Discount Factors in Continuous Time

Lorenzo Naranjo

Fall 2026

Marginal Utility of Consumption

Prices in Continuous Time

We model the price of risky assets as diffusions \frac{dS}{S} = \mu(\cdot) dt + \sigma(\cdot) dz where the drift \mu(\cdot) and the volatility of returns \sigma(\cdot) might depend on time and potentially other state variables.
The total instantaneous return of an asset is given by \frac{dS}{S} + \frac{D}{S} dt, where \frac{D}{S} is the dividend yield.

The Risk-Free Asset

A risk-free asset is either
- A security with constant price 1 that pays a dividend D = r^{f}.
- A security that pays no dividend but whose price satisfies \frac{dS}{S} = r^{f} dt.
In the first case the risk-free asset is a consol bond that pays continuously r^{f} dt in perpetuity.
In the second case the risk-free asset is a bubble like gold that grows deterministically at a rate of r^{f}.
We do not assume that r^{f} is constant, but follows a diffusion dr^{f} = \mu_{f}(\cdot) dt + \sigma_{f}(\cdot) dz_{f}.

The Fundamental Pricing Equation

We can rewrite the multiperiod pricing equation as S_{t} \beta^{t} u'(c_{t}) = \operatorname{E}_{t} \sum_{j=1}^{\infty} \beta^{t + j} u'(c_{t + j}) D_{t + j}.
The continuous time analog of the previous expression is S_{t} e^{-\delta t} u'(c_{t}) = \operatorname{E}_{t} \int_{0}^{\infty} e^{-\delta (t + s)} u'(c_{t + s}) D_{t + s} ds. where \beta = e^{-\delta}.
We see that \Lambda_{t} = e^{-\delta t} u'(c_{t}) is a continuous-time discount factor: S_{t} \Lambda_{t} = \operatorname{E}_{t} \int_{0}^{\infty} \Lambda_{t + s} D_{t+s} ds. \tag{1}

Some Manipulations

For small \Delta t we have that S_{t + \Delta t} \Lambda_{t + \Delta t} = \operatorname{E}_{t + \Delta t} \int_{0}^{\infty} \Lambda_{t + \Delta t + s} D_{t + \Delta t + s} ds.
Also, \begin{aligned} S_{t} \Lambda_{t} & = \operatorname{E}_{t} \int_{0}^{\infty} \Lambda_{t + s} D_{t + s} ds. \\ & = \operatorname{E}_{t} \int_{0}^{\Delta t} \Lambda_{t + s} D_{t+s} ds + \int_{\Delta t}^{\infty} \Lambda_{t + s} D_{t+s} ds. \\ & \approx \Lambda_{t} D_{t} \Delta t + \operatorname{E}_{t} \int_{0}^{\infty} \Lambda_{t + \Delta t + s} D_{t + \Delta t + s} ds. \end{aligned}

The Pricing Equation in Continuous Time

Subtracting both expressions \Delta S_{t + \Delta t} \Lambda_{t + \Delta t} \approx - \Lambda_{t} D_{t} \Delta t + (\operatorname{E}_{t + \Delta t} - \operatorname{E}_{t}) \int_{0}^{\infty} \Lambda_{t + \Delta t + s} D_{t + \Delta t + s} ds.
Taking expectations both sides and letting \Delta t \rightarrow 0 \operatorname{E}d(\Lambda S) + \Lambda D dt = 0. \tag{2}
In the following, we assume that \Lambda_{t} > 0, which is the case if \Lambda_{t} = e^{- \delta t} u'(c_{t}).

The Drift of the SDF in Continuous Time

The first definition of the risk-free asset implies \operatorname{E}d\Lambda + \Lambda r^{f} dt = 0, or \operatorname{E}\left(\frac{d \Lambda}{\Lambda}\right) = - r^{f} dt
The drift of the SDF in continuous time determines the equilibrium continuously-compounded risk-free rate.

Compensation for Time and Risk

Applying Ito’s lemma to \Lambda S d (\Lambda S) = S d\Lambda + \Lambda dS + d\Lambda dS, or \frac{d(\Lambda S)}{\Lambda S} = \frac{d\Lambda}{\Lambda} + \frac{dS}{S} + \frac{d\Lambda}{\Lambda} \frac{dS}{S}.
Taking expectations both sides, equation (2) implies that \operatorname{E}\left(\frac{dS}{S}\right) + \frac{D}{S} dt = r^{f} dt - \left(\frac{d\Lambda}{\Lambda}\right) \left(\frac{dS}{S}\right).

The Fundamental Pricing Equation

Property 1 Consider an asset S that follows a diffusion \frac{dS}{S} = \mu dt + \sigma dz. 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^{f}) dt = - \left(\frac{d\Lambda}{\Lambda}\right) \left(\frac{dS}{S}\right). \tag{3} In words, the risk-premium of the asset equals minus the covariance of the SDF and the asset’s returns.

A Simple Asset Pricing Model

SDF and Consumption Dynamics

Applying Ito’s lemma to \Lambda = e^{-\delta t} u'(c) we find \begin{aligned} d \Lambda & = \frac{\partial \Lambda}{\partial c} dc + \frac{1}{2} \frac{\partial^{2} \Lambda}{\partial dc^{2}} (dc)^{2} + \frac{\partial \Lambda}{\partial t} dt \\ & = e^{-\delta t} u''(c) dc + \frac{1}{2} e^{-\delta t} u'''(c) (dc)^{2} - \delta e^{-\delta t} u'(c) dt, \end{aligned} or \frac{d\Lambda}{\Lambda} = - \delta dt + \frac{1}{2} \frac{c^{2} u'''(c)}{u'(c)}\left(\frac{dc}{c}\right)^{2} + \frac{c u''(c)}{u'(c)} \frac{dc}{c}.
For power utility we have that \frac{d\Lambda}{\Lambda} = - \delta dt + \frac{1}{2} \gamma (\gamma + 1) \left(\frac{dc}{c}\right)^{2} - \gamma \frac{dc}{c}.

The Process for Consumption Growth

If we write \frac{dc}{c} = \mu_{c} (\cdot) dt + \sigma_{c}(\cdot) dz_{c}
Assuming power utility, we have that \frac{d\Lambda}{\Lambda} = \left(- \delta + \frac{1}{2} \gamma (\gamma + 1) \sigma_{c}^{2} - \gamma \mu_{c} \right) dt - \gamma \sigma_{c} dz_{c}. \tag{4}
We can now recover the risk-free rate dynamics in terms of consumption growth dynamics using r^{f} = - \frac{1}{dt} \operatorname{E}\left(\frac{d \Lambda}{\Lambda}\right).

Implications for the Interest Rate

The previous model implies that the instantaneous risk-free rate is given by minus the drift of d\Lambda/\Lambda, r^{f} = \delta + \gamma \mu_{c} - \frac{1}{2} \gamma (\gamma + 1) \sigma_{c}^{2}.
Real interest rates are high when:
- Impatience (\delta) is high.
- Expected consumption growth (\mu_{c}) is high
  - If agents expect consumption to go up, they need to save less, pushing bond prices down.
- Volatility of future consumption growth (\sigma_{c}) is low
  - If agents are less afraid of future consumption growth, they bid bond prices down \Rightarrow precautionary savings.

The Pricing of Risky Assets

Consider an asset paying a dividend flow D dt and following a diffusion \frac{dS}{S} = \mu_{S} dt + \sigma_{S} dz_{S} such that (dz_{S}) (dz_{c}) = \rho dt.
Equation (3) implies \mu_{S} + D / S - r^{f} = \gamma \rho \sigma_{c} \sigma_{p}
In this simple asset pricing model with power utility, the risk premium of any risky asset is higher when:
- Risk aversion (\gamma) is high
- The covariance of asset returns and consumption growth is high.

The Equity Premium Puzzle

Since |\rho| \leq 1 the previous expression implies \left|\frac{\mu_{S} + D / S - r^{f}}{\sigma_{p}} \right| \leq \gamma \sigma_{c}.
In the data, the Sharpe ratio of the market is around 0.5 whereas the standard deviation of consumption growth is around 0.01.
We need a RRA coefficient of at least 50 to explain the risk-premium of the market!

Generic SDFs in Continuous Time

The Payoff Space

Uncertainty is driven by K independent Brownian motions such that d\mathbf{z}' d\mathbf{z} = \mathbf{I} dt, where I is a K \times K identity matrix.
There are N \leq K securities whose price process follow a diffusion \frac{dS_{i}}{S_{i}} = \mu_{i} dt + \pmb{\sigma}_{i} d\mathbf{z}, \tag{5} where \pmb{\sigma}_{i} is a K \times 1 vector.
Each security pays continuously a dividend yield \delta_{i} dt.

Covariance Structure

Define \frac{d\mathbf{S}}{\mathbf{S}} = \left( \frac{dS_{1}}{S_{1}}, \frac{dS_{2}}{S_{2}}, \ldots, \frac{dS_{N}}{S_{N}} \right)' and denote by \mathbf{V} the N \times K matrix whose rows are given by \sigma_{i} defined in (5).
We have that \frac{d\mathbf{S}}{\mathbf{S}} = \pmb{\mu} dt + \pmb{\sigma} d\mathbf{z}, implying \left(\frac{d\mathbf{S}}{\mathbf{S}}\right) \left(\frac{d\mathbf{S}}{\mathbf{S}}\right)' = \pmb{\sigma} \pmb{\sigma}' dt.

A Potential SDF

The N \times N matrix \pmb{\sigma} \pmb{\sigma}' determines the instantaneous covariance of returns.
We can verify \frac{d\Lambda}{\Lambda} = - r_{f} dt - \left(\pmb{\mu} + \pmb{\delta} - r_{f} \pmb{\iota} \right)' \left(\pmb{\sigma} \pmb{\sigma}'\right)^{-1} \pmb{\sigma} d\mathbf{z} is an SDF that prices the N original assets correctly.
In the expression \pmb{\delta} denotes the vector of dividend yields \pmb{\delta} = \left(\frac{D_{1}}{S_{1}}, \frac{D_{2}}{S_{2}}, \ldots, \frac{D_{N}}{S_{N}}\right)'.

The Intertemporal CAPM

Linking the SDF to the Value Function

Let V(W, \mathbf{z}) denote the value function that the investor maximizes at each time.
- The level of wealth W is of course a state variable.
- The value function also depends on the risks to which all the assets are exposed, represented by \mathbf{z}.
In a more general setup, the marginal utility of consumption is the same as the marginal utility of wealth, i.e. u'(c) = V_{W}.
- The marginal value of any dollar must be the same in any use!
We can then write the SDF as \Lambda = e^{-\delta t} V_{W}(W, \mathbf{z}).

The SDF

Applying Ito’s lemma to \Lambda we find that: \frac{d\Lambda}{\Lambda} = (\cdot) dt + \frac{W V_{WW}(W, \mathbf{z})}{V_{W}(W, \mathbf{z})} \frac{dW}{W} + \frac{V_{W\mathbf{z}'}(W, \mathbf{z})}{V_{W}(W, \mathbf{z})} d\mathbf{z}. \tag{6}
Thus \operatorname{E}\left(\frac{dS}{S}\right) + \frac{D}{S} dt - r^{f} dt = \text{rra} \frac{dW}{W} \frac{dS}{S} - \frac{V_{W\mathbf{z}'}(W, \mathbf{z})}{V_{W}(W, \mathbf{z})} d\mathbf{z} \frac{dS}{S}.