Do option prices support the subjective probabilities of takeover completion derived from spot prices? Sergey Gelman March 2005

Transcription

1 Do option prices support the subjective probabilities of takeover completion derived from spot prices? Sergey Gelman March 2005 University of Muenster Wirtschaftswissenschaftliche Fakultaet Am Stadtgraben Muenster Germany 05segu@wiwi.uni-muenster.de Abstract This paper deals with the question, whether subjective probabilities of takeover completion, estimated from current observable spot-prices of target stock, from expected bid price and from current forecasted "stand-alone" price of target stock, enhance classical subjective probability forecasts, such as Black-Scholes model. Using nonparametric tests it can be shown, that the spot price model is more adequate to describe the option price distribution and hence subjective probability function of target stock price at the expected completion date.

2 1 Introduction The current probability investors attach to the success of an M&A deal can be estimated in several ways from the spot price of the target stock, its historical dynamics and the anticipated bid price (cf. Luo 2003). The problem is, that the reliability of such estimation can be hardly assessed based only on this data: probabilities are unique at any time point and there are no reasonable arguments why they should follow some parametric stochastic process, thus making ltering techniques useless. But the recent research inpricing derivatives (Ait-Sahalia andlo 1998, Jackwerth and Rubinstein 1996) enables to recover subjective (risk-neutral) probabilities of certain future states of nature from derivative prices. The case of an expected M&A deal is exactly a situation where we expect a well-speci ed event to occur at rather certain time point in nearest future. Therefore in this study the probability distribution derived from the spot-price model will be analytically compared to the one obtained from the derivatives market. In Section 2 I will brie y introduce an overview of the mixed distribution hypothesis derived from spot prices. In the next Section the technique of recovering probabilities from derivative pricing will be discussed. Thereafter Section 4 presents the methodology of the study. Section 5 reviews data, summarizes and discusses empirical results. Section 6 concludes. 2 Probability distribution derived from spot prices Since asset prices are viewed as discounted expected future cash ows, they can proxy under certain parametric assumptions the distribution of cash ows at a certain future time point. Especially in the case when there is only one payo expected at the future time point T the current spot price presents the investors forecast of the payo distribution at this time point T. In the literature it is often addressed as state-price-density. In case of expected M&A deal one can view the log target stock price p (t) as weighted average of the discounted anticipated (log) takeover price and of the expected price if no takeover takes place: p (t) = t fe t [v (T)] r t g + (1 t ) p (t); (1) where p (t) depicts the log price of stock under "stand-alone" scenario, E t [v (T)] represents the log expected at the current time point bid price to be paid at T; = T t stays for time left until the deal completion/termination, t represents the subjective probability of successful takeover completion and r t stands for cur- 1

3 rent risk free discount rate. Together with the parametric assumption about the fundamental process p (t): dp (t) = µ dt + ¾ p dw (t); (2) it induces the following expected state-price-probability (SPP) at the time point T for the price to lie in some interval A: ½ (1 t ) R SPP t (A) = f (p (T )) A dp (T ) + t if E t [v (T)] 2 A (1 t ) R f (p (T )) A dp (T ) if E t [v (T)] =2 A ; (3) where: f (p (T)) = Ã! 1 exp q2¼ ¾ [p (T ) (p (t) + µ )] 2 p 2 2¾ 2 p ; (4) is density of the price at the time point T under "stand-alone" scenario. As we see, the model described through Equations (1) and (2) forces SPPfunction to have a certain form. Unfortunately, all important parameters are variable in time: t ; f (p (T)) - since p (t) is variable. Even though the "stand-alone" price p (t) does follow parameterized stochastic process, we cannot reasonably de ne such for subjective probabilities of deal completion t, which are solemnly driven by irregularly arriving news. That s why there is no possibility of verifying the model based on spot prices. 3 Probability distribution derived from derivatives pricing In order to verify probabilities obtained from the model described by Equations (3) and (4), which implicates strong assumptions, in this study a nonparametric estimator of SPP will be introduced. It has become common place in nance literature, that prices of derivatives re ect the risk-neutral (under assumption of risk-neutrality of investors) 1 probability distribution of the underlying at the date of their expiration. Thus, the price of European call presents discounted expected value of the underlying asset in excess of the strike price at the execution date. Itis basedonthe theory-imposedrestriction, that the price of the call option C must be decreasing and convex function of the option s strike price K. Ruling out arbitrage opportunities yields linearity of pricing operator. From continuity and 1 In terms of Gelman (2004) model risk neutralitiy implies µ = r 2

4 linearity follows by the Riesz representation theorem (Ait-Sahalia and Lo, 1998) the existence of state-price density (SPD), which is denoted by f (P T jp t ; ;r t; ; ). Under these assumptions the call pricing function for an European call option at time t is given by: C (P t ;K; ;r t; ; ) = e rt,τ E t [max [P T K; 0]] (5) = e r t,τ Z +1 0 max (P T K; 0)f (P T jp t ; ;r t; ; ) dp T ; (6) where P t is the underlying stock price at date t, T = t + is the expiration date, _r t; the deterministic risk free interest rate for that maturity and stands for other factors. Note, that di erently to the previous section level prices instead of log prices are used inequation (5) and following. Further in the paper the conditioning information will be left implicit and f (P T ) will stand for f (P T jp t ; ;r t; ; ). One can also show, that the rst derivative of the call pricing function in respect to strike price should be negative but larger than e rt,τ; in order to rule out the arbitrage opportunities: Z (P t ;K; ;r t; ; ) = e r t,τ f (P T ) dp T ; K thus from the positivity of density and its integrability to one follows: e r (P t;k; ;r t; ; 0; (8) i.e. the call option price should fall with rising strike price, but the call price decrease should be not more than the discounted increase of the strike price. By di erentiating the call price function twice with respect to the strike price, one obtains, as in Breeden and Litzenberger (1978) and Banz and Miller 2 C (P t ;K; ;r t; ; 2 = e r t,τ f (P T ) 0 (9) which 2 C=@K 2 is proportional to a probability density function and therefore must be positive. Any local non-convexity of the call pricing function implies negative state prices, which existence violates the no arbitrage principle. The theory also imposes, that by no arbitrage the futures price F t; at t for delivery of the underlying asset at T = t + should be equal to the expected value of the underlying asset price at the time point T; and simultaneously the future value of the current spot price of the underlying asset: F t; = E t [S T ] = = S t e rt,τ : Z 1 0 P T f (P T ) dp T (10) 3

5 The Equations (5)-(10) provide a powerful tool to estimating state-price density (SPD) from derivative prices. The only problem is that strike prices are in reality not continuous. Furthermore, in certain situations, such as expected M&A deals, the probability of di erent states of nature is not continuous either. So instead of state-price density state-price probability of certain intervals should be estimated. Most of the studies in this area are devoted to estimating state-price density. The problem of obtaining continuous functions from discrete data was solved by using kernel regressions, as in Ait-Sahalia and Lo (1998) and similar methods. However, it demands large data samples. Ait-Sahalia and Lo dealt with that by pooling cross-section data, which implies time-stationarity of the underlying process. In case of an outstanding M&A deal this condition is explicitly violated for the target stock price. Thus, the simplest way (and, probably, the most correct one) is to approximate the partial di erentials from above by di erence equations. In particular, the lefthand side of the Equation (7) can be approximated through rst di erences: C (P t ;K; ;r t; ; ) K = C i+1 C i K i+1 K i ; (11) with a leading error of order K, and K 1 < K 2 < ::: < K n being di erent ascendingly ordered strike prices whereas C i denote corresponding prices of European call options: C i = C (K i ). In their turn, the discounted state-price densities from Equation (9)could be approximated through second di erences: e rt,τ f (P T ) ¼ 2 C (P t ;K; ;r t; ; ) ( K) 2 = C i+1 2C i + C i 1 ( K) 2 ; (12) with a leading error of order ( K) 2 : However, Equation (12) is valid only under assumption, that the discretization step K is constant. Even though the actual strike price step normally satis es these conditions, there are several violations observable, mainly by deep in-the-money or deep out-of-money options. The violation of continuity by probability function itself (cf. Equation (3)) however causes severe problems: one cannot estimate density, since the distribution is not everywhere continuous. Therefore the only possibility left is to estimate interval probabilities for su ciently small intervals through di erencing the cumulative distribution function: SPP (X i 1 < P T X i ) = F (X i ) F (X i 1 ); (13) where F (X i ) is a value of the (risk-neutral) cumulative distribution function at the point X i. However, the only X i -s on which one can explicitly get any data that allows to recover cumulative distribution function are the determined by an option exchange strike prices K i -s. Thus it is unfortunately impossible to estimate 4

6 probabilities of smaller intervals or intervals with other boundaries than K i -s. This may cause large errors of approximation if the discrete probabilities do not fall solely on the boundary values. This problem will be addressed later. Looking at Equation (7) one can suppose, that it is possible to recover cumulative distribution function from the rst di erence of the call prices in respect to strikes, since (in continuous case): F (K) = Z K 0 f (P T ) dp T = 1 = (P t;k; ;r t; ; Z 1 K f (P T ) dp T (14) 1 e rt,τ : Thus for the discrete case we approximately can estimate the value of cumulative distribution function as follows: F (K i ) ¼ 1 C i (P t ;K i ; ;r t; ; ) 1 K e ; (15) r t,τ where C i is a corresponding rst di erence of the call price. From Equation (15) one obtains for the state price probability of an interval (K i ;K i+1 ] by means of standard calculus: µ SPP (K i < P T K i+1 ) ¼ e rt,τ Ci+1 (P t ;K i+1 ; ;r t; ; ) C i (P t ;K i ; ;r t; ; ) : K (16) If using forward di erence formula, as in Equation (7), the SPP is biased downward: one gets lower probabilities for intervals where discrete events occur and higher probabilities in directly preceding intervals. However, by using central difference formula SPP is distorted from the interval with non-continuity to the both neighboring intervals. 4 Methodology According to the general theoretic assumptions of nance, stock prices follow geometric Brownian motion. Hence, the distribution of a stock price at some future date T is given in general case by following equation: Ã! 1 [p (T) (p (t) + r )]2 f (p (T )) = p exp ; (17) 2¼ ¾ 2 p Z SPP BS (A) = A 2¾p 2 Ã! 1 [p (T ) (p (t) + r )]2 p exp 2¼ ¾ 2 p 2¾p 2 dp (T ): (18) 5

7 The Equation (17) is the basis of the Black-Scholes model and will be addressed as Black-Scholes density further in the paper. 0,6 0,5 0,4 SPP 0,3 0,2 0,1 0 5/24/2004 6/01/2004 6/09/2004 6/17/2004 6/25/2004 7/05/2004 7/13/2004 7/21/2004 7/29/2004 8/06/2004 8/16/2004 8/24/2004 t K Figure 1: Black-Scholes interval probabilities of Aventis stock price on August, However, Black-Scholes distribution is symmetric, whereas based on the spotprice probabilities model one can expect strongly skewed to the left distribution with 2 local maxima: the lower one by the expected value of the "stand-alone" scenario and the upper one by the expected bid price. Thus, to support the spot-price probabilities model from Equations (1)-(3) the interval probabilities derived from option prices SPPt D (A i ) should be signi cantly higher in the interval where the expected bid price lies and signi cantly lower for the other intervals than the probabilities forecasted by the Black-Scholes model SPPt BS (A i ): SPPt D (A i ) SPPt BS (A i ) = a + u t (19) with a > 0 for v (T) 2 A i and a < 0 for v (T ) =2 A i. 5 Empirical study 5.1 Data For the empirical study I have chosen the acquisition of Aventis (target) by Sano (bidder) which has been accomplished in This particular deal was chosen not only because of actuality, but also since option prices for calls on Aventis for 6

8 the period of interest are available, whereas for the plenty of others M&A targets they are not existent or not accessible. The time-line of events: ² January, 26: Sano announces its bid o er for Aventis stock with bid price of 60,4 euros per share ² January, 28: the Board of Aventis declines the o er ² February, 5: management of Aventis announces intent of protective buying own stock ² February, 11: Sano repudiates speculative rumors about raising the original o er price ² April, 25: Sano raises bid o er price to 69 euros per share. The Board of Aventis shows approval of the o er ² August, 9: Sano o ciallyclaims to have acquiredmore than95% ofaventis stock Looking at the Aventis stock price dynamics it is obvious, that investors had anticipated the possible takeover before January 26. Moreover, they had also clearly anticipated that the original bid o er was to be corrected upwards. It is also noticeable, that at the 9th of August the success of the deal was for some reasons not absolutely certain. Therefore, for the parametric model I make following simpli cations: 1. Investors started to anticipate the deal on October 30, 2003 (to have su cient time bu er before the announcement) 2. The deal was successfully completed on August 31, Investors considered only the possibility of the deal completion on August 31, 2004 at the bid price 69 euros from the very beginning 5.2 Results and discussion For the means of the empirical study it is necessary to de ne the probabilities under the parametric model, described in Equations (3) and (4) we need to identify the parameters of the "stand-alone"-price process. Instead of de ning it as Brownian motion with drift, as in (2), it will be speci ed as "market model": dp (t) = a dt + b d (t) +¾ e dw 2 (t); (20) 7

9 where d denotes the di erence of the corresponding log market index (which is by coincidence the instantaneous rate of return on the said index). It is preferable to use the "market model" (20) compared to constant drift model (2) for the following reasons: 1. The empiricalinterdependence ofstockreturns withthe market indexchanges smaller than the value of the constant drift depending on what time frame is chosen 2. Unexpected strong all-market movements (meaning also strong changes in the "stand-alone" price) can be accounted for For Aventis stock the relevant market index is SFBF120, which includes 120 most liquid stocks on the Paris Stock Exchange (Aventis thereunder). On the 2-year time period surely preceding the point of rst anticipation a rather wellidenti ed "market model" can be estimated (t-statistics in parenthesis): dp (t) = 0:006 ( 0:89) + 1:02 (24:3) d (t) (21) R 2 = 57:3% Pr = 0:0000 Subsequently, the "stand-alone" price dynamics can be presented as follows: dp (t) = 1:02 d (t) +0:016 dw 2 (t): (22) The constant from Equation (21) can be neglected by modeling "stand-alone" price behaviour, since it is rather insigni cant. With the help of Equation (22) it is possible to quasi-estimate the not observable "stand-alone" price. However, to forecast the further development until the time point T some kind of constant drift is needed to be assumed. I suggest adopting the risk-neutrality assumption, meaning that the drift should be equal to the risk-free rate: µ = r. As proxy for the risk-free rate I take the mean between the German 10 year reference bond yield (about per year) and German interbank 3 month o ered rate (about per year): r ¼ 0:031(annualized) and hence for the daily data (by 250 working day a year) r ¼ 0: Also the daily standard deviation of returns is necessary for assessing the distribution under "stand-alone" scenario. For the same historical period as for the market model above we get ¾ p = 0:024369: Figure 2 shows observable and hypothetical development of the Aventis stock price under the model assumptions. Thus, to get the point probability of the price on August 31, 2004 to be exactly the expected bid price 69 euro is the fraction of the distance from the actual price to the "stand-alone" price to the distance between the both extreme theoretic scenarios. Thus our estimatedb t has the following path (Figure 3): 8

10 Certain completion AVENTIS aventis_alone /29/ /26/ /24/2003 1/21/2004 2/18/2004 3/17/2004 4/14/2004 5/12/2004 6/09/2004 7/07/2004 8/04/2004 9/01/2004 9/29/ /27/ /24/2004 Figure 2: Actual, forecasted "stand-alone" and "certain completion" paths of Aventis stock price alpha 1 alpha OBS 11/25/ /23/2003 1/20/2004 2/17/2004 3/16/2004 4/13/2004 5/11/2004 6/08/2004 7/06/2004 8/03/2004-0,2 Figure 3: Subjective probabilities of M&A success 9

11 1 0,8 SPP 0,6 0,4 0, K /24/2004 5/28/2004 6/03/2004 6/09/2004 6/15/2004 6/21/2004 6/25/2004 7/01/2004 7/07/2004 7/13/2004 7/19/2004 7/23/2004 7/29/2004 8/04/2004 8/10/2004 8/16/2004 8/20/2004 8/26/ t Figure 4: Interval probabilities of the Aventis price for August 31, Hence, according to the model from Section 1 we get state probability distribution for example ½ on January 30, R2004 (byb t = 0:7484): (1 0:7484) SPP t (A i ) = f (p (T )) A dp (T ) if ln(69) =2 A i 0: (1 0:7484) R A f (p (T )) ; dp (T ) if ln(69) 2 A i where f (p (T )) = Ã! 1 p exp [p (T ) (3: : )]2 2¼ 152 0: : Beginning May 24, the subjective probabilities predicted by the model, develop in the way, illustrated in Figure 4. Using non-parametric approach as in Equation (15) the probabilities can be theoretically estimated for the whole period from option prices. For the di erence between the probabilities recovered from the option prices and probabilities forecasted by Black-Scholes model we get: As one can see from Figure 5, probabilities recovered from option prices for the most part allow to reject Black-Scholes in favour of mixed distribution. The probability for the interval containing the expected takeover price (A 6 =]65; 70]) is signi cantly 9% higher than predicted by Black-Scholes. For the interval above the expected takeover price (A 7 =]70; 1]) it is signi cantly lower by remarkable 25%. The probabilities of intervals on the very low end of the distribution (A 1 = [0; 55];A 2 =]55; 57:5]) support the mixed distribution hypothesis as well, 10

12 55 57, , >70 a -0,085609*** -0,021784*** 0,021585*** 0,083530*** 0,168596*** 0,091859*** -0,251609*** Figure 5: Mean di erences between probabilities recovered from option prices and Black Scholes probabilities. since they are also signi cantly lower than Black-Scholes values. However, in several intervals not far below the expected takeover price the probabilities recovered from option prices are signi cantly higher as under null hypothesis, what is the opposite of what the mixed distribution hypothesis suggests. This fact can have several reasons. First, the assumptions could be too restrictive: investors may consider the possibility of delay of the deal or the possibility of lower takeover price, thus having lower expectations. Second, the probabilities derived from option prices are risk neutral, whereas the ones forecasted by the mixed distribution hypothesis are not. Therefore, if the investors have a very high degree of risk aversion they transform their original subjective probabilities into risk neutral ones in the way, that lower intervals become more probable. 6 Conclusion and further remarks This paper analyzes the advantages of the subjective probability estimates, derived from the mixed distribution hypothesis compared to Black-Scholes probability distribution. Using nonparametric approach risk-neutral probabilities are derived from option prices. It is shown, that Black-Scholes model does not t these risk-neutral probabilities, when a takeover of the observed stock is expected. The signi cant deviations from Black-Scholes distribution mostly support the suggestions of the mixed distribution model. However, in the area not far below the expected bid price one gets unsatisfying results for the mixed distribution model. It should be the focus of the further research to enhance the mixed distribution model by accounting for possible changes in deal conditions and for the risk aversion of investors. 11

13 References [1] Ait-Sahalia, Y., Lo A., Nonparametric estimation of state-price densities implicit in nancial asset prices. Journal of Finance 53, [2] Ait-Sahalia, Y., Lo A., Nonparametric risk management and implied risk aversion. Journal of Econometrics 94, [3] Ait-Sahalia, Y., Wang Y., Yared F., Do option markets correctly price the probabilities of movement of the underlying asset? Journal of Econometrics 102, [4] Ait-Sahalia, Y., Duarte J., Nonparametric option pricing under shape restrictions. Journal of Econometrics 116, [5] Black, F., Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy 81, [6] Breeden, D., Litzenberger, R., Prices of state-contingent claims implicit in option prices. Journal of Business 51, [7] Cox, J.C., Ross, S.A., The valuation of options for alternative stochastic processes. Journal of Financial Economics 3, [8] Gelman, S. Dynamics of target stock price by mergers and acquisitions under uncertainty of the deal completion. Presentations Journal of the Joint Conference of the European Economic Association and Econometric Society- European Meeting 2004 (temporary electronic publication; [9] Jackwerth, J.C., Rubinstein, M, Recovering probability distribution from option prices. Journal of Finance, 51, [10] Luo, Y., Do Insiders Learn from Outsiders? Evidence from Mergers and Acquisitions. AFA 2004 San Diego Meetings. < [11] Smith, G.D., Numerical solutions of partial di erential equations. Oxford University Press, London. 12