Asset Pricing in Continuous Time

Stochastic Foundations for Finance

Lorenzo Naranjo

Fall 2024

Independent Brownian Motions

  • If z_{1} and z_{2} are two Brownian motions independent from each other, then we have that \begin{aligned} \operatorname{E}(\Delta z_{1} \Delta z_{2}) & = \operatorname{E}(\Delta z_{1}) \operatorname{E}(\Delta z_{2}) = 0, \\ \operatorname{Var}(\Delta z_{1} \Delta z_{2}) & = \operatorname{E}(\Delta z_{1}^{2} \Delta z_{2}^{2}) - (\operatorname{E}(\Delta z_{1} \Delta z_{2}))^{2} \\ & = \operatorname{E}(\Delta z_{1}^{2}) \operatorname{E}(\Delta z_{2}^{2}) = (\Delta t)^{2}. \end{aligned}
  • Thus, in our computations we can assume that (dz_{1})(dz_{2}) = 0, whenever z_{1} and z_{2} are independent from each other.

Correlated Brownian Motions

  • Take z_{1} and z_{2} to be two independent Brownian motions and consider the process z_{3} = \rho z_{1} + \sqrt{1 - \rho^{2}} z_{2}.
  • This process is also a Brownian motion since (dz_{3})^{2} = (\rho dz_{1} + \sqrt{1 - \rho^{2}} dz_{2})^{2} = \rho^{2} (dz_{1})^{2} + (1 - \rho^{2}) (dz_{2})^{2} = dt, which guarantees that z_{3}(t) - z_{3}(s) \sim \mathcal{N}(0, t - s) whenever t > s.
  • We have also that dz_{3} dz_{1} = (\rho dz_{1} + \sqrt{1 - \rho^{2}} dz_{2}) dz_{1} = \rho (dz_{1})^{2} = \rho dt.
  • Thus, it is possible to create correlated Brownian motions out of independent ones.

Correlated Stock Price Processes

  • Now assume that you have two stocks such that \frac{dS_{i}}{S_{i}} = \mu_{i} dt + \sigma_{i} dz_{i}, where i = 1, 2 and dz_{1} dz_{2} = \rho_{1,2} dt.
  • Then we have that \begin{aligned} \operatorname{Var}\left(\frac{dS_{i}}{S_{i}}\right) & = \frac{1}{dt} \left(\frac{dS_{i}}{S_{i}}\right)^{2} = \sigma_{i}^{2}, \quad i = 1, 2, \\ \operatorname{Cov}\left(\frac{dS_{1}}{S_{1}}, \frac{dS_{2}}{S_{2}}\right) & = \frac{1}{dt} \left(\frac{dS_{1}}{S_{1}}\right) \left(\frac{dS_{2}}{S_{2}}\right) = \sigma_{1} \sigma_{2} \rho_{1, 2}. \end{aligned}
  • The instantaneous correlation of stock returns is therefore \rho_{1, 2}.

Multivariate Ito’s Lemma

  • Given a function F(X, Y, t), we can extend Ito’s lemma by taking into account the instantaneous covariance between X and Y. dF = F_{X} dX + F_{Y} dY + \frac{1}{2} F_{XX} (dX)^{2} + \frac{1}{2} F_{YY} (dY)^{2} + F_{XY} (dX)(dY) + F_{t} dt, where subindices of F denote partial derivatives with respect that variable or pair of variables.
  • Given dX = \mu_{X} dt + \sigma_{X} dz_{X} \quad \text{and} \quad dY= \mu_{Y} dt + \sigma_{Y} dz_{Y}, with (dz_{X})(dz_{Y}) = \rho_{XY} dt, we have that (dX)^{2} = \sigma_{X}^{2} dt, \quad (dY)^{2} = \sigma_{Y}^{2} dt, \quad \text{and} \quad (dX)(dY) = \sigma_{X} \sigma_{Y} \rho_{XY} dt.

Example

  • If F = X Y, we have F_{X} = Y, F_{Y} = X, F_{X Y} = 1, and F_{XX} = F_{YY} = F_{t} = 0.
  • Thus, dF = Y dX + X dY + dX dY, or \frac{dF}{F} = \frac{dX}{X} + \frac{dY}{Y} + \frac{dX}{X} \frac{dY}{Y}.

The Risk-Free Rate

  • The risk-free rate is defined as the drift of an asset M that has no instantaneous risk, i.e., \frac{dM}{M} = r dt.
  • This asset is typically known as the money-market account.
  • We do not assume that r is constant, but follows a diffusion dr = \mu_{r} dt + \sigma_{r} dz_{r}, where \mu_{r} and/or \sigma_{r} might depend on time and other state-variables.

Prices in Continuous Time

  • We model the price of risky assets as diffusions dP = \mu_{P} dt + \sigma_{P} dz_{P} where the drift \mu_{P} and the volatility of returns \sigma_{P} might depend on time and potentially other state variables.
    • For stocks we typically assume that \mu_{P} = \mu P and \sigma_{P} = \sigma P.
    • For options the drift depends on the \Delta, \Gamma and \Theta of the option.
  • If the asset pays a dividend flow D dt, the total instantaneous return of an asset is given by \frac{dP}{P} + \frac{D}{P} dt, where \frac{D}{P} is the dividend yield.

The Pricing Equation in Continuous Time

  • In the absence of arbitrage opportunities, there exists a stochastic discount factor (SDF) \Lambda > 0 such that \Lambda_{t} S_{t} = \operatorname{E}_{t} \left(\int_{t}^{T} D_{s} \Lambda_{s} ds + \Lambda_{T} S_{T} \right) \tag{1} which implies that \operatorname{E}d(\Lambda S) + \Lambda D dt = 0. \tag{2}
    • The total return of the discounted cash flows must be zero, otherwise there would be an arbitrage.
  • The stochastic discount factor \Lambda is also called the pricing kernel, state-price deflator, state-price density or marginal utility of consumption.

The Drift of the SDF

  • For the money-market account M we have that d\Lambda dM = 0 since M is locally riskless.

  • Thus, d(\Lambda M) = M d\Lambda + \Lambda dM = M d\Lambda + \Lambda r M dt,

  • The pricing equation (2) implies that \operatorname{E}d(\Lambda M) = 0, or \operatorname{E}\left(\frac{d \Lambda}{\Lambda}\right) = - r 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 dt = - \left(\frac{d\Lambda}{\Lambda}\right) \left(\frac{dS}{S}\right).

Property 1 (The Fundamental Pricing Equation) 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 = - \frac{1}{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 SDF Specification

  • Let \frac{d\Lambda}{\Lambda} = - r dt - \frac{\lambda_{m}}{\sigma_{m}} dz_{m}. \tag{4}
  • Consider a non-dividend paying stock such that \frac{dS_{i}}{S_{i}} = \mu_{i} dt + \sigma_{i} dz_{i}, and (dz_{i}) (dz_{m}) = \rho_{im} dt.
  • Equation (3) implies that \mu_{i} - r = \frac{\sigma_{i} \rho_{im}}{\sigma_{m}} \lambda_{m} = \beta_{i} \lambda_{m}. \tag{5}

Beta Pricing

  • Equation (5) says that the total expected return of the asset (\mu_{i} + q_{i}) over the risk-free rate (r) is equal to the asset’s beta with the SDF (\beta_{i}) times the SDF’s risk premium (\lambda_{m}).
  • The beta is a characteristic of the asset, whereas the SDF’s risk-premium is a characteristic of the market.
  • Note that \beta_{i} = \frac{\operatorname{Cov}\left(\frac{dS_{i}}{S_{i}}, \frac{d\Lambda}{\Lambda}\right)}{\operatorname{Var}\left(\frac{d\Lambda}{\Lambda}\right)}, so we recover the CAPM equation if dz_{m} represents a shock to the market portfolio.

Pricing a Call Option

  • Consider a non-dividend paying stock such that \frac{dS}{S} = \mu_{S} dt + \sigma_{S} dz.
  • Equation (3) implies that (\mu_{S} - r) dt = - \left(\frac{d\Lambda}{\Lambda}\right) \left(\frac{dS}{S}\right).
  • A call option written on S with strike K and expiring at T also satisfies equation (3), \operatorname{E}(dC) - r C dt = - \left(\frac{d\Lambda}{\Lambda}\right) (dC). \tag{6}

Black and Scholes Again

  • Applying Ito’s lemma to C(S, t) dC = C_{S} dS + \frac{1}{2} C_{SS} (dS)^{2} + C_{t} dt, we can compute \begin{aligned} \operatorname{E}(dC) & = \left(\mu_{S} S C_{S} + \frac{1}{2} \sigma_{S}^{2} S^{2} C_{SS} + C_{t}\right) dt, \\ \left(\frac{d\Lambda}{\Lambda}\right) (dC) & = S C_{S} \left(\frac{d\Lambda}{\Lambda}\right) \left(\frac{dS}{S}\right) = - (\mu - r) S C_{S} dt. \end{aligned}
  • Equation (6) implies the Black-Scholes PDE \mu_{S} S C_{S} + \frac{1}{2} \sigma_{S}^{2} S^{2} C_{SS} + C_{t} - r C = (\mu_{S} - r) S C_{S}.

The Risk-Neutral Measure

  • The SDF approach tells us how to adjust the underlying model parameters so that valuation is consistent with what risk-neutral agents would do to price risky assets.
  • In the risk-neutral world, \frac{dS}{S} = r dt + \sigma dz_{S}^{*} so that C = \operatorname{E}^{*} \left[\max(S_{T} - K, 0) e^{-r T}\right].
    • The asterisk indicates that the Brownian motion is not the same we started with, but a risk-adjusted Brownian motion that can be used to discount expected cash-flows at the risk-free rate.

Futures on the Stock

  • A futures contract fixes the price of an asset in future.
    • It does not require any investment, but positions are marked to market daily depending on how the futures price \varphi changes.
  • Thus, it must be the case that \operatorname{E}^{*} (d\varphi) = 0
    • Under suitable conditions, the futures price is a martingale under the risk-neutral measure.
  • The futures price of the stock with maturity T is then \operatorname{E}^{*}(S_{T}) = S_{0} e^{r T}.

A Unified Pricing Framework

  • The SDF pricing approach provides a unified framework to price stocks, bonds and derivatives.
  • Even though the risk-neutral pricing approach makes the pricing of derivatives easier, the SDF approach makes explicit the risk-correction that needs to be applied to any asset.
  • The previous analysis shows that the risk-premium of the stock’s returns is \mu - r whereas the risk-premium of option’s returns is equal to \mu_{C} - r = \frac{1}{dt} \operatorname{E}\left(\frac{dC}{C}\right) - r = (\mu_{S} - r) \frac{S C_{S}}{C}. \tag{7}
    • In general, S will be several times larger than C, so the option’s risk-premium gets amplified significantly compared to the stock risk-premium.

The Beta and Volatility of the Call

  • From equation (7) we can conclude \beta_{C} = \frac{S C_{S}}{C} \beta_{S}.
    • The beta of the call is much larger than the beta of the stock.
    • This is a consequence of the implicit leverage that the call carries.
  • Another way to see this is that Ito’s lemma implies that \sigma_{C} = \frac{S C_{S}}{C} \sigma_{S}.
  • Since in the Black-Scholes model the call and the stock are perfectly correlated, all the extra risk added by the leverage is passed to the call’s beta.

Example 1 Consider a non-dividend paying stock that currently trades for $100. The risk-free rate is 4% per year, continuously compounded and constant for all maturities. The instantaneous volatility of returns is 25% per year.

We saw in a previous example that for an at-the-money call expiring in 9 months we have that C_{S} = 0.5975 and C = 10.05. Thus, the current instantaneous volatility of the call is 0.25 \times \frac{100 \times 0.5975}{10.05} = 148.63\%.

Also, if the beta of the stock with the risk-factor is 1.2, the beta of the call is 1.2 \times \frac{100 \times 0.5975}{10.05} = 7.13.

Understanding Better the SDF

  • The expression provided in equation (4) might seem mysterious at first.
    • Where does the Brownian motion z_{m} come from?
    • How can we estimate \lambda_{m} and \sigma_{m} in practice?
  • In the following, a bold face letter is used to denote a vector or a matrix.
    • \mathbf{S} denotes the vector of security prices, \boldsymbol{\mu} denotes the vector of expected returns, \mathbf{z} is the vector of Brownian motions, \boldsymbol{\iota} is a vector of ones, \mathbf{I} is the identity matrix, \boldsymbol{\sigma} is a matrix of risk loadings, etc.
  • I use a single quote to denote the transpose of a vector or a matrix, so that \boldsymbol{\sigma}' is the matrix transpose of \boldsymbol{\sigma}, and if \mathbf{w} is a column vector, then \mathbf{w}' is a row vector.

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 traded securities whose price processes follow diffusions such that \frac{dS_{i}}{S_{i}} = \mu_{i} dt + \boldsymbol{\sigma}_{i} d\mathbf{z}, \tag{8} where \boldsymbol{\sigma}_{i} is a K \times 1 vector.
  • For notational simplicity, assume that dividends are reinvested continuously in each S_{i}, so that \mu_{i} represents the total return of each security.

The Vector of Instantaneous Returns

  • 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 \boldsymbol{\sigma} the N \times K matrix whose rows are given by \boldsymbol{\sigma}_{i} defined in (8).
  • We can write the vector of instantaneous returns as \frac{d\mathbf{S}}{\mathbf{S}} = \boldsymbol{\mu} dt + \boldsymbol{\sigma} d\mathbf{z}. \tag{9}

Covariance Structure

  • We can use equation (9) to show that \left(\frac{d\mathbf{S}}{\mathbf{S}}\right) \left(\frac{d\mathbf{S}}{\mathbf{S}}\right)' = \boldsymbol{\sigma} \boldsymbol{\sigma}' dt.
  • The N \times N matrix \boldsymbol{\sigma} \boldsymbol{\sigma}' determines the instantaneous covariance of returns.
    • We assume that (\boldsymbol{\sigma} \boldsymbol{\sigma}')^{-1} exists.
    • If not, some securities are redundant and can be eliminated from our analysis.
  • If K = N we say that the market is complete and therefore we assume that \boldsymbol{\sigma}^{-1} exists.
    • If not, remove the redundant securities from the analysis.

A Candidate SDF

  • We can verify \frac{d\Lambda}{\Lambda} = - r dt - \left(\boldsymbol{\mu} - r \boldsymbol{\iota} \right)' \left(\boldsymbol{\sigma} \boldsymbol{\sigma}'\right)^{-1} \boldsymbol{\sigma} d\mathbf{z} \tag{10} is an SDF that prices the N original assets correctly.
    • In the previous expression \boldsymbol{\iota} represents a vector of ones.
  • The SDF defined by (10) includes all possible sources of risk of the economy.

Market Completeness

  • If K = N the stochastic factor is unique and given by \frac{d\Lambda}{\Lambda} = - r dt - \left(\boldsymbol{\mu} - r \boldsymbol{\iota} \right)' \boldsymbol{\sigma}^{-1} d\mathbf{z}.
  • If K > N the market is incomplete and there are many other SDFs that will price the assets correctly.
    • It is possible to find Brownian motions orthogonal to \boldsymbol{\sigma} d\mathbf{z}, whose risks will not be priced.
    • All those Brownian motions are spanned by the null space of \boldsymbol{\sigma}.
    • The SDF presented in equation (10) is the only discount factor that can be achieved from a tradable strategy, though.

The Tangency Portfolio

  • We can show that the weights of the tangency portfolio are given by \mathbf{w}_{m} = \frac{\left(\boldsymbol{\mu} - r \boldsymbol{\iota} \right)' \left(\boldsymbol{\sigma} \boldsymbol{\sigma}'\right)^{-1}}{\left(\boldsymbol{\mu} - r \boldsymbol{\iota} \right)' \left(\boldsymbol{\sigma} \boldsymbol{\sigma}'\right)^{-1} \boldsymbol{\iota}}.
  • If we denote by m the value of the tangency portfolio, we must have that \frac{dm}{m} = \mathbf{w}_{m}' \boldsymbol{\mu} dt + \mathbf{w}_{m}' \boldsymbol{\sigma} d\mathbf{z} = \mu_{m} dt + \sigma_{m} dz_{m}, \tag{11} where \sigma_{m} = \sqrt{\mathbf{w}_{m}' \boldsymbol{\sigma} \boldsymbol{\sigma}' \mathbf{w}_{m}} and dz_{m} = \dfrac{\mathbf{w}_{m}' \boldsymbol{\sigma}}{\sqrt{\mathbf{w}_{m}' \boldsymbol{\sigma} \boldsymbol{\sigma}' \mathbf{w}_{m}}} d\mathbf{z}.

Interpretation of the Previous Result

  • Equation (11) shows that stochastic part of the tangency portfolio determines the stochastic part of the SDF defined in (4).
  • Moreover, we have that \mu_{m} - r = - \frac{1}{dt} \frac{d\Lambda}{\Lambda} \frac{dm}{m} = \frac{\lambda_{m}}{\sigma_{m}} \sigma_{m} = \lambda_{m}, so that \frac{d\Lambda}{\Lambda} = -r dt + \left(\frac{\mu_{m} - r}{\sigma_{m}}\right) dz_{m}.
    • The market price of risk of dz_{m} is indeed the maximum Sharpe ratio of the economy.

Market Incompleteness

  • If K > N, then it is possible to find a vector \boldsymbol{\zeta} such that \boldsymbol{\sigma} \boldsymbol{\zeta} = 0.
  • If we define the process \varepsilon as \frac{d\varepsilon}{\varepsilon} = - \boldsymbol{\zeta}' d\mathbf{z} and denote by \Lambda_{P} the stochastic discount factor defined in (10), any other SDF can be expressed as \frac{d\Lambda}{\Lambda} = \frac{d\Lambda_{P}}{\Lambda_{P}} + \frac{d\varepsilon}{\varepsilon}, since for any asset \left(\frac{dS}{S}\right) \left(\frac{d\varepsilon}{\varepsilon}\right) = - \boldsymbol{\sigma} d\mathbf{z} d\mathbf{z}' \boldsymbol{\zeta} = - \boldsymbol{\sigma} \mathbf{I} \boldsymbol{\zeta} dt = - \boldsymbol{\sigma} \boldsymbol{\zeta} dt = 0.

Implications of Market Incompleteness

  • The previous analysis shows that any SDF can be expressed as \frac{d\Lambda}{\Lambda} = - r dt - \boldsymbol{\lambda}' d\mathbf{z}, \tag{12} where \boldsymbol{\lambda} = \boldsymbol{\lambda}_{P} + \boldsymbol{\zeta} and \boldsymbol{\lambda}_{P} = \left(\boldsymbol{\mu} - r \boldsymbol{\iota} \right)' \left(\boldsymbol{\sigma} \boldsymbol{\sigma}'\right)^{-1} \boldsymbol{\sigma}.
  • Since we can only determine \boldsymbol{\lambda}_{P} from the traded assets, there are many prices of risk that will price all assets correctly.
  • If we want to price a new asset that is not traded yet, there will be many different prices that will be consistent with no-arbitrage.
    • Therefore, in an incomplete market the only way to know the price of this new asset is to start trading it!
    • The SDF framework will impose restrictions, though, on what prices are consistent with no-arbitrage.