Decision Making In A Put-Call Parity Framework: Evidence From The Game Show Deal Or No Deal. G. Glenn Baigent Carol M.

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1 Decision Making In A Put-Call Parity Framework: Evidence From he Game Show Deal Or No Deal G. Glenn Baigent Carol M. Boyer Jiamin Wang ABSRAC We propose put-call parity as a framework for analyzing decision making in the sense that it captures upside potential and downside risk relative to a fixed reference point or riskless asset. his framework is tested empirically using data from the game show Deal or No Deal where we find that contestant behavior generally reflects the coefficients with the correct signs as described by put-call parity but individuals place the greatest amount of weight on potential losses (put option values), indicating a tendency towards loss-aversion. JEL Classification: C70, C71, G13, and G19 Keywords: Decision making, option pricing, put-call parity. INRODUCION Consider a story of a wealthy man who observed someone buying a $1 Megamillions lottery ticket. he wealthy man learned that the jackpot would be split equally between any ticket-holders who chose the winning combination of numbers. After a short pause, the wealthy man purchased ten tickets with identical numbers. When asked why he did this he replied, If I have the winning combination of numbers, I m not sharing a significant proportion of the jackpot with another winner. echnically, the man was holding a call option on the lion s share of the jackpot with an exercise price of $0 because he had to pay nothing extra and had only to present the tickets to the lottery corporation to claim the prize. hat is, for a premium of nine dollars, which probably meant little to him because of the decreasing marginal utility of wealth, he could insure an asset with a potential value of millions of dollars. (As he is losing value on his jackpot he can redeem its value through an option to repurchase a portion of the jackpot at $0.) We propose that all decision making can be expressed in terms of options regardless of whether the choice is at a lottery, a buffet restaurant, or a capital investment project, and doing so provides insights into risk assessment and economic behavior. Viewing decision-making in the context of put-call parity is a significant departure from utility theory approaches and its well-documented deficiencies. o counter the shortcomings of utility theory, including the fact that many of the inputs are unobservable, Kahneman and versky (1979) advanced the behavioral sciences literature by defining prospect theory wherein individuals evaluate gains and losses rather than solving for the optimal solution of an expected utility function. Further, cumulative prospect theory postulates that individuals evaluate gains and losses relative to a reference point. We observe that this evaluation process is consistent with put-call parity as the call option captures the upside or potential gains and the put option captures the downside or potential losses, and the positive and negative deviations are relative to a datum, analogous to the exercise price of an option. We begin our analysis by reviewing Lee, Martin, and Senchak s (1982) proof of how any unknown outcome or gamble can be couched in terms of the put-call parity relationship. In this framework the call option captures the upside potential and the put option reflects the downside loss or risk so that any random outcome can be defined as a riskless quantity plus a contingent claim if the outcome is above the riskless quantity minus the potential loss if the outcome is below the riskless quantity. Moreover, we observe the similarity between the value function described by Kahneman and Baigent is an Associate Professor of Finance, Boyer is an Associate Professor of Finance, and Wang is an Associate Professor of Management. hey are all at Long Island University. Contact author is Baigent. heir s are glenn.baigent@liu.edu, carol.boyer@liu.edu, and jiamin.wang@liu.edu respectively. Journal of Financial and Economic Practice Page 72

2 versky and put-call parity. If each of the coefficients in versky and Kahneman s (1992) value function is constrained to unity and the reference point is greater than zero, then their prospect theory value function is linear and is identical to put-call parity. he link between prospect theory and option pricing has also been identified by Versluis, Lehnert, and Wolff (2008) who study option pricing in the context of prospect theory in an attempt to explain prices that diverge from those predicted by the Black-Scholes model. (See the appendix.) Our analysis assesses how individuals assign weights or probabilities to uncertain outcomes and in making decisions. In the context of put-call parity, we suggest that if risk-averse (risk-taking) behavior exists, it will be manifested through a greater emphasis on potential losses (gains), and that behavior will be reflected in the emphasis assigned to put (call) option values. he data used in our study was obtained from episodes of the game show Deal or No Deal that premiered in the Netherlands in he popularity of the game show quickly spread to other countries including the United Kingdom, the United States, Germany, Mexico, Australia, and Italy. Data from the Netherlands, Germany and the United States is used in our analysis. Given the characteristic of potentially large stakes gambles under uncertainty, there is a natural interest in studying the decisionmaking behavior of participants. Studies on this topic include Andersen et al (2006), Blavatsky and Progrebna (2006a, b), Mulino et al (2006), Bombardi and rebbi (2005), Deck et al (2006) and lastly Post et al (2008), however, none of these studies considers an option pricing framework for decision-making. As a brief description, in the US version of the game each contestant starts with 26 cases, each of which contains a monetary value ranging from small amounts ($0.01) to larger amounts ($1M). After choosing a case, the contestant enters different stages of the game. If the contestant chooses to play the game, Stage 1, (s)he must open six additional cases. After opening the six cases there is an offer from a banker. If the contestant chooses to continue to play, (s)he must then open five additional cases (Stage 2), which is followed by another offer from the banker. he contestant must open 4, 3, or 2 cases, before finally reaching a stage where (s)he can open one case at a time. Our analysis is restricted to decisionmaking in the one case at a time stages of the game. Our rationale for examining the one case at a time stages is two-fold. First, most games reach the latter stages so that study of the early rounds does not allow for many inferences regarding behavior probably and may be biased. Casual observation indicates that at least two rounds are played in the US version of the game before individuals accompanying the contestants are brought to the stage for introduction and input on whether to accept the deal offered by the banker. Second, we contend that earlier stages of the game have too many possible outcomes for contestants to make informed decisions. For example, after choosing one case and opening six, there are still nineteen cases left in play. he number of combinations or possible outcomes is 19! (19 5)! = 1,395, 360. Even at a later stage of the game where a contestant has only two cases to open the number of combinations is 7! (7 2)! = 42. Analyzing the probability of each of those outcomes along with their monetary value is most likely too difficult to be calculated with any degree of precision. [Consider Kyle s (1985) definition of precision, which is the inverse of the variance conditioned on the observation of a random variable.] In addition to limiting our analysis to the one case at a time rounds, the banker s offer is considered to be the riskless asset and contestants must evaluate potential gains and losses around that reference point. Post et al examine the regime to determine the bank offer as an expected outcome by using a dynamic model that estimates the rate at which the bank offer approaches the expected value. Our analysis is myopic and considers each round s decision independently from the previous with the exception of a correction for idiosyncratic errors. In the behavioral economics literature mental framing is defined as the way the problem is presented and for the game show contestants we have little to say on this matter since the presentation of the problem is straightforward: contestants know the cases that are remaining and the probability of opening each one (1/n). With respect to mental accounting we have a better story to tell. Mental accounting is the process by which individuals code, categorize, and evaluate outcomes with differing economic impact and it consists of two stages. In the editing stage of the process individuals simplify Journal of Financial and Economic Practice Page 73

3 complex problems into simpler sub-problems. In the context of using put-call parity, this is analogous to recognizing the probability of an increase or decrease in the next bank offer. his is followed by the evaluation stage in which contestants assess the economic impact of each outcome. hat is, contestants evaluate the impact of opening each of the remaining cases and the change to the next bank offer. For each observation in the data set we compute the call and put option values given the bankers offers and analyze the role of potential gains and losses through a probit model to determine if they are significant factors in making the choice between Deal and No Deal. he result that is consistent across all groups is that potential loss (measured by put option value) is a significant factor in decision making. German, American, and Dutch contestants all place significant emphasis on the probability of a negative change to the next bank offer (the riskless asset) so that loss aversion is evident. Regarding upside potential as measured by the value of call options, the results are somewhat mixed. German and American contestants place emphasis on upside potential but it is ignored by and statistically insignificant for Dutch contestants. Pairing contestants from Germany and the United States shows that the bank offer, the call option, and the put option are all significant, but the insignificance of upside potential for Dutch contestants occurs with so much magnitude that upside potential falls out of the decision making process when we aggregate the data and test the entire sample. In general, we find that contestants rationally evaluate the outcomes and probabilities of their decision, but there is a stronger tendency towards loss aversion. o reiterate, we note that potential downside loss is more statistically significant than upside gain potential. his finding is consistent with literature documenting asymmetric responses to outcomes. For example, Brown, Harlow, and inic s (1988) uncertain information hypothesis suggests that traders under-react to good news and overreact to bad news. he question is, does individual behavior reflect the probabilities and magnitudes of these outcomes a priori? Our framework for decision making allows for individuals to evaluate the probabilities and outcomes before they occur and that probabilities and magnitudes are reflected in call and put option values. he last portion of our study involves removing observations where there is no upside or downside potential. In total, 9.6% of the observations have a peculiarity represented by call and put option values of zero. If there is no downside risk then contestants should continue to play. Alternatively, if there is no upside potential then there is no reason to continue playing the game. We removed these observations and do not find an appreciable difference in the empirical results, although we suggest that the results are more robust. We propose that continuing to play in the absence of upside potential or ceasing play in the absence of downside potential are observations that need to be examined in the context of overconfidence or extreme risk (loss) aversion, and these behaviors can be identified through a put-call parity framework. Removing these observations does not appreciably change the empirical results, but leaves an avenue for further study into economic behavior or decision making. he remainder of this paper progresses as follows. We propose an option pricing framework for decision making in the context of any risky choice followed by an example of binomial option values and uncertain outcomes. he next section contains a description of the data and empirical results. We end with concluding remarks. AN OPION PRICING FRAMEWORK FOR DECISION-MAKING At each stage of the game, Deal or No Deal, contestants are presented with an offer from a banker, which serves as a risk-free asset because they can end the game with that dollar amount with probability 1.0. We denote the banker s offer by the quantity X. Following Lee, Martin, and Senchak (1982) we write the value of any uncertain outcome or gamble, V, at time as ( V X, 0) + Min( V X ) V = Max, (1) Journal of Financial and Economic Practice Page 74

4 Equation (1) shows the presence of a call option on the value of an asset (or the game in this case), through the first term. he second term has little intuitive meaning but is easily massaged so that its economic relevance is evident. aking the second term, Min( V, X ), and adding and subtracting X yields Min( V X, 0) + X. Recognizing that Min( V X, 0) Max( X V,0), and substituting this result into (1) yields V ( V X 0) Max( X V, ) = X + C P = X + Max, 0 (2) In (2), C represents a call option on continuing the game and captures the upside potential of the game, which is choosing cases with numbers lower than X, leaving the higher dollar amounts in play so that the next bank offer is greater than X. P captures the downside risk and is mathematically equivalent to the situation in which a contestant chooses a case with a value higher than X, leaving cases with lower amounts in play. Equation (2) is a restatement of the put-call parity relationship as shown by Stoll (1969). At each decision point in the game the banker s offer is X : If the contestant decides to no longer play, the values of C and P are zero, and the contestant earns X. However, if the contestant chooses No Deal and plays one more round, then the call and put option capture the upside and downside potential of the game. he call and put option values are based on all possible outcomes relative to the bank offer, X (reference point). he underlying assumption (or expectation) is that the next bank offer will reflect without bias the possible outcomes and their probabilities, but the evidence provided by Post et al indicates that this assumption is violated. Specifically, they solve for the time series coefficient or speed at which the bank offer approaches the average or expected bank offer and the equation they examine is, 9 r B ( x) = b r +1 xr + 1 where, b = b + ( b ) r + 1 r 1 r ρ (3a, b) Post et al find that the coefficients are in the vicinity of 0.8. Equations 3a and 3b, describe a serial relationship between bank offers and their empirical results reflect a convergence towards intrinsic value in much the same way that futures contract prices exhibit properties of normal backwardation or contango. In our analysis we assume that the coefficient b r +1 = 1 and that ρ = 0 across contestants. In this sense our modeling is myopic or that the bank offer will be unbiased. he only correlation (serial) that must be adjusted for is observations within contestants (clusters). We consider the bank offer to be myopic. herefore, we remove the observations where the decision to continue to play or not has been biased by bank offers that violate rules of unbiased expectations. DECISION HEORY AND BINOMIAL OPIONS In this section we compute the value of the game using two methodologies decision theory and option pricing. Computing the expected value of an event with good and bad outcomes is identical to computing the binomial option prices of the same uncertain set of outcomes. he importance of the approach is that it abstracts from the use of utility theory, upon which Post et al base their analysis. For illustrative purposes, consider an example from an episode in which the contestant was faced with the following cases to choose from, and a bank offer of X = $105, 000. he contestant has a 4/5 probability of increasing the bank offer by choosing a case with a value less than $105,000 and a probability of choosing a case with value greater than $105,000 of 1/5. he table below shows the expected gain in the value of the game if the cases with amounts shown in the left-hand column are chosen. Journal of Financial and Economic Practice Page 75

5 he average gain from choosing from the first group is $40, and the loss from choosing the $500,000 case is $77, Multiplying the average outcomes by their probabilities gives a change in the value of the game of (40, )(4/5)-(77,387.50)(1/5) = $17, so that the expected value of the game is $105,000+$17,090 = $122,090. (Despite the positive change in expected value, the contestant took the deal to forego the expected additional value.) Cox, Ross, and Rubenstein (1979) describe a framework in which there are two possible states of nature; the value of the underlying asset either increases or decreases. his describes the situation for a contestant on the game show Deal or No Deal in the sense that the next choice will either increase or decrease the banker s offer. Using the known up and down probabilities of p = 4 5 and 1 p = 1 5, the following results obtain. From able 3, the resulting call and put option values are: 4 C = [( )( 105,000) 105,000] + [ 0] = $32, P = 1 = 5 5 herefore, from (2), [ 105,000 ( )( 105,000) ] + ( 0) $15, V = X + C P = 105, , , = $122,090 As mentioned above, the contestant decided to forego the expected value added and accepted the bank offer of $105,000, evidence of loss-averse behavior. DAA AND EMPIRICAL RESULS In this section we describe the data used to test our model. he data includes the framing and accounting stages and we hypothesize that decisions are potential gains reflected in call options values and potential losses that are reflected by put options values. We use the same data set as Post et al (2008) with several exclusions but we add computed values of the call and put options. Full Sample Results Data used in our analysis is a subset of that used by Post et al because we do not analyze experimental data and we consider only those rounds in which a contestant opens a single case. here are three countries represented in the data: he Netherlands, Germany, and the United States. A logistic regression model was run for each country and then across all of the countries in the full sample. We also consider Germany and the United States together. he structure of the model is: Deal or No Deal = f (X, C, P). Where X is the bank offer, C is the computed call option value and P is the computed put option value. A complication with the data is that it is in clusters grouped by contestants. he potential serial correlation caused by clustering of contestant observations is corrected for and the MLE coefficients are presented in able 4a. he number of observations per contestant varies from 1 to n, depending on how many rounds the contestant plays. he optimization technique used is Fisher s scoring. he results indicate that German contestants place little emphasis on the bank offer and more weight on downside risk than upside potential in making a decision to continue playing or to stop and accept the bank offer. he low weight assigned to the bank offer is likely due to the German game having a smaller monetary scale. he Dutch contestants place weight on the bank offer and the downside risk, Journal of Financial and Economic Practice Page 76

6 but the upside potential represented by the call option values is insignificant. Contestants from the United States also give significant consideration to the bank offer but weight the upside potential more than participants from Germany. US contestants also weight the downside more than their German counterparts. When the data from Germany and the US are pooled, emphasis is placed on all three factors, but the most weight is placed on potential losses (put option value). Across all countries there is a significant emphasis on the bank offer, less weight on potential gains, and significant emphasis on potential losses. In aggregate we find that individuals consider upside and downside gains and losses consistent with put-call parity, but more emphasis is placed on potential losses so that loss-aversion is demonstrated. here is no house money effect observable in this data set and a reference point of $0 is not evident. Biases in the Bank Offer A critical assumption of our methodology is that the bank offer is rational and satisfies equation 3a. hat is, the call and put option values are computed around a reference point that is expected to reflect all possible outcomes. According to 3a, we should find that C i > 0 and P i > 0, i as long as the bank offer falls between the value of the lowest and highest cases remaining. However, when we review the computed call and put option values we find that in 9.6% of the observations, either the call or the put option value is equal to zero so that there is either no upside potential or downside risk. able 4b shows the empirical results after we remove the observations argued to contain biased bank offers. We observe little difference in the empirical results but suggest that they give a better representation of the coefficient values for the contestants. o emphasize the impact of the bank offer, consider an observed round in the German game where the bank offer was X = $ 600 and the remaining cases were $0.20, $20.00, and $ If the contestant chooses the case with $0.20 then the expected offer is $510.00; if the $20.00 case is chosen the expected offer is $500.10; and if the $ case is chosen the expected offer is $ Under assumptions of unbiased expectations regarding the bank offer there is no probability of the bank offer exceeding $600.00, therefore, the contestant should stop playing and accept the bank offer, but this is not what occurred. he contestant ignored the risk and continued playing, eventually winning $750. (he contestant chose the $20 case and the bank offer was $750 and that offer was accepted.) When there is no downside risk and no upside potential there is a perturbation of the decision-making process. hat is, if we determine that P = 0 there is no downside risk to be avoided so that a contestant should continue to play. Alternatively, if C = 0, the contestant should statistically cease playing. If a contestant continues to play when (s)he should not, (s)he must be exhibiting risk taking behavior or perhaps overconfidence. Conversely, if a contestant does not continue to play when (s)he should, (s)he may be exhibiting risk avoidance behavior or have a significant coefficient of risk aversion. CONCLUSION We propose a put-call parity framework for decision making as being consistent with prospect theory in the sense that individuals are able to frame the problem and then proceed to a mental accounting stage to evaluate potential outcomes. Our hypothesis is that this information is summarized in call and put option values around a fixed point of reference. We test this hypothesis using a subset of the data used by Post et al (2008) and examine only the one case at a time rounds of the game show Deal or No Deal. We find that the greatest emphasis is placed on potential losses, captured by put option values, so that loss aversion is more descriptive of contestant behavior. We also suggest that when decisions are couched in terms of call and put option values (or upside and downside potential), deviations from the pattern of behavior predicted by put-call parity can be identified and examined to gain further insights into decision making. Journal of Financial and Economic Practice Page 77

7 able 1 Case Amount Probability $50 1/5 $400 1/5 $10,000 1/5 $100,000 1/5 $500,000 1/5 able 2 Case Opened Expected Change to $105,000 Offer $50 $47, $400 $47, $10,000 $45, $100,000 $22, $500,000 -$77, able 3 State of Nature Probability ( R) E Call Option Put Option Choose lower than $105K 4/ % 32, Choose higher than $105K 1/ % 0 $15, Journal of Financial and Economic Practice Page 78

8 able 4a Empirical Results Full Sample Country Number of Observations Number of Contestants Intercept X C P Germany E-6 (0.9062) 6.5E-5 (0.0890) -5.4E-4 (0.0433) Netherlands (0.0005) 1.4E-5 (0.1307) -5.57E-6 (0.6050) -1.7E-4 (0.0100) United States E-6 (0.1188) 1.5E-5 (0.0633) -1.1E-4 Germany & USA E-6 (0.0409) 1.8E-5 (0.0404) -1.4E-4 All Countries E-6 (0.0238) 8.67E-6 (0.2475) -1.3E-4 Note: Coefficients are shown above with p-values in parentheses. able 4b Empirical Results Reduced Sample Country Number of Observations Number of Contestants Intercept X C P Germany E-6 (0.8492) 5.8E-5 ( ) -4.5E-4 (0.0406) Netherlands (0.0008) 1.4E-5 (0.1279) -5.44E-6 (0.6068) -1.6E-4 (0.0096) United States E-6 (0.1213) 1.5E-5 (0.0701) -1.1E-4 Germany & USA E-6 (0.0378) 1.8E-5 (0.0371) -1.3E-4 All Countries E-6 (0.0244) 8.56E-6 (0.2412) -1.2E-4 Note: Coefficients are shown above with p-values in parentheses. Journal of Financial and Economic Practice Page 79

9 REFERENCES Allais, M., 1953, Le comportement de l homme rationnel devant le risque: critique des postulats et axiomes de l école Américaine, Econometrica, 21, Andersen, S, G. Harrison, M. Lau, and E Rutstrom, 2006, Dynamic choice behavior in a natural experiment. University of Durham working paper in economics and finance 06/06. F. Black and M. Scholes, 1973, he pricing of options and corporate liabilities, Journal of Political Economy, Blavatsky, P and G. Pogrebna, 2006a, Loss aversion? Not with half-a-million on the table! University of Zurich working paper 274. Blavatsky, P and G. Pogrebna, 2006b, Risk aversion when gains are likely and unlikely: Evidence from a natural experiment with large stakes. Zurich IEER working paper 278. Bombardi, M. and F. rebbi, 2005, Risk aversion and expected utility theory: A field experiment with large and small stakes. Brown, K., W. V. Harlow, and S. inic, 1988, Risk aversion, uncertain information, and market efficiency. Journal of Financial Economics, 22, Cox, J, S. Ross, and M. Rubenstein, 1979, Option pricing: A simplified approach. Journal of Financial Economics, 7, Daniel, K., J. Hirschleifer, and A. Subrahmanyam, 1998, Investor psychology and security market under- and overreactions, Journal of Finance, 53, Deck, C. A., J. Lee, and J. A. Reyes, Risk attitudes in large stake gambles: Evidence from a game show. De Roos, N. and Y. Sarafidis, 2006, Decision making under risk in Deal or No Deal. Kahneman, D. and versky, A., 1979, "Prospect heory: An analysis of decision under risk," Econometrica, 47, Kyle, A., 1985, Continuous auctions and insider trading, Econometrica, 53, W. Y. Lee, J. D. Martin, and A. J. Senchack, 1982, he case for using options to evaluate salvage values in financial leases, Financial Management, Mulino, D., R. Scheelings, R. D. Brooks, and R. W. Faff, 2006, An empirical investigation of risk aversion and framing effects in the Australian version of Deal or No Deal. H. R. Stoll, 1969, he relationship between put and call option prices, Journal of Finance, versky A. and D. Kahneman, 1992, Advances in Prospect heory, Cumulative Representation of Uncertainty, Journal of Risk and Uncertainty, 5, Versluis. C.,. Lehnert, and C. Wolff, 2008, A prospect approach to option pricing, Working paper. Journal of Financial and Economic Practice Page 80

10 APPENDIX he value function with a non-zero reference point presented by versky and Kahneman (1992) is v ( x) = a ( x RP) λ( RP x) b if if x RP x < RP Where x is the outcome, RP is the reference point, and λ, a, and b are coefficients that describe weights and the curvature of the value function. If we make all of the coefficients equal to 1.0 we have, v ( x) x RP = ( RP x) if if x RP x < RP v is the outcome if x RP (call option) and its probability less the outcome if x < RP and its probability (put option), all of which is relative to a reference point. Intuitively, the value of the game ( x) Journal of Financial and Economic Practice Page 81