Deriving the Multidimensional Dynamics of Discounted Prices
Summary
The document derives the stochastic dynamics of a vector of asset prices after discounting by the accumulated risk-free rate. Starting from price dynamics with a vector drift and a matrix of Brownian exposures, it defines the discount factor and applies Itô's lemma to the product of each price and that factor. Since the discount factor has finite variation, the derivation retains the drift and diffusion terms without an additional quadratic-variation contribution.
The key algebraic step is expressing the price vector as its componentwise diagonal matrix multiplied by a vector of ones. This puts the short-rate adjustment into the same diagonal-matrix form as the drift, yielding discounted drift equal to the original drift less the risk-free rate for each asset; the diffusion exposure remains unchanged apart from discounting the price scale. The result is a standard multidimensional discounted-price equation. The derivation assumes the stated price process and a scalar short rate shared across assets, and it does not discuss estimation or empirical validation.
Key ideas
- Discounted prices are obtained by multiplying each asset price by the common stochastic discount factor.
- Itô's lemma applied to the price and finite-variation discount factor produces drift and diffusion terms without a cross-variation term.
- Writing the price vector as its diagonal matrix times a vector of ones aligns the rate adjustment with the vector drift.
- Discounting subtracts the short rate from each asset's drift while preserving the form of the volatility term.
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# Dynamics of discounted prices (multi-dimensional)
# Dynamics of discounted prices (multi-dimensional)
My objective is to find the dynamics of the discounted prices, given by $\mathbf{y}_{t} = \mathbf{P}_{t}\mathrm{e}^{-\int^{t}_{0} r_{s} ds}$. I know the dynamics should be $d\mathbf{y}_{t} = \mathrm{diag}(\mathbf{y}_{t})[(\mu_{t} - r_{t}\mathbf{1})dt + \sigma_{t}d\mathbf{W}_{t}]$, where $\mu_{t}$ is a vector, $\mathbf{1}$ is a vector of ones, and $\sigma_{t}$ is a matrix.
My approach is as follows. Define $f(t,\mathbf{y}_{t},D_{t}) = \mathbf{P}_{t}D_{t}$, where $D_{t} = \mathrm{e}^{-\int^{t}_{0}r_{s}ds}$. The dynamics of $D_{t}$ is: $dD_{t} = -D_{t}r_{t}dt$. Using Itô's Lemma, we get $$\begin{align*} d\mathbf{y_{t}} &= 0dt + D_{t}d\mathbf{P}_{t} + \mathbf{P}_{t}dD_{t} + \frac{1}{2} \cdot 0(d\mathbf{P}_{t})^{2} + \frac{1}{2} \cdot 0 (dD_{t})^{2} + \mathbf{1} (d\mathbf{P}_{t})(dD_{t})\\ &= D_{t}d\mathbf{P}_{t} + \mathbf{P}_{t}dD_{t} \end{align*}.$$
It is given that the dynamics of $\mathbf{P}_{t}$ is: $d\mathbf{P}_{t} = \mathrm{diag}(\mathbf{P}_{t})[\mu_{t}dt + \sigma_{t}d\mathbf{W}_{t}]$, again $\mu_{t}$ is a vector and $\sigma_{t}$ is a matrix. Inserting the $\mathbf{P}_{t}$ and $D_{t}$ dynamics we get
$$\begin{align*} d\mathbf{y}_{t} &= D_{t}d\mathbf{P}_{t} + \mathbf{P}_{t}dD_{t}\\ &= D_{t}\left[\mathrm{diag}(\mathbf{P}_{t})[\mu_{t}dt + \sigma_{t}d\mathbf{W}_{t}]\right] + \mathbf{P}_{t}\left[-D_{t}r_{t}dt\right]\\ &= D_{t}\mathrm{diag}(\mathbf{P}_{t})\mu_{t}dt - \mathbf{P}_{t}D_{t}r_{t}dt + \mathrm{diag}(\mathbf{P}_{t})D_{t}\sigma_{t}d\mathbf{W}_{t}\\ &= \mathrm{diag}(\mathbf{y}_{t})\mu_{t}dt - \mathbf{y}_{t}r_{t}dt + \mathrm{diag}(\mathbf{y}_{t})\sigma_{t}d\mathbf{W}_{t}. \end{align*}$$ This is where I'm stuck and it is probably some linear algebra rule that I'm missing, when deriving the relevant derivatives. To get the right result I need $\mathbf{y}_{t}r_{t}dt$ to be $\mathrm{diag}(\mathbf{y}_{t})r_{t}\mathbf{1}dt$.
Any help would be appreciated.
## Answer by John Stevens (score 0, accepted)
https://quant.stackexchange.com/a/71777
The realisation we need to make is that $\mathbf{P}_{t} = \mathrm{diag}(\mathbf{P}_{t})\mathbf{1}$. Doing this, we get
$$\begin{align*} d\mathbf{y}_{t} &= D_{t}d\mathbf{P}_{t} + \mathbf{P}_{t}dD_{t}\\ &= D_{t}\left[\mathrm{diag}(\mathbf{P}_{t})[\mu_{t}dt + \sigma_{t}d\mathbf{W}_{t}]\right] + \mathbf{P}_{t}\left[-D_{t}r_{t}dt\right]\\ &= D_{t}\mathrm{diag}(\mathbf{P}_{t})\mu_{t}dt - \mathbf{P}_{t}D_{t}r_{t}dt + \mathrm{diag}(\mathbf{P}_{t})D_{t}\sigma_{t}d\mathbf{W}_{t}\\ &= D_{t}\mathrm{diag}(\mathbf{P}_{t})\mu_{t}dt - \mathrm{diag}(\mathbf{P}_{t})\mathbf{1}D_{t}r_{t}dt + \mathrm{diag}(\mathbf{P}_{t})D_{t}\sigma_{t}d\mathbf{W}_{t}\\ &= \mathrm{diag}(\mathbf{P}_{t}D_{t})\mu_{t}dt - \mathrm{diag}(\mathbf{P}_{t}D_{t})\mathbf{1}r_{t}dt + \mathrm{diag}(\mathbf{P}_{t}D_{t})\sigma_{t}d\mathbf{W}_{t}\\ &= \mathrm{diag}(\mathbf{y}_{t})\mu_{t}dt - \mathrm{diag}(\mathbf{y}_{t})r_{t}\mathbf{1}dt + \mathrm{diag}(\mathbf{y}_{t})\sigma_{t}d\mathbf{W}_{t}\\ &= \mathrm{diag}(\mathbf{y}_{t})[(\mu_{t} - r_{t}\mathbf{1})dt + \sigma_{t}d\mathbf{W}_{t}] \end{align*}$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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