Estate of Andrew J. McKelvey v. Commr. of Internal Revenue

Jon O. Newman, Circuit Judge: This appeal concerns somewhat unusual financial instruments known as variable prepaid forward contracts ("VPFCs"). A VPFC is an agreement between a short party (typically, the shareholder of a large *28 quantity of low-basis, appreciated stock) and a long party (typically an investment bank). The long party agrees to pay the shareholder a substantial sum of money equal to the value of the stock discounted to present value. In exchange, the shareholder agrees to deliver to the long party on a specified settlement date up to a maximum number of shares of stock (or their cash equivalent), the exact number to be determined by the price of the shares on a specified valuation date. The short party also agrees to secure its delivery obligation with the maximum number of shares to be delivered at settlement. A VPFC usually sets a floor price and a cap price that limit the number of shares to be delivered in the event that the share price on the valuation date is below the floor price, above the cap price, or between them. The issues on this appeal arise because a shareholder, after executing two similar VPFCs with two financial institutions, paid substantial sums of money to each institution to obtain an extension of the settlement date and, more significantly, the valuation date. There are two precise issues. The first is whether, with respect to each contract, the extensions resulted in a short-term capital gain. The Commissioner of Internal Revenue ("Commissioner") contends that a short-term capital gain occurred because either (1) the extension of the valuation date resulted in an exchange of property with a more valuable new contract replacing the original contract or (2) a termination of the delivery obligation occurred because the obligation in the first contract to deliver shares on the original settlement date was extinguished. The second issue is whether, with respect to each contract, the extension of the valuation date also resulted in a long-term capital gain. The Commissioner contends that the execution of each new contract resulted in a constructive sale of the shares pledged as collateral to secure the obligation of the new contract. His reason for this claim is that, on the date of the new contract, the share price of the stock pledged as collateral was so far below the floor price that there was no more than a fifteen and thirteen percent probability, respectively, for each contract that the share price would reach that floor price and therefore, under each contract, the shareholder would almost certainly be required to deliver the maximum number of collateralized shares. As a result, the Commissioner contends, the number of shares to be delivered at settlement was "substantially fixed" within the meaning of 26 U.S.C. § 1259 (d)(1) on the date of each new contract, resulting in a long-term capital gain on shares constructively sold. These rather esoteric issues arise on an appeal by the Commissioner from the May 22, 2017, decision of the United States Tax Court (Robert P. Ruwe, Judge) rejecting the Commissioner's claims to collect $41,257,103 from the estate of Andrew J. McKelvey ("Estate") for both short- and long-term capital gain taxes alleged to have been incurred by the decedent in 2008. Background McKelvey, who died on November 27, 2008, was the founder and principal shareholder of Monster Worldwide, Inc. ("Monster"), a publicly traded company that maintains a website, monster.com, which helps job-seekers find jobs. In 2007, McKelvey executed two VPFCs, one with Bank of America, N.A. ("BofA") as long party and another with Morgan Stanley & Co. International plc ("MSI") as long party. The BofA VPFC. Under the BofA contract, which became effective September 11, 2007, BofA agreed to pay McKelvey *29 $50,943,578.31 on September 14, 2007; he agreed to pledge 1,765,188 shares of Monster stock to secure his obligation to BofA; and he agreed to deliver to BofA up to 1,765,188 shares of Monster stock (or the cash equivalent) at settlement. Settlement was to be made by delivering to BofA up to ten percent of the 1,765,188 shares on each of ten consecutive weekdays between September 11 and 24, 2008. At the close of trading on the NASDAQ on September 11, 2007, the price of Monster stock was $32.91. The contract provided that the actual number of shares to be delivered on each of the ten settlement dates would be determined in one of three ways, depending on the closing price of Monster stock on each of the ten dates. If the closing price on a settlement date was less than (or equal to) $30.4610 ("BofA floor price"), the number of shares to be delivered on each of the ten dates would be 176,519 (ten percent of 1,765,188). 1 If the closing price on a settlement date was more than the BofA floor price but less than (or equal to) $40.5809 ("BofA cap price"), the number of shares to be delivered on each of the ten dates would be a fraction of 176,519: the numerator of the fraction would be the BofA floor price and the denominator would be the Monster stock closing price. If the closing price on a settlement date was more than the BofA cap price, the number of shares to be delivered would be a more complicated fraction of 176,519: the numerator of the fraction would be the closing price minus the difference between the BofA cap price and the BofA floor price, and the denominator would be the closing price. These three methods of determining the number of shares to be delivered at settlement would yield curious results. To illustrate these results, it will be convenient to ignore the fact that ten percent of the total number of the 1,765,188 shares would be delivered on each of ten consecutive weekdays and consider the collateralized shares as a bloc. If the closing price was equal to, or any price below, the floor price, the number of shares to be delivered would always be the total number of shares pledged as collateral, which would be the maximum number of shares required to be delivered at settlement. If the closing price was between the floor price and the cap price, the number of shares to be delivered would decline from 1,765,188 the closer the closing price was to the cap price. The decline would end when the closing price equaled the cap price, at which point the number of shares to be delivered would be 1,324,993 (1,765,188 times 30.4610/40.5809). 2 If the closing price was any price above the cap price, the number of shares to be delivered would increase from 1,324,993 and continue to increase the more the closing exceeded the cap price. The increase would be continuous as the closing price increased and the number of shares to be delivered approached the total number of the collateralized shares, but the number of shares to be delivered would never exceed that maximum total number. These effects are illustrated in the following table, showing an example of how many shares of a 1,000-share bloc would be delivered at various closing prices: some below or equal to the floor price of 30.5 (rounded), some between the floor price and the cap price of 40.6 (rounded), and some above the cap price. The table also shows the fraction used to *30 determine the number of shares to be delivered. closing price: 20 25 30.5 35 40 40.6 45 50 60 fraction: 30.5/35 30.5/40 30.5/40.6 34.9/45 39.9/50 49.1/60 shares delivered: 1000 1000 1000 850 763 751 776 798 818 Under the BofA contract, McKelvey had the option to settle the contract with the "cash equivalent" no matter which of the three methods for determining the number of shares to be delivered was applicable. The cash equivalent for each share to be delivered was 105 percent of the closing share price three trading days prior to the valuation date for the first portion of the collaterized shares to be delivered, which was September 11, 2008. 3 Of course, had McKelvey used the cash equivalent option (for both the BofA and MSI contracts), he would have had to pay a substantial sum of money. On July 24, 2008, two months before the ten settlement dates, McKelvey paid BofA $3,477,949.92 to amend the BofA contract by extending the original settlement dates, which also served as valuation dates, from ten consecutive weekdays in September 2008 to ten consecutive weekdays in February 2010 ("amended contract"). No other terms of the 2007 BofA contract were changed. On the date of the BofA extension, the closing price of Monster stock was $18.24. The MSI VPFC. Under the MSI contract, effective September 24, 2007, MSI agreed to pay McKelvey $142,626,185.80 on September 27, 2007; he agreed to pledge 4,762,000 Monster shares to secure his obligation to MSI; and he agreed to deliver to MSI up to 4,762,000 Monster shares (or the cash equivalent) on September 24, 2008. At the close of trading on the NASDAQ on September 24, 2007, the price of Monster stock was $33.47. The contract provided that the actual number of shares to be delivered on the settlement date would be determined in one of three ways depending on the average of the closing prices of Monster stock on ten valuation dates ("average price"). If the average price was less than (or equal to) $30.894 ("MSI floor price"), the number of shares to be delivered would be 4,762,000. If the average price was more than the MSI floor price but less than (or equal to) $35.772 ("MSI cap price"), the number of shares to be delivered would be a fraction of 4,762,000, the numerator of the fraction to be the MSI floor price and the denominator to be the average price. If the average price was more than the MSI cap *31 price, the number of shares to be delivered would be a more complicated fraction of 4,762,000: the numerator of the fraction would be the average price minus the difference between the MSI cap price and the MSI floor price, and the denominator would be the average price. These three methods of calculation yielded precisely the same curious results described above with respect to the BofA contract when the closing price was equal to or below the MSI floor price, above the MSI cap price, or between them. Under the MSI contract, like the BofA contract, McKelvey had the option to settle the contract with the "cash equivalent" no matter which of the three methods for determining the number of shares to be delivered was applicable, but the calculation of the cash equivalent differed from the BofA contract. Under the MSI contract, the cash equivalent was the number of shares to be delivered multiplied by the closing price of Monster stock on the last of the ten averaging dates. 4 That date was September 24, 2008. On July 15, 2008, two months before the settlement date, McKelvey paid MSI $8,190,640 to amend the MSI contract by extending the original settlement date from September 24, 2007, to January 15, 2010, and to extend the dates on which the average price would be determined from ten consecutive weekdays in September 2008 to ten consecutive weekdays in January 2010 ("amended contract"). No other terms of the 2007 MSI contract were changed. On the date of the MSI extension, the closing price of Monster stock was $17.28. On the dates of the extensions of both the BofA and MSI contracts, the value of McKelvey's Monster shares was about $114 million. If McKelvey had delivered his Monster shares on those dates instead of extending the settlement and valuation dates of the VPFCs, he would have realized a substantial capital gain. Settlement of amended contracts . After McKelvey's death, the Estate settled the amended BofA contract by delivering 1,757,016 Monster shares to BofA on May 8, 2009, 5 and settled the amended MSI contract by delivering 4,762,000 Monster shares to MSI on August 5, 2009. Both the original VPFCs and the amended contracts provided for expedited settlement in the event of various occurrences including McKelvey's death. The parties make no claim that the expedited settlements have any significance to the issues on appeal. The Estate obtained a stepped-up basis for the Monster shares. See 26 U.S.C. § 1014 (a)(1). To recapitulate: by executing both VPFCs in September 2007, McKelvey received *32 about $194 million, 6 pledged about 6.5 million Monster shares, 7 then worth about $218 million, 8 and agreed to deliver one year later between about 5.4 million 9 and 6.5 million Monster shares (then worth between about $181 million 10 and $218 million). Ten months later, McKelvey paid $11,668,590 to execute amended contracts, which extended the settlement dates and the valuation dates that would determine the number of shares to be delivered at settlement. The Estate settled the amended contracts by delivering 6,519,016 Monster shares, which the Commissioner states were worth about $88 million, to BofA and MSI. Neither McKelvey nor the Estate paid any income taxes with respect to the Monster shares. McKelvey's 2008 income tax return. McKelvey's 2008 federal income tax return, filed by the executor of his Estate, reported no income attributable to the execution of the amended contracts. The Estate's reason for not reporting any short-term capital gain was its view that the extensions of the settlement and valuation dates did not result in a taxable exchange of the original VPFCs for the amended contracts. The Estate's reason for not reporting any long-term capital gain was its view that such a gain could not have occurred until the amended contracts were settled by delivery of Monster shares to BofA and MSI, and, by that time, the shares had acquired a stepped-up basis following McKelvey's death, see 26 U.S.C. § 1014 (a)(1), and the stock price had declined between the date of death and the settlement date. The Commissioner's deficiency determination. The Commissioner determined a deficiency of more than $41 million in McKelvey's 2008 federal income tax based on his determination that McKelvey realized a capital gain of more than $200 million when he executed the VPFC extensions in 2008. This deficiency was based on two separate determinations. First, McKelvey realized a short-term capital gain because the extensions of the settlement and particularly the valuation dates resulted in taxable exchanges of the original VPFCs for the more valuable amended contracts, which the Commissioner deemed to be "forward contracts" within the meaning of 26 U.S.C. § 1259 (d)(1). 11 Second, McKelvey realized a long-term *33 capital gain because the number of shares to be delivered at settlement of these forward contracts was "substantially fixed" within the meaning of subsection 1259(d)(1), resulting in constructive sales of the Monster shares that he had pledged as collateral under what the Commissioner deemed to be forward contracts. We set forth in more detail the Commissioner's rationale for these claims of capital gains below when we consider his arguments on this appeal. The Tax Court decision . McKelvey's estate commenced a Tax Court action to challenge the Commissioner's determinations. On joint motion of the parties, the case was decided without trial based on stipulated facts. See Estate of McKelvey v. Commissioner , 148 T.C. No. 13 , 2017 WL 1402129 , at *1 (2017) (" TC op."). The Tax Court began its consideration by noting that the execution of the VPFCs in 2007 did not result in recognition of any capital gains and would not result in any capital gains until the VFPCs were settled. The VPFCs were "open" transactions, i.e. , the identity of and the number of shares to be delivered at settlement was not substantially fixed because the taxpayer could substitute cash or non-collaterized stock to satisfy his delivery obligations and the amount of cash or stock to be delivered depended on the stock price at settlement. The Commissioner had previously acknowledged that VPFCs did not incur capital gains when executed. That position conformed to Revenue Ruling 2003-7, 2003- 1 C.B. 363 (2003). The Tax Court then noted that the ultimate issue to be decided was "what tax consequences, if any, occurred when [McKelvey] extended the settlement and averaging dates of the original VPFCs." TC op. 14, 2017 WL 1402129 , at *4. Judge Ruwe observed that neither party had cited any decisions considering the tax consequences of extending VPFC valuation dates, and the case appeared to be one of first impression in the Tax Court. It is in this Court as well. The Tax Court ruled in favor of the taxpayer on all issues. With respect to the claimed short-term capital gain, the Court held that the execution of the contracts was not a taxable "disposition of property" under 26 U.S.C. § 1001 because the VPFCs were not "property" to the taxpayer at the time they were exchanged for the amended contracts. See id. at 18-23 , 2017 WL 1402129 , at *6-*8. The Court explained that, at the time the amended contracts were signed, McKelvey had received the cash prepayment due him under each VPFC and "had only obligations under the contracts-and obligations are not property ... and therefore section 1001 is inapplicable." Id . at 20, 2017 WL 1402129 , at *7. Having concluded that the VPFCs were not property on the date of the amended contracts, the Tax Court did not consider the possibility that the execution of the amended contracts resulted in short-term capital gain on the theory that the obligations of the VPFCs had been terminated by the execution of the amended contracts. See 26 U.S.C. § 1234A(1). 12 With respect to the claimed long-term capital gain, the Tax Court ruled that the amended contracts did not result in the constructive sale of the collateralized Monster shares under 26 U.S.C. § 1259 . See *34 TC op. at 35-36, 2017 WL 1402129 , at *11-*12. The Court stated that the "open transaction treatment" under the VPFCs "continued" under the amended contracts, see id . at 36, 2017 WL 1402129 , at *12, which the Court did not regard as forward contracts under section 1259(d)(1). The Tax Court also noted that because shares other than the Monster shares pledged as collateral could be used to settle the amended contracts, McKelvey "had the discretion to settle the VPFCs using stock with a higher or lower basis than the stock pledged as collateral." Id . at 30, 2017 WL 1402129 , at *10. Discussion I. Standard of Review This Court reviews de novo Tax Court decisions rendered on a stipulated record. See General Electric Co. v. Commissioner , 245 F.3d 149 , 154 (2d Cir. 2001). Generally, the Commissioner's determinations in a notice of deficiency are presumed correct, and the taxpayer bears the burden of proving them incorrect by a preponderance of the evidence. See Tax Ct. Rule 142(a) ; Welch v. Helvering , 290 U.S. 111 , 115, 54 S.Ct. 8 , 78 L.Ed. 212 (1933). II. Short-Term Gain We agree with the Tax Court that McKelvey did not incur a short-term capital gain on the basis of the Commissioner's claim that replacement of the VPFCs with the amended contracts was an "exchange of property." 26 U.S.C. § 1001 (c). At the time the VPFCs were extended, McKelvey no longer had any rights in the contracts that could constitute property. He had already received the $194 million prepayments from the banks, and nothing else was owed to him. He had only the obligation to deliver Monster shares (or their cash equivalent) to the banks in September 2008. As the Tax Court explained, "obligations are not property." TC Op. at 20, 2017 WL 1402129 , *7. Nevertheless, the Commissioner has an alternative claim that McKelvey realized a short-term gain because his obligation under each VPFC was terminated when he executed the amended contracts. "Gain ... attributable to the cancellation ... or other termination of ... a right or obligation ... with respect to property which is ... a capital asset in the hands of the taxpayer ... shall be treated as gain ... from the sale of a capital asset." 26 U.S.C. § 1234A(1) ; see Pilgrim's Pride Corp. v. Commissioner , 779 F.3d 311 , 317 (5th Cir. 2015) (interpreting section 1234A(1) to mean that "[c]apital gain or loss results from the termination of contractual or derivative rights with respect to capital assets"). Although the Tax Court did not consider the termination-of-obligation argument, both parties agree that this Court may consider it. The Commissioner asserts that the termination issue "was placed squarely before the Tax Court by the [E]state itself," Brief for Commissioner at 52, and the Estate "does not dispute that this Court may consider the Commissioner's new arguments, given that the Estate explained below why the extensions did not result in a termination of Mr. McKelvey's obligations," Brief for Estate at 27. The parties differ, however, on whether the amended contracts accomplished a termination of McKelvey's obligations under the VPFCs. Normally, we would remand that issue in its entirety to the Tax Court, but because Judge Ruwe's opinion rejected a premise of the Commissioner's termination argument, we will consider the issue in part. The Commissioner contends, and the Estate disputes, that the VPFCs executed in 2007 were replaced by amended *35 contracts executed in July 2008. The Tax Court rejected this premise of the Commission's termination argument by stating, "[T]here is no merit to [the Commissioner's] contention that the extended VPFCs should be viewed as separate and comprehensive financial instruments." T.C. op. at 36, 2017 WL 1402129 , at *12. This statement was made in the course of rejecting the Commissioner's claim that the extension of the valuation dates resulted in a constructive sale of the collateralized shares. We agree with the Commissioner that extension of the valuation dates resulted in amended contracts that replaced the original contracts. The new valuation dates determined the share price upon which the number of shares to be delivered at settlement would be calculated, and these dates were seventeen months later than the dates for the original BofA contract and sixteen months later than the dates of the original MSI contract. As the Commissioner argues, "By extending the valuation dates, the parties fundamentally changed the bets that the VPFCs represented, from bets on the value of Monster stock in September 2008 to bets on the value of Monster stock in January and February 2010." Brief for Commissioner at 36. As the Estate acknowledged in the Tax Court, "a 'sufficiently fundamental or material change' to an original contract that results in 'a change in the fundamental substance of the original contract' will be considered an exchange of the original contract for the amended contract." Tax Court Brief for Estate at 43 (quoting Rev. Rul. 90-109, 1990- 2 C.B. 191 (1990) ). Extending the valuation dates was a fundamental change. The new valuation dates in the amended contracts resulted in new contracts just as new expiration dates for option contracts result in new option contracts. The active trading of option contracts based on significant differences in expiration dates demonstrates that the options market regards different expiration dates as constituting different option contracts. As the report of the Commissioner's expert witness, Dr. Henrick Bessembinder, illustrates, on Sept. 11, 2007, the effective date of the BofA VPFC, call options for Monster stock with a strike price of $35 could be purchased for $0.35 if the expiration date was September 22, 2007, but cost $2.55 if the expiration date was January 19, 2008, and cost $6.10 if the expiration date was January 17, 2009. In the pending case, McKelvey paid the banks approximately $11 million to obtain the new valuation dates. Obviously, he did not think he was making insignificant changes. Whether the replacement of the obligations in the original VPFCs with the obligations in what we hold are new contracts satisfies the criteria for a termination of obligations that gives rise to taxable income, presumably capital gain, and the amount of such gain are issues that we leave for determination in the first instance by the Tax Court on remand. 13 III. Long-Term Capital Gain The Commissioner renews on this appeal the argument he made to the Tax Court: the execution of the 2008 contracts extending the valuation dates resulted in the constructive sale of the shares pledged as collateral. *36 The Commissioner bases his claim of long-term gains on a statutory ground and a legal contention. The Commissioner's statutory ground is that a constructive sale under 26 U.S.C. § 1259 occurs when a taxpayer holds an "appreciated financial position" in stock and enters into a "forward contract to deliver the same or substantially identical property," 26 U.S.C. § 1259 (c)(1)(C), and a "forward contract" is defined as "a contract to deliver a substantially fixed amount of property (including cash) at a substantially fixed price," id . § 1259(d)(1) (emphasis added). 14 There is no dispute that on the dates of the amended contracts all of McKelvey's Monster shares were an "appreciated financial position." 15 The Commissioner's legal contention, the disputed issue on the constructive sale portion of this appeal, is that the amount of Monster shares to be delivered at settlement of each amended contract was "substantially fixed" on the date when each amended contract was executed. His rationale is that, because the closing price of Monster stock on that date had fallen so far below the floor price of each contract ("deep in the money" in stock market parlance), there was only a remote chance that the price would recover and exceed the floor price by the valuation date. Based on this circumstance, the Commissioner contends that on the execution date of the amended contracts it was virtually certain that on the settlement date McKelvey would have to deliver all of the collateralized shares pledged under each amended contract, i.e. , 1,765,188 shares to BofA and 4,762,000 shares to MSI, which the contracts required if the Monster stock price closed below the floor price. That virtual certainty, the Commissioner concludes, means that, the amount of property to be delivered at settlement was "substantially fixed" within the meaning of subsection 1259(d)(1) and therefore the collateralized shares had been constructively sold. The key step in the Commissioner's claim of constructive sales is his reliance on the remoteness of the possibility that the price of Monster stock would recover and exceed the floor price by the valuation date of each amended contract. He bases his reliance on Dr. Bessembinder's report (the "Report"). The Report used the so-called Black-Scholes formula, a formula widely used for determining the value of option contracts. 16 The Black-Scholes formula uses probability analysis, which, in addition to being used to price options, can also be used to determine the probability *37 that a stock will reach a certain price by a certain date. The formula uses several factors: (1) the market price of the underlying stock on the valuation date, (2) the risk-free interest rate on the valuation date, (3) the period between the purchase of the option and the expiration, (4) the option strike price, (5) the volatility of the rate of change in the spot price of the underlying stock, and (6) the dividend yield. Using the Black-Scholes formula, the Report stated that for the BofA amended contract "the probability that the settlement price on the expiration date would be greater than the floor price was approximately [ sic ] 14.90% immediately after the extension [of the valuation date], as compared to 52.78% when the contract was originated" and that the comparable figures for the MSI amended contract were 12.87% as compared to 53.62%. Joint App'x at 199. Whether probability analysis may be used to determine that an amount of property is "substantially fixed" for purposes of subsection 1259(d)(1) is a novel question. Obviously, the modifier "substantially" informs us that the amount need not be exactly fixed and that Congress contemplated some leeway. A clear example of an amount substantially fixed would be an amount within a narrow range of limits. In the pending case, the amount is claimed to be substantially fixed for a different reason: the contract's amount of shares to be delivered is fixed whenever the closing price on the valuation date is below or equal to the floor price, and on the valuation date there was a very low probability that the closing price would reach the floor price before the settlement date. Although the Report presents the probability (for each contract) that the closing price will be equal to or above the floor price on the valuation date, we think the matter should be analyzed by using the reciprocals of the Report's percentages: there was a probability of 85.10% and 87.13% for the BofA and MSI amended contracts, respectively, that the closing price would be below the floor price on the settlement date. 17 The arithmetic is the same with either form of expression, but "substantially" in this context is better understood to mean substantially certain that the closing price will be below the floor price, rather than how unlikely it is that the closing price will equal or exceed the floor price. Neither party cites a decision on the use of probability analysis to determine whether an amount has been "substantially fixed" for purposes of subsection 1259(d)(1). Relevant, however, is Progressive Corp. v. United States , 970 F.2d 188 (6th Cir. 1992). That case concerned a corporate taxpayer that bought shares of stock and simultaneously sold call options with respect to the shares. 18 Call options are options enabling the option buyer to buy a stock at a specified price (the strike price) at any time before the option expires. When the taxpayer in Progressive sold the call options, they were "in the money," meaning that the strike price was below the market trading price. The spread between the strike and market prices gave the option buyer an opportunity to make an immediate profit by exercising the options at the strike price and *38 selling the stock at the market price (as long as the spread exceeded the purchase price of the options). In Progressive the Commissioner had asked the District Court to decide whether the call options "were so deep-in-the-money" that, from the option seller's standpoint (the taxpayer), each option "was the equivalent of a contractual obligation to sell" because it was "virtually certain that the purchasers of the call options would exercise them" promptly and take their quick profits. Id . at 193. That mattered because an immediate sale would reduce the taxpayer's holding period of the stock to zero, see 26 U.S.C. § 246 (c)(3), a consequence that would deprive it of a claimed inter-corporate dividend exclusion, see id . § 246(c)(1)(A). See Progressive, 970 F.2d at 189-90 . The District Court had not decided whether the spread was so great that exercise of the options was virtually certain, and the Sixth Circuit remanded that issue to the District Court for its determination. See id . at 194. The Sixth Circuit's analysis and disposition is relevant to our case because the District Court was asked to decide how likely it was that the option buyer would immediately exercise its purchase right. Or, to frame the issue in terms of Dr. Bessembinder's analysis, the issue was whether the spread created so high a probability of the option buyer immediately exercising its rights that the option seller's obligation to sell was "virtually certain." Progressive differs from our case in two respects. The probability to be determined needed to meet the high standard of "virtual certainty" rather than "substantially fixed," and the probability concerned action to be taken on the basis of a market price at the time the option was written, rather than at a future evaluation date when the amended contract would be settled. Nevertheless, meeting the Sixth Circuit's standard on the date the options were written would require consideration of a probability, i.e. , the likelihood that the option buyer would then exercise its rights. Using probability analysis to decide in the pending appeal the likelihood that a stock will not reach a floor price, thereby affecting tax consequences, is neither explicitly authorized nor prohibited by any relevant statute. And although Congress authorized the issuance of "necessary or appropriate" regulations to implement the constructive sale statute, see 26 U.S.C. § 1259 (f), and the relevant Senate report contemplated that the Treasury Department would do so, see S. Rep. 105-33 at 126 (1997), no such regulations have been issued. Nevertheless, we are persuaded to accept probability analysis in this context. Tax laws are to be applied with an eye to economic realities. See, e.g ., Frank Lyon Co. v. United States, 435 U.S. 561 , 573, 98 S.Ct. 1291 , 55 L.Ed.2d 550 (1978) (economic realities of transaction to be considered); Greene v. United States , 79 F.3d 1348 , 1356 (2d Cir. 1996) ( 26 U.S.C. § 1256 enacted "to harmonize tax treatment of commodity futures contracts with the economic realities of the marketplace"). Virtually all stock transactions rest on the market's (albeit differing) perceptions of the probabilities of share price movement, both the direction and extent of such movement. Probabilities are an economic reality affecting such transactions, and we see no reason why they should not affect the tax consequences of them. Illustrating the point in a context especially relevant to this appeal is the pricing of stock options. Whether or not all traders of options know it, a major determinant of option prices that are bid and asked every day in options markets is the Black-Scholes formula, the same formula that Dr. Bessembinder used to determine the *39 probabilities in the pending case. See footnote 14 , supra . So the economic reality pertinent to this case is not only the use of probability analysis in general but the use of the widely accepted Black-Scholes probability formula in particular. A further consideration guides our resolution of this issue. A taxpayer holding a large bloc of appreciated securities and wishing to diversify his portfolio faces the prospect of a considerable capital gain if he sells his shares. Executing a VPFC provides him with the immediate cash that a sale would produce (but no immediate capital gain in view of Rev. Rul. 2003-7 ). Because financial institutions are unlikely to set settlement dates much later than execution dates (witness the one- and one-and-a-half-year intervals in the contracts in this case), a taxpayer wishing to obtain his up-front payment without having to settle and incur a large capital gain will want to proceed, as McKelvey did, by executing amended contracts extending his settlement and valuation dates. This device is so alluring that he will be willing to pay substantial sums, in this case $11 million, to obtain the extended dates, and financial institutions, as this case shows, will be willing to extend the dates at an appropriate price. A taxpayer and his VPFC long party can often be expected to repeat these extensions for the taxpayer's life, knowing that at his death the shares will have a stepped-up basis in the hands of his estate. The up-front payment will have been received without ever incurring the capital gains tax that would have been due had the payment resulted from a sale of the stock. In this case that payment was $194 million, and thus far, no capital gains taxes have been paid. The Internal Revenue Code should not be readily construed to permit that result. We must acknowledge, however, that using probability analysis to prevent capital gain avoidance in this case does not affect all amended VPFCs but only those amended to become forward contracts where the number of shares to be delivered at settlement is substantially fixed because of a share price significantly below the floor price. Nevertheless, despite the somewhat limited frequency of situations in which amendment of the valuation date of a VPFC will create liability for capital gains taxes, we conclude that probability analysis may be used for such a purpose. The question remains in this case whether the 85 and 87 percentages of probability are sufficiently high (or the 15 and 13 percentages are sufficiently low) to show that the low share price at execution of each amended contract rendered the amount of shares to be delivered at settlement "substantially fixed." No bright line need be established. The percentages are very high, and the share prices yielding these percentages were so low as to be barely more than half of the floor prices. Dr. Bessembinder's report noted that even in the unlikely event that the share price would slightly exceed the floor price on the amended valuation date of each contract, an increase to $31 a share would decrease the number of shares to be delivered at settlement by less than 50,000 shares, less than 0.8 percent of the approximately 6.5 million total of collateralized shares, hardly a "significant variation." S. Rep. 105-33, at 125-26 (1997). So while the probability that McKelvey would have to deliver the total number of collateralized shares was 85 and 87 percent under the two contracts, the probability that he would have to deliver a number of shares close to the total, which would still be a substantially fixed amount, was even higher. The taxpayer had the burden to prove the determinations in the Commissioner's *40 notice of deficiency erroneous, see T.C. Rule 142(a) ; Welch v. Helvering , 290 U.S. 111 , 115, 54 S.Ct. 8 , 78 L.Ed. 212 (1933). The Estate presented no evidence to challenge any of Dr. Bessembinder's data or calculations. On this record, we see no basis to conclude that the amount of shares to be delivered at settlement was not "substantially fixed" on the dates each contract was amended. Constructive sales of the collateral shares therefore resulted. In rejecting the Commissioner's constructive sale contention, the Tax Court did not reach the issue of whether the amount of shares to be delivered at settlement was "substantially fixed." Instead, Judge Ruwe, at least implicitly, rejected the Commissioner's constructive sale claim because the amended contracts did not require McKelvey "to deliver the same or substantially identical property" as the collateralized shares. 26 U.S.C. § 1259 (c)(1)(C). We say "implicitly" because Judge Ruwe did not say that subsection 1259(c)(1)(C) was inapplicable. But he did say that (1) "the extensions [of the valuation dates] did not clarify the uncertainty of which property [McKelvey] would ultimately deliver to settle the contracts," and (2) "[McKelvey] had the discretion to settle the VPFCs using stock with a higher or lower basis than the stock pledged as collateral." T.C. op. at 30, 2017 WL 1402129 , at *10. Thus, the Tax Court appears to have rejected the Commissioner's constructive sale claim because the shares to be delivered at settlement did not have to be the same as, or substantially identical to, the shares pledged as collateral. Somewhat surprisingly, neither party explicitly considers this aspect of the Tax Court's ruling. The Commissioner grounds his constructive sale argument solely on the theory, which we accept, that the amount of shares (not the identity of shares) to be delivered at settlement was "substantially fixed" because of the depressed price of Monster stock. The Estate grounds its opposition to a constructive sale on two arguments. First, the Estate contends that "Mr. McKelvey did not enter into new contracts at the time of the extensions," Brief for Estate at 47, leaving the original contracts "open," id . We have rejected that argument. Second, the Estate contends that, even if the amended contracts were new contracts, there would not be a constructive sale because the amended contracts "would not constitute forward sales of a substantially fixed amount of property under section 1259." Brief for Estate at 48 (emphasis added; capitalization altered). Disputing the Commissioner's probability analysis, the Estate asserts, "[T]he chance that Monster stock would rebound to above the floor price of the VPFCs before the extended expiration was certainly not remote." 19 Brief for Estate at 52. We have rejected that argument. Expanding its second argument, the Estate contends that any fixation of the amount of shares to be delivered was not established by the "terms" of the contracts. Brief for Estate at 48-53. But the contract terms, by focusing on closing prices at settlement and keying the number of deliverable shares to the relation of those prices to the floor and cap prices, necessarily require consideration of what those prices would be. Perhaps both sides plausibly believe that it is the "substantially fixed price" language of subsection 1259(d)(1) that controls the constructive sale issue. Or they more plausibly believe that the "same or substantially identical property" language *41 of subsection 1259(c)(1)(C) means that the property to be delivered must have the same value as the appreciated position. It is clear that McKelvey's option to settle with shares other than the collateralized shares required him to deliver property of equal value. Moreover, the Tax Court's observation that McKelvey could have settled with shares having a different basis than the collateralized Monster stock would affect the amount of capital gain arising from a constructive sale, but not whether a constructive sale occurred. In any event, we decide the constructive sale issue as the parties have presented it and conclude that constructive sales of the collateralized shares occurred. Conclusion The decision of the Tax Court is reversed, and the case is remanded for (1) determination, in light of this opinion, of whether the termination of obligations that occurred when the amended contracts were executed resulted in taxable short-term capital gains, and (2) calculation of the amount of long-term capital gains that resulted from the constructive sales of the collateralized shares. José A. Cabranes, Circuit Judge, concurring: I agree with the Court's conclusion that McKelvey, as issuer of the nondebt financial instruments in this case, did not exchange property when he modified his contracts with the banks because he held no property interests under the contracts at the time of modification. I write separately to stress that this conclusion does not affect, by implication or analogy, the existing application of Treasury Regulations section 1.1001-3 to holders and issuers of debt instruments. Section 1.1001-3 sets forth principles for determining when the modification of a debt instrument is sufficiently "significant" to constitute a taxable event. These principles, as I understand them, apply to both the holder-obligee and the issuer-obligor of the instrument. See, e.g. , Rev. Rul. 2004-37, 2004- 1 C.B. 583 (applying the principles of section 1.1001-3 to require the issuer of a recourse note to recognize gain resulting from the modification of the note). Footnotes [1] The number of shares for the first two dates was 176,518 and for the next eight days was 176,519 so that the total equaled 1,765,188. This slight variation applied to all three methods of determining the number of shares to be delivered at settlement. [2] See footnote 9, infra . [3] This simplified explanation is derived from several provisions of the original BofA contract, all of which were carried forward into the amended contract with the valuation dates extended. "If Party B [McKelvey] elects Cash Settlement ... Party B shall pay the Preliminary Forward Cash Settlement Amount to Party A [BofA] on the Preliminary Cash Settlement Payment Date." The Preliminary Forward Cash Settlement Amount is defined as "The sum of all the Daily Preliminary Forward Cash Settlement Amounts," and the "Preliminary Cash Settlement Payment Date" is defined as "The Currency Business Day immediately following the Preliminary Cash Settlement Pricing Date." The Daily Preliminary Forward Cash Settlement Amount is defined as "105% of the Forward Cash Settlement Amount that would apply if the Valuation Date were the Preliminary Cash Settlement Pricing Date." The Preliminary Cash Settlement Pricing Date is defined as "The third Scheduled Trading Day immediately prior to the Scheduled Valuation Date for the Component with the earliest scheduled Valuation Date." The earliest scheduled Valuation Date for the first component, i.e ., 10 percent of the collateralized shares was September 11, 2008. [4] This simplified explanation is derived from two sources. The original MSI contract, with the valuation date extended by the amended contract, provided that "If Cash Settlement applies, then on the Cash Settlement Payment Date, Counterparty [McKelvey] shall pay to MSI plc the Forward Cash Settlement Amount," which "shall be determined in accordance with the Equity Definitions" of the International Swaps and Derivatives Assn., Inc. ("ISDA"). Under the relevant Equity Definitions, Forward Cash Settlement Amount means "an amount equal to the Number of Shares to be Delivered ... multiplied by the Settlement Price," § 8.5(f), ISDA, "2002 ISDA Equity Derivatives Definitions" 25 (2002), and Settlement Price means "the price per Share ... as of ... the Valuation Date, § 7.3(a), id . at 22. The MSI contract specified that the Valuation Date was September 24, 2008. [5] The parties do not explain why the total number of shares delivered to BofA at settlement, 1,757,016, was slightly less than the anticipated total number of shares, 1,765,188, to be delivered if the closing price was below the floor price in the BofA contract, which it was. [6] $50,943,578.31 BofA prepayment plus $142,626,185.80 MSI prepayment equals $193,569,564.11. [7] 1,765,188 shares in the BofA contract plus 4,762,000 shares in the MSI contract equals 6,527,188 shares. [8] 6.5 million times $33.47, the closing price of Monster stock on Sept. 24, 2007, equals $217,555,000. [9] Under the BofA and MSI contracts, McKelvey was obligated to deliver the minimum number of shares at settlement if the closing price at that time equaled the cap price. To determine the minimum number of shares in that circumstance, the applicable fraction for the BofA contract is $30.4610 (floor price)/$40.5809 (cap price), which is the smallest applicable fraction under the contract, applied to the number of shares pledged in the BofA contract, 1,765,188, which yields 1,324,993 shares. To determine the minimum number of MSI shares, the applicable fraction is 30.894 (floor price)/$35.77 (cap price) applied to the number of shares pledged in the MSI contract, 4,762,000, which yields 4,112,636 shares. Adding 1,324,993 to 4,112,636 yields 5,437,629 shares. The Commissioner's briefs to the Tax Court reported that the minimum number of shares to be delivered would be about 5.4 million shares. [10] 5.4 million times $33.47 equals $180,738,000. [11] Subsection 1259(d)(1) provides: "Forward contract. -The term 'forward contract' means a contract to deliver a substantially fixed amount of property (including cash) for a substantially fixed price." [12] Subsection 1234A(1) provides: "Gain or loss attributable to the cancellation ... or other termination of ... a right or obligation ... with respect to property which is ... a capital asset in the hands of the taxpayer ... shall be treated as gain or loss from the sale of a capital asset." [13] The parties recognize that this case concerns contracts that are non-debt instruments, and we make no implication as to the tax consequences of fundamental changes in debt instruments. [14] The relevant Senate report explains the definition of forward contract from the opposite perspective, explaining that "a forward contract providing for delivery of an amount of property, such as shares of stock, that is subject to significant variation under the contract terms does not result in a constructive sale." S. Rep. 105-33, at 125-26 (1997) (emphasis added). [15] "Appreciated financial position" generally means "any position with respect to any stock ... if there would be gain were such position sold ... at its fair market value." 26 U.S.C. § 1259 (b)(1). [16] The Black-Scholes formula, published in 1973 by three economists, Fischer Black, Myron Scholes, and Robert Merton, is "perhaps the world's most well-known options pricing model." Jean Folger, Options Pricing: Black-Scholes Model , https://www.investopedia.com/university/options-pricing/black-scholes-model.asp (last visited July 8, 2018). For their work, Scholes and Merton were awarded the 1997 Nobel Prize in Economics (Black was ineligible for the award because he had died, but the Nobel committee acknowledged his role). See id . The extremely complicated formula is shown in Folger, Figure 4, along with a typical calculator that can be used to apply the formula to the relevant factors, id ., Figure 5. See also Black-Scholes model , https://en.wikipedia.org/wiki/Black-Scholes_model (last visited July 8, 2018). [17] The Commissioner also uses the reciprocal of Dr. Bessembinder's percentage, stating that "there was a greater than 85% chance that there would be zero variation" in the number of shares to be delivered at settlement. Reply Brief for Commissioner at 32 (emphasis in original). [18] As the Sixth Circuit explained, the corporate taxpayer made two sets of complicated arrangements, Progressive , 970 F.2d at 190 , but only the Court's treatment of the call options in the second set is relevant to the pending appeal. [19] If the price of Monster shares had closed above the floor price and McKelvey had settled the contracts before his death, he would have been entitled to an adjustment in light of the previous taxation of constructive sales. See 26 U.S.C. § 1259 (a)(2), (e)(1).