Why are there different definitions of admissibility in the literature, and why do we need admissibility?

Wikipedia essentially defines an admissible trading strategy as a stochastic process $$H = (H_t)_{t\geq 0}$$ such that the associated value process $$\int H(u) d S(u)$$ is lower bounded. As I understand it, this rules out doubling strategies, since the value process is then not allowed to take arbitrary values in $$(-\infty,0)$$. Other books, e.g. "Martingale Methods in Financial Modelling" by Rutkowski and Musiela, define a trading strategy $$\phi$$ as being admissible (wrt a measure $$P^\ast$$) if the associated discounted value process is a martingale (wrt $$P^\ast$$), see for example Definition 3.1.4 on page 91.

First of all, I am a little confused at the different definitions of admissibility and how they are connected, but my understanding is that the concept of admissibility is used, generally, to avoid arbitrage in a financial model. But isn't the Wikipedia definition of admissibility superfluous? I mean, it rules out doubling strategies, but so does the existence of an equivalent martingale measure, which we need anyway, right? So why include the lower bound on the value process if we avoid doubling strategies using the first fundamental theorem of asset pricing?

For the second definition of admissibility: As far as I understand, we have that (under some conditions on the model) there exists an equivalent martingale measure $$Q$$ if and only if the discounted value process is a martingale under $$Q$$. I suppose this is a consequence of the discounted price process being a martingale under $$Q$$. Why then do we need to define the admissible strategies as those strategies $$\phi$$ under which the discounted value process is a martingale? Again, can we not just show that there is an equivalent martingale measure, and since this rules out arbitrage, we can consider any self-financing trading strategy we want? Even doubling strategies? But if we can consider doubling strategies, how can the model be arbitrage free?

• Why does the existence of an equivalent martingale measure rule out doubling strategies? Feb 14 at 2:28
• I thought it was because doubling strategies are arbitrage strategies, and an equivalent martingale measure rules out arbitrage strategies.
– xy z
Feb 14 at 2:35
• What's the definition of "arbitrage strategy" you're using? Feb 14 at 3:02
• A self-financing trading strategy $\phi$ is called an arbitrage opportunity if the associated value process $V(\phi)$ satisfies $V_0(\phi) = 0, \ P(V_T(\phi) \geq 0) = 1, \ P(V_T(\phi) > 0) > 0$.
– xy z
Feb 14 at 3:11
• Hm...where did you find that definition? Typically the definition requires $\phi$ to be admissible for some definition of "admissible" stronger than just self-financing, such as the ones mentioned in your question. Otherwise, most classical models (such as Black-Scholes) fail to be arbitrage free due to the existence of doubling strategies. Feb 14 at 3:22

Why do we need admissibility?

The problem is that you need extra conditions for the first fundamental theorem of asset pricing to hold; the existence of an equivalent martingale measure is not enough if you don't restrict the set of trading strategies you can use. Here, I give an example of an arbitrage strategy in the Black-Scholes model.

Consider the market model given by:

$$dS_t = S_t dB_t$$

where $$B_t$$ is a Brownian motion and the interest rate is zero. In particular, this is the Black-Scholes model with $$\mu = r = 0$$ and $$\sigma = 1$$. Notice that here the stock price is already a martingale, so an equivalent martingale measure exists and is simply the physical measure. Now, consider the process given by $$M_t = \int_0^t \frac{1}{\sqrt{1 -s}} dB_s \qquad 0 \leq t < 1$$ whose quadratic variation is $$\langle M \rangle_t = \int_0^t \frac{1}{{1 -s}} ds = \log \frac{1}{1-s}$$ If we take the inverse function $$\rho (s) = \langle M \rangle_s^{-1} = 1 - e^{-s}$$ and perform the time-change $$W_t = M_{\rho (t)}$$, we readily observe that $$\langle W \rangle_t = t$$. By Lévy's characterisation theorem, $$W$$ is a Brownian motion. Now, define the stopping time $$\tau$$ by: $$\tau = \inf \{t > 0 \, : M_t =1\} = \inf \{t > 0 \, : W_{\log \frac{1}{1-t}} =1\}$$ where we note that $$\tau < 1$$ almost surely, by virtue of the Brownian motion $$W$$ hitting every point in finite time. Equipped with this stopping time, awe are now ready to set up our arbitrage strategy.

Let $$\epsilon_t$$ and $$\eta_t$$ be your positions in the risky and riskless instruments, respectively, given by: \begin{align*} \epsilon_t &= \frac{1}{S_t \sqrt{1-t}} 1(t \leq \tau) \\ \eta_t &= -\epsilon_t S_t + \int_0^t \epsilon_s S_s dB_s \end{align*} Then, the value process is given by: $$V_t = \epsilon_t S_t + \eta_t \underbrace{B_t}_{=1} =\int_0^t \epsilon_s S_s dB_s = \int_0^t \frac{1}{\sqrt{1-s}} 1(s \leq \tau) dB_s = M_{t \land \tau}$$

Notice that this strategy is self-financing, as $$dV_t = \epsilon_t dS_t = \epsilon_t dS_t + \eta_t \underbrace{dB_t}_{=0}$$.

Finally, we exhibit arbitrage, since $$V_0 = 0$$ and $$V_1 = M_{1 \land \tau} = 1$$.

Moral: you truly need admissibility of trading strategies to be able to use the fundamental theorem of asset pricing.

• I see. So more precisely the FFTAP should read ''there are no admissible arbitrage strategies iff there exists an EMM?" I will accept your answer.
– xy z
Feb 14 at 22:08
• @xyz indeed. For a statement in full generality, I recommend checking out Thr Mathematics of Arbitrage by Delbaen and Schachermayer Feb 15 at 5:46