Deriving the Risk-Neutral Martingale Property of the Normalized Gains Process

# Normalized Gains Process is a Q-Martingale - Proof and Intuition


# Normalized Gains Process is a Q-Martingale - Proof and Intuition












I'm trying to work the proof that the normalized gains process, $G^z_t = \frac{S_t}{B_t}+\int^t_0\frac{1}{B_s}dD_s$ is a Q-martingale under Q (the risk-neutral measure). I'll show what I've worked through, but it feels like I'm just invoking circular logic, so I'm not sure if I've followed the right path. Also, I have a question regarding the intuition of the whole thing, and also a practical question related to the Wiener process under Q that I'll place at the end. This will probably be long, so many thanks to anyone who takes the time to lend any insight.

My proof:

First, note that $\frac{1}{B_t}$ can be thought of as $e^{-rt}$, if we think of $B_0$ as \$1 in a bank account. Furthermore, we can also think of $dD_t=qS_tdt$, with q the continuously compounded dividend yield. Finally, also let the normalized price process $\frac{S_t}{B_t}=Z_t$. Now, by Ito, $\frac{S_t}{B_t}$, or equivalently $S_te^{-rt}$ differentiates to $-r\frac{S_t}{B_t}dt+\frac{1}{B_t}dS_t$. Assuming a GBM for $dS_t=\alpha S_tdt+\sigma S_tdW_t$, we are led to the following for $Z_t$ and $G^z_t$:

$$dZ_t=Z_t(\alpha-r)dt+Z_t\sigma dW_t$$

$$dG^z_t=Z_t(\alpha-r+q)dt+Z_t\sigma dW_t$$

Now, by Girsanov, let $dQ=LdP$ where $dL_t=L_t\phi dW_t$, we have $dW_t=\phi_tdt+dW^Q_t$. Plugging this into the dynamics of $dS_t$ we have

$$dS_t=S_t(\alpha+\sigma\phi_t)dt+S_t\sigma dW^Q_t$$

Making this the Q-dynamics for the stock price.

Furthermore, plugging this into our result for $G^Z_t$ we are led to $$dG^Z_t=Z_t(\alpha-r+q+\sigma\phi_t)dt+Z_t\sigma dW^Q_t$$

Now, to be a martingale, we need no drift, that is $a-r+q+\sigma\phi_t=0$, or $\alpha+\sigma\phi_t=r-q$. From this we see that we can go back to the Q-dynamics for the stock price and write

$$dS_t=S_t(r-q)dt+S_t\sigma dW^Q_t$$

...and this is where I sort of end. I don't know if I've proved what was meant to be proved or not. I know that I've ended up with the known Q-dynamics result, which is encouraging.

Finally, for a practical question - is there any difference in simulating $W^Q_t$ in practice versus $W_t$? For instance, suppose I decide to price a derivative via Monte Carlo simulation in excel. Would it be the exact same function (i.e. norm.inv(rand(0,1))?

***Update - ok, I think I took it a step further and have it now. So, now that I basically rigorously derived the Q-dynamics of the stock price price process in the last line, let's go back to the fact that the normalized gains process is $G^z_t = \frac{S_t}{B_t}+\int^t_0\frac{1}{B_s}dD_s$, or equivalently $$G^z_t=e^{-rt}S_t+\int^t_0e^{-rs}S_sqds$$

Let's focus on the undifferentiated part, that is, $e^{-rt}S_t$, call it $\tilde{S}_t$. By Ito, we have

$$d\tilde{S}_t=-re^{-rt}S_tdt+e^{-rt}dS_t$$

Plugging in the before derived Q-dynamics for $dS_t$, we have

$$d\tilde{S}_t=-r\tilde{S}_tdt+e^{-rt}[(r-q)S_tdt+\sigma S_tdW^Q_t]<=>$$

$$d\tilde{S}_t=(r-r-q)\tilde{S}_tdt+\sigma\tilde{S}_tdW^Q_t<=>$$

$$d\tilde{S}_t=-q\tilde{S}_t+\sigma\tilde{S}_tdW^Q_t$$

Now, going back to the full equation for the gains process, we have

$$dG^z_t=-q\tilde{S}_tdt+q\tilde{S}_tdt+\sigma\tilde{S}_tdW^Q_t<=>$$

$$dG^z_t=\sigma\tilde{S}_tdW^Q_t$$

Proved Q-martingale