The Pricing of Options and Corporate Liabilities
Journal of Political Economy, 1973 · paper · Economics
Derives a theoretical valuation formula for options from the principle that correctly priced options allow no sure profit from hedged portfolios of options and stock. Notes the formula also applies to corporate liabilities such as stock, bonds and warrants.
Checked: PDF from cs.princeton.edu, fetched 5 Oct 2026. Page numbers are those of that PDF.
3 findings
Substance Equation (27) is printed "u(x,t) = −xN(−d1) + ce^{−rt*}N(−d2)"; substituting (13) into (26) gives the discount factor e^{r(t−t*)}, not e^{−rt*}.
- p. 12 Equation (27)
u(x,t) = −xN(−d1) + ce^{−rt*}N(−d2)
- p. 12 Equation (26)
u(x,t) = w(x,t) − x + ce^{r(t−t*)}
- p. 9 Equation (13)
w(x,t) = xN(d1) − ce^{r(t−t*)}N(d2)
Judge's note: Read on the page images of PDF 12 and PDF 9. Equation (13): w = xN(d1) − ce^{r(t−t*)}N(d2); equation (26): u = w − x + ce^{r(t−t*)}. Hence u = −x[1−N(d1)] + ce^{r(t−t*)}[1−N(d2)] = −xN(−d1) + ce^{r(t−t*)}N(−d2). The exponent in (27) clearly reads −rt* in the image, which agrees only at t = 0; the paper never sets t = 0, the left side is u(x,t), d1 and d2 are 'as in equation (13)', and the sentence right after (27) again writes ce^{r(t−t*)}.
Second check: another model agreed
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Substance "they will receive nothing or half of the difference between the old equity and half the exercise money" does not follow from the preceding sentence (half of the new equity minus the exercise money); it should be half of the difference between the old equity and the exercise money.
- p. 14 Warrant Valuation, second paragraph on the page
nothing or half of the difference between the old equity and half the exercise money
- p. 14 Preceding sentence
half of the new equity, minus the exercise money
- p. 14 End of first paragraph on the page
the new equity of the company, which is equal to the old equity plus the exercise money
- p. 14 Warrant Valuation, second paragraph
half of the difference between the old equity and half the exercise money
Judge's note: Read on the page image of PDF 14. The paper defines new equity = old equity + exercise money and says holders get half the new equity minus the exercise money: (E+K)/2 − K = (E−K)/2. The printed wording is (E − K/2)/2 = E/2 − K/4. Example with E = 100, K = 20: 40 versus 45. The next sentence (options on a share of equity, equity divided by twice the number of shares, at half the stated exercise price) gives E/(2N) − k/2 per warrant, which matches (E−K)/2 and not the printed sentence. No alternative definition of 'exercise money' is given in the paper.
Second check: another model agreed
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Reference or label Reference list: Lintner, "Rev. Econ. and Statis. 47 (February 1965): 768-83." carries the same page range as Mossin, "Econometrica 34 (October 1966): 768-83."
- p. 18 References, Lintner
47 (February 1965): 768-83.
- p. 18 References, Mossin
Econometrica 34 (October 1966): 768-83.
- p. 18 References, Mossin
(October 1966): 768-83.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that Lintner's article in the Review of Economics and Statistics 47 (February 1965) is on pp. 13-37 and that 768-83 is Mossin's range. On the page image of PDF 18 both entries do print 768-83; two articles in different journals and years with one page range, and a February issue reaching p. 768, make the Lintner range the likely copy error. The correct pages cannot be established from the files.
Second check: another model agreed
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6 minor (typos and the like; most checked by the judge only)
Wording "the number of options given by expression (1) will change by an amount Δx": it is the value of that number of options (1/w1 options, each changing by w1Δx) that changes by Δx, not the number.
- p. 6 Paragraph after expression (1)
number of options given by expression (1) will change by an amount Δx
Judge's note: Read on the page image of PDF 6. (1/w1)·w1Δx = Δx is the change in value of the option position; the next sentence ("change in value of a short position in 1/w1 options") shows this is what is meant, and the change in the number of options is treated separately in the following paragraph. Loose wording, meaning recoverable.
Second check: another model agreed
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Typo Text says "let us refer to the solid line (T2) in figure 1", but in Fig. 1 as printed T2 is a dashed curve and no curve is solid.
- p. 6 Paragraph beginning 'To illustrate the formation of the hedged position'
solid line (T2) in figure 1
- p. 3 Figure 1
T2
- p. 6 p. 641, hedged-position illustration
let us refer to the solid line (T2) in figure 1
- p. 3 p. 638, Fig. 1
FIG. 1.-The relation between option value and stock price
Judge's note: Viewed p. 638 (PDF 3) and p. 641 (PDF 6). In Fig. 1, T1 and T3 are dotted, T2 is drawn with long dashes, A and B with thin dashes; nothing is solid. The curve is named, so the reference is unambiguous; description-versus-drawing mismatch only (possibly the draughtsman's rendering of a line the authors drew solid).
Second check: none (judge only)
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Typo Footnote 8: "developed by Treynor (1961b); Sharpe (1964), Lintner (1965), and Mossin (1966)" has a semicolon after the first item of a comma-separated list.
- p. 10 Footnote 8
Treynor (1961b); Sharpe (1964), Lintner (1965), and Mossin (1966)
Judge's note: Enlarged the footnote on PDF 10: the mark after "(1961b)" is a semicolon, the other separators are commas. It may be deliberate (setting the unpublished memorandum apart from the published papers), so either form could be legitimate; cosmetic.
Second check: none (judge only)
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Typo Inconsistent hyphenation on p. 645: "capital asset pricing model" (twice) and "capital-asset pricing model" (twice); also "capital-asset pricing model" on p. 646.
- p. 10 An Alternative Derivation, first paragraph
the “capital asset pricing model.”
- p. 10 An Alternative Derivation, second paragraph
The capital-asset pricing model says that
Judge's note: Seen on the page image of PDF 10: the first paragraph of 'An Alternative Derivation' and the opening of the second have no hyphen; later in the second and in the third paragraph the hyphen appears. PDF 11 has the hyphenated form. Consistency point only.
Second check: none (judge only)
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Typo "Then the expected return on the option and the stock are:" is followed by equation (16) for the stock, E(Δx/x), and (17) for the option, E(Δw/w), the reverse of the order named.
- p. 11 Sentence before equations (16) and (17)
Then the expected return on the option and the stock are:
Judge's note: Seen on the page image of PDF 11. The equations are self-identifying through Δx/x and Δw/w and the sentence does not say 'respectively', so nothing is misstated; order of presentation only.
Second check: none (judge only)
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Typo Reference list: "Portfolio Theory and Capital Markets: New York: McGraw-Hill, 1970." has a colon after the book title where the other book entries have a period.
- p. 19 References, Sharpe 1970
Portfolio Theory and Capital Markets: New York: McGraw-Hill, 1970.
Judge's note: Enlarged the entry on PDF 19: the mark after 'Markets' is a colon. Other book entries (Churchill, Cootner, Fama and Miller, McKean, Thorp and Kassouf) end the title with a period before the place of publication.
Second check: none (judge only)