Deriving Dividend Dynamics under the Risk-Neutral Measure

Summary

The document asks how to change the dynamics of a dividend stream from the physical measure to the risk-neutral measure when the dividend shock and stochastic discount factor are driven by correlated Brownian motions. It gives a geometric diffusion for dividends, a diffusion for the discount factor, and a present-value expression under the physical measure that includes the covariance term involving correlation, dividend volatility, and market price of risk.

The author proposes that the risk-neutral dividend drift should be reduced by that covariance term while the volatility stays unchanged, and asks how Girsanov’s theorem produces this result. The document does not provide a derivation or a response, so the proposed drift should be treated as a question rather than a verified conclusion. Its setup illustrates why changing measure for correlated risks requires accounting for their covariance; it does not discuss additional assumptions, such as whether the stated discounting and valuation expressions are internally consistent.

Key ideas

• Changing from the physical to the risk-neutral measure changes the drift of the dividend process.
• The dividend shock is correlated with the Brownian motion driving the stochastic discount factor.
• The question proposes a risk-neutral drift adjustment that depends on correlation, volatility, and the market price of risk.
• The document asks for a Girsanov derivation but does not supply or verify one.

Tags

# Dividend Dynamics under Q Measure / Using Girsanov Theorem with Covariance


# Dividend Dynamics under Q Measure / Using Girsanov Theorem with Covariance












I want to find the value of a dividend stream. I can do it under the P-measure, but now I would also like to do it under the Q-measure but cant figure out how to derive the dynamics of the dividend under Q.

I have the dynamics of the dividend stream and the stochastic discount factor with two dependent Brownian motions :

$$ dD_t=D_t[\alpha dt + \beta dW_t] $$ $$ dH_t=-H_t[r dt + \theta dB_t] $$

where $ \theta$ is the market price of risk and the instantaneous covariance is $ <W_t,B_t>=\rho$.

I want to calculate the value of the dividend stream ($V_t$). Under the P-measure I arrive at:

$$ V_t=E^P[\int_{t}^{T}H_t D_t ds] = \frac{1}{\alpha -r-\rho \beta \theta} D_t [exp[(T-t)(\alpha-r-\rho \beta \theta)] - 1] $$ (Which I think is correct. I happily provide more steps if asked for)

Now I want to calculate the same under the Q-measure: $$ V_t=E^Q[\int_{t}^{T} exp(-r) D_t ds] $$

The result should obviously be the same, but I cant figure out how to derive the dynamics of $D_t$ under the risk-neutral measure: $$ dD_t = D_t[\alpha^Q dt + \beta^Q dW_{t}^{Q}] $$ , where $\alpha^Q$ and $\beta^Q$ are placeholders.

I think the solution should be: $ dD_t = D_t[(\alpha-\rho \beta \theta) dt + \beta dW_{t}^{Q}] $, since this would lead to the same result as under P, but I don't see how to get there. So far I never used Girsanovs Theorem in a case where covariance was involved, so I don't understand where $\rho$ comes from.

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