Option pricing by the arbitrage method

  • Bernard Dumas
  • Blaise Allaz
Part of the The Current Issues in Finance Series book series (CIF)

Abstract

The method of pricing securities by absence of arbitrage is applicable to all secondary securities. As options are the oldest and most commonly encountered example of a secondary security, we shall use this example to illustrate the method, and our account therefore begins by describing these financial instruments. This is the subject of Section 6.1. The pricing method, which is the subject of this chapter, 1 is applied to options in Sections 6.2 and 6.3. These two sections differ in the hypothesis used concerning the passage of time, Section 6.2 introducing the hypothesis, intended here as no more than a teaching tool, that time is divided into discrete periods, whereas, in Section 6.3, we make the interval separating two instants tend towards zero in order to arrive at continuous time. Section 6.4 addresses the problems which arise when implementing the pricing method. Section 6.5 is an assessment of the empirical validity of pricing formulae. Section 6.6 extends the discussion beyond options to consider other examples of secondary securities, or securities derived from other securities, to reveal how wide the field of application of the method studied here can be.

Keywords

Income Hull Posit Volatility Univer 

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Notes to Chapter 6

  1. 1.
    It is not one of our objectives to explain investment strategies on the options market. This is the subject of specialized works, for example: [McMillan 1986].Google Scholar
  2. 2.
    For a classic empirical test of put-call parity, see [Stoll 1969].Google Scholar
  3. 3.
    If the rate is 6% per year, the unitary period in question is in the order of a month.Google Scholar
  4. 4.
    Readers with a background in economics will recognize here a form of ‘Arrow-Debreu price’: see [Debreu 1984] Chapter 7.Google Scholar
  5. 5.
    The generalization of this technique to two state variables enables us to extend its field of application; see [Boyle 1988].Google Scholar
  6. 6.
    See Section 7.5.2.Google Scholar
  7. 7.
    The passage to continuous time is not the only way of obtaining the Black-Scholes formula. Some hypotheses, covering not only the probability distribution of prices, but also the probability distribution of investors’ future marginal utility, yield the same formula in discrete time; see [Brennan 1979] or [Rubinstein 1976].Google Scholar
  8. 8.
    A similar formula exists for European puts. It may be deduced from the formula below by noting, as we have already done, that buying a call attached to the sale of a put where both are of the European type and have the same exercise price, equates to a forward purchase of the underlying stock (a relationship known as parity). No analogous formula exists for American options.Google Scholar
  9. 9.
    If r is a continuously compounded interest rate.Google Scholar
  10. 10.
    Which means ‘assuming the exercise takes place’.Google Scholar
  11. 11.
    This rate of return is, as it were, ‘replaced’ in the formula by the risk-free interest rate.Google Scholar
  12. 12.
    See [Cox and Rubinstein 1985], pages 287 to 317.Google Scholar
  13. 13.
    α represents the instantaneous expected return on the option, covering the infinitely small time interval between now and the immediate future. Rubinstein obtained a formula giving the expected rate of return of the option over a finite period of time; see [Cox and Rubinstein 1985], page 324.Google Scholar
  14. 15.
    [Black 1989].Google Scholar
  15. 16.
    Although they are not without importance in practice, we have given no space here to the somewhat unrealistic assumptions which pertain to the institutional framework of the market and which make possible complete and exact arbitrage (see the introduction to Part II).Google Scholar
  16. 17.
    See the price limit in the appendix, or Chapter 7, to understand the nature of the analogous process in continuous time.Google Scholar
  17. 18.
    Some very specific trinomial processes certainly do exist which, when the time interval tends towards zero, also lead to the Black-Scholes formula. However this ‘generalization’ is of little interest; the only trinomial processes to which this applies are those which differ less and less from binomial processes as the time divisions become smaller and smaller.Google Scholar
  18. 19.
    The price of a share also jumps on the day of a dividend payout. We shall return to the effect of dividends when discussing the exercise of options before maturity.Google Scholar
  19. 20.
    The prospect of a jump adds to the volatility of the underlying asset. The price comparison we are making extends to unchanged total volatility. Thus we compare the price of an option with a possible jump of the underlying asset to that of an option with no jump but with a higher σ.Google Scholar
  20. 21.
    Such a configuration appeared clearly over the days and weeks following the stock market crash of 19th October, 1987. Prices reflected the expectation of a second crash. See [Bates 1991].Google Scholar
  21. 22.
    This is a Poisson process.Google Scholar
  22. 23.
    To read further on this subject, see [Ball and Torous 1983, 1985], [Jones 1984], and [Bates 1988a,b].Google Scholar
  23. 24.
    This observation is equally applicable to stock options or options on currencies or bonds.Google Scholar
  24. 25.
    [Brenner and Subrahmanyam 1988] point out that for options at parity (more precisely, according to their definition, those whose exercise price is such that: S = Ke rt), we have: C ͌ 0.398 S σ √t, with the result that the price of the option rises linearly with the volatility.Google Scholar
  25. 26.
    In this case, two approaches may be considered. Either we assume that the fluctuations in volatility are a source of risk which is diversifiable and which, consequently, is not remunerated on the financial market by any risk premium: this is the approach taken by [Hull and White 1987], [Johnson and Shanno 1987] and [Scott 1987]. This clashes with the empirical findings of [Christie 1982] according to which the volatilities of different shares tend to vary together. Or we impose restrictions not only on the joint probability distribution of S and σ, but even on the joint distribution including S, σand the marginal utility of the investors, which provides an explicit formulation of the risk premiums attached to the volatility of the volatility: this is the approach taken by [Wiggins 1987], [Nelson 1991] and [Melino and Turnbull 1991]. Whether we use one approach or the other, we must specify the behaviour of the volatility over time: we thus often use an ARCH (AutoRegressive Conditional Heteroskedasticity or autoregressive conditional variance) process as do [Scott 1987] and [Nelson 1991]. The empirical relevance of this type of process is tested by [Akgiray 1989] for shares of stock.Google Scholar
  26. 27.
    [Black 1976a] or [Black 1989].Google Scholar
  27. 28.
    See a corporate finance textbook such as [Brealey and Myers 1982], Chapter 12; see also Chapter 8 of the present work. An analogous empirical phenomenon also appears to exist on the currency market; the explanation is to date unknown. However, for currency the effect is much less discernible since its volatility also varies autonomously according to the economic context and the prospects of intervention by central banks on the foreign exchange market.Google Scholar
  28. 29.
    If the volatility of the underlying asset changes deterministically by being purely a function of time, the Black-Scholes formula remains applicable provided it is slightly amended.Google Scholar
  29. 30.
    We shall address later the problems of estimating volatility which occur when volatility is assumed constant. Regarding the more complex problem of estimating the two coefficients a and γ of the CEV model, let us mention here the work of [Tucker et al. 1988] and [Gibbons and Jacklin 1988]. [Bates 1988] measured the degree of asymmetry of the distribution of the underlying asset by comparing the prices of call options with those of put options.Google Scholar
  30. 31.
  31. 32.
    Conversely, [Rubinstein 1983] considers the case of a firm possessing a portfolio of risky assets (whose variance is constant) and risk-free assets. When the value of the risky asset falls, its part in the total asset also falls so that the total volatility falls. The value of the firm follows a ‘displaced diffusion process’. Such behaviour of the total risk of the firm does not apparently conform to reality, but it gives rise to an explicit option pricing formula.Google Scholar
  32. 33.
    Which may also be found in [Cox and Rubinstein 1985] page 414.Google Scholar
  33. 34.
    Economic theory gives only the relative prices, or prices of objects of choice relative to one another, the concept of the absolute price of a commodity item having no meaning since a transaction always consists of exchanging one commodity item for another. Rather than calculating all the relative prices of commodities taken two at a time, which would be superfluous, the economist will always choose a single reference commodity item, called the numeraire, which is used as a basis for pricing, relative to which we determine the prices of the other commodities and whose price is arbitrarily set at the value of 1. The numeraire commonly chosen is a unit of consumption available on the current date or, in monetary models, a nominal currency unit. The change of measurement unit of which we are speaking here is the same thing as changing the numeraire. The numeraire used in the previous developments was the ECU (real or nominal) of today’s date t. The new numeraire used by Merton is a bond with a face value of 1 ECU repayable at the maturity date of the option.Google Scholar
  34. 35.
    The value of the European call option SN(d 1) − Ke rtN(d 2) has a lower bound imposed by SKe rt, and is therefore always greater than the value of an exercised option SK.Google Scholar
  35. 36.
    For stock options, the case of a dividend paid continuously is neither relevant nor useful. However, it becomes very relevant when we consider options on currency or commodities. A call option on a foreign currency can be compared to the direct holding of the currency which would allow a continuous income to be received in the form of interest: this is the dividend in question. A perfect symmetry characterizes currency options: a call option on the dollar is identical to a put option on the ECU: continuous interest is received or constitutes an opportunity cost on both sides, depending on whether one has cash in hand or not.Google Scholar
  36. 37.
    See [Grabbe 1983] and [Carr 1988]. The procedure used is once again that used by [Merton 1973] for a random rate of interest and a nondividend-paying option (see Section 6.4.3). The random dividend and the random rate of interest play perfectly symmetrical roles and can be treated symmetrically. The volatility σ must therefore be that of the forward price of fixed maturity coinciding with that of the option.Google Scholar
  37. 38.
    If a call option on a foreign currency is exercised, the currency is received and may then be invested, allowing the foreign rate of interest to be earned continuously. As we have pointed out, this is an example of a dividend paid continuously by the underlying asset.Google Scholar
  38. 39.
    Exercise prior to maturity is not the only circumstance where the lifetime of an option is cut short. Some operations carried out by the firm issuing the underlying asset may have the same effect; mergers and acquisitions may be mentioned here. Incorporations of reserve may also change the nature of the option if by virtue of its definition contract the option is not ‘protected’.Google Scholar
  39. 40.
    We will not be addressing the problems linked to transactions costs, margin calls and taxes. On these matters see respectively [Leland 1985], [Heath and Jarrow 1987] and [Dammon and Green 1987].Google Scholar
  40. 41.
    See [Dahlquist and Björck 1974]. Convergence is extremely rapid.Google Scholar
  41. 42.
    See [Chiras and Manaster 1978] or [Maloney and Rogalski 1989]. Further [Latané and Rendleman 1976] and [Ghiras and Manaster 1978] show that the implicit volatility arising from the quoted option prices is a better predictor of future volatility than is past volatility. Preliminary tests carried out by Augros (see [Augros 1987], page 168) on volatilities observed on the Paris Stock Exchange do not appear to confirm the results obtained in the United States.Google Scholar
  42. 43.
    See [Patell and Wolfson 1979]. On the fluctuation of volatilities on the Paris Stock Exchange and their consistency across maturities, see [Augros 1987], page 162 onwards.Google Scholar
  43. 44.
    The Black-Scholes formula gives a value of ∂C/∂σ which is particularly high in the vicinity of the exercise price. The price of an option which is currently at parity therefore provides a better estimate of the volatility of the underlying asset: see [Beckers 1981].Google Scholar
  44. 45.
    For more detail, see [Cox and Rubinstein 1985], page 278, or [Black 1976].Google Scholar
  45. 46.
    See [Cox et al. 1981].Google Scholar
  46. 47.
    See [Stoll 1988] and [Stoll and Whaley 1990].Google Scholar
  47. 48.
    First appearing in 1982, several dozen types of index contract are quoted today in the United States, Britain, Australia and Canada, including index options specialized by industrial sector and options on stock exchange index futures contracts. In Paris, the opening in 1989 of an options contract on the GAC index marked an additional modernizing step. On maturity of the index options, settlement is purely in cash, on the basis of the level of the index noted: the contract does not provide for any delivery of securities.Google Scholar
  48. 49.
    See [Stoll and Whaley 1986] and [Day and Lewis 1988].Google Scholar
  49. 50.
    To make things worse, the time necessary to calculate the index makes it necessary to grant a delay in the exercise decision. See [Evnine and Rudd 1985].Google Scholar
  50. 51.
    See Section 6.6.6.2.Google Scholar
  51. 52.
    At the time of writing, no contract of this type exists where the primary underlying asset is a security price; the only existing contracts cover indices of security prices. They therefore simultaneously have properties linked to their character as index contracts (see above) and properties linked to their character as futures contracts. On the other hand, options do exist on currency futures (see [Ogden and Tucker 1988]) and on commodities futures (see [Jordan et al. 1987]).Google Scholar
  52. 53.
    The properties of options on futures contracts were addressed for the first time by [Black 1976]; the most complete treatment may be found in [Ramaswamy and Sundaresan 1985]; [Whaley 1986] is the best empirical study to date.Google Scholar
  53. 54.
    See [Margrabe 1978] or [Stulz 1982].Google Scholar
  54. 55.
    See [Dumas 1980], [Cox and Jacquillat 1984], [Moussi and Roger 1985].Google Scholar
  55. 56.
    Difficulties of implementation are numerous. Since the Black-Scholes formula is based on the same idea, we already know the limitations (see Section 6.4): random interest rate, variable volatility, price jumps. Nor must we forget transactions costs, which may rapidly become significant, anomalies of prices on the index futures market which we have already noted and the fact that the portfolio to be insured does not have the same composition as the market index relating to the available futures contracts.Google Scholar
  56. 57.
    Some commentators have accused portfolio insurers of precipitating the crash of 19th October, 1987. On this subject, an original thesis is expounded by [Grossman 1988]. See also the special edition of the Journal of Economic Perspectives (Summer 1988) on this issue.Google Scholar
  57. 58.
    The Moivre-Laplace theorem is usually stated (see [Métivier 1979], page 208) for the case where p = constant. We are applying here a generalization of this theorem.Google Scholar

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Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Bernard Dumas
    • 1
  • Blaise Allaz
    • 1
  1. 1.School of ManagementGroupe HECFrance

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