External writer-extendible options: pricing and applications

Transcription

1 Eternal writer-etendible options: pricing and applications Jian WU * ABSRAC his article aims to eamine a new type of eotic option, namely the eternal writeretendible option. Compared to traditional options, such option has two characteristics: first, its maturity is etended by the optionwriter for a given period as soon as the option is not inthe-money at the initial maturity date, without any additional premium payment from the optionholder; second, once etended, the initial underlying asset is replaced by a new underlying asset until the etended maturity date. We derive closed-form valuation formulas for this kind of European-style call and put options, and show that firms may use these instruments in their warrant issues as well as in their risk management. Keywords: Valuation of eotic options; etendible options; writer-etendible options; multiasset options; eternal writer-etendible options. Journal of Economic Literature Classification: G3, G34 * WU is an Associate Professor of Finance from the Rouen School of Management. She would like to thank the participants at the 006 Eastern Finance Association Conference, the 006 Western Decision Sciences Institute Annual Meeting, and the 006 Global Conference on Business and Economics for their helpful comments. Comments could be sent to the following address: Jian.wu@groupe-esc-rouen.fr, or Jian WU, Rouen School of Management, rue du Maréchal Juin BP 5, 7685 Mont-Saint-Aignan Cede, France. Phone: Fa:

2 . Introduction o make stock issues or bond issues more attractive, warrants, namely traditional call options on the stock of the issuing firm, are typically attached to stocks or bonds to be issued. o make warrant issues still more attractive, additional clauses, such as the etension clause, are added to warrant contracts. Originally, the etension clause was designed to protect investors against a stock market crisis. For eample, some warrants have the possibility to be etended if the benchmark inde decreases more than 5% within the last month before the maturity date. Such etension clauses constitute effectively a sort of anti-crash instrument, as they work only when stock markets fall within a short time. Compared to a traditional warrant, an etendible warrant has at least two advantages. First, by etending the maturity of the contract, it reduces the warrantholder s non-eercise risk. Second, with a higher price than a plain-vanilla warrant, it permits the issuing firm to collect more funds. However, the etension clause may be used in a more general contet by making the etension condition less restrictive. For instance, the warrant can be etended as soon as the underling stock price falls below the strike price at the maturity date. In this way, investors are not only protected against a significant decrease of the stock price within a short time, but also against a progressive, but continual decrease along the maturity of the contract. Longstaff 990 distinguishes the buyer-etendible option or simply the etendible option from the writer-etendible option. For the first one, the etension decision is made by the optionholder with the payment of an additional premium, whereas for the second one, the etension decision is made by the optionwriter when the option is at-the-money or out-ofthe-money at the maturity date, without any additional premium payment from the optionholder.

3 On the basis of etendible options, optionholder s non-eercise risk can be reduced even more by replacing the initial underlying asset by another one over the etended period. In order that such replacement makes sense, the new underlying asset should not be perfectly correlated with the initial one. For eample, within the framework of a warrant, the new underlying asset could be the stock price of a parent or subsidiary of the initial underlying stock. In this way, we are led to the conception of a new type of option, entitled eternal etendible options. In fact, as traditional etendible options Longstaff, 990, the maturity date of these options can be postponed for a specified period. Unlike traditional etendible options, the underlying asset of these options is replaced by another one over the etended period. In the sense that two or more underlying assets are involved in the option contract, eternal etendible options are also similar to multi-asset options Margrabe, 978; Stulz, 98; Johnson, 987. his article aims to study eternal writer-etendible options, henceforth called eternal etendible options. he mechanism of such options is as follows: at the initial maturity date, the option is eercised as a plain-vanilla European option if it is in-the-money; if not, it is etended as a new European option with an etended maturity date and a new underlying asset. In Section, the closed-form valuation formulas are derived within the framework of the model of Black & Scholes for this type of European options, and on this basis, analytical properties are analyzed. In Section 3, two eamples are presented as corporate applications of these options. he fourth and last section summarizes the main results obtained and presents concluding remarks.

4 . Pricing eternal etendible options.. Valuation framework Option valuation is made in the framework of the model of Black and Scholes 973, generalized by Harrison and Kreps 979, Harrison and Pliska 98. We assume that the prices of the initial underlying asset, S, and of the second underlying asset, S, are two basic assets whose prices follow a joint geometric Brownian motion GBM: ds t S t ds t S t dt t dw ~ dt t dw ~ a b Where t is time, W ~ and W ~ correlation coefficient between W ~ and W ~, with are two standard Wiener processes, and is the instantaneous dw ~ dw ~ t t dt. In Equations a and b, and can be interpreted as the epected total returns resulting from capital gain as well as dividend income on the underlying assets, and as the continuous dividend yields, and as the instantaneous volatility rates. Furthermore, we designate the riskless interest rate as r and we assume that the valuation is made at time 0. For simplicity, S i 0 is noted as S i. 3

5 .. Eternal etendible calls... Analytical pricing formula We designate the value at time t of an eternal etendible call option as WECOS t, S t, K, t, K, t, where and represent the initial and the etended maturity dates, K and K represent the initial and the new strike prices, and S and S represent the initial and the new underlying assets. At time, if S > K, then the call option is eercised as a plain-vanilla call option; on the contrary, if S K, then the call option is etended by the optionwriter and no additional amounts are paid by the optionholder. he payoff of the option at is: S K WECO S,S,K, 0,K, C S,K, if S K if S K Where C S,K, represents the value of a traditional European-style call option with an underlying asset price S, a strike price K, and a maturity. he right-hand term of Equation can be transformed as follows: S K S K C S,K, S K 3 Where Condition is an indicator function, with if the condition is met Condition 0 if not. In Equation 3, the first component is the payoff at the maturity date of a traditional call option with an underlying asset price S, a strike price K, and a maturity date, whereas the second component is the payoff of the etension clause. hus, the value of the eternal etendible call option is the sum of the value of the traditional European-style call option and that of the etension clause. As the Black and Scholes formula derives the value of a traditional European-style call option, the valuation problem consists in valuing the etension clause. According to the valuation theory of contingent securities, the value at time t with t [0, 4

6 ] of the etension clause, designated as WECCS t, S t, K, t, K, t, can be written as: WECC S t,s t,k, t,k, t e r t EP* C S,K, S K Where E P* [.] is the mathematical epectation with the risk-neutral probability P*. As we assume that the valuation is made at time 0, we have: WECC r EP* C S,K, S K S,S,K,,K, e 4 o get the closed-form formula from Equation 4, the mathematical development could be realized in three steps. In the first step, the value at of the etended call option, C S,K,, can be written with its closed-form epression thanks to the Black & Scholes formula. Namely, we have: C S,K, e r N Z K e N Z S S S 5 Where Na is the cumulative probability of the standard normal density in the u a e interval [, a], with N a du, and S ln r K Z S. As a result, the valuation problem consists in calculating the mathematical epectation of a function of S and S. In the second step, S and S can be written as a function of their respective initial levels, S and S : S Se r / 6a 5

7 S Se r / 6b Where, follows a standard bivariate normal distribution with a correlation coefficient, which is the same as the correlation coefficient between W ~ and W ~. 3 In the third and last step, C S,K,, S, and S are replaced by their epressions written in Equations 5, 6a, and 6b, respectively. hen, the right-hand term of Equation 4 can be transformed into a function of bivariate normal distributions. hese developments lead to for more details, see Appendi: WECC S,S,K,,K, Se N, ; r K e N, ; 7 Where N a, b ; is the cumulative probability of the standard bivariate normal density with correlation coefficient for the rectangular region [, a] [, b], with u * uv v ep a b * N a,b; * dvdu, * S ln r K, S ln K r, and. In Equation 7, the first component represents the present value of the average price of the second underlying asset when the option is etended and eercised at the etended maturity date, whereas the second one represents the present value of the second strike price in the same case. he value of the etension clause can be delimited by some rational bounds. For eample, it can never be negative, but is smaller than C S,K,, the value of the traditional European-style call option with the second underlying asset, the second strike 6

8 price, and the etended maturity date. In other words, the value of the eternal etended call option is bigger than that of the initial traditional call option, but smaller than the sum of the initial traditional call option and the etended call option. he etension clause has a certain number of interesting special cases. For eample, if the prices of the two underlying assets are perfectly correlated i.e., = and the two strike prices are the same i.e., K = K = K, then the closed-form formula derived by Longstaff 990 can be obtained. If S 0 and K, then the value of the etension clause tends towards zero, as the probability of the etension approaches zero. If S 0 and K, then the value of the etension clause tends towards C S,K,, as the probability of the etension approaches one. If the prices of the two underlying assets are independent i.e., = 0, then the value of the etension clause is equal to the multiplication of the value of the etended call option and the probability of the etension.... Analytical properties In deriving comparative statics, we focus especially on the differences between the classical etension clause i.e., without changing the underlying asset over the etended period and the eternal etension clause i.e., with a new underlying asset over the etended period. It is also noteworthy that the computation of the cumulative probability of the standard bivariate normal density is based on Drezner s algorithm Drezner, 978. As indicated in Longstaff 990, the classical etension clause is not always a monotone increasing function of S, the initial underlying asset price. In fact, an increase in S reduces the probability of the etension at, and on the other hand, increases the value of the 7

9 etended option at. his is no longer the case with the eternal etension clause, which is a monotone decreasing function of S cf. Figure, as an increase in S reduces only the probability of the etension, without having any effect on the etended option whose underlying asset is no more S. his is effectively the most important advantage that eternal etendible call options have compared to classical etendible options: even if S tends towards zero, as soon as S does not approach zero, the value of the option tends towards the value of the etended option, rather than zero. Insert Figure here As for the classical etension clause, the eternal etension clause is not always a monotone function of the volatility of the initial underlying asset price. In fact, a high volatility may lead to a high level of S, which leads to a low probability of the etension, as well as a low level of S, which leads to a high probability of the etension. As a result, may have a positive or negative effect on the value of the etension clause. As for the classical etension clause, an increase in has an indeterminate effect on the value of the eternal etension clause cf. Figure. In fact, on the one hand, a longer duration of gives more chance for the initial underlying asset to eceed the initial strike price, which reduces the probability of the etension. On the other hand, a longer duration of gives more chance for the new underlying asset to eceed the new strike price, which increases the value of the etended option. As a result, may have a positive or negative effect on the value of the etension clause. Insert Figure here As for a traditional European-style call option, the value of the eternal etension clause is a 8

10 monotone increasing function of S cf. Figure 3, and, but a monotone decreasing function of K and. However, its sensitivity to these variables is lower than that for the traditional option, as the probability of the etension is smaller than one. Insert Figure 3 here As shown in Figure 4, the value of the eternal etension clause is a monotone decreasing function of the correlation coefficient between the two underlying asset prices. In fact, what finally accounts for the etension clause is the value of the etended option at its maturity date. For this reason, analysis should be concentrated on the case when S eceeds K at the etended maturity date: if S and S are positively correlated i.e., > 0, when S rises, S also tends to rise, which reduces the probability of the etension, and so the value of the etension clause; on the contrary, if S and S are negatively correlated i.e., <0, when S rises, S rather tends to decrease, which increases the probability of the etension, and so the value of the etension clause. As a result, a positive correlation between S and S is more favorable for the etension of the initial option and for the eercise of the etended option. When = +, the value of the eternal etension clause reaches its minimum, which is also the value of the etension clause of a traditional etendible option. On the contrary, when =, it reaches its maimum. Insert Figure 4 here.3. Eternal etendible puts.3.. Analytical pricing formula We designate the value at time t of an eternal etendible put option as WEPOS t, S t, 9

11 K, t, K, t. At time, if S < K, then the put option is eercised as a plainvanilla put option; on the contrary, if S K, then the put option is etended by the optionwriter and no additional amounts are paid by the optionholder. he payoff of the option at time is: K S WEPO S,S,K, 0,K, P S,K, if S K if S K 8 Where P S,K, represents the value of a traditional European-style put option with an underlying asset price S, a strike price K, and a maturity. We designate the value at t of the etension clause as WECPS t, S t, K, t, K, t. Proceeding as before, the value of the etension clause can be derived as follows: WECP S,S,K,,K, Se N, ; r K e N, ; 9 As for the call option, the value of the etension clause for the put option should be superior or equal to zero, but inferior to P S,K,, the value of the traditional European-style put option with the second underlying asset, the second strike price, and the etended maturity date. It also has a certain number of interesting special cases. For eample, if the prices of the two underlying assets are perfectly correlated i.e., = and the two strike prices are the same i.e., K = K = K, then the closed-form valuation formula derived by Longstaff 990 can be obtained. If S and K 0, then the value of the etension clause tends towards zero, as the probability of the etension approaches zero. If S and K 0, then the value of the etension clause tends towards P S,K,, as the etension probability approaches one. If the prices of the two underlying assets are independent i.e., = 0, then the value of the etension clause is equal to the multiplication of the value of the etended 0

12 put option and the probability of the etension..3.. Analytical properties As for call options, similar properties can be derived for the etension clause for eternal etendible put options. More precisely, unlike the etension clause of traditional etendible put options, the value of the eternal etension clause is a monotone increasing function of S, but may increase or decrease with and. As a traditional put option, the value of the eternal etension clause is a monotone increasing function of K,,, and, and is a monotone decreasing function of S. As for call options, the value of the eternal etension clause for put options decreases with, the correlation coefficient between S and S. 3. Eamples of eternal etendible options Eternal etendible options can be used by firms in different contets. In this section, two applications are presented, one for corporate warrants and the other for corporate risk management. 3.. Corporate warrants Corporate warrants constitute an important financing tool for firms Smith, 977. o make warrants even more attractive for investors, the eternal etension clause may be added to traditional warrants. In fact, to protect warrantholders against a general fall that is common for the whole stock market, the etension clause can be introduced as a first measure. For eample, the CAC40 inde fell 7% in 994, whereas it rose 0.5% in 995 and 5% in 996.

13 In such a contet, the etension, for the duration of one year, of a warrant epiring in 995 gives warrantholders an additional period to eercise their options. Furthermore, to protect warrantholders against a fall that is rather specific to the stock price of the issuing firm, it may be useful to change the underlying stock price over the etended period. For eample, though the CAC40 inde rose 0% within the first semester of 996, the real estate sector fell 0.8% and the financial sector rose only 0.56%. In this case, for investors holding warrants issued by a bank, it is more interesting to change the initial underlying stock by a new stock that is in an industrial sector, such as automobiles, petroleum, or pharmaceuticals. Compared to a traditional etendible warrant, an eternal etendible warrant has at least three advantages. First, by changing the underlying asset over the etended period, it reduces warrantholders non-eercise risk, which is not only due to the systematic risk related to the whole stock market, but also due to the specific risk related to the issuing firm. Second, with a higher price than a traditional etendible warrant, it permits the issuing firm to collect more funds. It is noteworthy that the underlying asset substitution over the etended period is particularly interesting for small firms, which are still at their start-up stage. In fact, even though the growth of these firms needs to be financed, their high risk dissuades investors from buying their stocks or warrants. In case these firms eperience difficulties, the substitution of their stocks by those of their mother firms 3 may reassure investors, and such reassurance may facilitate firms financing operation. he third and last advantage is that, for eisting stockholders of the issuing firm, the dilution effect is inferior or equal to that resulting from a traditional etendible warrant. In fact, when the warrant is eercised at its initial maturity date, both of these two warrants lead to dilution and the dilution effect is the same; when the warrant is etended, but the etended warrant is not eercised at the etended maturity, neither of these warrants leads to dilution; when the warrant is etended, and the

14 etended warrant is eercised at the etended maturity, the traditional etendible warrant leads to dilution for the stock of the issuing firm, whereas the eternal etendible one does not, as in the second case the stock of the issuing firm is no more involved during the etended period Risk management Industrial firms are eposed to the risk resulting from fluctuation in both raw material prices and product prices. Let us take the eample of a firm whose activity consists in transforming crude oil into a refined product. Assume that the firm plans to buy a certain quantity of crude oil at a future date, and that the refined product will be sold on the market at another future date, with > > 0. he firm is eposed to two risks at two different dates, namely the increase in the buying price of crude oil at, and/or the decrease in the selling price of the refined product at. Assume that and are so close that crude oil price at and the refined product price at are significantly correlated, for eample, with a correlation coefficient that is higher than o hedge the firm s risks, one classical solution is to buy two traditional options, namely a call option on crude oil price with as maturity and a put option on the refined product price with as maturity. However, this solution over-hedges the firm s risks insofar as, in most cases, the firm cannot eercise both of these two options. In fact, if crude oil price increases at, then the refined product price tends to increase at. In this case, the firm can eercise its call option on crude oil price at, but is not able to eercise its put option on the refined product price at. On the contrary, if crude oil price decreases at, then the refined product price tends to decrease at. In this case, the firm will not be able to eercise its call 3

15 option on crude oil price at, but can eercise its put option on the refined product price at. he firm needs a solution that is more appropriate to its situation. In fact, what it needs is an option that works as a traditional call option on crude oil price with as maturity if the call option is in-the-money at ; as soon as the call option is not in-the-money at, it is transformed into a traditional put option on the refined product price with as maturity. he payoff at of the option can be written as:,s K P S,K, S K ma 0 0 he last formula can be transformed as follows:,s K P S,K, P S,K, S K ma 0 In Formula, the first component represents the payoff at of the traditional call option on S with as maturity, the second component represents the value at of a traditional put option on S with as maturity, and the third component represents the value of the etension clause of an eternal etendible put option. his means that the new solution is the same as the classical one, ecept that the etension clause of an eternal etendible put option is sold in addition. In fact, if the call option on crude oil price is in-the-money at its maturity, then the firm eercises its call option and sells at the same time its put option, as it needs no more to be protected against the decrease of the refined product price at ; on the contrary, if the call option on crude oil price is at-the-money or out-of-the-money at its maturity, then the firm, not being able to eercise its call option, holds on to its put option, as it needs to be protected against the decrease of the refined product price at. he price difference between the classical hedging solution and the new one is the value of the eternal 4

16 etension clause. According to Formulas 9 and, the closed-form epression of the value of the new option contract can be written as: C S,K, S e N, ; r K e N, ; 4. Discussions In this article, a new type of eotic options, called eternal writer-etendible options, is designed. hese options have two characteristics compared to traditional ones. First, their maturity is etended for a given period by the optionwriter without any additional payment from the optionholder as soon as the option is not in-the-money at the initial maturity date. Second, once etended, their initial underlying asset is replaced by a second underlying asset until the etended maturity date. Such options enable optionsholders to reduce their noneercise risk due to a general decline related to the whole stock market as well as a fall that is rather specific to the initial underlying stock. Within the framework of the model of Black and Scholes 973, the closed-form valuation formulas have been derived for European call options as well as for European put options. It has also been shown that such options can be used by firms in their warrant issues and in their risk management. 5

17 Appendi: Pricing the etension clause of the eternal writer-etendible call option his appendi aims at deriving the analytical pricing formula at time 0 of the etension clause of an eternal etendible call option. he payoff at of the etension clause is: WECC r EP* C S,K, S K S,S,K,,K, e 4 where E P* [.] is the mathematical epectation with the risk-neutral probability P*, and C S,K, e r N Z K e N Z S S S 5 where Z S S ln K r From Equations 4 and 5, we have: WECC S,S,K,,K, WECC WECC A where WECC WECC r e EP* S e N Z S S K r r e EP* K e N ZS S K A A3 he pricing of the etension clause consists now in deriving the closed-form epression of WECC and WECC. For this, we write: Z S A4 6

18 7 / Z S A5 K S A6 A.. Closed-form epression of WECC From Equations A, A4, A6, and 6b, we have: P* N e E e S WECC A7 As, follows a standard bivariate normal distribution with a correlation coefficient, the last mathematical epectation can be written by using the density function of,, which is: v uv u ep p u,v A8 We have: Integral e S WECC A9 where dvdu v uv u ep v N e Integral v A0 We replace the standard normal distribution function N[.] by an integral of the standard normal density function as follows:

19 a N a e w / dw A We have: Integral e v v e w / u uv v ep dw dvdu A In the last equation, we need to withdraw the variable v from the superior bound of the integral relative to w. For this, we resort to a variable change, namely w w' v, which leads to: Integral 3 / ep X dw' dv du A where X u u w' w' w' u v A3 In order to withdraw the integral relative to the variable v from Equation A by the fact / e that v dv N, we need to resort to the following variable change: v' v w' u A4 8

20 his variable change leads to: where Integral ep X dw' du A5 X u u w' w' A6 o transform the integral into a standard bivariate normal distribution function relative to u and w, we need to use the following variable changes: u ' u A7 w'' w' A8 hese variable changes lead to: Integral e u' u' w'' w' ' ep dw'' du' A9 Equations A9 and A9 lead to: e N, ; WECC S A0 where a N a,b; * u * uw w ep b * dwdu * A 9

21 0 A.. Closed-form epression of WECC From Equations A3, A5, and A6, we have: P* r / N E Ke WECC A By using the standard normal density function of,, we have: Integral Ke WECC r A3 where dvdu v uv u ep v / N Integral A4 By replacing the standard normal distribution function N[.] by an integral of the standard normal density function, we have: dvdu v uv u ep dw e Integral v / / w A5 In the last equation, we need to withdraw the variable v from the superior bound of the integral relative to w. For this, we resort to a variable change, namely, v w ' w, which leads to: dv du dw' ep Y Integral / 3 A6

22 where Y u uw' w' w' u v A7 In order to withdraw the integral relative to the variable v from Equation A6 by the fact / e that v dv N, we need to resort to the following variable change: v' v w' u A8 his variable change leads to: Integral u uw' w' ep dw' du A9 Equations A3 and A9 lead to: r N, ; WECC Ke A30 Equations A, A0, and A30 lead to: WECC S,S,K,,K, r Se N,; K e N, ; A3

23 References Black, F., & Scholes, M he pricing of options and corporate liabilities. Journal of Political Economy, 8, Drezner, Z Computation of the bivariate normal integral. Mathematics of Computation, 3, Harrison, J. M., & Kreps, D Martingales and arbitrage in multi-period securities markets. Journal of Economic heory, 0, Harrison, J. M., & Pliska, S. 98. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and heir Applications,, Johnson, H Options on the maimum or the minimum of several assets. Journal of Financial and Quantitative Analysis,, Longstaff, F. A Pricing options with etensible maturities: analysis and applications. Journal of Finance, 3, Margrabe, W he value of an option to echange one asset for another. Journal of Finance, 33, Smith, C. W. Jr Alternative methods for raising capital: rights versus underwritten offerings. Journal of Financial Economics, 5, Stulz, R. M. 98. Options on the minimum or the maimum of two risky assets: analysis and applications. Journal of Financial Economics, 0, 6 85.

24 Liste of figure captions Figure : Option value in function of S Figure : Option value in function of Figure 3: Option value in function of S Figure 4: Option value in function of 3

25 Once issued, the warrants are detached from stocks or bonds to be traded independently. For instance, in France, holders of warrants epiring at the end of 987 were not able to eercise their options due to the stock market crash in October. o reduce their losses, companies having issued these warrants asked the COB or Commission des Opérations de Bourse the equivalent Security Echange Committee in France to give its permission to etend the warrant contracts. Such request was systematically rejected by the COB due to the fact that no etension clause had been planned in the contracts. Since then, some companies have added to their warrant contracts an etension clause in order to protect warrantholders against an eventual stock market debacle. 3 Legally, this mechanism is possible. For eample, the warrants issued in June 996 by Northumbrian Water Group NWG gave to their holders the right to buy a stock of Lyonnaise des Eau, the mother firm of NWG, rather than a stock of NWG. 4 In other words, for an eternal etended warrant, the eercise of the etended warrant leads to dilution for the new underlying stock, rather than dilution for the initial underlying one. 4