Discount Factors in Continuous Time
Price Processes
We work in a probability space (\Omega, \mathcal{F}, \operatorname{P}) where all randomness is generated by K independent Brownian motions \mathbf{B} = \begin{pmatrix} B_{1} & B_{2} & \ldots & B_{K} \end{pmatrix}' satisfying (d\mathbf{B}) (d\mathbf{B})' = \mathbf{I} dt, where \mathbf{I} is the K \times K identity matrix. The K shocks capture distinct sources of uncertainty in the economy — technology, demand, interest rates, and so on.
Although the Brownian motions in \mathbf{B} are independent, we can always construct a correlated Brownian motion by taking a linear combination. For example, let Z = \rho B_{1} + \sqrt{1 - \rho^{2}} B_{2} where |\rho| \leq 1. Then Z is a Brownian motion since (dZ)^{2} = \rho^{2} dt + (1 - \rho^{2}) dt = dt, and it is instantaneously correlated with B_{1} since (dZ)(dB_{1}) = \rho (dB_{1})^{2} = \rho\, dt. More generally, any Brownian motion correlated with B_{1}, \ldots, B_{K} takes the form Z = \frac{1}{\sqrt{\mathbf{a}' \mathbf{a}}} \mathbf{a}' \mathbf{B} for some vector \mathbf{a}. This means that by driving different assets with overlapping linear combinations of \mathbf{B}, we can capture any pattern of co-movement among returns.
We model the price of a risky asset as a diffusion \frac{dS}{S} = \mu_{S}(\cdot)\, dt + \sigma_{S}(\cdot)\, dB_{S}, where the drift \mu_S(\cdot) and return volatility \sigma_S(\cdot) may depend on time t, the state \omega \in \Omega, or other state variables, and B_S is a Brownian motion driven by some linear combination of \mathbf{B}.
We also assume a risk-free money-market account \beta that earns the continuously-compounded rate r. Starting from \beta_{0}, \frac{d\beta}{\beta} = r\, dt, \tag{1} which solves to \beta_{t} = \beta_{0} \exp\!\left( \int_{0}^{t} r_{s}\, ds \right). \tag{2} The rate r need not be constant; in term-structure models it follows its own diffusion dr = \mu_{r}(\cdot)\, dt + \sigma_{r}(\cdot)\, dB_{r}, in which case \beta_t depends on the entire path of rates from 0 to t.
The Dividend-Reinvested Price
The total instantaneous return of an asset is given by \frac{dS + Ddt}{S} = \frac{dS}{S} + \frac{D}{S} dt, where D/S is the dividend yield. Rather than modeling dividends as a separate cash payment, we model them as generating new shares at rate D/S per unit time — as if every dividend is immediately reinvested in the stock. This allows us to track total wealth without separately accounting for dividend payments. To see this, let \frac{dX}{X} = \frac{D}{S} dt. \tag{3} Here X represents the number of new shares accruing to the owner of the stock determined by the dividend yield. Note that X_{t} = X_{0} \exp\left( \int_{0}^{t} \frac{D_{u}}{S_{u}} du \right), \tag{4} so the total number of shares grows exponentially with a growth rate equal to the dividend yield of the asset. In other words, X_{t} keeps track of the total number of shares at each point in time.
The dividend-reinvested asset price is then P = X S, where P denotes the total value of this investment given by the number of shares times the price per share. We can find the dynamics of P by applying Ito’s product rule. Since X is locally predictable (it has no Brownian d\mathbf{B} term), its cross-variation with S is zero (dX dS = 0), yielding: \frac{dP}{P} = \frac{dS}{S} + \frac{dX}{X} = \frac{dS}{S} + \frac{D}{S} dt. The dynamics of P equal the total return on the asset: capital gains dS/S plus the dividend yield D/S.
The Pricing Equation
We seek a strictly positive process \Lambda — the cumulative SDF — such that \Lambda P is a martingale for every dividend-reinvested price P. This is the continuous-time counterpart of the one-period condition \Lambda_t P_t = \operatorname{E}_t[\Lambda_{t+1} P_{t+1}]. The martingale property requires a zero conditional expected instantaneous increment: \operatorname{E}_t[d(\Lambda P)] = 0. \tag{5} To express this in terms of the observable price S and the dividend flow, note that dP/P = dS/S + (D/S)\,dt, so by Ito’s product rule \frac{d(\Lambda P)}{\Lambda P} = \frac{d\Lambda}{\Lambda} + \frac{dP}{P} + \frac{d\Lambda}{\Lambda}\frac{dP}{P} = \frac{d(\Lambda S)}{\Lambda S} + \frac{D}{S}\,dt, where the last step uses dP/P = dS/S + (D/S)\,dt and the fact that the cross term (d\Lambda/\Lambda)(D/S\,dt) vanishes. Thus 5 is equivalent to \operatorname{E}_t[d(\Lambda S)] + \Lambda D\, dt = 0. \tag{6}
The risk-free rate. The money-market account \beta pays no dividends, so (6) gives \operatorname{E}_t[d(\Lambda \beta)] = 0. By Ito’s product rule, d(\Lambda \beta) = \Lambda\,d\beta + \beta\,d\Lambda, and since d\beta = r\beta\,dt, \operatorname{E}_t\!\left(\frac{d\Lambda}{\Lambda}\right) + r\,dt = 0 \implies \operatorname{E}_t\!\left(\frac{d\Lambda}{\Lambda}\right) = -r\,dt. The drift of the SDF equals minus the risk-free rate. This holds even when r is stochastic, as long as it is adapted to the filtration.
The risk premium. For a risky asset S, Ito’s product rule gives \frac{d(\Lambda S)}{\Lambda S} = \frac{d\Lambda}{\Lambda} + \frac{dS}{S} + \frac{d\Lambda}{\Lambda}\frac{dS}{S}. Taking conditional expectations and using \operatorname{E}_t[d(\Lambda S)]/(\Lambda S) = -(D/S)\,dt from (6) and \operatorname{E}_t(d\Lambda/\Lambda) = -r\,dt: \operatorname{E}_t\!\left(\frac{dS}{S}\right) + \frac{D}{S}\,dt = r\,dt - \frac{d\Lambda}{\Lambda}\frac{dS}{S}. The cross-term (d\Lambda/\Lambda)(dS/S) is the instantaneous quadratic covariation. Because it is the product of two Brownian increments, it is entirely deterministic and of order dt, so it passes through the conditional expectation unchanged. The right-hand side is therefore minus the instantaneous covariance between the SDF and the asset return. This gives the fundamental pricing equation:
Property 1 Consider an asset S that follows a diffusion \frac{dS}{S} = \mu\, dt + \sigma\, dB. If the asset pays a dividend yield q = D / S, and there are no arbitrage opportunities, it must be the case that (\mu + q - r)\, dt = - \left(\frac{d\Lambda}{\Lambda}\right) \left(\frac{dS}{S}\right). \tag{7} In words, the risk premium of the asset equals minus the covariance of the SDF and the asset’s returns.
The Market Price of Risk
The drift of the SDF is pinned at -r\,dt by the money-market account, as we derived above. The diffusion component, however, is not uniquely determined by the absence of arbitrage — no-arbitrage only requires that the SDF prices every traded asset correctly. Writing the d\mathbf{B} exposure of the SDF as -\mathbf{\lambda}', the general form is \frac{d\Lambda}{\Lambda} = - r \, dt - \mathbf{\lambda}' d\mathbf{B}, \tag{8} where \mathbf{\lambda} = (\lambda_1, \lambda_2, \ldots, \lambda_K)' \in \mathbb{R}^{K} is the market price of risk vector. The k-th component \lambda_k measures the instantaneous risk premium per unit of exposure to Brownian shock B_k: an asset that loads positively on B_k commands a higher expected return proportional to \lambda_k.
Suppose there are N \leq K securities, each driven by the common shock vector \mathbf{B}: \frac{dS_{i}}{S_{i}} = \mu_{i} dt + \pmb{\sigma}_{i}' d\mathbf{B}, \tag{9} where \pmb{\sigma}_{i} is a K \times 1 vector of return exposures, and each security pays a continuous dividend yield q_{i}\,dt. Applying the fundamental pricing equation (7) to asset i and using (8) with (d\mathbf{B})(d\mathbf{B})' = \mathbf{I}\,dt: Stacking all N assets, with \pmb{\sigma} denoting the N \times K matrix whose i-th row is \pmb{\sigma}_{i}' and \mathbf{q} = (D_1/S_1,\, \ldots,\, D_N/S_N)': Each asset’s risk premium equals its exposure vector \pmb{\sigma}_i dotted with \mathbf{\lambda}. The N \times N instantaneous return covariance matrix is \pmb{\sigma}\pmb{\sigma}'\,dt.
When the number of traded assets is less than the number of underlying risk sources (N < K), the market is incomplete. In this scenario, (11) is an underdetermined system: it has N equations but K unknowns, and therefore infinitely many solutions. Economically, this non-uniqueness arises because some risks cannot be hedged by trading the available assets.
Mathematically, any two valid solutions differ by a vector \mathbf{\nu} that lies in the null space of \pmb{\sigma} (i.e., \pmb{\sigma}\mathbf{\nu} = \mathbf{0}). Adding this “unpriced” risk \mathbf{\nu} to \mathbf{\lambda} changes the SDF but leaves the risk premiums of all N assets unchanged, since \pmb{\sigma}(\mathbf{\lambda} + \mathbf{\nu}) = \pmb{\sigma}\mathbf{\lambda}.
Among all these valid solutions, the minimum-norm solution is particularly important: \mathbf{\lambda}_{\min} = \pmb{\sigma}' \left(\pmb{\sigma} \pmb{\sigma}'\right)^{-1} \left(\pmb{\mu} + \mathbf{q} - r \pmb{\iota}\right). \tag{12} This is the unique \mathbf{\lambda} that prices all N assets and lies entirely in the row space of \pmb{\sigma}. Because the null space and the row space are orthogonal, every admissible solution can be decomposed into orthogonal components: \mathbf{\lambda} = \mathbf{\lambda}_{\min} + \mathbf{\nu}, \qquad \mathbf{\nu} \in \mathrm{Null}(\pmb{\sigma}). By the Pythagorean theorem, the squared norm of any valid market price of risk is: \lVert \mathbf{\lambda} \rVert^{2} = \lVert \mathbf{\lambda}_{\min} \rVert^{2} + \lVert \mathbf{\nu} \rVert^{2}. Since \lVert \mathbf{\nu} \rVert^{2} \geq 0, adding any unhedged risk strictly increases the overall norm. Thus, \mathbf{\lambda}_{\min} represents the least volatile pricing kernel that successfully prices the available assets. By contrast, when markets are complete (N = K and \pmb{\sigma} is invertible), the null space is trivial (\mathbf{\nu} = \mathbf{0}) and the SDF is uniquely determined as \mathbf{\lambda} = \pmb{\sigma}^{-1}(\pmb{\mu} + \mathbf{q} - r\pmb{\iota}).
The Hansen-Jagannathan Bound
This minimum-norm SDF provides a fundamental limit on asset returns. Let \sigma_i denote the instantaneous return volatility of asset i. Since the asset’s return variance is (\pmb{\sigma}_i' d\mathbf{B})^2 / dt = \pmb{\sigma}_i' \pmb{\sigma}_i, we have \sigma_i \equiv \lVert \pmb{\sigma}_i \rVert > 0.
From (10), the absolute risk premium of asset i is |\mu_i + q_i - r| = |\pmb{\sigma}_i' \mathbf{\lambda}|. Because the unpriced risk \mathbf{\nu} is orthogonal to the asset’s exposures (\pmb{\sigma}_i'\mathbf{\nu} = 0), we can write this strictly in terms of the minimum-norm solution: \pmb{\sigma}_i'\mathbf{\lambda} = \pmb{\sigma}_i'\mathbf{\lambda}_{\min}.
Dividing by the volatility \sigma_i and applying the Cauchy-Schwarz inequality (|\mathbf{a}'\mathbf{b}| \leq \lVert\mathbf{a}\rVert \lVert\mathbf{b}\rVert) yields: \left| \frac{\mu_i + q_i - r}{\sigma_i} \right| \leq \lVert \mathbf{\lambda}_{\min} \rVert. \tag{13} This is the continuous-time counterpart of the Hansen-Jagannathan bound. It states that the Sharpe ratio of any traded asset (the left side) cannot exceed the norm of the market price of risk. In other words, \lVert \mathbf{\lambda}_{\min} \rVert is the sharp upper bound on the maximum attainable Sharpe ratio.
Equality in (13) holds only for an asset whose diffusion vector is perfectly proportional to \mathbf{\lambda}_{\min} (i.e., an asset that is perfectly negatively correlated with the minimum-norm pricing kernel). In the data, the U.S. equity market has historically delivered a Sharpe ratio of roughly 0.5 per year. Therefore, to be empirically plausible, any asset pricing model must feature a market price of risk satisfying \lVert \mathbf{\lambda}_{\min} \rVert \geq 0.5.