# Predicting completion of cash acquisitions using option implied risk-neutral probabilities

## Estimate risk-neutral probabilities and the rational for its application. Empirical results of predictive power assessment for risk-neutral probabilities as well as their comparisons with stock-implied probabilities defined as in Samuelson and Rosenthal.

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National Research University - Higher School of Economics

International College of Economics and Finance

Predicting completion of cash acquisitions using option implied risk-neutral probabilities

Author:

Elizaveta Andreeva

Supervisor:Dr. Sergey Gelman

Moscow. June 16, 2015

# Recalling PDF properties and rearranging the equation we can express the first-order derivative as a function of the cumulative density function of the risk-neutral distribution and get a direct dependents between “exercise price delta” and risk-neutral CDF:

As CDF is always less or equal to 1 it follows:

# No-arbitrage condition then is that first derivative of the call price function with respect to strike price should be negative but greater than . In other words, price of a call should be a decreasing function of strike price, but the fall should be less or equal to the discounted value of the strike price increase.

Expressing risk neutral CDF and differentiating once again will yield us PDF that is proportionate to the second derivative of option's price to strike price:

# Data Selection

risk neutral samuelson Rosenthal

Percentile

5%

25%

50%

75%

95%

Deal Duration

35

54

94

161

366

# Table 2 a). Information on 5 largest deals.

Target Company

Target Ticker

Acquirer Company

Target Equity Value, mln \$

APC

BHP Billiton Ltd

44 603

Alcoa Inc.

AA

Rio Tinto plc.

27 329

Dell Inc.

DELL

Silver Lake Management LLC (MBO)

21 073

HJ Heinz Co

HNZ

Berkshire Hathaway Inc.; 3G Capital Inc.

23 576

Genzyme Corp

GENZ

Takeda Pharmaceutical Co Ltd

21 237

Goodrich Corp

GR

United Technologies Corp

16 513

Whole Foods Market Inc.

WFM

Kohlberg Kravis Roberts & Co and Bain Capital

15 947

Life Technologies Corp

LIFE

Thermo Fisher Scientific Inc.

13 641

Sara Lee Corp

SLE

JBS SA; Blackstone Group LP

13 425

Motorola Mobility Holdings Inc.

MMI

12 938

# Table 2 b). Information on 5 largest deals.

Target Ticker

Announcement date

Resolution date

Offer price, \$per share

Target price before announcement, \$ per share

Target Price, Completion Date, \$ per share

Success

Failure

APC

30.12.10

15.03.12

90

68,5

31,4%

AA

03.05.11

11.09.12

25,5

16,438

55,1%

DELL

05.02.13

29.10.13

13,88

12,8

13,86

8,7%

HNZ

14.02.13

07.06.13

72,5

60,7

72,49

19,5%

GENZ

14.11.10

14.03.12

82

72,18

13,6%

GR

21.09.11

26.07.12

127,5

86,76

127,48

47,0%

WFM

18.08.11

18.08.12

90

31,2746

187,8%

LIFE

15.04.13

03.02.14

76,13

71,11

76,04

7,1%

SLE

18.12.10

14.03.12

21

17,43

20,5%

MMI

15.08.11

22.05.12

40

38,13

39,98

4,9%

# Table 4. Brier score

Brier Score B=B1+B2-B3

P-value

Base rate B1

Calibration rate B2

Resolution rate B3

Week 1

0,165

0,013

0,174

0,063

0,072

Week 2

0,163

0,0096

0,174

0,069

0,081

Week 3

0,156

0,004

0,174

0,063

0,081

Week -3

0,135

0,000

0,174

0,059

0,099

Week -2

0,126

0,000

0,174

0,054

0,102

Week -1

0,126

0,000

0,174

0,052

0,101

Total 6 weeks

0,145

0,000

0,174

0,058

0,088

# Table 5. Calibration tests

Probability category

0,05

0,15

0,3

0,5

0,7

0,9

Observed frequency

Week 1

0,25

0,18

0,63

0,86

0,95

1,00

Week 2

0,22

0,17

0,64

0,91

0,94

1,00

Week 3

0,29

0,17

0,56

0,90

0,97

1,00

Week -3

0,08

0,14

0,61

0,83

1,00

1,00

Week -2

0,08

0,13

0,64

0,80

1,00

1,00

Week -1

0,08

0,14

0,68

0,72

1,00

1,00

P-value t-test

Week 1

0,01

0,77

0,00

0,00

0,00

0,08

Week 2

0,02

0,87

0,00

0,00

0,00

0,07

Week 3

0,00

0,87

0,00

0,00

0,00

0,06

Week -3

0,60

0,96

0,00

0,00

0,00

0,05

Week -2

0,60

0,84

0,00

0,02

0,00

0,03

Week -1

0,66

0,96

0,00

0,06

0,00

0,03

P-value LR test

Week 1

0,06

0,77

0,00

0,00

0,00

N/A

Week 2

0,08

0,87

0,00

0,00

0,00

N/A

Week 3

0,04

0,87

0,01

0,00

0,00

N/A

Week -3

0,63

0,96

0,00

0,00

N/A

N/A

Week -2

0,63

0,84

0,00

0,02

N/A

N/A

Week -1

0,68

0,96

0,00

0,05

N/A

N/A

Week 1

Week 2

Week 3

Week-3

Week-2

Week-1

Average 6 weeks

(with constant)

# Pseudo

32,9%

38,1%

40,4%

53,0%

53,9%

55,1%

53,3%

P-value

0,00

0,00

0,00

0,00

0,00

0,00

0,00

Coefficient

3,96

4,41

4,65

5,55

5,26

5,61

6,37

Constant

-0,9

-1,13

-1,12

-1,48

-1,41

-1,47

-1,76

Risk-neutral probabilities (without constant)

P-value

0,00

0,00

0,00

0,00

0,00

0,00

0,00

Coefficient

2,19

2,32

2,18

2,40

2,43

2,37

2,38

Number of observations

129

129

129

129

129

129

129

Comparative analysis of option-implied and stock-implied forecasts

We continue the analysis of forecasts' performance by comparing their predictive power with that of “naпve” probabilities derived from stock prices. Recall that “naпve” probabilities are defined, as in Samuelson and Rosenthal (1986) and given by:

where -price of the stock at time t, -fallback price, -offer price per share, (T-t)-time to deal resolution, - risk-free rate for the appropriate period

Fitting a regression for a fallback price of failed deals on pre-announcement price and offer price yields the following result (standard deviation of the coefficient in parenthesis), suggesting that pre-announcement price and offer bid predict fallback price quite well:

(0,07)

Substituting the obtained fallback price estimates into the probability formula we obtain the weekly “naпve” probability forecasts for our sample of 129 deals and then compare them with option-implied probability estimates. For many of the deals stock-implied probabilities were outside the desired [0;1] range and, thus, those deals had to be excluded from the comparative analysis. Table 7 summarizes the results (pseudo- and p-values for coefficients) of cross-sectional probit regressions for each week and for the 6-week average.

The results of the comparisons are mixed. For the first 3 weeks after announcement and for the week that is 3 weeks before resolution risk-neutral forecasts generate, on average, larger pseudo , suggesting to have higher predictive power than “naпve” probabilities. However, the situation is reversed, as resolution date approaches. 2 weeks before resolution “naпve” probability estimates experience a significant jump of their predictive power and start to outperform risk-neutral probability forecasts (pseudo of 50,3% compared to 44,4% for week -2 and 63,0% compared to 47,6% for week -1 respectively). For the average of the 6-week period risk-neutral forecasts, indeed, outperform “naпve” ones in terms of predictive quality (pseudo of 41,8% compared to 36,8%).

Joint regression of deal outcome on both probability estimates suggest that predictive power of the model is significantly higher when both forecasts are used in combination. Both estimates tend to be significant, with the exception for week -3 and week -1 for which the hypothesis of no significance is rejected at 1% level for “naпve” and option-implied forecasts respectively. All in all, no straightforward answer on whether option-implied probabilities outperform “naпve” ones can be given. For the period right after the deal announcement option market tends to react more wisely, implying better predictive power of risk-neutral probabilities. Closer to resolution, however, stock market revises its expectations and stock price movements become more informative. But option-implied probabilities still add significant value to forecasting deal outcome, especially when used in combination with stock-implied probabilities.

Table 3. Probit regression output

Week 1

Week 2

Week 3

Week-3

Week-2

Week-1

Average 6 weeks

Risk-neutral probabilities

Pseudo

23,2%

33,4%

34,7%

39,6%

44,4%

47,6%

41,8%

P-value

0,00

0,00

0,00

0,00

0,00

0,00

0,00

"Naпve" probabilities

Pseudo

24,3%

26,0%

23,7%

26,9%

50,3%

63,0%

36,8%

P-value

0,00

0,00

0,00

0,00

0,00

0,00

0,00

Joint regression

Pseudo

38,4%

57,6%

48,6%

46,2%

69,0%

72,3%

59,9%

P-value "naпve" probabilities

0,00

0,00

0,00

0,04

0,00

0,00

0,00

P-value risk-neutral probabilities

0,00

0,00

0,00

0,00

0,00

0,05

0,00

Number of observations

74

73

78

86

89

89

103

Deal examples

Let's now take a closer look at some of the deals from the sample. We first consider the bid by ConAgra to acquire Ralcorp that ultimately failed. This deal also provides an example of divergence between option-implied and stock-implied probability estimates and how it changed over time. The deal was announced on 29th of April 2011 and the stock market reacted positively, indicating 75,5% success probability for the first week after the announcement. Ralcorp shares traded at and above \$86 offer bid. However, this finding contradicted the unsupportive reception of the offer by the Ralcorp's board. In the article published by the New York Times on 4th of May it was outlined that Ralcorp commented that the offer “is not in the best interest of shareholders” and adopted a shareholder rights plan. The option market, on the contrast, showed little reaction to the announcement and risk-neutral probability of success was estimated to be 25,2%. Offer was withdrawn on 19th of September. By that time bid price was raised to \$94 dollars per share. Option-implied success probability dropped to 17,2% two weeks before the withdrawal and then to 2,8% one week before the withdrawal. Stock market still over predicted the success probability, estimating it to be 51,1% two weeks before the resolution. However, during one week before the withdrawal the gap between option-implied and stock-implied probability estimates shrank with stock market indicating probability of success to be 12,4%. Daily forecasted success probabilities for post-announcement and pre-resolution periods are shown in Figure 2.

The acquisition of Ariba, provider of cloud-based collaborative commerce applications, by SAP AG in 2012 is the example of a successful deal for which “naпve” probability estimates outperformed the risk-neutral ones for the period of 3 weeks after the announcement. On 22th of May 2012 SAP AG, the largest maker of enterprise-applications software, announced to acquire Ariba Inc. for the price of \$45 per share. This offer corresponded to 15% premium compared to average price of Ariba's 2 weeks before the announcement. Market reacted with a price increase to \$45 and the stock continued to trade approximately at the offer price for the following 3 weeks. The probability of success estimated from stock prices was 99,6%, 92,1% and 86,4% for weeks 1,2 and 3 respectively. Option market, on the contrary, didn't react as sharply and estimated the success probability only at 63,8%, 72,8% and 77,6% for the above mentioned time periods. However, option market predictions improved significantly and converged to those of the stock market closer to resolution. One week before the resolution risk-neutral probability of success equalled to 90,5% while “naпve” method forecasted 92,0%. Figure 3 represents daily probability forecasts for both methods. Another important thing to notice is that we detect higher volatility for risk-neutral forecasts.

Figure 2. Post-announcement and pre-resolution option-implied and stock-implied probabilities for Ralcorp.

Figure 3. Post-announcement and pre-resolution option-implied and stock-implied probabilities for Ariba.

Merger arbitrage and excess returns

Let's now briefly consider practical application of the obtained risk-neutral probabilities to investment decisions and merger arbitrage. Recall that merger arbitrage (for cash deals) is a strategy associated with buying target company's stock as soon as possible after the announcement and selling it at the resolution date. We define the excess return on a portfolio of stocks as the difference between its return and the return on Hedge Fund Merger Arbitrage index provided by HFR database. This index aggregates the performance of merger arbitrage strategies of the whole hedge fund industry and is assumed to be a benchmark that carries the comparable level of risk. Table 4 summarizes information of excess returns associated with different portfolios. Based on the chosen sample equally weighted portfolio that is comprised of stocks that exhibited option-implied probability of success above 0,6 during first week after announcement generated the return of 4,2%, compared to 0,4% return of HFRX Merger Arbitrage index (Portfolio 4). If the investor didn't bother with analysing success probability and simply invested equal shares in all target companies after the deal's announcement the return would have been 2,7% compared to 0,3% HFRX Merger Arbitrage index return (Portfolio 1). Thus, the excess return for “high probability strategy” exceeds the one of “simple risk arbitrage strategy” by 1,5 percentage points. Portfolios that put weights on “high probability” stocks in proportion of 2 to 1 and 10 to 1 compared to “low probability” stocks generate the excess return of 2,4% and 2,6% respectively (Portfolios 2 and 3). Thus, based on the chosen sample one can infer that the optimal strategy would be to invest in “high probability” stocks only as this strategy generates higher excess returns.

Table 4. Excess returns for different merger arbitrage strategies.

Return, %

HFRX Merger Arbitrage Index return, %

Excess return, %

Portfolio 1

2,7%

0,3%

2,4%

Portfolio 2

2,7%

0,3%

2,4%

Portfolio 3

2,9%

0,3%

2,6%

Portfolio 4

4,2%

0,4%

3,8%

# References

1) Ait-Sahalia, Y., Lo A., 1998. Non-parametric estimation of state-price densities implicit in financial asset prices, Journal of Finance 53, 499-547.

2) Ait-Sahalia, Y., Lo A., 2000. Non-parametric risk management and implied risk aversion, Journal of Econometrics 94, 9-51.

3) Ait-Sahalia, Y., Wang Y., Yared F., 2001. Do option markets correctly price the probabilities of movement of the underlying asset?, Journal of Econometrics 102, 67-110.

4) Baker, Malcolm, and Serkan Savaєoglu, 2002, Limited arbitrage in mergers and acquisitions, Journal of Financial Economics 64, 91-115.

5) Barone-Adesi, Giovanni, Keith C. Brown, and W.V. Harlow, 1994. On the Use of Implied Volatilities in the Prediction of Successful Corporate Takeovers, Advances in Futures and Options Research, 7, 147-165.

6) Barraclough, Kathryn, David T Robinson, Tom Smith, and Robert E Whaley, 2013. Using option prices to infer overpayments and synergies in M&A transactions, Review of Financial Studies 26, 695-722.

7) Basset, Gibert W. Jr., 1997. Nonparametric bounds for the probability of future prices based on option values, IMS Lecture Notes - Monograph Series, 31.

8) Bliss, Robert R., and Nikolaos Panigirtzoglou, 2002.Testing the stability of implied probability density functions, Journal of Banking and Finance 26, 381-422.

9) Breeden, D., Litzenberger, R., 1978. Prices of state-contingent claims implicit in option prices. Journal of Business 51, 621-651.

10) Brown, Keith C., and Michael V. Raymond 1986: Risk Arbitrage and the Prediction of Successful Corporate Takeovers, Financial Management, 15, 54-63.

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