STOCK OPTION PRICING

Field of the Invention

The invention relates to methods for the valuation and pricing of tradable financial instruments, and has particular application to financial derivatives in futures markets, such as call and put options, and especially to so-called "European-Style" options.

Background and Prior Art Known to the Applicant

Financial derivatives, where options to buy or sell a stock at a future date, and for a given price, are well known. A challenge for the financial markets has always been to determine, in a rational way, the value — and therefore the price - of such an option, hi the 1970's, Fischer Black and Myron Scholes developed an equation, known now as the "Black-Scholes" formula, to calculate the value of European call and put options

(Black, F., and M. Scholes. The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81 (1973), 637-654.). This formula is now widely used by the financial markets.

Underlying the Black-Scholes formula is an assumption that variations in stock prices follow a Gaussian distribution. However, evidence shows that the distributions of price returns are not Gaussian, but that they diminish slowly at the high end - the so-called "fat tails". Section 1 of the Technical Annex to this specification describes the background to the problem more fully, whilst Section 4 provides a summary of recent literature in the field.

Whilst the Black-Scholes formula has served the financial community well, there are occasions where the mismatch between the statistical reality of the market behaviour and the assumptions underlying the Black-Scholes formula leads to erroneous valuations. It is an object of the present invention to attempt to provide a more robust method for valuing financial derivatives.

Summary of the Invention

Accordingly, the invention provides a method of estimating the price, C(x; t), of a European-Style call option on a stock, for a given log price, x = log(S _t ), at the present time, t, and at future time, T, comprising the steps of:

(a) estimating the drift, a , of the log stock price, log(S _t );

(b) estimating the tail index, D μ , and the volatility, σ , of the fluctuations of the log stock price; (c) computing the price of the said call option using equation [122].

The invention also provides a method of estimating the price, C(x; t), of a European-Style call option on a stock, for a given log price, x = log(S _t ), at the present time, t, and at future time, T, comprising the steps of: (a) estimating the drift, a , of the log stock price, log(S _t );

(b) estimating the tail index, D μ , and the volatility, σ , of the fluctuations of the log stock price;

(c) computing the price of the said call option using equations [168] and [169].

The invention also provides a method of estimating the prices C(S \ , t) of European-Style call options on a portfolio of correlated stocks, comprising the steps of:

(a) estimating the parameters of the multivariate distribution of fluctuations of the stock prices, S ₁ ;

(b) computing the prices of the said call options using equation [192].

The invention also provides a method of estimating the prices C(S _t ,t) of European-Style call options on a portfolio of uncorrelated stocks, comprising the steps of:

(a) estimating the parameters of the multivariate distribution of fluctuations of the stock prices, S ₁ ; (b) computing the prices of the said call options using equation [194].

The invention also provides a method of estimating the prices of a European-Style put option, comprising the steps of:

(a) estimating the price of a corresponding call option, according to either claim 1 or claim 2;

(b) computing the price of the said put option using the principle (known per se) of call-put parity.

The invention also provides a method of estimating the prices of a European-Style put option according to claim 5, wherein the relationship between the put price of the stock at time t, P(x,t), and the call price of the stock at time t, C(x,t), is given by:

P(x, 0 = C(x,t) -S _t +K exp(-r(r-0) where:

S is the stock price K is the strike price r is the riskless interest rate

T is the expiry time of the option.

Included within the scope of the invention is a method of estimating the prices of European-Style call, or put options substantially as described herein, with reference to and as illustrated by any appropriate combination of the accompanying drawings.

Also included within the scope of the invention are methods for estimating prices of call or put options using equations functionally equivalent to those described above.

Details of the invention are provided in the following Technical Annex, in a form that will enable the skilled addressee to implement the invention. In particular, section 3.7 of the Technical Annex provides details of embodiments of the invention showing how parameters for the distribution of fluctuations of stock prices may be calculated.

Whilst the embodiments described relate to non-dividend paying stocks, it will be apparent to the skilled addressee how to make the necessary amendments in these circumstances. (For details, see e.g. "Investments", William F. Sharpe, Gordon J. Alexander and Jeffery V. Baley (Eds.), ISBN 013011507X). The use of the methodology for dividend-paying stocks is also, therefore, within the scope of the present invention.

Further areas of applicability of the present invention will become apparent from the detailed description provided hereinafter in the Technical Annex. It should be understood that the detailed description and specific examples, while indicating the preferred embodiment of the invention, are intended for purposes of illustration only, since various changes and modifications within the spirit and scope of the invention will become apparent to those skilled in the art from this detailed description.

In some rare circumstances, determination of the value of the tail index, D μ, might lead to a value of 2 for this parameter. In this case, the stochastic variation of the stock is Gaussian distributed, and the methodology reduces to the familiar and known Black- Scholes equation. Embodiments of the disclosed methodology having this parameter value are therefore specifically disclaimed from the scope of the invention.

It should be noted that, whilst the total value of the portfolio is real, the option prices computed from equation (122) are complex numbers. It follows from equation (129) that this property ensures that the portfolio is "risk free". From the definition of the portfolio in equation (46) it should be clear that the real part of the option price has a meaning of the price.

It should further be noted that all formulas in the specification take into account the so- called stylised facts of financial data:

(a) The Hurst exponent, i.e. the exponent related to the time dependence of the expectation value of the difference of the extremal values of prices in the time interval t, is different from two and is a fractional number. This may be included by noting that for a stable process L _t the expectation value, E |L _t+τ - LtP is proportional to τ™ ^/μ ' and independent of the time t, providing β<μ.

(b) Time series exhibit volatility clustering, i.e. the standard deviation of log prices as a function of time exhibits sudden outbursts at different time scales. Each outburst is followed by a region of increased fluctuations that endure some time

(the volatility has a memory). These outbursts of volatility are related to large fluctuations of the log-price.

For estimation of the parameters of multivariate distribution of fluctuations on portfolios of stocks (see Section 3.7 of Technical Annex), many approximate methods for estimating the correlation parameters in such multivariate stock data are available. (See, for example, "Limit Distributions for Sums of Independent Random Vectors", Meerschaert M, Scheffler H-P, John Wiley and Sons 2001) These methods are based on knowledge of the asymptotic limit of distributions of sample covariance matrices in the limit that the size of the matrices tends to infinity. In this limit we know exactly the distributions of the estimators for our parameters and thus these estimators can be used to assess the parameters of the multivariate distribution, |σ .

The term "stock" is used in this specification to indicate a tradable asset with a fluctuating price; the term, and scope of the claims, thus includes both shares and commodities.

TECHNICAL ANNEX

Abstract

We model the logarithm of the price (log-price) of a financial asset as a sum of independent, identically distributed random variables obtained by projecting an operator stable random vector with a scaling index matrix J£ onto a non-random vector. The scaling index E_ models prices of the individual financial assets (stocks, mutual funds, etc.). We find the functional form of the characteristic function of real powers of the price returns and we compute the expectation value of these real powers and we speculate on the utility of these results for statistical inference. Finally we consider a portfolio composed of an asset and an option on that asset. We derive the characteristic function of the deviation of the portfolio, S^ , defined as a temporal change of the portfolio diminished by the the compound interest earned. We derive pseudo-differential equations for the option as a function of the log-stock-price and time and we find exact closed- form solutions to that equation. These results were not known before. Finally we discuss how our solutions correspond to other approximate results known from literature,in particular to the well known Black & Scholes equation.

Key words and phrases: Option pricing, heavy tails , operator stable, fractional calculus.

1 Introduction

Early statistical models of financial markets assumed a Gaussian distribution [I]. However, there is evidence [2] that price returns are not Gaussian distributed and that the distributions diminish slowly in the high end (fat tails). Except at very high frequencies or short times ([2]), a better statistical description for many

financial assets is provided by a model where the logarithm of the price is a heavy tailed one-dimensional Levy /i-stable process [3, 4, 5, 6]. Since the tail parameter μ that measures the probability of large price jumps will vary from one financial asset to the next, a model based on operator stable Levy processes [7] is appropriate. This model allows the tail index to differ for each financial asset in the portfolio. Hence we formulate a model where the log-price is the sum of a large number of projections of multi-dimensional operator stable jumps onto a predefined direction (this projection determines the portfolio mix). The cumulative probability distribution of the log-price diminishes as a mixture of power laws. Thus the higher-order moments of the distribution may not exist and the characteristic function of the distribution may not be analytic.

In section (2) we formulate the model of the financial asset and recall some facts from the theory of stochastic processes. In particular we derive new results concerned with statistical inference in operator stable populations. In section (3) we focus on a portfolio that also includes an a stock and an option and we derive a new pseudo-differential equation for the dependence of the option on the stock price and on time. An essential feature of our approach is that we only assume the temporal change (deviation) of the portfolio is a martingale. We do not assume the individual asset prices within the portfolio are martingales as is usually the case. Thus our model is essentially different from the common approaches in mathematical finance [8] where the price of the stock is assumed to have finite moments. In section 4 we discuss, in more detail, these differences from models evaluated by other authors. We have postponed the literature review to the final section purposedly. We want to demonstrate in this way that our approach is new and exact when compared to existing methods that axe only approximations usually derived under inconsistent assumptions. The conclusions are summarized in section 5.

2 The model of the stock market

In this section we define the model. In the following we recall certain known properties of operator stable distributions and we derive Fourier transforms of real powers of projections of operator stable vectors onto a non-random vector. In subsections (2.2) and (2.3) we derive Fourier transforms of operator stable random vectors for particular forms of parameters of the distribution.

2.1 The basic properties and new results

Let log(St) be the logarithm of the price of the portfolio (log-price) at time t. We assume that the temporal change of the log-price is composed of two terms, a deterministic term and a fluctuation term viz: dlog{S _t ) — \og{S _i+dt ) - log(S _t ) = adt + σ ■ dL _t (1)

The parameters a e R (the drift) and the elements of the D dimensional vector σ := (σi, .. . , σ _∑> ) (the portfolio mix) are assumed to be non-random constants. The random vector L _t is (strictly) operator stable, meaning that it is an operator-normalized limit of a sum of some independent, identically distributed (ϋd) random vector price jumps X ₂ . We have

= J dkδ{k + ι)P _5.lit [k) = J dkδ(k + τ)i> _ff _ _Ldi (k) = e- ^dt ^-^ (53)

K -ι+K

In the first equality in (52) we inserted a delta function into the definition of the expectation value, in the second equality we used the integral representation of the delta function, in the first equality in (53) we integrated over z and ξ and we used (238), in the second equality we shifted the integration line by using the Gauchy theorem applied to a rectangle [-R, R] U R + ι[0, l] U -ι + [R, R] V R - τ[0, 1] in the limit R → oo and in the last equality we used (9) and (7).

In financial mathematics one constructs martingales from the price of the stock in two ways, viz by subtracting the sum of conditional means E [S _t — St- at \St- _d t] over t from S _t (natural martingale) or by transforming the log-characteristic function of the jumps in order to ensure the absence of drift (Esscher transformed martingale) ⁴ . It will be possible to construct a natural martingale however we will proceed in a different way.

We will therefore construct a zero-expectation value stochastic process (47) as a linear combination (46) of two stochastic processes St and C(S _t ;t) that have both non-zero expectations values. For this purpose we will analyze the probability distribution of the deviation variable S _j and work out conditions for the option price such that the conditional expectation value E S^ \S _t \ is equal zero. Now we compute the deviation of the portfolio:

= N ₃ (S _t+dt - S _t ) + (C _t+dt - Ct) + V(i) (1 - e ^Tdt ) (55)

= NsSt (e ^adt+s - ^l« - l) + (C(W; t + dt) ~ Ct) + V{t) (1 - e ^τdt ) (56)

= NsSt (e" ^dt+ *- ^Z« - ή + ^dt+ ∑ (e ^« *"* ^£ * - l) ^" + V(t) (l - e"*) (57)

In (57) we have expanded the price of the option in a Taylor series to the first order in time and to all orders in the price of the option. In that we have assumed that the price of the option is a perfectly smooth function of the price of the stock. This may limit the class of solutions. In particular, solutions may exist, where the price of the stock is a function satisfying the Holder condition: for any St+ _dt and St, a constant A and a Holder exponent A 6 [0, 1) and thus the price of the option can be expanded in a fractional Taylor series [14] in powers of St+dt — St. We will seek for these solutions in future work.

The process ©[ ' is a sum of infinitely many terms that have non-zero expectation values. We could compute its expectation value directly using (51) and re-sum the series. However, the characteristic function of the process T)\ conditioned on the value of the process means that we propagate the process St by an infinitesimal value dt and we compute the characteristic function of the increment and we require the zero value derivative of the characteristic function to be equal zero. This technique is not new, see discussion about solving master equations of Markov processes in

4 Recent literature

In an interesting paper, Hurst, Platen and Rachev [21] model the change of the logarithm of the stock price as a one-dimensional /z-stable process with mean zero. The process is obtained by subordinating a Wiener process to another process, namely an intrinsic time process), that is μ/2-stable with nonnegative increments. It is argued that when the stock price drift equals the riskless rate of interest there exists an equivalent probability measure such that the stock price process is a martingale. Under this new measure the price of the option equals the expectation value of the discounted payoff of the option at maturity. This expectation value is then computed by conditioning on the intrinsic time process and amounts to pricing the European style option by averaging the Gaussian Black & Scholes formula with respect to the time to maturity T — t which is also a μ/2-stable random variable.

In our opinion this particular change of probability measure is unrealistic and made essentially to manipulate the equations governing the stock price process using the rich mathematical machinery of martingales. However this has come at the expense of a modification of the underlying model (1) without explicitly stating the fact. Furthermore even though the model has been so changed, the authors state "..the volatility smile ..indicates that the assumptions of the Black and Scholes option pricing model are not fulfilled" . This fact is well known. Rather the conclusion should be that the modified model examined by the authors does not fit the data.

An exponential from a μ-stable Levy process (or of its many dimensional operator stable equivalent) is not a martingale. If the stock price is modeled as such it has a time-dependent expectation value (see (53)) . In our model we make no assumptions about the relationship between the drift a and the riskless rate of interest r and we nevertheless obtain an equation (80) that has solutions for all times t. Furthermore it appears that the solution of equation (80) is a good approximation to the solution of the risk hedging problem because, as shown in section (3.2.1), it ensures that the unconditional expectation value of the portfolio grows exponentially with time with finite modulations and thus shows that all markets characterized by log-stable fluctuations are complete, meaning that risk can be perfectly hedged, in contrary to what is claimed in [H]. An alternative approach is taken by Cartea and Howison [23] who analyse a model where the stock price is driven by damped one dimensional stable fluctuations. ⁵ . American perpetual put options (to be exercised at any time including T → oo) are priced as expectation values of the discounted payoff at maturity for the case where the fluctuations are maximally positively skewed. An infinite order PDE of the same form as the first equality in (80) is obtained. The coefficients are related to integer moments of AS and the existence of these coefficients is ensured by multiplying parameters σ and λ by At to some power. However as with result

⁵ This is a continuous time stochastic process whose log-characteristic function is that of a stable process multiplied by a decaying exponential with rate 1/λ

discussed above, these authors also show that the model yields "the volatility smile encountered.... when the Black and Scholes framework is employed". They further state that "the skew obtained in the implied volatility is a consequence of the absence of normality in the underlying stochastic process St" .

In our opinion these statements reveal that the approximations made to the original model are not adequate and in introducing their damping approximation the authors have lost the ability of Levy stable processes to deal correctly with non-normal fluctuations.

We have analyzed the model (1) without any modifications and the coefficients θn,χ)μ in (80) differ from those of Cartea and Howison (see last equation on page 23 in [23]).

McGulloch [24] has analyzed a model where the stock price is a ratio of two iid logarithmically-stable maximally negatively skewed random variables, namely the marginal utility of the asset, U2, and the marginal utility of the numeraire in which the asset is priced, U1. A Risk Neutral Measure (RNM) is introduced as the probability density of the stock price multiplied by a ratio of two expectation values: 1) expectation value of the numeraire conditioned on the value of the stock price and 2)the unconditional expectation value of the numeraire. The RNM is derived from general properties of stable variables and it is shown that the RNM itself is in fact not log-stable but is a convolution of two functions, one of which is log-stable. Call and put options are priced by Fourier transforming the discounted expectation values of maturity payoffs with respect to the RNM and inverting the results numerically. The author states that "it is not clear whether the (different option pricing) formulas give equivalent option values ..".

However, in the generic setting of operator stable fluctuations the RNM may not exist because the expectation value of symmetric stable fluctuations is zero EUi = 0. It is therefore not clear whether this concept may be used in the generic setting that we analyze in this paper. Since the author constructs the stock price as a ratio of two log-stable variables, the stock price itself is certainly log-stable and therefore the authors' option pricing procedure should appear as a special case of our formula (80) for some particular value of the stable index E. However, it is not clear if every stable variable can be represented as a difference of two extremal-ly skewed random variables though.

5 Conclusions

We have applied the technique of characteristic functions to the problem of pricing an option on a stock that is driven by operator stable fluctuations. We have developed a technique to ensure that the expectation value of the portfolio grows exponentialy with time. In doing this we have not, unlike other authors, made any assumptions about the analytic properties of the log-characteristic function of the stock price process. Instead we have expressed all results in terms of the characteristic function of the operator stable fluctuation L ₁ .

Subsequent to successful numerical tests, we ought then to be able to price analytically not only European options but also exotic options with a finite number of different exercise times. This should also allow us to price American style options by allowing the number of exercise times to become infinite

We may also compute the 99th percentile of the probability distribution of the deviation of the portfolio (Value at Risk) as a function of σ and of the log-characteristic function φ of the random vector L ₁ . The Value at Risk will be expressed as an integral equation involving the conditional characteristic function of the portfolio deviation (69). The resulting integrals will be carried out by means of the Cauchy complex

) )

[12] Bertoin J, Levy processes as Markov processes, in: Levy processes Cambridge University Press, 1996

[13] Boyarchenko, Svetlana I. and Levendorskϋ, Sergei Z., General Option Exercise Rules, with Appli- cations to Embedded Options and Monopolistic Expansion (October 30, 2005). Available at SSRN: http://ssrn.com/abstract=838624

[14] Samko S G, Kilbas A A, Marichev O I, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach Science Publishers S.A. 1993

[15] Dzherbashyan M M, Nersesyan A B, The criterion of the expansion of the functions to the Dirichlet series, Izv. Akad. Nauk Armyan. SSR Ser. Fiz.-Mat. Nauk, 11, no 5, 85-108

[16] Option Pricing for Gaussian, for non-Gaussian fluctuations, and for a fluctuating variance, chapters 20.4.3 — 20.4.5, 1416-1428 in: Klemert H, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific Publishing Co., Singapore 3rd edition (2004)

[17] Redner S, A guide to first passage processes, Cambridge University Press, 2001

[18] Kleinert H, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific Publishing Co., Singapore 3rd edition (2004)

[19] Kleinert H, Option Pricing from Path Integral for Non-Gaussian Fluctuations. Natural Martingale and Applications to Truncated Levy Distributions, preprint cond-mat/0202311

[20] Cont R and Tankσv P, Risk neutral modelling with exponential Levy processes, 353-379 in: Financial Modelling with Jump Processes, Chapman & Hall, Financial Mathematics Series, 2004

[21] Hurst S R, Platen E and Rachev S T, Option Pricing for a LogStable Asset Pricing Model, Mathematical and Computer Modelling 29, 105-119 (1999)

[22] Rachev S and Mittnik S, Stable Paretian Models in Finance, John Wiley & Sons 2000

[23] Cartea A ₁ Hσwinson S, Distinguished Limits of Levy Stable Processes, and Applications to Option Pricing, Oxford Financial Research Centre, No 2002mf04.

[24] McCulloch H J, The Risk-Neutral Measure and Option Pricing under Log-Stable Uncertainty, Econometric Society 2004 North American Winter Meetings 428, Econometric Society

Figure Captions

Figure 1: The integrand of the integral representation of the coefficients & _k , _d> in (84).

Figure 2: The same as in Fig. 1 but for the integral representation (85) for m = 2.

Figure 3: The coefficients ^Q k,φ/ \<?\ ^Dμ computed from the integral representation (84) as a function of k for Dμ = 1.9, 1.8, 1.7, 1.6, 1.5 from below to the above respectively.