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In the derivation of the Black-Scholes formula given by Joshi (extract below), he says $C(t,S_t)/B_t$ is a martingale. Why?
I understand this can be deduced from the Black-Scholes PDE since the drift term is equal to zero. But how can he deduce $C(t,S_t)/B_t$ is a martingale before we have derived the Black-Scholes PDE.
It is important to note that he says: "In the risk-neutral world, $\frac{C(t,S_t)}{B_t}$ is a martingale." That is true by definition of what the risk-neutral measure is, also called martingale measure for exactly that reason.
A risk-neutral measure is defined such that asset prices deflated by the numeraire (unit with which prices are measured) are martingales. In your example, the standard numeraire is used: a bank account which is continuously reinvested at the risk-free rate. There are plenty of sources that discusses the (almost) equivalence of the existence of a risk-neutral measure and the absence of arbitrage (or more precisely: No free lunch with vanishing risk). The most comprehensive book on these issues is by Delbaen and Schachermeyer, but it's mathematically very demanding. I personally like the book by Duffie, but opinions about the books vary. A good compromise between mathematical completeness and financial intuition is the book by Björk.
In Joshi's Book "The Concepts and Practice of Mathematical Finance", the whole chapter 6 is devoted to this topic.
In the Black-Scholes world, it is assumed that the option value $C(t, S_t)$ is replicable by an admissible self-financing trading strategy $\phi$, where $\phi_t=(\alpha_t, \beta_t)$. That is, \begin{align*} C(t, S_t) = \alpha_t B_t + \beta_t S_t, \end{align*} and \begin{align*} dC(t, S_t) = \alpha_t dB_t + \beta_t dS_t. \end{align*} Since $dB_t = rB_t dt$, and $dS_t = S_t(rdt + \sigma dW_t)$, then \begin{align*} d\bigg(\frac{C(t, S_t)}{B_t} \bigg) &= \frac{dC(t, S_t)}{B_t}-\frac{C(t, S_t)}{B_t^2}dB_t\\ &=\beta_t \frac{dS_t}{B_t} - \beta_t \frac{S_t}{B_t^2}dB_t\\ &=\beta_t \frac{S_t(rdt + \sigma dW_t)}{B_t} - \beta_t \frac{S_t}{B_t}rdt\\ &=\sigma\beta_t\frac{S_t}{B_t}dW_t. \end{align*} Therefore, $\{C(t, S_t)/B_t \mid 0\leq t \leq T\}$ is a martingale.
Let's go to Chapter 6 in the book where he talks about risk-neutrality. In the chapter he proved the existence of a risk-neutral measure. Now, let's stop and think what this means.
It means our hedging portfolio for the option can't beat the risk-free rate. Although not exactly mathematically correct, you can think C/B as a ratio of the option price to the zero-coupon-bond. If this ratio has a drift (i.e: a trend), the option (i.e: heading portfolio) performs better than the zero-coupon-bond. Obviously, it'll create an arbitrage opportunity. Mathematically, this is simply a martingale under the risk-neutral measure.
Now, we know the option price must be a martingale under the risk-neutral measure and we also know that the drift is zero, it's not hard to see why Mark Joshi derived the way he did in the book.
Not sure whether author explains that, but when talking about replicating portfolios, the discounted portfolio must be a martingale for non-arbitrage conditions. That's a very important fact in the theory of risk-neutral pricing. More details are in Shiryaev's "Essentials of Stochastic Finance", and maybe in some probability theory oriented books on the topic, e.g. Musiela and Rutkowski.
