For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). I know that, by virtue of Girsanov's theorem, the Brownian motion under the risk-neutral measure is defined by $$dB_t^{\Bbb Q} = \lambda dt + dB_t$$ where $\lambda$ is the unique market price of risk, or the so-called Sharpe ratio.

Under the risk-neutral measure, any non-dividend paying stock price process $S_t$ thus follows $$\frac{dS_t}{S_t} = rdt + \sigma_SdB_t^{\Bbb Q}.$$

However, in Kerry Back's A Course in Derivative Securities page 220, the author claimed without a proof that the instantaneous rate of return for a call option on the stock price $C_t$ is also $r$, i.e. $$\frac{dC_t}{C_t} = rdt + \sigma_C d B_t^{\Bbb Q}$$ where $\sigma_C$ is some stochastic process that we're not interested in. The author make crucial use of the above formula (i.e. the drift of $C_t$ is $rC_tdt$) to derive the BS PDE.

Question: is it true that under the risk neutral measure, any non-dividend paying asset price $X_t$ must have its instantaneous rate of return equal to $r$? If so, what would be a rigorous explanation for this?

Edit: Antoine is spot on. Under the risk neutral measure, any discounted asset price $Y_t=e^{-rt}X_t$ must be a martingale or equivalently an Ito integral without drift. Hence $$\frac{dY_t}{Y_t}=\sigma_Y dB_t^{\Bbb Q}.$$ where $\sigma_Y$ can be a quite general stochastic process. On the other hand, by the compounding rule of Ito processes, $$\frac{dY_t}{Y_t}=-rdt+\frac{dX_t}{X_t}$$ Therefore it follows $$\frac{dX_t}{X_t}=rdt+\sigma_Y dB_t^{\Bbb Q}.$$


Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.

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    $\begingroup$ Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks. $\endgroup$ – Vim Feb 7 '19 at 11:35

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