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We have developed an extended model for stock price behaviour that is able to accommodate fat-tailed distributions with support as large as
. The “homogeneously saturated” (HS) model avoids exponential price changes for large fluctuations by means of a saturation parameter. In the limit where the saturation parameter is zero, the standard model of stock price behaviour (
*i.e.*, geometric Brownian motion) is recovered. We compare simulated stock price series generated for both the standard and HS model for the DJIA and five random stocks from the NYSE and NASDAQ exchanges. We find that in all cases, the HS model provides a better fit to the observed price series than the standard model. This has implications to many areas of finance including the Black-Scholes formula for option pricing.

The standard model of stock price behaviour generates prices through geometric Brownian motion with a deterministic drift rate [

d S ( t ) d t = α S ( t ) + σ S ( t ) f ( t ) (1)

where S ( t ) is the price of the stock at time t, α is the drift rate, σ is the volatility of the stock price and f ( t ) is a zero mean, normally distributed, stochastic, uncorrelated in time, noise driving term. Using η ( t ) = α + σ f ( t ) , we can simplify Equation (1) to

d S ( t ) d t = η ( t ) S ( t ) . (2)

Integrating Equation (2) gives the predicted price of the stock at a later time t:

S s ( t ) = S 0 e ω ( t ) , (3)

where the subscript s denotes the standard model, S 0 is the price of the stock at t = 0 , and for clarity, we have made the substitution

ω ( t ) = ∫ 0 t η ( t ′ ) d t ′ . (4)

Equations (3) and (4) suggest that the price of the stock depends exponentially on the integral of the noise driving term, f ( t ) . Since this term is assumed to be normally distributed, the probability of a large price is essentially zero and the predicted price remains bounded. If, however, the underlying noise is not normally distributed, Equation (3) might predict wild price swings that are unrealistic and not observed on the market.

Stock returns are generally assumed to follow a normal distribution, in part owing to mathematical simplicity. It has been known for some time, however, that this assumption is not supported by actual stock prices (e.g., [

A simulation of stock prices using Equation (2) and drawing η ( t ) from different distributions is given in

this comparison is the Student’s T distribution (see Appendix A), where the “fatness’’ of the tails is governed by the shape parameter ν . It is clear from

There exist several approaches to pricing stocks when the noise driving term is a fat-tailed distribution. One approach is to modify the tails of the distribution such that the contributions far into the tails are negligible while not affecting significantly the central portion of the distribution, which fits well the observed data (e.g. [

A different approach to pricing stocks when the underlying distribution is fat-tailed is to allow for saturation of the stock price by depletion of the resource that supports the price (i.e., by depletion of the reservoir of money that is available to purchase the stock). This is the approach we investigate in this paper. This “homogeneously saturated” (HS) model for the price of a stock is constructed and compared to the standard model to gain insight into the pricing of financial assets when the underlying distribution is fat-tailed.

The aim of this section is to develop a pricing model which can handle fat-tailed distributions as the noise driving term. Let M ( t ) be the amount of money available in a reservoir to buy the stock, N the rate at which money is added to the reservoir, γ S ( t ) M ( t ) the rate at which money is removed from the reservoir due to the purchase of the stock, and ζ M ( t ) the rate at which money is removed from the reservoir due to the purchase of other goods. We then have

d d t M ( t ) = N − γ S ( t ) M ( t ) − ζ M ( t ) + δ f ( t ) (5)

where δ f ( t ) is a noise driving term. It is interesting to note that the noise in the homogeneously saturated model is ascribed to fluctuations in the amount of money available to invest in the stock. All parameters ( ζ , N , γ ) in Equation (5) have a time dependence, but it is assumed that these parameters vary slowly enough that they can be treated as constants. As well, we assume the system is in

steady state, such that d d t M ( t ) = 0 , leading to

M ( t ) = α + ρ f ( t ) 1 + λ S ( t ) (6)

where we have made the substitutions: α = N / ζ , ρ = δ / ζ and λ = γ / ζ .

The next step in our derivation is inspired from laser physics, where coupled rate equations are used to describe the interaction between the laser output (analogous to S ( t ) in our case) and the inversion (analogous to M ( t ) in our case) [

d d t S ( t ) = M ( t ) S ( t ) . (7)

The validity of Equation (7) will be determined by how well it fits the available data. Using Equation (6) in (7) gives

d d t S ( t ) = α S ( t ) + ρ f ( t ) S ( t ) 1 + λ S ( t ) . (8)

Making the substitution η ( t ) = α + ρ f ( t ) we have

d d t S ( t ) = η ( t ) S ( t ) 1 + λ S ( t ) . (9)

The price of the stock in our HS model is therefore

S ( t ) = S 0 e ω ( t ) e λ ( S ( t ) − S 0 ) = S s ( t ) e λ ( S ( t ) − S 0 ) . (10)

The HS model is similar in form to the standard model, Equation (3), except for the e λ ( S ( t ) − S 0 ) term in the denominator of the former. Indeed, when λ = 0 the HS model reduces to the standard model. The S ( t ) dependency in the denominator of Equation (10) effectively saturates the price of the stock; without it wealth would not be conserved and S ( t ) could continually increase exponentially with time.

To demonstrate the behaviour of the saturation parameter, we generate simulated prices over a 1000 day period using the HS model, Equation (9), by adding different and independent one day solutions for S ( 1 ) for each t to obtain the price for the next day, i.e., S ( t + 1 ) = S ( t ) + S t + 1 ( 1 ) , for t = 0 , ⋯ , 1000 with S ( 0 ) = 1000 and S t + 1 ( 1 ) being the one day solution over the time interval t to t + 1 . These generated price series are shown in ^{t}^{h} value from the PRNG sequence for a Student’s T distribution with shape parameter ν = 2.5 , scale parameter β = 0.01 , and drift μ = 0.001 . When λ is zero, the price of the stock increases exponentially with the noise driving term and leads to large price swings. The case λ = 0 is equivalent to the standard model with a fat-tailed Student’s T noise source. When λ > 0 , the price of the stock approximately linearly follows the noise; i.e., it becomes saturated. Variations in the price of the stock are increasingly damped with higher values of λ . See Appendix B for a justification of the approach S ( t + 1 ) = S ( t ) + S t + 1 ( 1 ) .

To reiterate, λ describes the rate at which money is removed from the money reservoir owing to purchase of the stock. λ = 0 (as in the standard model) means that buying the stock has no effect on the money supply; essentially the standard model assumes an infinite reservoir of money is available to buy the stock. From Equation (9), it can be observed that the reciprocal of the saturation parameter, λ − 1 , has the same units as the stock price S ( t ) . One can identify λ − 1 as the stock price at saturation. When S ( t ) = λ − 1 , the instantaneous rate of change with time of S ( t ) is one-half of what it would be with λ = 0 ; see Equation (9). For S ( t ) ≪ λ − 1 , the reservoir of money is not depleted (or saturated) by the rate of transactions, and the rate of change of S ( t ) is similar in magnitude to the standard model. For S ( t ) ≫ λ − 1 , the reservoir of money that is available to purchase the stock is saturated (or depleted) by the rate of transactions and the time rate of change of S ( t ) is greatly diminished.

The goal in developing the HS model is to provide an extension to the standard model that can accommodate assets whose returns are fat-tail distributed. In this section we compare the HS model to the standard model using real data in an effort to corroborate our claim.

Our metric for how close simulated prices match the observed will be the mean absolute percentage error (MAPE) ( [

M ( A , O ) = 100 × 1 n ∑ t = 1 n | A t − O t A t | (11)

where n is the number of simulated days, A t is the observed price on day t, and O t is the simulated price on day t. Of course since we are drawing randomly from distributions, any given simulated price series will yield a different M. We therefore create a set of k = 100,000 simulated price series for each model and compare the average M from those trials. That is, we compare

M avg , D ( λ ) = 1 k ∑ i = 1 k M ( SIM D ( λ ) , OBS ) (12)

where D is the distribution (either N for normal or ST for Student’s T), SIM D ( λ ) i is a simulated price series obtained using Equation (9) (for a given λ and distribution), and OBS is the observed price series.

As an example, we use closing values of the Dow Jones Industrial Average (DJIA) for 1000 days starting on January 1 2010 as our observed price series. The returns along with the Normal and Student’s T best fit are shown in the first panel of

We begin by determining the saturation parameter, λ 0 , that gives the best fit between the Student’s T simulated price series and the observed DJIA price series. We do this by using Equation (12) as a function of λ in a standard optimization technique (i.e., Brent’s Method [

• M avg , N ( 0 ) (i.e., standard model, Equation (9) with λ = 0 and η drawn from the normal distribution with parameters given in

• M avg , S T ( 0 ) (i.e., Student’s T distribution directly inserted in the standard

Normal | Student’s T | ||||||
---|---|---|---|---|---|---|---|

σ ( × 10 3 ) | μ ( × 10 3 ) | χ 2 | ν | β ( × 10 3 ) | μ ( × 10 3 ) | χ 2 | |

DJIA | 7.18 (0.82) | 0.82 (0.69) | 3.28 | 2.65 (1.55) | 5.72 (0.79) | 0.85 (0.85) | 1.81 |

TRP | 9.56 (0.69) | 0.43 (0.56) | 1.23 | 4.99 (2.34) | 8.40 (0.59) | 0.33 (0.48) | 0.57 |

PAGG | 10.75 (1.05) | 0.58 (0.85) | 2.31 | 3.87 (2.70) | 9.13 (1.13) | 0.42 (0.90) | 1.60 |

VBF | 6.09 (0.80) | 0.04 (0.71) | 4.15 | 1.69 (0.95) | 4.66 (0.74) | -0.04 (0.59) | 1.96 |

PNW | 8.86 (0.78) | 0.71 (0.65) | 1.88 | 3.88 (2.49) | 7.63 (0.81) | 0.46 (0.90) | 1.15 |

HBCP | 10.61 (1.11) | 0.37 (0.92) | 2.70 | 2.61 (1.41) | 8.49 (1.10) | 0.14 (0.86) | 1.51 |

model).

• M avg , S T ( λ 0 ) (i.e., HS model, Equation (9) with λ = λ 0 and η drawn from the Student’s T distribution with parameters given in

The results for the DJIA are given in the first row of

To test these ideas further, we have randomly selected five stocks from the NYSE and NASDAQ exchanges; TRP, PAGG, VBF, PNW and HBCP. The results from these additional stocks are given in

In 4 out of the 6 cases, the standard model performs relatively poorly. The exceptions are TRP with M avg = 19.3 % and VBF with M avg = 13.5 % . For TRP this is expected since its returns are actually well fit by a normal distribution. VBF, on the other hand, is poorly fit by a normal distribution so its low M avg is surprising.

In all cases, except marginally PNW, simply substituting a fat-tailed distribution into the standard model results in a higher M avg than the standard model. This, as we mentioned previously, is due to the large price swings inherent

Stock | λ 0 | M avg , N ( 0 ) | M avg , S T ( 0 ) | M avg , S T ( λ 0 ) |
---|---|---|---|---|

DJIA | 0.000204 | 33.6 (3.3, 196.0) | 42.4 (4.3, 118.2) | 7.5 (2.7, 177.6) |

TRP | 0.037544 | 19.3 (4.0, 155.3) | 20.3 (4.2, 187.7) | 11.9 (2.9, 38.7) |

PAGG | 0.266817 | 34.4 (4.5, 271.2) | 31.3 (5.4, 331.1) | 8.0 (5.8, 17.1) |

VBF | 0.134320^{1} | 13.5 (3.1, 94.4) | 42.1 (4.7, 500699.3) | 13.2 (2.6, 627.2) |

PNW | 0.022735 | 20.5 (3.3, 166.1) | 20.2 (3.8, 200.8) | 13.3 (2.7, 48.2) |

HBCP | 0.157830 | 19.4 (4.0, 144.0) | 28.8 (5.2, 991.1) | 19.1 (2.9, 270.8) |

in fat-tailed distributions, and is the motivation for this paper.

Finally in all 6 cases, the HS model performs better than the standard model at predicting future stock prices. The most surprising result here is that for VBF the HS model ( M avg = 13.2 % ) outperforms the standard model ( M avg = 13.5 % ) by only a slight margin. This is unexpected since VBF returns are fit much better by a fat ( ν = 1.69 ) Student’s T distribution. A clue may lie in the exceptionally high maximum M score for VBF (~600%) indicating that large price fluctuations are still present. This, however, is not nearly as large as the maximum M score of ~500,000% for a Student’s T distribution in the standard model without saturation. This may signal a limitation of the HS model in that although the addition of the saturation parameter, λ , reduces the effects of large price swings for fat-tailed distributions, it does not eliminate them altogether. This limitation is most pronounced for exceptionally fat-tailed distributions where the probability of large returns is likely.

The standard model, Equation (1), is the most popular model for stock price behaviour. It has wide reaching influence, most notably in the development of the Black-Scholes formula for option pricing ( [

• Daily stock returns are indeed better fitted by a fat-tailed distribution (Student’s T in this case) than by a normal (Gaussian) distribution.

• The standard model, in most cases, performs poorly when predicting future stock prices.

• Fat-tailed distributions, when substituted directly into the standard model ( λ = 0 ), perform poorly when predicting future stock prices.

• A non-zero saturation parameter, λ , when added to the standard model is able to effectively suppress large price swings inherent in fat-tailed distributions for daily returns. λ represents the fraction of money removed from the reservoir due to the purchase of the stock. In the standard model λ = 0 which implies an infinite amount of money available for the purchase of the stock.

• The HS model ( λ ≠ 0 ) developed in this paper is able to incorporate fat-tailed distributed returns, and consistently outperforms the standard model when predicting future stock prices.

We have shown that a model for stock price behaviour using fat-tailed distributions with support as large as [ − ∞ , + ∞ ] for daily returns is attainable. This model is realistic in that it allows for a finite supply of money to be saturated by the net rate of transactions and does not require truncation or capping [

This work was funded in part by the Natural Sciences and Engineering Research Council of Canada.

Koning, N., Cassidy, D.T. and Ouyed, R. (2018) Extended Model of Stock Price Behaviour. Journal of Mathematical Finance, 8, 1-13. https://doi.org/10.4236/jmf.2018.81001

A Student’s T distribution is a fat-tailed distribution with a probability density function

f ( t ) = 1 β π ν Γ ( ν + 1 2 ) Γ ( ν 2 ) ( 1 + ( t − μ ) 2 ν β 2 ) − ν + 1 2 (A.1)

where Γ is the gamma function, μ is the mean, β is the scale parameter, and ν is the shape parameter.

σ = β ν ν − 2 for ν > 2 .

We wish to show that solutions S ( ⋅ ) to the differential Equation (9) have the property S ( t + 1 ) = S ( t ) + S t + 1 ( 1 ) .

Add a second argument to S ( t ) and w ( t ) to indicate the time origin:

w ( a , b ) = ∫ a b η ( ξ ) d ξ (B.1)

and

S ( a , b ) = S ( a , a ) e w ( a , b ) e x p ( λ ( S ( a , b ) − S ( a , a ) ) ) . (B.2)

S ( a , a ) equals S ( a ) and is the value of S at a. S ( a , b ) is the value of S ( t ) | t = b given the initial value at t = a was S ( a ) = S a .

Start with S ( a , b ) + S ( b , c ) and use Equation (B.2) to simplify:

S ( a , b ) + S ( b , c ) (B.3)

= S a e w ( a , b ) exp ( λ ( S ( a , b ) − S a ) ) + S b e w ( b , c ) exp ( λ ( S ( b , c ) − S b ) ) (B.4)

= exp ( λ ( S ( b , c ) − S b ) ) S a e w ( a , b ) + exp ( λ ( S ( a , b ) − S a ) ) S b e w ( b , c ) exp ( λ ( S ( a , b ) + S ( b , c ) − ( S a + S b ) ) ) (B.5)

= S b e w ( b , c ) S ( b , c ) S a e w ( a , b ) + S a e w ( a , b ) S ( a , b ) S b e w ( b , c ) exp ( λ ( S ( a , b ) + S ( b , c ) − ( S a + S b ) ) ) (B.6)

= S a S b e w ( a , c ) ( 1 S ( b , c ) + 1 S ( a , b ) ) exp ( λ ( S ( a , b ) + S ( b , c ) − ( S a + S b ) ) ) (B.7)

= ( S ( a , b ) + S ( b , c ) ) e w ( a , c ) ( S a S b S ( a , b ) S ( b , c ) ) exp ( λ ( S ( a , b ) + S ( b , c ) − ( S a + S b ) ) ) . (B.8)

But S b = S ( a , b ) , by definition, since S b is the value of S ( t ) at time t = b , a < b < c , given that the initial value for the time series is S a . Then,

S ( a , b ) + S ( b , c ) (B.9)

= ( S ( a , b ) + S ( b , c ) ) e w ( a , c ) ( S a S ( b , c ) ) exp ( λ ( S ( b , c ) − S a ) ) (B.10)

or

S ( b , c ) = S a e w ( a , c ) exp ( λ ( S ( b , c ) − S a ) ) . (B.11)

S ( b , c ) is the value of S ( t ) | t = c given a value of S b at time t = b . The

interim value b does not appear in Equation (B.11). If one replaces S ( b , c ) in Equation (B.11) with S ( a , c ) = S c , since the initial value for the time series is S a , then one has

S ( a , b ) + S ( b , c ) = S ( a , c ) = S c = S a e w ( a , c ) exp ( λ ( S c − S a ) ) . (B.12)

This completes the demonstration and illustrates a useful property of the homogeneously saturated solution. It is possible to simulate a time series by adding increments to the series or by solving for the value over the full time interval. If one chooses to add increments, then one must be careful to use the appropriate initial condition for the increment. This appropriate initial condition is the value just before the increment starts.

^{1} M avg , S T ( λ 0 ) does not yield a minimum value for the VBF data. This is because the low ν parameter for VBF ensures that extreme price swings are realized even when the saturation parameter is large (see text). Essentially λ = ∞ , a straight line, gives the lowest average. The minimum M was used in the optimization to find λ 0 for this case only.