Abstract
The method of pricing financial securities in discrete time, studied in the previous chapter, makes use of an arbitrary concept: that of the unitary holding period of financial securities. In fact, we considered two successive instants t and t + 1, put into place an arbitrage portfolio at instant t, then examined the rebalancing of this operation at instant t + 1, the interval of time between t and t + 1 being fixed and given. A procedure of this type excludes the possibility of rebalancing at an intermediate instant. In practice, however, investors are free to operate on the financial market at any time.1 A more natural procedure therefore involves using a continuous timescale. It is in this spirit that we showed in the previous chapter what the price limit of an option would be if we indefinitely reduced the interval of time between two successive instants. This procedure is, however, indirect. We shall now study, and apply to option pricing, a mathematical technique allowing us to reason directly in continuous time: the technique of stochastic differential calculus.
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Notes to Chapter 7
Of course they are sometimes prevented from doing so by transactions costs and by difficulty in finding a counterpart. But arbitrarily fixing the interval of time between two successive transactions will obviously not take account of these two phenomena.
Intuitively, we denote by random ‘process’ a random variable whose probability distribution changes over time. In more rigorous terms, this is a family of random variables indexed by a parameter. This parameter, in all the applications we consider, will be time. It belongs to the set of non-negative real numbers. The set of processes indexed by a real number is constructed from the set of processes indexed by an integer in the same manner as we construct the set of real numbers from the set of integers. We arrive at a fundamental theorem whose existence is due to Kolmogorov.
Discontinuous price processes are dealt with by [Cox and Ross 1976].
In precise terms, a process is said to be continuous when the probability that a trajectory of this process is a continuous function of time is equal to 1.
Cf. [Feller 1966], page 210, or [Métivier 1979] page 214.
[Bachelier 1900], [Einstein 1905, 1906].
See [Karatzas and Shreve 1988], page 82.
Reminder: a sum of multivariate normal random variables is a normal random variable.
And when the time difference between instants t and t + 1 is equal to one unit of time.
Here, we assume a division into equal intervals to keep the explanation simple. In fact, this is not necessary. The more general condition commonly imposed on a partition t i. i = l...n of the interval of time [t, t + τ](t n = t + τ) is that: Maxi. |t i − t i −1 → 0 when n → ∞.
See [Duffie 1988], p. 248 et seq., [Karatzas and Shreve 1988], p. 66 et seq. According to Donsker’s theorem, there is convergence towards a Brownian even if the probability distribution of the increases z t + 1−z t is not normal. By virtue of the Central Limit theorem, it is sufficient that the increases are independent and of the same variance.
From now on, the symbol σ2 indicates the variance per unit of time.
It is customary, in fact, to keep stochastic differential equations such as [1.7] in a form containing the differential elements dt and dz. It is not good practice to produce ‘derivatives’ by dividing both sides of Equation 1.7 by dt. The reason for this precaution is (see Appendix 1) that the trajectories of a Brownian are extremely erratic (non-smooth) and non-differentiable.
See Appendix 2 for the definition of the stochastic integral; in particular condition A2.4.
Cf [Arnold 1974], pages 105 ff.
There is, however, a significant difference between these two classes of process. Diffusion processes are, by definition, Markovian processes, whose movement depends only on the current value of the variable, whilst the class of Itö processes may be widened: a( ) and σ( ) may depend on past values of x.
But, purely on account of the fact that the drift depends on x, the variance of an increase in x over a finite interval of time will also depend on x.
And by using the expression of (dx) 2 which results from this: (dx) 2 = σ2 (x,t)dt.
An Itö’s Lemma also exists for functions of several random processes y =f(x 1, x 2, t). It brings in the covariance between the random parts of x 1 and x 2.
It is positive in Figure 7.4.
We should also ensure that the functions α and σ are such that stochastic differential equation 2.1 possesses one and only one solution, that is to say that they satisfy the growth condition and the Lipschitz condition.
When reproducing a call option b is negative. In fact, this is then a borrowing position on the risk-free security.
The true mathematical meaning of Equation 2.6 is that of the corresponding integral equation giving the increase in value of the portfolio over an interval of time [0, τ]: \( V_\tau - V_0 = \int_0^\tau {h_t dS_t } + \int_0^\tau {b_t D_t rDt = \int_0^\tau {\left[ {h_t S_t \alpha _t + b_t D_t r} \right]dt + } } \int_0^\tau {h_t S_t \sigma _t dz_t } \)It is the first term on the right-hand side of the first line of this equation, or the last term of the second line, which requires the stochastic integral with respect to a Brownian to be defined (see Appendix 2). For this integral to exist, the integrability condition [A2.4] must be imposed on the portfolio: As we show in Appendix 2, it is this condition which prevents the portions invested in the risky security from growing indefinitely and which forbids strategies of the ‘doubling strategy’ type (see the introduction to Part II). From an economic point of view, this is a condition of solvency.
This is a notable advantage of this argument over Black and Scholes’ original argument [Black and Scholes 1973]. Moreover, the present argument, as opposed to that of Black and Scholes, does not assume a priori that the price of the option is a twice differentiable function of the price of the underlying security. Here we constructed, without referring to the option, a twice differentiable function V. We then showed that the price of the option had to be identical to that of this function V.
See [Karatzas and Shreve 1988], page 267.
The change from drift α to drift r of the underlying asset induces a change in the probability distribution of the future values of S. If we know the ‘true’ probability distribution of these future values in the presence of drift a, we may obtain the ‘adjusted’ distribution resulting from a drift equal to r by means of the Girsanov theorem: see [Karatzas and Shreve 1988], page 190 or [Lipster and Shiryayev 1977], page 232.
Holding and storage costs may occur in relation to options on commodity or currency.
If we were in the presence of a tax system hitting dividends and capital gains differently, we could be less definite and assume that the drop in price is proportional to the dividend: we would then have to specify a coefficient of proportionality.
The opposite specification, in which the amount of dividend D actually paid on the date T is limited by the total assets available, can only be solved by numerical methods (Section 7.4).
See Geske’s article [Geske 1979a], or [Cox and Rubinstein 1985], page 414.
The ‘ordinary’ Geske formula, giving the price of an option on an option, calls in the bivariate normal distribution function, which must be calculated by numerical integration. The valuation of complex compound options ‘on several levels’ (an option on an option on an option...) requires numeric calculation of the multivariate normal law, which is extremely laborious.
An option gives the right to receive one security in exchange for another. Most often, one of the two securities is cash: a call is the right to receive a security in exchange for cash; a put is the right to receive cash in exchange for a security. We are speaking here of the dividend attached to the security that is received.
Recall that some options obey partial differential equation 3.3 rather than Equation 2.3.
Clearly, it is not the cash received itself which brings about the exercise before maturity. Without the plan to reinvest this cash, there would be no hurry to receive it.
Remember that the path of S is continuous. We assume here that: S(0) >S*(0), otherwise the option is exercised at instant 0.
The effective value of the put is then given by the curve to the right of point B and, to the left of this point, by the segment giving the exercise price S − K. This generally results in a kink in B.
We are reproducing here the reasoning of [Merton 1973], note 60.
See [Dumas 1991].
Note, however, that we obtained condition 4.4 from the condition: ∂P/∂S* = 0, which is of the marginal type. We assume here a small movement of the exercise frontier.
In the conclusion to this chapter, we review again the cases in which explicit, exact or approximate solutions are feasible and desirable.
That is, we count the time backwards, j = 0 corresponding to the date of (terminal) maturity of the option.
The reader will find a complete treatment of this method in [Smith 1978].
This is known as a tridiagonal structure.
It should be added that the finite difference method is applicable to American options. Exercise before maturity is taken into account by an additional stage of calculation: after calculating the prices of the non-exercised option on line j + 1, as shown, we need only compare them with the exercised price and keep the larger of the two before moving to the next time stage.
This method is known as an ‘implicit method’ as it calculates derivatives from values which are still unknown, which must be obtained by solving a system of simultaneous equations. The ‘explicit’ method would evaluate these derivatives at instant j, at which time all the values are already known. After substituting into the differential equation, the unknown value of C at point i at instant j + 1 would appear only once in the ith equation, in the term C t. We could then solve each equation separately. This would be simpler, but the method is unfortunately often unstable (numerical errors will be magnified when moving from one line to the next). Furthermore, this method would introduce a similar bias, but with the opposite sign to that which we are about to describe in respect of the implicit method. We shall see that this bias may be eliminated by using an average of the implicit evaluation and the explicit evaluation of the derivatives with respect to S.
Expressions 5.1 represent, it will be recalled, (see previous note) the implicit method.
The derivatives estimated at instant j would be obtained by expressions analogous to [5.1] where j + 1 had been replaced by j.
See also [Smith 1978].
Here we give what is known as the Euler approximation. We may also simulate, with greater precision, the process solution to a stochastic differential equation using Runge-Kutta stochastic approximation. See [Rumelin 1982].
Most spreadsheets and statistical software are programmed with a function to generate numbers at random.
On the Monte Carlo technique applied to options, see, for example, [Boyle 1987].
The same method may be applied to the binomial technique of Chapter 6. The binomial technique is implemented twice: once for the main problem which we wish to solve and once for the control problem. Comparing the binomial solution of the control problem with its exact solution then provides an indication as to the error made in the binomial solution of the main problem and allows this to be rectified. See [Hull and White 1988].
We leave aside the formulae which apply to cases where the underlying asset may undergo jumps (mentioned in Chapter 6) or formulae linked to American options, the only known formulae being those which apply to American options with infinite maturity.
The ‘variation’ of a path is the length of the line which draws it.
We assume here a partition into equal intervals for ease of explanation. In fact, this is not necessary. See note 10 above.
Using slightly stronger hypotheses, it may even be shown that this equation is true almost certainly (that is with a probability equal to 1). See [Lipster and Shiryayev 1988], page 89.
See Equation A2.4.
This may be found, for example, in [Karatzas and Shreve 1988], page 137 or [Lipster and Shiryayev 1988], page 103.
See the much more precise definition of ‘adapted’ processes in the books already cited.
See [Dybvig and Huang 1988].
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Dumas, B., Allaz, B. (1996). Option pricing in continuous time. In: Financial Securities. The Current Issues in Finance Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7116-6_7
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