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Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete.

(a) $\beta_{t}=e^{t}, S_{t}=B_{t}+1$

(b) $\beta_{t}=e^{t}, S_{t}=e^{t+\int_{0}^{t} s d B_{s}}$

(c) $\beta_{t}=e^{t}, S_{t}=e^{t+\int_{0}^{t} B_{s} d s}$


Here's my approach to these questions:

$$\text { (a) } \frac{S_{t}}{\beta_{t}} =\frac{1+B_{t}}{e^{t}} =e^{-t}+e^{-t} B_{t}$$

$e^{-t}$ is determimistic, but not constant and hence this model admits arbitrage and is not complete. Is this right logic?

$$\text { (b) } \quad \frac{S_{t}}{\beta_{t}}=\frac{e^{t} e^{\int_{0}^{t} s d B_{s}}}{e^{t}} = e^{\int_{0}^{t} s d B_{s}} $$

How do I proceed from here? I'm unsure what to do.. same with (c) as well.

$$\text { (c) } \quad \frac{S_{t}}{\beta_{t}}=\frac{e^{t} e^{\int_{0}^{t} B_{s}ds}}{e^{t}} = e^{\int_{0}^{t} B_{s}ds} $$

How do I apply Girsanov's theorem to (b) and (c)? I'm not sure how to prove it for the last two, any help would be appreciated thank you

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First, let's check if these models are abritrage free. The first fundamental theorem of asset pricing says that if there exists an equivalent probability measure under which $\frac{S_t}{\beta_t} = e^{-t}S_t$ is a martingale, then the market is arbitrage free, so we will check whether such an equivalent martingale measure exists. This is where we will use Girsanov's theorem, which states that if $Z_t = \exp\left(\int_0^t \theta_s dB_s - \frac 12 \int_0^t \theta_s^2 ds\right)$ and $d\tilde{\mathbb{P}} = Z_T d\mathbb{P}$, then $\tilde{B}_t = B_t - \int_0^t \theta_s ds$ is a Brownian motion under $\tilde{\mathbb{P}}$. We could also write this as $d\tilde{B}_t = dB_t - \theta_t dt$.

In a), we use Ito's lemma to compute $d(e^{-t}S_t) = e^{-t}(dS_t - S_tdt) = e^{-t}(dB_t - S_tdt)$. We want this to be a martingale under $\tilde{\mathbb P}$, so we want $d\tilde B_t = dB_t - S_tdt$. This suggests setting $\theta_t = S_t$ for all $t$ in Girsanov's theorem, i.e. define $Z_t = e^{\int_0^t S_sdB_s - \frac 12 \int_0^tS_s^2ds}$ and $d\tilde{\mathbb{P}} = Z_T d\mathbb{P}$. Then $d(e^{-t}S_t) = e^{-t}d\tilde B_t$ is a martingale under $\tilde{\mathbb{P}}$, so this model is arbitrage free.

In b), we compute \begin{align*}d(e^{-t}S_t) &= d(e^{\int_0^t s dB_s}) \\ &= e^{\int_0^t s dB_s}(tdB_t + \frac 12 t^2 dt) \\ &= te^{\int_0^t s dB_s}(dB_t + \frac 12 t dt).\end{align*} We again want to find a probability measure that makes this a martingale, so we want $d\tilde B_t = dB_t + \frac 12 t dt$. This suggests setting $\theta_t = -\frac 12 t$ in Girsanov's theorem, so define $Z_t := \exp\left(-\frac 12 \int_0^t s dB_s - \frac 18 \int_0^t s^2 ds\right)$ and $d\tilde{\mathbb{P}} := Z_T d\mathbb{P}$. Then $d(e^{-t}S_t) = te^{\int_0^t s dB_s}d\tilde B_t$ is a martingale under $\tilde{ \mathbb{P}},$ so this model is also arbitrage free.

In c), we compute \begin{align*}d(e^{-t}S_t) &= d(e^{\int_0^t B_sds}) \\ &= e^{\int_0^t B_sds}B_tdt.\end{align*} No matter how we change the measure, we cannot make this a martingale because there is no $dB_t$ term. Hence this model is not arbitrage free.

Now we want to check if these models are complete. Typically the definition of complete requires that the model be arbitrage free, so we can rule out c) immediately. The second fundamental theorem of asset pricing says that an arbitrage free model is complete if and only if the equivalent martingale measure is unique. In a) and b) we saw that there was only a single choice of $\theta_t$ to make $e^{-t}S_t$ a martingale, so both of these models are also complete. A good rule of thumb is that models are complete when there are the same number of risky assets as sources of uncertainty (i.e. Brownian motions).

EDIT: I should probably close a small gap in my answer to c). The first fundamental theorem of asset pricing doesn't have a (simple) converse, so the fact that there isn't an equivalent martingale measure does not imply there is an arbitrage. Instead, we can explicitly construct an arbitrage strategy. If we consider the wealth $X_t$ of an investor who holds $\Delta_t$ shares of stock at time $t$, then their wealth dynamics are $$dX_t = \Delta_t dS_t + (X_t - \Delta_t S_t)dt = (\Delta_t S_t (1+B_t) + (X_t-\Delta_t S_t))dt = (X_t + \Delta_t S_t B_t)dt.$$ Setting $\Delta_t = \operatorname{sgn}(B_t)$ then gives $dX_t = (X_t + S_t |B_t|)dt$. Since the drift is non-negative and is strictly positive whenever $B_t \ne 0$, we conclude this is an arbitrage because starting with $X_0 = 0$ we end up with $X_T \ge 0$ a.s. and $\mathbb{P}(X_T > 0) > 0$.

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    $\begingroup$ Hmm, seems then that my answer is wrong? Curious to find out my mistake... for example, what would be wrong with the first answer I gave for model a? $\endgroup$ Aug 4 '21 at 16:53
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    $\begingroup$ @JanStuller You showed that $e^{-t}S_t$ isn't a martingale under the original probability measure, but the condition to rule out arbitrage is that there exists an equivalent probability measure that makes it a martingale. It's fine for the discounted stock process to have a drift, it just can't have a risk-free drift. $\endgroup$ Aug 4 '21 at 16:59
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    $\begingroup$ Thank you. I assumed the model is stated under the risk-neutral measure already. It wasn't clear to me that it was under the physical measure. $\endgroup$ Aug 4 '21 at 19:07
  • $\begingroup$ By definition, the discounted stock process is a martingale under any risk-neutral measure. A risk-neutral measure may not exist, however, which is why the fundamental theorems of asset pricing relate existence of a risk-neutral measure to absence of arbitrage. $\endgroup$ Aug 4 '21 at 19:12
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    $\begingroup$ If $f(t,S_t) = e^{-t} S_t$, then should it not be (by Ito's lemma ) $df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial S_t} dS_t + \frac{1}{2}\frac{\partial^2 f}{\partial S_t^2}(dS_t)^2 = -e^{-t}S_t dt + e^{-t} dS_t = e^{-t}(dS_t - S_t dt)$? $\endgroup$
    – RRL
    Aug 4 '21 at 19:42
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I assume all three models are stated under the money-market measure: then there is no arbitrage if the discounted pay-off is a martingale under the money-market Numeraire. Therefore to show no arbitrage for all three models, we would want to show that:

$$\mathbb{E}\left[\frac{S_t}{\beta_t}|\mathcal{F_0}\right]=\frac{S_0}{\beta_0}$$

Model a:

$$\frac{S_0}{\beta_0}=\frac{1+B_0}{e^0}=1$$

$$\mathbb{E}\left[\frac{S_t}{\beta_t}|\mathcal{F_0}\right]=\mathbb{E}[e^{-t}+e^{-t}B_t|\mathcal{F_0}]=e^{-t}+e^{-t}\mathbb{E}[B_t|B_0]=e^{-t}\neq1$$

Therefore this model admits arbitrage as you've rightly pointed out.

Model b:

$$\frac{S_0}{\beta_0}=1$$

$$\mathbb{E}\left[\frac{S_t}{\beta_t}|\mathcal{F_0}\right]=\mathbb{E}\left[\frac{e^{t+\int_0^thdB_h}}{e^t}|\mathcal{F_0}\right]=\mathbb{E}\left[e^{\int_0^thdB_h}|\mathcal{F_0}\right]$$

Now by Ito formula, $\int_{h=0}^{h=t}hdB_h=tB_t-\int_{h=0}^{h=t}B_hdh$, and this quantity is Normally distributed with expectation of zero and variance of $\frac{1}{3}t^3$ (by Ito Isometry, see end of post*). So we know that $e^{\int_0^thdB_h}$ is log-normally distributed, and we can write:

$$\mathbb{E}\left[e^{\int_0^hhdW_h}\right]=e^{0+0.5*\frac{1}{3}t^3}=e^{\frac{1}{6}t^3}\neq1$$

So again, this model is not arbitrage-free.

Model c:

$$\frac{S_0}{\beta_0}=1$$

By Ito formula, $\int_{h=0}^{h=t}B_hdh=tB_t-\int_{h=0}^{h=t}hdB_h$. This has expectation of zero again and variance is again $\frac{1}{3}t^3$ (see here), so that:

$$\mathbb{E}\left[e^{\int_{h=0}^{h=t}B_hdh}\right]=e^{\frac{1}{6}t^3}\neq1$$

So this model is not arbitrage-free.

*Ito Isometry states that, for any adapted Stochastic process $X_t$:

$$\mathbb{E}\left[\left(\int_{h=0}^{h=t}X_hdB_h\right)^2\right]=\mathbb{E}\left[\int_{h=0}^{h=t}X_h^2dh\right]$$

So we have ($X_t=t$):

$$\mathbb{E}\left[\left(\int_{h=0}^{h=t}hdB_h\right)^2\right]=\mathbb{E}\left[\int_{h=0}^{h=t}h^2dh\right]=\left[\frac{1}{3}h^3\right]_{h=0}^{h=t}=\frac{1}{3}t^3$$

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  • $\begingroup$ Shouldn't we also check this condition under stock numeraire?in a) the stock cant be numeraire because it can be negative but I wonder for b) and c) $\endgroup$
    – emot
    Aug 4 '21 at 15:20
  • $\begingroup$ @emot We've shown the model admits arbitrage under the money market numeraire in all 3 cases, no need to check also for the stock Numeraire. $\endgroup$ Aug 4 '21 at 16:16
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    $\begingroup$ For the models to be arbitrage-free, the discounted stock price should be a martingale under the risk-neutral measure. By the way the problem is framed, it seems to me that the market models have been provided under the physical measure. This explains the difference between this answer and the other answer. $\endgroup$ Aug 4 '21 at 16:59
  • $\begingroup$ @Jose Avilez isn't it possible that the model was stated under measure with stock as numeraire therefore $\frac{B_t}{S_t}$ will be a martingale, but $\frac{S_t}{B_t}$ won't be a martingale in that measure? When we check if only $\frac{S_t}{B_t}$ is a martingale, we are implying that dynamics were specified under bank account numeraire. $\endgroup$
    – emot
    Aug 4 '21 at 17:26
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    $\begingroup$ @Jose Avilez I know that there is misunderstanding and the task was to find risk neutral measure. I initially assumed that the models were stated in equivalent martingale measure and the task was to verify if that was correct. Therefore it is possible that the models were defined such that $\frac{B_t}{S_t}$ is martingale i.e. under equilvalent martingale measure with stock numeraire. Then veryfing if $\frac{S_t}{B_t}$ is a martingale is not enough, because under that measure, only $\frac{B_t}{S_t}$ is a martingale. $\endgroup$
    – emot
    Aug 4 '21 at 18:48

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