Discount Factors in Continuous Time
Every finance course starts by pricing a payoff as its expected value discounted at the interest rate. Written in the right way, this familiar rule says that discounted prices are martingales, which is the reason martingales appear throughout finance. Replacing the deterministic discount by a random one, a stochastic discount factor (SDF), extends the rule to risky assets without changing its form. Carrying it to continuous time then requires the change in the product of two Itô processes, the discount factor and the price, which is where Itô’s lemma for products comes in.
These notes follow these steps from discrete to continuous time. In continuous time the pricing equation becomes a statement about drifts: the SDF has drift equal to minus the risk-free rate, and the risk premium of any asset equals minus the instantaneous covariance of its return with the SDF. The notes on derivatives pricing use this result to construct the risk-neutral measure. For a complete treatment of the discrete-time theory, see Cochrane (2009), and for continuous time, Duffie (2001).
The Pricing Equation in Discrete Time
Discounted Prices Are Martingales
Consider an asset with price p_{t} that pays x_{t + 1} = p_{t + 1} + D_{t + 1} at time t + 1, the price at which it can be sold plus any dividend. Write \operatorname{E}_{t}(\cdot) for \operatorname{E}(\cdot \mid \mathcal{F}_{t}). If investors care only about expected payoffs, or if the payoff is known in advance, the price is the expected payoff discounted at the interest rate r, p_{t} = \frac{\operatorname{E}_{t}(x_{t + 1})}{1 + r}. Dividing both sides by (1 + r)^{t} gives \frac{p_{t}}{(1 + r)^{t}} = \operatorname{E}_{t}\left(\frac{x_{t + 1}}{(1 + r)^{t + 1}}\right), \qquad \text{or} \qquad \operatorname{E}_{t}\left(\frac{x_{t + 1}}{(1 + r)^{t + 1}} - \frac{p_{t}}{(1 + r)^{t}}\right) = 0. Measured in dollars of time zero, the value of the asset has zero expected change. For an asset without dividends, x_{t + 1} = p_{t + 1}, and the discounted price p_{t} / (1 + r)^{t} is a martingale. A fair price in finance is the counterpart of a fair game in the notes on discrete-time martingales: no strategy based on past information can make the discounted value grow on average.
Random Discount Factors
Discounting expected payoffs at the interest rate ignores risk. Investors value a dollar more in a recession, when it is scarce, than in a boom, so an asset that pays off mostly in good times is worth less than its expected payoff discounted at r. To capture this, replace the deterministic discount 1 / (1 + r)^{t} with a strictly positive random process \Lambda_{t}, the SDF, and require \Lambda_{t} p_{t} = \operatorname{E}_{t}\left(\Lambda_{t + 1} x_{t + 1}\right), \qquad \text{or} \qquad \operatorname{E}_{t}\left(\Lambda_{t + 1} x_{t + 1} - \Lambda_{t} p_{t}\right) = 0. \tag{1} The equation has exactly the same form as before, and with \Lambda_{t} = 1 / (1 + r)^{t} it reduces to discounting at the interest rate. The same process \Lambda prices every asset, and in the absence of arbitrage such a strictly positive process exists. Applied to a risk-free bond, (1) gives \operatorname{E}_{t}(\Lambda_{t + 1}) / \Lambda_{t} = 1 / (1 + r), so on average \Lambda still discounts at the risk-free rate, and its randomness adjusts the price for risk.
Reinvesting Dividends
Dividends prevent \Lambda_{t} p_{t} itself from being a martingale, but we can remove them by reinvesting. Suppose we hold X_{t} shares at time t and use the dividend D_{t + 1} on each share to buy more shares at the price p_{t + 1}, so that X_{t + 1} = X_{t} \left(1 + \frac{D_{t + 1}}{p_{t + 1}}\right). The value of the position is V_{t} = X_{t} p_{t}, and V_{t + 1} = X_{t + 1} p_{t + 1} = X_{t} (p_{t + 1} + D_{t + 1}) = X_{t} x_{t + 1}. Multiplying (1) by X_{t}, which is known at time t, \Lambda_{t} V_{t} = \operatorname{E}_{t}\left(\Lambda_{t + 1} V_{t + 1}\right). The value of a dividend-reinvested position, multiplied by the SDF, is a martingale. This is the form of the pricing equation that carries over most easily to continuous time.
Prices and Dividends in Continuous Time
In continuous time, uncertainty is driven by one or more Brownian motions, and \mathcal{F}_{t} is the information they generate up to time t. The price of a risky asset follows an Itô process \frac{\mathop{}\!\mathrm{d}S_{t}}{S_{t}} = \mu_{t} \mathop{}\!\mathrm{d}t+ \sigma_{t} \mathop{}\!\mathrm{d}B_{t}, where the expected return \mu_{t} and the volatility \sigma_{t} are adapted processes. The geometric Brownian motion of the previous notes is the case of constant coefficients. Most often I will drop the time subscripts and write \mathop{}\!\mathrm{d}S / S = \mu \mathop{}\!\mathrm{d}t+ \sigma \mathop{}\!\mathrm{d}B.
I also use \operatorname{E}_{t} to extract the drift of a process, so that \operatorname{E}_{t}\left(\frac{\mathop{}\!\mathrm{d}S_{t}}{S_{t}}\right) = \mu_{t} \mathop{}\!\mathrm{d}t, and many times I will just write \operatorname{E}(\mathop{}\!\mathrm{d}S / S) = \mu \mathop{}\!\mathrm{d}t, with the understanding that the expectation is there to extract the drift of the diffusion.
Dividends. The asset pays dividends at the rate D_{t} per unit of time, so its total instantaneous return is \frac{\mathop{}\!\mathrm{d}S + D \mathop{}\!\mathrm{d}t}{S} = \frac{\mathop{}\!\mathrm{d}S}{S} + \frac{D}{S} \mathop{}\!\mathrm{d}t, where \delta = D / S is the dividend yield. Reinvesting dividends works as in discrete time. The dividends on X shares over \mathop{}\!\mathrm{d}t amount to X D \mathop{}\!\mathrm{d}t dollars, which buy X D \mathop{}\!\mathrm{d}t/ S new shares, so \frac{\mathop{}\!\mathrm{d}X}{X} = \frac{D}{S} \mathop{}\!\mathrm{d}t, \qquad X_{t} = X_{0} \exp\left(\int_{0}^{t} \frac{D_{u}}{S_{u}} \mathop{}\!\mathrm{d}u\right). The number of shares grows at a rate equal to the dividend yield. The value of the reinvested position is V = X S, and since X has no diffusion term, the product rule from the notes on stochastic calculus gives \frac{\mathop{}\!\mathrm{d}V}{V} = \frac{\mathop{}\!\mathrm{d}S}{S} + \frac{\mathop{}\!\mathrm{d}X}{X} = \frac{\mathop{}\!\mathrm{d}S}{S} + \frac{D}{S} \mathop{}\!\mathrm{d}t. The dynamics of V combine the capital gains on the stock with its dividend yield.
The money-market account. There is a risk-free rate r_{t}, which may be random and may follow its own diffusion, as in models of the term structure. A money-market account \beta earns this rate, \frac{\mathop{}\!\mathrm{d}\beta_{t}}{\beta_{t}} = r_{t} \mathop{}\!\mathrm{d}t, \qquad \beta_{t} = \beta_{0} \exp\left(\int_{0}^{t} r_{s} \mathop{}\!\mathrm{d}s\right). Over the next instant the return on \beta is known, even when r_{t} itself changes randomly over time.
The Pricing Equation in Continuous Time
From Discrete to Continuous Time
Over a short interval of length \Delta t, the discrete-time pricing equation for a dividend-reinvested position reads \operatorname{E}_{t}\left(\Lambda_{t + \Delta t} V_{t + \Delta t} - \Lambda_{t} V_{t}\right) = 0. Letting \Delta t \to 0 gives \operatorname{E}_{t} \mathop{}\!\mathrm{d}(\Lambda_{t} V_{t}) = 0, \tag{2} which we usually write as \operatorname{E}\mathop{}\!\mathrm{d}(\Lambda V) = 0. In discrete time we never needed to expand \Lambda_{t + 1} V_{t + 1} - \Lambda_{t} V_{t}. In continuous time both \Lambda and V are Itô processes, and to use the pricing equation we need the change in their product, which the product rule from the notes on stochastic calculus provides. Its cross term (\mathop{}\!\mathrm{d}\Lambda)(\mathop{}\!\mathrm{d}V) has no counterpart when the discount factor is deterministic, and it is the source of risk premia. The pricing equation says that the process \Lambda V has zero drift, so it is a local martingale in the sense of the notes on stochastic calculus. The notes on derivatives pricing assume conditions under which it is in fact a martingale.
It is often convenient to state the pricing equation in terms of the price S and the dividend D. By the product rule in percentage form, \begin{aligned} \frac{\mathop{}\!\mathrm{d}(\Lambda V)}{\Lambda V} & = \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} + \frac{\mathop{}\!\mathrm{d}V}{V} + \left(\frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda}\right) \left(\frac{\mathop{}\!\mathrm{d}V}{V}\right) \\ & = \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} + \frac{\mathop{}\!\mathrm{d}S}{S} + \left(\frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda}\right) \left(\frac{\mathop{}\!\mathrm{d}S}{S}\right) + \frac{D}{S} \mathop{}\!\mathrm{d}t\\ & = \frac{\mathop{}\!\mathrm{d}(\Lambda S)}{\Lambda S} + \frac{D}{S} \mathop{}\!\mathrm{d}t, \end{aligned} where the second line uses (\mathop{}\!\mathrm{d}\Lambda)(\mathop{}\!\mathrm{d}t) = 0. Multiplying by \Lambda V = X \Lambda S and dividing by X > 0, the condition \operatorname{E}\mathop{}\!\mathrm{d}(\Lambda V) = 0 is equivalent to \operatorname{E}\mathop{}\!\mathrm{d}(\Lambda S) + \Lambda D \mathop{}\!\mathrm{d}t= 0. \tag{3} This is the continuous-time version of (1): the expected change in the discounted price plus the discounted dividend is zero.
The Risk-Free Rate
The money-market account pays no dividends, so (3) implies \operatorname{E}\mathop{}\!\mathrm{d}(\Lambda \beta) = 0. Since \beta has no diffusion term, \mathop{}\!\mathrm{d}(\Lambda \beta) = \Lambda \mathop{}\!\mathrm{d}\beta + \beta \mathop{}\!\mathrm{d}\Lambda, and \operatorname{E}\left(\frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda}\right) = -\operatorname{E}\left(\frac{\mathop{}\!\mathrm{d}\beta}{\beta}\right) = -r \mathop{}\!\mathrm{d}t. The drift of the SDF determines the risk-free rate, just as \operatorname{E}_{t}(\Lambda_{t + 1}) / \Lambda_{t} = 1 / (1 + r) in discrete time. The rate r need not be constant, since it can be any adapted process.
Risk Premia
Applying the product rule to \Lambda S, \frac{\mathop{}\!\mathrm{d}(\Lambda S)}{\Lambda S} = \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} + \frac{\mathop{}\!\mathrm{d}S}{S} + \left(\frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda}\right) \left(\frac{\mathop{}\!\mathrm{d}S}{S}\right). Taking the drift of both sides, using (3) and the drift -r \mathop{}\!\mathrm{d}t of the SDF, \operatorname{E}\left(\frac{\mathop{}\!\mathrm{d}S}{S}\right) + \frac{D}{S} \mathop{}\!\mathrm{d}t= r \mathop{}\!\mathrm{d}t- \left(\frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda}\right) \left(\frac{\mathop{}\!\mathrm{d}S}{S}\right).
The product (\mathop{}\!\mathrm{d}\Lambda / \Lambda)(\mathop{}\!\mathrm{d}S / S) is the instantaneous covariance of the two percentage changes, computed with the multiplication table. An asset that pays off poorly when the SDF is high, that is, in bad times, has a negative covariance with the SDF and must offer an expected return above the risk-free rate. If the SDF has no diffusion term, the covariance is zero and every asset earns the risk-free rate, as in the discrete-time rule we started from.
Examples
The fundamental pricing equation becomes concrete once we specify the SDF. In each example, the SDF is an Itô process of the form \mathop{}\!\mathrm{d}\Lambda / \Lambda = -r \mathop{}\!\mathrm{d}t+ (\text{diffusion terms}), and the diffusion terms determine which risks earn a premium.
Example 1 Suppose the SDF and the stock are driven by the same Brownian motion, \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} = -r \mathop{}\!\mathrm{d}t- \lambda \mathop{}\!\mathrm{d}B, \qquad \frac{\mathop{}\!\mathrm{d}S}{S} = \mu \mathop{}\!\mathrm{d}t+ \sigma \mathop{}\!\mathrm{d}B, and the stock pays no dividends. Then (\mathop{}\!\mathrm{d}\Lambda / \Lambda)(\mathop{}\!\mathrm{d}S / S) = -\lambda \sigma \mathop{}\!\mathrm{d}t, and (4) gives \mu - r = \lambda \sigma, or \lambda = \frac{\mu - r}{\sigma}. The volatility of the SDF is the instantaneous Sharpe ratio of the stock. With \lambda = 0.4 and \sigma = 0.25, the stock must offer a risk premium of \mu - r = 0.10 per year. Every asset driven by the same Brownian motion must have the same Sharpe ratio \lambda. This is the setting of the notes on derivatives pricing.
Example 2 Suppose the SDF is driven by B_{1} and the stock by B_{2}, where the increments of the two Brownian motions have correlation \rho: \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} = -r \mathop{}\!\mathrm{d}t- \lambda \mathop{}\!\mathrm{d}B_{1}, \qquad \frac{\mathop{}\!\mathrm{d}S}{S} = \mu \mathop{}\!\mathrm{d}t+ \sigma \mathop{}\!\mathrm{d}B_{2}. The multiplication table gives (\mathop{}\!\mathrm{d}B_{1})(\mathop{}\!\mathrm{d}B_{2}) = \rho \mathop{}\!\mathrm{d}t, so the risk premium of a non-dividend-paying stock is \mu - r = \lambda \rho \sigma. Only the part of the stock’s volatility that is correlated with the SDF earns a premium. A stock whose risk is unrelated to the SDF, \rho = 0, earns the risk-free rate however volatile it is, and a stock with \rho < 0, which pays off well when the SDF is high, has an expected return below the risk-free rate.
Where does the SDF come from? In a consumption-based model, it is proportional to the marginal utility of consumption, so it is high in states where consumption is low and an extra dollar is most valuable (Lucas 1978).
Example 3 In a consumption-based model, the SDF is the marginal utility of consumption discounted at a rate of time preference \phi. With power utility, \Lambda_{t} = e^{-\phi t} C_{t}^{-\gamma}, where \gamma is the coefficient of relative risk aversion. Suppose aggregate consumption follows a GBM, \mathop{}\!\mathrm{d}C / C = \mu_{C} \mathop{}\!\mathrm{d}t+ \sigma_{C} \mathop{}\!\mathrm{d}B_{C}. The example on powers of a GBM in the notes on modeling stock prices, with a = -\gamma, together with the factor e^{-\phi t}, gives \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} = \left(-\phi - \gamma \mu_{C} + \tfrac{1}{2} \gamma (\gamma + 1) \sigma_{C}^{2}\right) \mathop{}\!\mathrm{d}t- \gamma \sigma_{C} \mathop{}\!\mathrm{d}B_{C}. The drift of the SDF is -r, so the risk-free rate is r = \phi + \gamma \mu_{C} - \tfrac{1}{2} \gamma (\gamma + 1) \sigma_{C}^{2}. For a stock with volatility \sigma whose return has correlation \rho with consumption growth, Example 2 with \lambda = \gamma \sigma_{C} gives the risk premium \mu + \delta - r = \gamma \rho \sigma_{C} \sigma. The risk premium is proportional to risk aversion and to the covariance of the stock’s return with consumption growth.
The last example is the consumption CAPM of Breeden (1979). Aggregate consumption is smooth, with a volatility of only a few percent per year, so explaining the historical equity premium with this formula requires an implausibly high risk aversion. This is the equity premium puzzle of Mehra and Prescott (1985).