Method and system for modeling financial markets and assets using fractal activity time

A method for computing a financial value of at least one financial instrument, the at least one financial instrument having a price history representable at least in part by a random process, the method comprising:

• providing a sequence of data points derived from values of the random process obtained from a number of observations thereof;
• computing at least one of a sequence of squares of the data points and a sequence of magnitudes of the data points;
• computing an autocorrelation function of said at least one of a sequence of squares and a sequence of magnitudes;
• and
• if the autocorrelation function does not decrease by more than a threshold factor over a range no greater than a threshold number of observations of the random process, performing the steps of:
• using the sequence of data points to generate time-parameter data corresponding to an approximation of a fractal activity process;
• using the time-parameter data to determine whether the random process exhibits at least one of fractal scaling and self-similarity;
• and
• if the random process exhibits at least one of fractal scaling and self-similarity, determining that a model based upon the time-parameter data is valid for modeling the random process for computing the financial value of the at least one financial instrument.

A method for computing a financial value of at least one financial instrument, the at least one financial instrument having a price history representable at least in part by a random process, the method comprising:

• providing time-parameter data corresponding to an approximation of a fractal activity time process;
• using the time-parameter data as a time parameter of a model of the random process;
• and
• using the model for modeling the random process for computing the financial value of the at least one financial instrument.

A method for using a computer to compute a financial value of at least one financial instrument, the at least one financial instrument having a price history representable at least in part by a random process, the computer having a memory and a processor in communication with the memory, the method comprising:

• storing, in the memory, time-parameter data corresponding to an approximation of a fractal activity time process;
• using the processor to analyze the time-parameter data to allow determination of whether the random process exhibits at least one of fractal scaling and self-similarity;
• and
• if the random process exhibits at least one of fractal scaling and self-similarity, determining that a model based upon the time-parameter data is valid for modeling the random process for computing the financial value of the at least one financial instrument.

A method for using a computer to compute a financial value of at least one financial instrument, the at least one financial instrument having a price history representable at least in part by a random process, the computer having a memory and a processor in communication with the memory, the method comprising:

• storing, in the memory, a sequence of data points derived from values of the random process obtained from a number of observations thereof;
• using the processor to compute at least one of a sequence of squares of the data points and a sequence of magnitudes of the data points;
• using the processor to compute an autocorrelation function of said at least one of the sequence of squares and the sequence of magnitudes;
• and
• if the autocorrelation function does not decrease by more than a threshold factor over a range no greater than a threshold number of observations of the random process, performing the steps of:
• using the processor to generate, from the sequence of data points, time-parameter data corresponding to an approximation of a fractal activity time process;
• using the processor to analyze the time-parameter data to allow determination of whether the random process exhibits at least one of fractal scaling and self-similarity;
• and
• if the random process exhibits at least one of fractal scaling and self-similarity, determining that a model based upon the time-parameter data is valid for modeling the random process for computing the financial value of the at least one financial instrument.

A system for computing a financial value of at least one financial instrument, the at least one financial instrument having a price history representable at least in part by a random process, the system comprising:

• a memory for storing a sequence of data points derived from values of the random process through a number of observations thereof;
• and
• a processor in communication with the memory programmed to compute at least one of a sequence of squares of the data points and a sequence of magnitudes of the data points, to compute an autocorrelation function of said at least one of the sequence of squares and the sequence of magnitudes, to determine whether the autocorrelation function decreases by more than a threshold factor over a range no greater than a threshold number of observations, to generate using the sequence of data points time-parameter data corresponding to an approximation of a fractal activity time process, wherein the processor is programmed to generate the time-parameter data if the processor determines that the autocorrelation function does not decrease by more than said threshold factor over said range, the time-parameter data being used to make a determination of whether the random process exhibits at least one of fractal scaling and self-similarity the determination being made if the processor determines that the autocorrelation function does not decrease by more than said threshold factor over said range, the determination of whether the random process exhibits at least one of fractal scaling and self-similarity being for determining whether a model based upon the time-parameter data is valid for modeling the random process for computing the financial value of the at least one financial instrument.

A method for computing a financial value of at least one financial instrument, the at least one financial instrument having a price history representable at least in part by a random process, the method comprising:

• a. deriving a vector u from return data {Pkδ, k=0, 1, . . . } of said random process, the return data {Pkδ, k=0, 1, . . . } representing at least a portion of the price history of the at least one financial instrument. where k is a vector index and δ is a time step size;
• b. calculating logarithms of respective elements of u to derive a vector v=log u;
• c. calculating differences of pairs of elements log Pkδ and log P(k−1)δ of v to thereby generate a vector x representing a data set {Xkδ, k=0, 1, . . . }, wherein Xkδ=log Pkδ−log P(k−1)δ, wherein kδ represents time, and wherein elements within each of said pairs are separated by a lag of δ;
• d. calculating absolute values of elements of x to thereby generate a vector abs(x);
• e. generating data representing an autocovariance function of x, comprising:
• i. selecting a first plurality of values of an autocovariance lag parameter n, and
• ii. based upon each of said first plurality of values of n, calculating an autocovariance acf(x,n);
• f. determining, from said autocovariance function of x, how rapidly said autocovariance function of x diminishes with respect to n;
• g. generating data representing an autocovariance function of the absolute value of x abs(x), comprising:
• i. selecting a second plurality of values of n, and
• ii. based upon each of said second plurality of values of n, calculating acf(abs(x), n);
• h. determining, from said autocovariance function of abs(x), how rapidly said autocovariance function of abs(x) diminishes with respect to n;
• i. generating a standardized array z=((x)−mean(x))/(var(x))1/2, wherein mean(x) denotes a mean of x, and wherein var(x) denotes a variance of x;
• j. calculating an exponent test function mean((abs(z))r), for a plurality of values of r, wherein r is increased until the test function begins to increase rapidly with respect to r, thereby determining a maximum exponent value p for which the exponent test function converges;
• k. calculating a fractal activity time vector w=cumsum(z2) representing a data set {{circumflex over (T)}kδ, k=1,2, . . . }, wherein cumsum(z2) denotes a cumulative sum of z2 with respect to index k;
• l. calculating a model test function Ŝq(m) for a plurality of combinations of values of m and q, wherein 0<q≦p, comprising, for each of said plurality of combinations of m and q, performing the steps of:
• i. generating a vector um=diff(w, lag=m), where diff(w, lag=m) represents a data set {{circumflex over (T)}kδ−{circumflex over (T)}(k−m)δ},
• ii. generating a vector vm=abs((um)−mδ), and
• iii. calculating Ŝq(m)=mean ((vm)q);
• m. using values of Ŝq(m) obtained from step l to determine linearity of log Ŝq(m) with respect to log m, to thereby determine whether the random process exhibits fractal scaling;
• n. calculating Eq=mean(abs(u1−δ))q for a plurality of values of q, wherein u1 is the vector um with m=1;
• o. using values of Ŝq(m) obtained from step l, and values of Eq obtained from step n, to determine linearity of Fq=log Ŝq(m)−log Eq with respect to q, to thereby determine whether the random process exhibits self-similarity;
• and
• p. confirming constancy of a slope of Fq with respect to q, in order to verify that the fractal activity time vector w accurately models a time parameter T in an equation Pt=Po exp[μt+σW(T)], where Pt represents a value of the random process at time t, Po is an initial value of Pt, μ and σ are constants, and W(T) is a standard Brownian motion function of T, and the financial value of the at least one financial instrument at time t comprises Pt.

A method for computing a financial value of at least one financial instrument, the at least one financial instrument having a price history representable at least in part by a random process, the method comprising:

• providing time parameter data having a distribution represented by a probability density of the form g v ⁢ ( x ) = ( ( v - 2 ) / 2 ) v / 2 Γ ⁢ ( v / 2 ) ⁢ x - v / 2 - 1 ⁢ e - ( v - 2 ) / 2 ⁢ x ,
• where x>0, v>2, Γ is the gamma function, v is a fixed parameter estimated from the time parameter data and x is a dummy variable defining the gv(x) over the interval (0, ∞);
• using the time-parameter data to determine whether the random process exhibits at least one of fractal scaling and self-similarity;
• and
• if the random process exhibits at least one of fractal scaling and self-similarity, determining that a model based upon the time-parameter data is valid for modeling the random process for computing the financial value of the at least one financial instrument.

A method for using a computer to compute a financial value of at least one financial instrument, the at least one financial instrument having a price history representable at least in part by a random process, the computer having a memory and a processor in communication with the memory, the method comprising:

• storing, in the memory, time-parameter data having a distribution represented by a probability density of the form g v ⁢ ( x ) = ( ( v - 2 ) / 2 ) v / 2 Γ ⁢ ( v / 2 ) ⁢ x - v / 2 - 1 ⁢ e - ( v - 2 ) / 2 ⁢ x ,
• where x>0, v>2, Γ is the gamma function, v is a fixed parameter estimated from the time parameter data and x is a dummy variable defining gv(x) over the interval (0, ∞);
• using the processor to analyze the time-parameter data to allow determination of whether the random process exhibits at least one of fractal scaling and self-similarity;
• and
• if the random process exhibits at least one of fractal scaling and self-similarity, determining that a model based upon the time-parameter data is valid for modeling the random process for computing the financial value of the at least one financial instrument.

A system for computing a financial value of at least one financial instrument, the at least one financial instrument having a price history representable at least in part by a random process, the system comprising:

• a memory for storing time-parameter data having a distribution represented by a probability density of the form g v ⁢ ( x ) = ( ( v - 2 ) / 2 ) v / 2 Γ ⁢ ( v / 2 ) ⁢ x - v / 2 - 1 ⁢ e - ( v - 2 ) / 2 ⁢ x ,
• where x>0, v>2, Γ is the gamma function, v is a fixed parameter estimated from the time parameter data and x is a dummy variable defining gv(x) over the interval (0, ∞);
• and
• a processor in communication with the memory programmed to analyze the time-parameter data to allow determination of whether the random process exhibits at least one of fractal scaling and self-similarity, the determination of whether the random process exhibits at least one of fractal scaling and self-similarity being for determining whether a model based upon the time-parameter data is valid for modeling the random process for computing the financial value of the at least one financial instrument.

A method for valuating a call option for a traded financial asset where time parameter data generated from price data of the asset over time exhibits at least one of fractal scaling and self similarity, the call option giving a holder the right, but not the obligation, to purchase the asset for an exercise price K at a maturity time t1, the method comprising the steps of determining the value C(t,t1,K) of the call option at time t using the relation

• C(t,t1,K)=PtE[Φ(d(t1,t))]−K exp(r(t−t1))E[Φ(d(t1,t)−σ(Tt1−Tt)1/2],
• where Φ ⁢ ( x ) = 1 2 ⁢ π ⁢ ∫ - ∞ x ⁢ e - 1 2 ⁢ u 2 ⁢ ⁢ ⅆ u
• is the distribution function of the standard normal and r is a fixed interest rate for bonds and d ⁡ ( t 1 , t ) = log ⁡ ( P t ⁢ / ⁢ K ) + r ⁡ ( t 1 - t ) + 1 2 ⁢ σ 2 ⁡ ( T t 1 - T t ) σ ⁡ ( T t 1 - T t ) 1 2
• with the expectation being taken over the distribution Tt1−Tt, where
• Tt1−Ttdt1−t+(t1−t)κ(T1−1),
• where d denotes equality in distribution and κ is a fractal scaling parameter, T1 having a scaled generalized inverse Gaussian distribution having a probability density of the form g v ⁢ ( x ) = ( ( v - 2 ) / 2 ) v / 2 Γ ⁢ ( v / 2 ) ⁢ x - v / 2 - 1 ⁢ e - ( v - 2 ) / 2 ⁢ x ,
• where x>0, v>2, Γ is the gamma function and v is a fixed parameter estimated from the time parameter data and x is a dummy variable defining gv(x) over the interval (0, ∞) .

A method for valuating a put option for a traded asset where time parameter data generated from price data of the asset over time exhibits at least one of fractal scaling and self similarity, the put option giving a holder the right, but not the obligation, to sell the asset at an exercise price K at a maturity time t1, the method comprising the steps of determining the price P(t,t1,K), at time t, of the put option using the relation

• Pt+P(t,t1,K)−C(t,t1,K)=Ke−r(t1t),
• where r is a fixed interest rate for bonds, Pt is the price of the asset at time t and C(t,t1,K) is the value of a call option for the same asset at time t, the call option giving a holder the right, but not the obligation, to purchase the security for an exercise price K at a maturity time t1, and is given by the relation
• C(t,t1,K)=PtE[Φd(t1,t))]−K exp(r(t−t1))E[Φ(d(t1,t)−σ(Tt1−Tt)1/2],
• where Φ ⁢ ( x ) = 1 2 ⁢ π ⁢ ∫ - ∞ x ⁢ e - 1 2 ⁢ u 2 ⁢ ⁢ ⅆ u
• is the distribution function of the standard normal, r is a fixed interest rate for bonds and d ⁡ ( t 1 , t ) = log ⁡ ( P t ⁢ / ⁢ K ) + r ⁡ ( t 1 - t ) + 1 2 ⁢ σ 2 ⁡ ( T t 1 - T t ) σ ⁡ ( T t 1 - T t ) 1 2
• with the expectation being taken over the distribution Tt1−Tt, where
• Tt1−Ttdt1−t+(t1−t)κ(T1−1),
• where d denotes equality of distribution and κ is a fractal scaling parameter, T1 having a scaled generalized inverse Gaussian distribution having a probability density of the form g v = ( ( v - 2 ) / 2 ) v / 2 Γ ⁢ ( v / 2 ) ⁢ x - v / 2 - 1 ⁢ e - ( v - 2 ) / 2 ⁢ x ,
• where x>0, v>2, Γ is the gamma function, v is a fixed parameter estimated from the time parameter data and x is a dummy variable defining gv(x) over the interval (0, ∞).