Dividend-Paying Asset Dynamics Under a Change of Numeraire
Summary
The document asks how to derive risk-neutral dynamics for a dividend-paying asset when the bank account is the numeraire. It outlines the martingale condition for the discounted asset plus accumulated discounted dividends, then applies Itô’s formula and Girsanov’s theorem to solve for the measure change in a geometric Brownian motion setting.
The question becomes more involved when the asset is foreign: converting both the asset price and dividends into domestic currency introduces exchange-rate terms, so the drift may no longer factor into a single asset-price term. The author suggests deriving the domestic asset dynamics from the separate foreign asset and exchange-rate processes. No worked resolution or empirical evidence is included, and the discussion flags that the dividend martingale condition has assumptions, such as a bank account with only a time component.
Key ideas
- The discounted asset plus discounted dividends is presented as a martingale under the bank-account measure, subject to assumptions.
- Itô’s formula and Girsanov’s theorem can be used to infer the asset’s dynamics under the new measure.
- Domestic-currency conversion adds exchange-rate effects to both asset and dividend dynamics.
- Deriving the foreign asset and exchange-rate dynamics separately may simplify the domestic-currency calculation.
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# Finding the dynamics of a dividend paying asset under arbitrary numeraire
# Finding the dynamics of a dividend paying asset under arbitrary numeraire
Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding EMM $Q^B$. From Björk's book I have that $$\frac{S(t)}{B(t)}+\int_0^t \frac{dD(s)}{B(s)}$$ must be a martingale under $Q^B$ (actually this is not the entirely true for all cases, but if $B$ only has the dt-term it will be). Then I could find the dynamics of $S$ under the new measure by first finding the dynamics of the expression above, i.e. $$d\Big(\frac{S}{B}\Big) + \frac{dD}{B}$$ using Ito's formula. When the dynamics are in GBM-form I will neatly be able to factor out the $S/B$-term. Then, after using Girsanov Theorem, $dW = \varphi + dW^{Q^B}$, I would end up with a dt-term $$\frac{S}{B}\big((\alpha - r + \delta) + \varphi (\sigma +\delta)\big)$$ which must be equal to 0 since it should be a martingale. Here I have assumed the following dynamics of the processes
$dB = rBdt$
$dS= \alpha S dt + \sigma S dW$
$dD = \delta S dt$
Then I could easily solve for $\varphi$ and then the dynamics under the new measure follows. However, now assume the stock is traded on a foreign market with corresponding exchange rate $E$, then I'd like to determine $S_d$ (which is the process of $S$ in domestic currency) defined by $S_d = S \cdot E$. Analogously we have the dividend process $D_d$. In this case $S_d$ will still have dynamics on GBM-form with different parameters. The $D_d$ term will change quite dramatically though (I get $dD_d = (\delta S_d + D_d \mu_E)dt + \sigma_E D_d dW$, where $\mu_E, \sigma_E$ are the parameters in the GBM of the exchange rate). Since not all terms include $S_d$ I will not be able to factor that out when I solve for $\varphi$ as in the previous case. So I can't find the new dynamics in this manner. Is there something I'm missing or could I approach this problem differently?
Edit: Maybe it would be easier to find the dynamics of just $S$ and $E$ under the new measure, and then find the dynamics of $S_d$ by $S_d=S \cdot E$? I could find the dynamics of $E$ by the fact that $E/B$ must be a martingale under the measure (is this true?).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.