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# 统计代写|鞅论代写Martingale Theory代考|SF2971 The Frictionless Market Assumption

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## 统计代写|鞅论代写Martingale Theory代考|The Frictionless Market Assumption

For subsequent usage, a market is defined to be collection $((\mathbb{B}, \mathbb{S}), \mathbb{F}, \mathbb{P})$ representing the stochastic processes for the traded assets, the market’s information set, and the statistical probability measure. The underlying state space and $\sigma$-algebra, $(\Omega, \mathscr{F})$, are always implicit in this collection and not included. A market is always assumed to be frictionless, with the exception of the admissibility condition (unless otherwise indicated), and competitive.

The assumption of frictionless markets implicitly appears in the definition of the set of admissible s.f.t.s. $\mathscr{A}(x)$ in that, except for the admissibility constraint, the accumulated value of the trading strategy has no additional adjustments for other frictions, e.g. transaction costs, taxes, indivisible shares, explicit short sale constraints, or margin requirements. As noted previously, the admissibility condition is, in fact, a trading constraint imposed on the aggregate value of all shorts in the s.f.t.s.

The assumption of competitive markets also implicitly appears in the definition of $\mathscr{A}(x)$ because the price processes $\left(\mathbb{B}_u, \mathbb{S}_u\right)$ do not depend on the trading strategy $\left(\alpha_0, \alpha\right)$. The trader is a price-taker because there is no quantity impact on the price processes from trading the shares $\left(\alpha_0, \alpha\right)$.

Although a misnomer, the convention in the literature is to still call a market frictionless if the only restriction imposed is the admissibility condition. Because admissibility is needed to exclude doubling strategies, its imposition is thought to be very mild. It is also standard in portfolio optimization problems, in the context of a frictionless market, to impose an analogous constraint that a trader’s wealth is always nonnegative (see Part II of this book).

For the remainder of the book, a market is always assumed to be frictionless in the sense just discussed. It is important to keep this misnomer in mind when using the phrase “frictionless markets” in the subsequent models. Doing so one can more easily understand why asset price bubbles often exist as an implication of the model. For example, in the asset price bubbles Chap. 3 this clarifies why asset price bubbles exist in a frictionless and competitive market where there are no arbitrage opportunities (to be defined). Second, in the portfolio optimization Chaps. 10-12, this clarifies how an optimal wealth and consumption path can exist in the presence of asset price bubbles in a frictionless and competitive market. And finally, in Chaps. 13-16 that study economic equilibrium, this also clarifies how asset price bubbles can exist in a frictionless and competitive market rational equilibrium.

## 统计代写|鞅论代写Martingale Theory代考|Change of Numeraire

Normalization by the money market account, which is a change of numeraire, simplifies the notation and is almost without loss of generality. The lost of generality is that the set of trading strategies $\mathscr{A}(x)$ after the change of numeraire may differ from the set of trading strategies before due to the modified integrability conditions needed to guarantee that the relevant integrals exist. This section presents the new notation and the evolutions for the mma and the risky assets under this change of numeraire.

Let $B_t=\frac{\mathbb{B}_t}{\mathbb{B}_t}=1$ for all $t \geq 0$, this represents the normalized value of the money market account (mma).

Let $S_t=\left(S_1(t), \ldots, S_n(t)\right)^{\prime} \geq 0$ represent the risky asset prices when normalized by the value of the mma, i.e. $S_i(t)=\frac{\mathbb{S}_i(t)}{\mathbb{B}_t}$. Then,
$$\begin{gathered} \frac{d B_t}{B_t}=0 \quad \text { and } \ \frac{d S_t}{S_t}=\frac{d \mathbb{S}_t}{\mathbb{S}_t}-r_t d t \end{gathered}$$
Proof Using the integration by parts formula Theorem 3 in Chap. 1, one obtains (dropping the t’s)
$$d\left(\frac{\mathbb{S}}{\mathbb{B}}\right)=\frac{1}{\mathbb{B}} d \mathbb{S}+\mathbb{S} d\left(\frac{1}{\mathbb{B}}\right)=\frac{d \mathbb{S}}{\mathbb{S}} \mathbb{B}-\frac{\mathbb{S}}{\mathbb{B}} \frac{d \mathbb{B}}{\mathbb{B}} .$$
The first equality uses $d\left[\mathbb{S}, \frac{1}{\mathbb{B}}\right]=0$, since $\mathbb{B}$ is continuous and of finite variation (use Lemmas 2 and 7 in Chap. 1).
Substitution yields
$d S=\frac{d S}{S} S-S \frac{d \mathbb{B}}{\mathbb{B}}$. Algebra completes the proof.

## 统计代写|鞅论代写Martingale Theory代考|Change of Numeraire

$$\frac{d B_t}{B_t}=0 \quad \text { and } \quad \frac{d S_t}{S_t}=\frac{d \mathbb{S}_t}{\mathbb{S}_t}-r_t d t$$

$$d\left(\frac{\mathbb{S}}{\mathbb{B}}\right)=\frac{1}{\mathbb{B}} d \mathbb{S}+\mathbb{S} d\left(\frac{1}{\mathbb{B}}\right)=\frac{d \mathbb{S}}{\mathbb{S}} \mathbb{B}-\frac{\mathbb{S}}{\mathbb{B}} \frac{d \mathbb{B}}{\mathbb{B}} .$$

$d S=\frac{d S}{S} S-S \frac{d \mathbb{B}}{1 \mathrm{~B}}$. 代数完成了证明。

## MATLAB代写

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