (c) relationship between spread, volatility, and trading volume, (d) fat

tails of return probability distribution. It appears that now we can add the (e) volatility smile and

term structure character to this list. What is interesting is that all of these features result from a

single framework without having to add any of them externally.

Option pricing procedure in this framework has flexibility that allows to price options depending

on trader’s risk-aversion level and intention with respect to the options. This can be useful in

option market making and risk management.

Quantum theory of price dynamics originated from attempts to describe spread and its

characteristics. As such it allows to model securities with limited liquidity. Option pricing

mechanism based on quantum framework must therefore be feasible for pricing options on such

securities.

Lastly, it is interesting to see what consequences the disorderly nature of probability distribution

may have on options that are sensitive to such nature. We expect that the most affected are the

path-sensitive exotic options, and options with early execution features.

5. References

[1] J. Sarkissian, “Quantum Theory of Securities Price Formation in Financial Markets” (April

13, 2016). Available at SSRN: http://ssrn.com/abstract=2765298

[2] J. Sarkissian, “Coupled mode theory of stock price formation”, available at

arXiv:1312.4622v1 [q-fin.TR], (2013)

[3] J. Sarkissian, “Spread, Volatility, and Volume Relationship in Financial Markets and Market

Maker's Profit Optimization” (June 23, 2016). Available at SSRN:

http://ssrn.com/abstract=2799798

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