Applying Bachelier Part 1: Option Value

Applying Bachelier Part 1: Option Value Ian A Thomson1 1 Introduction This paper provides a review of the Louis Bachelier two parameter option pricing model23 adapted for the modern o tra ts, i parti ular the Chi ago optio . This enables basic model properties to be analysed and the odel s key elements to be defined being the intrinsic value and the instability or option risk premium. Further the paper reviews the option contract value, vis-a-vis the option contract premium or price, and based on this defines certain rational expectations propositions to demonstrate its operation. The purpose is to provide the groundwork for applying this form of option model to other financial questions which are briefly outlined in the implications section, and include:    defining basic contract forms and market design, enabling analysis of the degree or nature of speculative aspects of trading. equity or contingency valuation and implications thereof to capital structure, market pri e eha iour, a d the o ept of shareholder alue a i isatio real options, including comparison to Binomial and risk neutral approaches These are proposed to be developed in further papers. 1 I express my gratitude to Prof Patrick Harvey of Chinese University of Hong Kong for his ongoing encouragement in developing this paper. Any factual, mathematical or other errors or false comment are solely the responsibility of the author. 2 The modern construct was developed in modern finance by Case Sprenkle (Sprenkle, 1964) and James Boness (Boness, 1964), and then subsequently adapted in the Black-Scholes Merton model (Black & Scholes, 1973) (Merton, 1973). This latter construct of the model incorporated the risk free discount based on the perfect hedge and portfolio replication arguments. These arguments are shown by the author, (Thomson, 2016), to be dependent on a static form of the model and not applicable in the dynamic setting necessary to justify a perfect hedge assumption. This further leads to various demonstrable price behaviour anomalies, and as a result the paper utilises the asset discount or expected rate of return, . A separate paper will be prepared exploring this further. 3 Note Leonard R. Higgins, The Put-and-Call (Higgins, 1906), gives a clear definition of the Option mechanism and that in verbal terms with numerical analysis based on the period 1887 to 1894 of the risk dispersal not dissimilar to that done by Bachelier with a practical verbal outline of the various types of options and trading strategies in the London option market over equities and Consols. Note the original publication date is 1896, pre edi g Ba helier s dissertatio ork Higgins work was applied to Wall Street by S.A. Nelson The ABCs of Options and Arbitrage (Nelson, 1904) Thomson, IA 2018 1 of 22 Applying Bachelier Part 1: Option Value 2 The Bachelier Option Price Model 2.1 Price path 2.1.1 Opening comments Bachelier applies a two parameter model of asset price motion to the optio o tra t s underlying asset, 4, which in simple terms tracks the price change of asset, , based on the mean return expectation with a stochastic dispersal. This price path can be described variously in arithmetic, geometric and continuously compounding terms with a normal or a log-normal dispersal price path5. Bachelier applied the arithmetic form with an absolute coefficient for risk dispersal given a normal distribution in his practical numerical illustrations – this is not unreasonable in terms of the market behaviour sampled and nature of the futures linked option contract he valued - and does not limit generalisation of Ba helier s approach. Modern option models describe the price path in exponential terms with a log-normal distribution applying a relative measure of the standard deviation. This approach later adapted by Black-Scholes with the risk free application can be shown to be an approximation of the Bachelier model6. 2.1.2 Price Path Mathematically, the price path for asset, , is described for a time period, ,in arithmetic terms as: Where ( √ ) (1) For simplicity in this paper assumes that the options and underlying assets are for asset, – the subscript being unstated. However, for returns the asset notation is referenced using the subscript notation, as in , this enables clarity where differentiation from the risk free specification, or other forms of return are required – refer footnote 2. 5 Note, it can be shown that due to the non-orthogonal or angular price path for a finance asset with respect to the necessarily vertical price tick (i.e. up and down) that even though described by a normal dispersal pattern the finance prices naturally exhibit a non-normal price dispersal driven by the underlying expected return of the financial asset in time, this idea is to be presented separately. 6 This paper will apply the exponential or continuous compounded model, except as noted. 4 Thomson, IA 2018 2 of 22 Applying Bachelier Part 1: Option Value is the spot price of the asset, , underlying the contingency is the mean price path described by the motion of the asset, , through time is the standard deviation of the price motion or dispersal around the price path for asset, , which expands by the square root of the discrete change in time, √ is the Wiener process describing the stochastic movement in time for the asset, Note, the Wiener stochastic process takes a nil value for the expected price given the dispersal is normally distributed – described by the standard deviation parameter. 2.1.3 Summarising expected price, Form price path, i.e. Normal Arithmetic Exponential ( √ √ Log-normal ) 2.2 Bachelier Option Pricing Model - General Form The general model developed by Louis Bachelier7 is applied here to the Chicago option contract is stated as:8 (2) is the stochastic probability of Bachelier expands into two components, first, the implicit probability the curvature correction at the boundary or exercise point, and, second, . This is necessary given the nature of the implicit probability behaviour, giving, Note, Bachelier developed the odel for aluatio of Paris o tra ts ased o a u derl i g futures contract over French Rentes as outlined in the following section. In this paper the implied discount rate/yield is that of the underlying asset – justification for this is discussed in a separate paper. 8 Notation of time is the time of valuation or pricing, 0 representing the time of issuance. is the time of exercise or contract maturity is the term to maturity, being on issuance, or at an intermediate point. 7 Thomson, IA 2018 3 of 22 Applying Bachelier Part 1: Option Value (3) , the intrinsic probability at the mean expectation of , the curvature correction or implied instability premium9, related to the dispersal measure for the exercise value relative to the underlying asset price. Expanding, the pricing model, in present value terms we get: (4) The model can therefore be seen as the addition of two components: , value of the implicit probability in the premium , value of the curvature correction or instability element reflecting the right to exercise, which is clear when compared to the Put-Call Parity – futures relationship. This form provides significant insights to financial questions as will be seem in the final section of the paper where a few implications are briefly outlined. 2.3 Bachelier Option Pricing Model The model documented by Louis Bachelier was based on pri i g Parisia option contracts. These contracts were short term, typically one to two months, giving a right to take up a futures contract on the penultimate trading day over French Rentes as traded on the Paris Exchange in the late 19th Century. These contracts accrued the interest coupons to the underlying party (holder of the futures contract), with the option premium, specified by the market, being payable on exercise at maturity. These contracts were priced at issuance agai st a true arket futures pri e, (being the quoted price adjusted for coupons passed to holder) against a set of fixed premium, thus the key issue was the negotiated contract price to be paid on exercise, 9 Bachelier terms this the i sta ilit Thomson, IA 2018 oeffi ie t 4 of 22 . Hence, Bachelier s Applying Bachelier Part 1: Option Value focus in pricing was on the expected spread, pricing forward to maturity or exercise based on the spread given on set contract premiums. This model format is adapted here to reflect the modern option contract approach with negotiable premium and applying futures prices: Where ( ) ; (5) (3) √ ; Bachelier used an arithmetic form of the Gaussian Normal Distribution with the absolute measure of risk, , being the given value. expands at √ 10. equates to and hence the denominator Note this form has an implied weakness as the dispersal relates to the exercise price so does not reflect changing underlying asset values and changing time measures for return and standard deviation. The latter is overcome by breaking i to it s component elements, and adapting a log normal price change approach. 2.3.1 Chicago Call Option For the Chicago call option contract where the option contract premium is paid on purchase or issuance and the exercise price is paid at maturity, this can be stated as follows. Initially, in the arithmetic terms, per Bachelier: ⁄ √ ; (6a) then expanding and simplifying we get a simple expansion11 ( ) √ 10 (6b) For information, the Leonard Higgins measure of risk data expands on this basis. (Higgins, 1906) Bachelier uses the arithmetic form of the model, which works for the short term single period analysis where payments are singularly at maturity. However, the return and dispersal parameters need adjusting in this model to reflect the implied compounding in application for longer periods of time. The pricing relationship between arithmetic and continuous compounding models for a set investment terms is addressed separately 11 Thomson, IA 2018 5 of 22 Applying Bachelier Part 1: Option Value This form is important as it gives us a clear picture of the form of this form, being in essence the intrinsic or net payoff value of the option subject to the probability of exercise driven by the mean price behaviour of the underlying asset, plus an instability element driving dispersal of the outcome at the right to exercise. This breakdown is absent from the standard form of the option pricing model in finance literature. The separation of the elements is the core benefit of the Bachelier model which then drives various implications across the almost ubiquitous issue of business investment options and valuation. Turning to a continuous compounding definition of the boundary measure we have: ⁄ √ ; such that, (7a) (7b) Note, this form does not easily simplify on differentiation. But, does not require the Black Scholes difference in boundary in terms of 12, notated here, . Looking at this differently the option compensates the writer for the probability adjusted potential value loss implied in the current underlying asset price and the price path potential, and for issuing the other party the right to undertake the transaction. The holder pays this compensation to receive the right to exercise. 2.3.2 Put Option The Put Option contract price in these terms is the inverse of the Call in terms of the right to exercise, hence 12 refer (Thomson, 2016) Thomson, IA 2018 6 of 22 Applying Bachelier Part 1: Option Value ̈ (8) ̈ being the stochastic probability of Them, expanding the probability, ̈ ; gives (9) Again this represents compensation for the option writer s loss given the value of the underlying asset falls below the exercise price plus a premium for volatility or risk. 2.3.3 Futures & Put Call Parity Simply the Put Call Parity is demonstrated as the futures price equates with the Call premium less the Put premium (10) Thus, expanding and, simplifying by recognising that as the raw probability and crucially the instability premia eliminate, we thus have (11) Note, the simple exposure of the rate applied in the market. This enables assessment of market returns being applied. This is the price of the related futures contract, and at maturity this simplifies down to the payoff on the futures contract. It is clear from this how this relationship works being a sum Thomson, IA 2018 7 of 22 Applying Bachelier Part 1: Option Value of probabilities, equalling one, by the value difference between the agreed exercise price and the underlying asset value, the risk or volatility premia being eliminated. 3 Option Contract Value This section reviews the option contract value as distinct from the Option contract price or premium which the standard option model determines. The issue is to determine the value of the asset being acquired through payment of the premiums or contract price and the defined payment on exercise, given the probability of exercise at any instant. This assists with a fuller understanding and analysis of the different forms of contract and is absent in most of the literature, possibly due to pricing anomalies caused by the risk free return assumption.13 3.1 Call Option Contract Value, ̅ This can be presented in both present value and probability of exercise terms. The latter is more fully expounded upon in this paper. 13 Note this analysis can be reflected in Black-Scholes Merton model terms, subject to the pricing anomaly created by the application of the risk free rate. Thomson, IA 2018 8 of 22 Applying Bachelier Part 1: Option Value 3.1.1 In present value terms In simple terms we can state the value of the option contract is the sum of the exercise price plus the option premium. ⏞ (12) Noting, key value properties: ⏞ , such that, ⏞ ⏞ ⏞ , by definition such that, For comparison, where the model applies the risk free rate then a property is for ⏞ The property for the exercise price continues to hold such that: ⏞ 3.1.2 In probability of exercise terms For a call option the value, ̅ (Value), at time is given simply by adding the value of the exercise price at the mean probability of exercise ̅ (13) That is, the value a Call Option is the Call Option premium or contract price, plus the exercise price agreed to be paid on exercise adjusted for the mean or intrinsic probability of exercise.14 Expanding we have ̅ Combining and eliminating Common terms, i.e. 14 This is a different notion to the previous section and reflects the value at any point in the option term. Thomson, IA 2018 9 of 22 Applying Bachelier Part 1: Option Value ̅ (14) That is, the Call Option value equates to the Underlying Asset Price by the mean probability of exercise, plus the instability adjustment. Alternatively, by rearrangement of terms as the Underlying Asset Price adjusted for the full probability less an instability element adjusted Exercise Price. ̅ , that is ̅ (15) Further, restating in arithmetic terms allows the simplicity of the proposition to be observed: ̅ √ (16) That is, the Call Option Contract Value equates to the Underlying Asset price by mean probability of exercise, plus the dispersal or exercise instability premium. A clear price path implication exists in this formulation that the value is driven by the underlying asset return, in that: Thomson, IA 2018 10 of 22 Applying Bachelier Part 1: Option Value ̅ [ √ ] √ (17) Stating this above equation further enhances the model proposition and enables determination of a rational pricing rule based on expectations. 3.1.3 At Maturity or Exercise By extension we can review the valuation at Maturity or option exercise. As the value has crystallised and the option exercised or not and assuming the option premium is invested at the underlying asset yield we have: ̅ ; or ̅ (18a) (18b) 3.1.4 Rational Expectation for Call Option Pricing From this we can develop the rational expectation based on the proposition that the writer or seller of a call option will only write or sell should they be no worse off, or by taking the risk they add value. That is we have, at the time of issuance or sale: (19a) and that the expectation for maturity or expected exercise date is: (19b) That is the parties expect the value of the call premium invested at the underlying asset return plus the exercise price will be equal to or exceed the expected share price at maturity. Thomson, IA 2018 11 of 22 Applying Bachelier Part 1: Option Value This is a more complete approach from the standard option payoff/profit propositions, however can be readily adapted for the writer of a call option (19c) Note this expectation holds irrespective of whether the option is exercised. The call option writer expects to profit or be no worse off due to writing the option. For a call option the return rate used is critical, for instance a risk free return, sufficient to meet the expectation given , fails to generate a value ,. 3.2 Put Option Contract Value 3.2.1 In present value terms In simple terms we can state the value of the option contract is the exercise price less the option premium. ⏞ (20) Noting, key value properties: ⏞ , such that, ⏞ ⏞ , by definition such that, ⏞ For comparison, where the model applies the risk free rate then a property is for ⏞ The property for the exercise price continues to hold such that: ⏞ 3.2.2 In probability of exercise terms Similarly to Call Options, the Put Option value is the Put Option premium plus the value given up by the holder – namely the underlying asset price in probability adjusted terms. ̅ Thomson, IA 2018 (21a) 12 of 22 Applying Bachelier Part 1: Option Value Expanding the terms ̅ ̅ (21b) Which while similar to the call is clearly different in that the core value element is the Exercise Price, and hence the ceiling on Put Option Value. 3.2.3 At Maturity or Exercise This is a similar proposition to the Call option, ̅ ; or ̅ (22a) (22b) 3.2.4 Rational Expectation for Put Option Pricing From this we can develop the rational expectation proposition that the writer or seller of a put option will only write or sell the option should this create value by taking the risk on. That is, at the time of issuance or sale: (23a) and that the expectation is (23b) Thomson, IA 2018 13 of 22 Applying Bachelier Part 1: Option Value That is the parties expect the value of the exercise price less the put premium invested at the underlying asset return will be equal to or less than the expected share price at maturity, being the value given up by the writer. The writer receives the share plus the put premium and pays the exercise price, so creates value where the latter is less than the former. This is a more complete approach from the standard option payoff/profit propositions, and can be readily adapted for the writer of a put option (23c) Note this expectation holds irrespective of whether the option is exercised. The put option writer is expected to profit or be no worse off due to writing the option. For a put option the return rate used is less critical as the proposition is closed on the upper and lower boundary, as illustrated. 4 The Put Call Parity in Contract Value terms Reviewing the Put-Call Parity in this case we can see the impact of the probability element with the instability element eliminating as per standard Put-call giving the futures contract price as the difference between the asset and exercise price. ̅ ̅ ̅ ̅ Thomson, IA 2018 (24) 14 of 22 Applying Bachelier Part 1: Option Value Thus as we move to the extremes As ̅ As At ̅ ̅ ̅ ̅ , and likewise ̅ 4.1 A Call plus a Put This section illustrates the behaviour where we hold both the Put and the Call. Initially in Contract Value terms: ̅ ̅ Simplifying gives ̅ ̅ (25) Secondly, in terms of the option premiums we have Thomson, IA 2018 15 of 22 Applying Bachelier Part 1: Option Value (26) Hence as, Thomson, IA 2018 16 of 22 Applying Bachelier Part 1: Option Value 5 Risk Premium Properties As noted previously, a key aspect of the Bachelier model is that the dispersal or instability premium, or instability coefficient as labelled by Bachelier, is broken out as a separate component within the probability determination in the model. Following are some basic properties of this element. The behaviour of this element has significant implications for applying this model in other contexts, so has been a missing element in finance discussion. For oth the Put a d the Call the instability pre iu is gi e : (27) For the arithmetic formulation this simplifies to √ which highlights aspect of the behaviour, and led to critiques of the Bachelier model. However, the latter used a probability defined around the Exercise Price, possible due to the futures nature of the contract, so his formulation collapses to At behaviour of Thomson, IA 2018 √ . the model exhibits the special property of .at √ , due to the , and represents the peak value of the risk premium. 17 of 22 Applying Bachelier Part 1: Option Value Importantly the instability component of the premium compensates or corrects for the loss on the underlying simple probability element implicit value of, compared with the . Further, when comparing to the latter we can see the fact the instability premium, net of the simple probability component and the implicit value, peaks sharply at . This particular aspect of the model has critical implications for financial investment and management behaviour in various applications of the model contexts. 5.1 Price behaviour for dispersal & time In simple terms, as time of maturity, , increases the dispersal increases, and the peak increases, plus the dispersal shifts left with an increased skew. As the standard deviation, , increases the dispersal increases and the peak increases. Thomson, IA 2018 18 of 22 Applying Bachelier Part 1: Option Value 6 Implications 6.1 Model Form As noted the Bachelier model takes a different form from what has become the traditional model in modern finance. The key feature is that the model breaks into two elements related to the two price path parameters assumed. First, the mean return tied to the implicit value or mean probability of exercise; second, the instability element related to standard deviation measure of dispersal. (4) (9) A useful outcome of this form of pricing model is the ability to clearly define an expected value for the option at maturity or the expected date of exercise. ̅ ̅ (14) (21b) This allows clear statements on the rational expectations for option price and option value can be made. Note this works for risk neutral and risk averse pricing scenarios. for a call (21a) for a put (23a) and that the expectation for maturity or expected exercise date is: for a call (19b) for a put (23b) This model supports the proposition in a risk averse pricing market that the return appropriate in the model is the underlying asset yield. Thomson, IA 2018 19 of 22 Applying Bachelier Part 1: Option Value 6.2 Implications for Contract Design A key issue in documenting this paper is developing an understanding of the basic forms of contract being used by the two primary pricing models. These models reflecting contemporary option contracts traded in Paris and then in Chicago markets.  The Parisian option being in essence based on futures pricing with all payments being made at maturity. Due to its nature this option contract tends to be driven by the  instability element The Chicago option being set with a range of exercise prices in and out of the market, with the exercise price paid at maturity or exercise, and the premium compensating the writer for lost opportunity and instability or exercise risk in is paid on issuance or purchase. In a following paper the author will address the continuums of contract structure and define the pricing propositions implied, the continuum or risk or the speculation involved and discuss options for market operation and contract design to create different exposures to speculation and the risk elements. In building this paper the definition of an option will be necessarily addressed with a restatement of the standard dictum. 6.3 Equity Value and related Issues In discussing options it is well recognised that the shares in a company can be considered as an option over the business value. Given this, the above model provides some forms to define the nature of the equity share contract, expected behaviours in the market and by managers dependent on the companies finance position. This then has implications for capital structure of companies as to when different forms of financing will be attractive. In addition, the approach to executive decisions will be shown to impact corporate value dependent on points in the business life cycle, business capital structure and funding cycle which will be raised. A further aspect that the author expects to address is question of the impact that maximising shareholder value has on corporate and business outcomes. The model enables a focus on certain aspects of this issue which may be counter intuitive. Thomson, IA 2018 20 of 22 Applying Bachelier Part 1: Option Value 6.4 Real Options Given the above the author also will address applying the form of the Bachelier model to real options. This enables further aspects of the business decision to be drawn out and a clearer understanding to be defined. 7 Closing Comment In closing, the paper has worked through a range of structural valuation issues and defined some additional rational expectations criteria for consideration. The paper emphasises the different pricing form applying Louis Bachelier option pricing model, and as a result has identifies a range of implications thereof which will be expanded upon in further papers expanding on this subject. Ian Thomson July 2018 Thomson, IA 2018 21 of 22 Applying Bachelier Part 1: Option Value 8 References Bachelier, L. (1900). Theorie de la Speculation. Paris: Gauthier-Villars. Bachelier, L. (1964). Theory of Speculation, (1900). In P. H. Cootner, The Random Character Of Stock Market Prices (A. J. Boness, Trans., pp. pp17-78). cambridge, massachusetts, USA: The M.I.T. Press. Bachelier, L. (2011, - -). The Theory of Speculation. (version 1). (D. Mayer, Trans.) -, -, -: email references uwa.edu.au. Retrieved March 29, 2016 Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81 (3), 637-654. Boness, A. J. (1964, April). Elements of a Theory of Stock-Option Value. Journal of Political Economy, 72(2), 163-175. Cootner, P. H. (1964). the random character of stock market prices. Cambridge, Massachuesetts: The MIT Press. Higgins, L. R. (1906). The Put-and-Call. London: Effingham Wilson. Merton, R. C. (1973, Spring). Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4(1), 141-183. Nelson, S. (1904). The ABCs of Options and Arbitrage. New York: S.A. Nelson. Sprenkle, C. M. (1964). Warrant Prices as Indicators of Expectations and Preferences. In C. P. H, & C. P. H (Ed.), the Random Character of Stock Market Prices (pp. 412-474). Cambridge, Massachusetts: The Massachuesetts Institute of Technology. Thomson, I. A. (2016, April). Option Pricing Model: comparing Louis Bachelier with BlackScholes Merton. Thomson, IA 2018 22 of 22