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Our delta-hedging strategy was a bit too simple as it consists only in buying or selling the asset at $t=0$ and holding it until the maturity. A more involved strategy is to buy (or sell) units of the asset at a date $t_{k}$ until a next date $t_{k+1}$. Let us compute the value of our delta-hedged portfolio at the maturity $T$. At $t_{k}$, the portfolio value $\pi_{t_{k}}$ is
$$\pi_{t_{k}}=\left(\pi_{t_{k}}-H_{t_{k}} S_{t_{k}}\right)+H_{t_{k}} S_{t_{k}}$$
where $H_{t_{k}}$ is the number of shares held at time $t_{k}$. Although this expression seems algebraically trivial, its financial interpretation is important: the term $H_{t_{k}} S_{t_{k}}$ is the value at $t_{k}$ of a position consisting of $H_{t_{k}}$ units of the asset. The term $\pi_{t_{k}}-H_{t_{k}} S_{t_{k}}$ represents the cash part invested in a bank account. The variation of our portfolio between $t_{k}$ and $t_{k+1}$ is then
\begin{aligned} \delta \pi_{t_{k}} &=\left(\pi_{t_{k}}-H_{t_{k}} S_{t_{k}}\right) r \delta t+H_{t_{k}} \delta S_{t_{k}} \ &=\pi_{t_{k}} r \delta t+H_{t_{k}}\left(\delta S_{t_{k}}-S_{t_{k}} r \delta t\right) \end{aligned}
with $\delta S_{t_{k}} \equiv S_{t_{k+1}}-S_{t_{k}}, \delta t=t_{k+1}-t_{k}$ small enough. As no cash is injected between $t_{k}$ and $t_{k+1}$, our portfolio is called self-financing. By setting $\bar{\pi}{t{k}} \equiv e^{-r t_{k}} \pi_{t_{k}}$ and $\bar{S}{t{k}} \equiv e^{-r t_{k}} S_{t_{k}}$, we obtain the variation of the discounted portfolio
$$\delta \tilde{\pi}{t{k}}=H_{t_{k}} \delta \tilde{S}{t{k},}, \delta \tilde{S}{t{k}} \equiv \tilde{S}{t{k+1}}-\tilde{S}{t{k}}$$
Here the state of information evolves over time and is described by a filtration $\mathcal{F}=\left(\mathcal{F}{t{1}}, \ldots, \mathcal{F}{t{n}}\right)$ where the $\sigma$-algebra $\mathcal{F}_{t}$ is the set of events that will be known to be true or false. We take here $\mathcal{F}{t{k}}=\sigma\left(S_{0}, \ldots, S_{t_{k}}\right)$ the natural filtration. $H_{k}=H_{k}\left(S_{0}, \ldots, S_{t_{k}}\right)$ is adapted, i.e., a measurable function with respect to $\mathcal{F}{t{k}}$ : we don’t look into the future. If we now assume that the trader sells an option with payoff $F_{T}$ at the price $C$ at $t=0$ and then delta-hedges his position at the intermediate dates $t_{0} \equiv 0<t_{1}<\ldots<t_{n} \equiv T$, we get
$$e^{-r T} \pi_{T}=-e^{-r T} F_{T}+C+\sum_{k=0}^{n-1} H_{t_{k}}\left(S_{0}, \ldots, S_{t_{k}}\right) \delta \bar{S}{t{k}}$$
By playing the same game as in Theorem 1.1, we obtain the dual expression:

## 统计代写|离散时间鞅理论代写martingale代考|Black–Scholes replication

Here we assume some familiarity with stochastic analysis. However, this section is not needed for the rest of the book and therefore can be skipped (see however the expression of the Black-Scholes formula). We consider that $S_{t}$ is modeled by a log-normal process under $\mathbb{P}^{\text {hist. }}$ :
$$\frac{d S_{t}}{S_{t}}=\mu d t+\sigma d W_{t}^{\mathrm{P}^{\text {hint }}}$$
$\mathcal{M}{\infty}$ corresponds to the set of $\mathbb{Q}$-martingale measure equivalent to Phist $^{\text {. From }}$ the Girsanov theorem (see e.g. $[130]), \mathcal{M}{\infty}$ reduces to a singleton $\left{\mathbb{Q}^{\mathrm{BS}}\right}$ under which
$$\frac{d S_{t}}{S_{t}}=r d t+\sigma d W_{t}^{\mathrm{Q}^{\mathrm{BS}}}$$ We conclude that there is a unique arbitrage-free price (independent of $\mu$ compare with formula (1.6)):
$$C=\mathbb{E}^{\mathrm{Q}^{\mathrm{BS}}}\left[e^{-r T} F_{T}\right]$$
We deduce also that the payoff can be dynamically hedged:
$$-e^{-r T} F_{T}+C+\int_{0}^{T} \partial_{S_{t}} \mathbb{E}^{\mathbb{Q}^{\mathrm{BS}}}\left[e^{-r(T-t)} F_{T} \mid S_{t}\right] d \tilde{S}{t}=0, \quad \text { Phist }^{-a . s .}$$ Note that for a call payoff $F{T}=\left(S_{T}-K\right)^{+}$, we obtain the Black-Scholes formula.

We assume that $T$-Vanilla options on each asset are traded on the market. They are specified by a payoff $\lambda\left(S_{T}\right)$ at a maturity $T$. In practice, these Vanilla payoffs can be replicated by holding a strip of put/call $T$-Vanillas through the Taylor expansion formula [38]:
\begin{aligned} \lambda\left(S_{T}\right)=\lambda\left(S_{0}\right)+\lambda^{\prime}\left(S_{0}\right)\left(S_{T}-S_{0}\right) &+\int_{0}^{S_{\mathrm{a}}} \lambda^{\prime \prime}(K)\left(K-S_{T}\right)^{+} d K \ &+\int_{S_{0}}^{\infty} \lambda^{\prime \prime}(K)\left(S_{T}-K\right)^{+} d K \end{aligned}
where $\left(K-S_{T}\right)^{+}\left(\right.$resp. $\left.\left(S_{T}-K\right)^{+}\right)$is the payoff of a put (resp. call). Derivatives $\lambda^{\prime \prime}(K)$ are understood in the distribution sense. We then assume that the pricing operator $\Pi[\cdot]$ (used by market operators to value Vanillas) is linear meaning that
$$\Pi\left[\sum_{i} \lambda_{i}\left(S_{T}-K_{i}\right)^{+}\right]=\sum_{i} \lambda_{i} \Pi\left[\left(S_{T}-K_{i}\right)^{+}\right]$$

Moreover, from the no-arbitrage condition, we should have that
$$\Pi[1]=e^{-r T}, \quad \Pi\left[S_{T}\right]=S_{0}$$
Also, still from the no-arbitrage condition, $\Pi\left[\left(S_{T}-K\right)^{+}\right]$should be nonincreasing, convex with respect to $K$ and $\Pi\left[\left(S_{T}-K\right)^{+}\right] \geq\left(S_{0}-K e^{-r T}\right)^{+}$. From Riesz’s representation theorem (with the additional requirement that the market price of a call option with strike $K$ goes to 0 as $K \rightarrow \infty$ ), this implies that there exists a probability $\mathbb{P}^{m k t}$ such that
$$C(K) \equiv \Pi\left[\left(S_{T}-K\right)^{+}\right]=\mathbb{E}^{\mathbb{P}^{\mathrm{mkt}}}\left[e^{-r T}\left(S_{T}-K\right)^{+}\right]$$
with $\mathbb{E}^{\text {prkt }}\left[e^{-r T} S_{T}\right]=S_{0}$.
Below and in the rest of the book, for the sake of simplicity, we take $r=0$. This can be easily relaxed by including in the formulas below a multiplicative factor $e^{-r T}$.

From the linear property, the market price of the payoff $\lambda\left(S_{T}\right)$, inferred from market prices of put/call options, is
\begin{aligned} \Pi\left[\lambda\left(S_{T}\right)\right]=\mathbb{E}^{\mathrm{P}^{\mathrm{mkt}}}\left[\lambda\left(S_{T}\right)\right] &=\lambda\left(S_{0}\right)+\int_{0}^{S_{0}} \lambda^{\prime \prime}(K) \mathbb{E}^{\mathbb{P}^{\mathrm{mkt}}}\left[\left(K-S_{T}\right)^{+}\right] d K \ &+\int_{S_{0}}^{\infty} \lambda^{\prime \prime}(K) \mathbb{E}^{\mathrm{P}^{\mathrm{mkt}}}\left[\left(S_{T}-K\right)^{+}\right] d K \end{aligned}

## 离散时间鞅理论代考

d圆周率吨ķ=(圆周率吨ķ−H吨ķ小号吨ķ)rd吨+H吨ķd小号吨ķ =圆周率吨ķrd吨+H吨ķ(d小号吨ķ−小号吨ķrd吨)

d圆周率~吨ķ=H吨ķd小号~吨ķ,,d小号~吨ķ≡小号~吨ķ+1−小号~吨ķ

## 统计代写|离散时间鞅理论代写martingale代考|Black–Scholes replication

d小号吨小号吨=μd吨+σd在吨磷暗示

d小号吨小号吨=rd吨+σd在吨问乙小号我们得出结论，存在一个独特的无套利价格（独立于μ与公式（1.6）比较）：

C=和问乙小号[和−r吨F吨]

−和−r吨F吨+C+∫0吨∂小号吨和问乙小号[和−r(吨−吨)F吨∣小号吨]d小号~吨=0, 费斯特 −一个.s.请注意，对于电话收益F吨=(小号吨−ķ)+，我们得到 Black-Scholes 公式。

λ(小号吨)=λ(小号0)+λ′(小号0)(小号吨−小号0)+∫0小号一个λ′′(ķ)(ķ−小号吨)+dķ +∫小号0∞λ′′(ķ)(小号吨−ķ)+dķ

C(ķ)≡圆周率[(小号吨−ķ)+]=和磷米ķ吨[和−r吨(小号吨−ķ)+]

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