Abstract
A simple formula is developed for the valuation of uncertain income streams consistent with rational investor behavior and equilibrium in financial markets. Applying this formula to the pricing of an option as a function of its associated stock, the Black–Scholes formula is derived even though investors can trade only at discrete points in time.
This paper is reprinted from Bell Journal of Economics, 7 (Autuamn 1976), No. 2, pp. 407–425.
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Notes
- 1.
Extension of the single-price law of markets to portfolios of securities (i.e., convex combinations of dividends) is inconsistent with transaction costs that vary with the scale of investment in any security. However, it insures that any pricing formula for arbitrary securities will possess the “portfolio property” that it applies to arbitrary portfolios as well.
- 2.
If it were also assumed that a security existed insuring a non-negative rate of return to which all investors had access (i.e., cash), then 0 < Z[s(t)] < 1 and indeed ∑ s(t) Z[s(t)] ≤ 1 for all dates. Moreover, R 1 ≤ R 2 ≤ R 3 ….
- 3.
These probabilities need not be held homogeneously by all investors, or indeed by any investor. They may be viewed as a purely mathematical construct. However, every state must be interpreted as possible so that π[s(t)] > 0.
- 4.
Beja (1971) has developed similar results to Equation (41.1).
- 5.
By definition, \({\mu}_{g} \equiv E(g),\ {\mu}_{y} \equiv E(y),\ {\sigma}_{\mathit{gy}} \equiv \mathit{Cov}(g,\ y),\ {\mu}_{\mathit{gy}} \equiv E(\mathit{gy})\).
- 6.
From much of the literature on “efficient markets,” it may have been expected that security rates of return should follow a random walk irrespective of the stochastic process of dividends. For example, see Granger and Morgenstern (1970, p. 26). Clearly, this is not true for default-free pure-discount bonds. Moreover, I have argued elsewhere (1974b) that in an important case, the rate of return of the market portfolio will be serially independent if and only if the rate of growth of aggregate consumption (i.e., the social dividend) is serially independent. This result is generalized in Section III.
- 7.
Samuelson (1973) has derived a more general proposition that, unlike the simple case here, does not require stationary discount rates over time. However, like the present case, it does require that discount rates be nonstochastic functions of time; that is, future interest rates are assumed known in advance. I have shown elsewhere (1974a) that this may lead to unrealistic implications.
- 8.
This last condition would be satisfied if the financial market were complete, or if, as in the familiar mean-variance model, a riskless security exists and all investors divide their wealth after consumption between it and the market portfolio. The condition is required for Pareto-efficient exchange arrangements.
- 9.
If all investors were identical this assumption is trivially satisfied. I have developed elsewhere (1974a, 1974b) more general sets of conditions. Perhaps the most appealing is the case where investors are heterogeneous with respect to the scale and composition of endowed resources, lifetime, time-preference, beliefs, and whether proportional risk aversion is increasing, constant, or decreasing. Additive generalized logarithmic utility is the only homogeneity requirement.
- 10.
In the limiting case of \(\mathrm{b} = 1,\ {U}_{t}(\widetilde{{C}}_{t}) = {\rho}_{1}{\rho}_{2}\cdots {\rho}_{t}\log \widetilde{{C}}_{t}\).
- 11.
Specifically, it follows from Rubinstein (1974b), that
$$\widetilde{{k}}_{t}^{-1} = 1 + {\rho}_{t+1}^{B}\widetilde{{E}}_{t}\left ({r}_{\mathit{Ct}+1}^{1-b}\right ) + {\rho}_{t+1}^{B}{\rho}_{t+2}^{B}\widetilde{{E}}_{t}\left ({r}_{\mathit{Ct}+1}^{1-b}{r}_{\mathit{Ct}+2}^{1-b}\right ) + \ldots \,,$$where \(B \equiv {b}^{-1},\ \widetilde{{r}}_{\mathit{Ct}} \equiv \widetilde{{C}}_{t}/\widetilde{{C}}_{t-1}\), and expectations are assessed with respect to information available at date t. When the rate of growth of per capita consumption follows a random walk, then k t is nonstochastic, since
$${k}_{t}^{-1} = 1 + {\rho}_{t-1}^{B}{E}_{t}\left ({r}_{\mathit{Ct}+1}^{1-b}\right ) + {\rho}_{t+1}^{B}{\rho}_{t+2}^{B}{E}_{t}\left ({r}_{\mathit{Ct}+1}^{1-b}\right ){E}_{t}\left ({r}_{\mathit{Ct}+2}^{1-b}\right ) + \ldots $$Moreover, it is easy to see when b = 1 that k t is nonrandom even if the rate of growth of per capita consumption does not follow a random walk.
- 12.
See Kraus and Litzenberger (1976) for similar remarks.
- 13.
Merton (1973a) mentions that Samuelson and Merton (1969) have uncovered yet a third set of assumptions for Equation (41.4): (a) the average investor has CRPA, (b) only three securities exist – a default-free bond, the option, and its associated stock, and (c) the net supply of both the options and the bonds is zero.
- 14.
Merton (1973a) has shown that rational nonsatiated investors will not exercise a call option prior to expiration if it is properly payout-protected and if the strike price is fixed. He also proves this will not generally be true for puts.
- 15.
- 16.
Merton (1973a) has relaxed the stationarity conditions on the interest rate and allows the standard deviation of the logarithm of one plus the rate of return of the associated stock (σ) to be a known nonstochastic function of time; here σ can be stochastic as well, extending the model to more realistic stochastic processes. See Rosenberg (1972).
- 17.
The Samuelson-Merton (1969) assumptions (see Footnote 14) are clearly a special case. Their assumptions (b) and (c) imply that \({Y}_{t} =\widetilde{{R}}^{-b}\), since the associated stock is the market portfolio. The risk neutrality justification is a degenerate special case where \(\widetilde{{Y}}_{t}\) is nonstochastic. It should also be noted that the premise of Theorem 5 is stronger than necessary. We actually require a random variable \(\widetilde{{Y}}_{t}\) shared only by the option, its associated stock, and the default-free bond, which is jointly lognormal with \(\widetilde{{S}}_{t}\). In effect, as in the Black-Scholes analysis, Equation (41.5) will apply even if S and R Ft are not in equilibrium with the other securities in the market. Finally, agreement on Var(log S t ) is not required if heterogeneous investor beliefs can be meaningfully aggregated; see Rubinstein (1976).
- 18.
See Samuelson and Merton (1969). The corollary generalizes Merton (1973a, p. 171) by allowing the dividend yield to depend on time. In a succeeding working paper, Geske (1976) has further generalized the option pricing model for underlying stock with a stochastic but lognormal dividend yield. Since a perfect hedge cannot be constructed in this case, even if investors can make continuous revisions, his generalization requires the preference-specific approach developed in this paper.
- 19.
To my knowledge, this property of bivariate normal variables was first noted in Rubinstein (1971). Stein (1973) has independently made its discovery.
- 20.
The general formula for the nth order central moment of a lognormal variable is
$$E\left [{(X - {\mu}_{x})}^{n}\right ] = {({\mu}_{x})}^{n}\left [{\sum \nolimits}_{r=0}^{n} \frac{n} {r!(n - r)!}{(-1)}^{n-r}{e}^{\frac{r(r-1){\sigma}_{x}^{2}} {2}}\right ]$$
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Acknowledgment
Research for this paper was supported in part by a grant from the Dean Witter Foundation. The author would like to acknowledge comments on the original version of this paper from John Cox, Robert Geske, Robert Merton, and Stephen Ross.
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Rubinstein, M. (2010). The Valuation of Uncertain Income Streams and the Pricing of Options. In: Lee, CF., Lee, A.C., Lee, J. (eds) Handbook of Quantitative Finance and Risk Management. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77117-5_41
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