随机微积分作业代写stochastic calculus代考| PRICING OF CONTINGENT CLAIMS

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微积分作业代写calclulus代考|Derivatives of securities paying dividends

2.c Derivatives of securities paying dividends. So far we have assumed that the stock $S$ does not pay a dividend. This was important in the definition of a selffinancing trading strategy, which in turn is the basis for arbitrage arguments. When considering opportunities for riskless profit the dividend stream accruing to the holder of an asset paying dividends must be taken into account.

Let us assume now that the asset $S$ pays a dividend continuously at the yield $q(t)$ assumed to be a deterministic function of time. In other words we assume that the dividend process $D_{t}$ satisfies
$$d D_{t}=q(t) S_{t} d t .$$
Here $D_{t}$ is to be interpreted as the total amount of dividend received up to time $t$. An example of such an asset is a foreign currency. In this case $q(t)$ is the foreign risk-free rate.

A trading strategy $\phi_{t}=\left(K_{t}, H_{t}\right)$ holding $K_{t}$ units of the riskless bond and $H_{t}$ units of the dividend paying stock $S$ at time $t$ will now be called sel $f$-financing, if the corresponding portfolio price process $V_{t}(\phi)=K_{t} B_{t}+H_{t} S_{t}$ satisfies
$$d V_{t}(\phi)=K_{t} d B_{t}+H_{t} d S_{t}+q(t) H_{t} S_{t} d t$$
Set $Q(t, T)=\int_{t}^{T} q(s) d s, A_{t}=e^{-Q(t, T)}$ and $S_{t}^{1}=A_{t} S_{t}=S_{t} e^{-Q(t, T)}$. Then $A_{t}$ is a nonstochastic, bounded variation process satisfying $d A_{t}=q(t) A_{t} d t$ and $A_{T}=1$. Using the stochastic product rule and 1.a.eq.(1) we have
\begin{aligned} d S_{t}^{1} &=A_{t} d S_{t}+S_{t} d A_{t}=A_{t} d S_{t}+q(t) A_{t} S_{t} d t \ &=A_{t} S_{t}\left[\mu(t) d t+\sigma(t) d W_{t}\right]+q(t) A_{t} S_{t} d t \ &=S_{t}^{1}\left[(\mu(t)+q(t)) d t+\sigma(t) d W_{t}\right] . \end{aligned}
Thus $S_{t}^{1}$ follows a geometric Brownian motion with time dependent nonstochastic parameters. We can thus view $S_{t}^{1}$ as the price process of a new stock. Let $\psi_{t}=$ $\left(K_{t}, H_{t} / A_{t}\right)$ be a trading strategy taking positions in the bond and the new stock $S_{t}^{1}$. The corresponding price process $V_{t}(\psi)$ satisfies
$$V_{t}(\psi)=K_{t} B_{t}+\left(H_{t} / A_{t}\right) S_{t}^{1}=K_{t} B_{t}+H_{t} S_{t}=V_{t}(\phi)$$
that is, $\phi$ and $\psi$ have the same price process. From $(0)$ it follows that $d S_{t}^{1}=$ $A_{t}\left(d S_{t}+q(t) S_{t} d t\right)$. If $\phi$ is self-financing, then
\begin{aligned} d V_{t}(\psi) &=d V_{t}(\phi)=K_{t} d B_{t}+H_{t}\left(d S_{t}+q(t) S_{t} d t\right) \ &=K_{t} d B_{t}+\left(H_{t} / A_{t}\right) d S_{t}^{1} . \end{aligned}
This is the self-financing condition for the trading strategy $\psi$ if the new stock $S_{t}^{1}$ is assumed to pay no dividend. Let us call the asset $S^{1}$ the dividend-free reduction of $S$. Then we have seen:

微积分作业代写calclulus代考|Equivalent martingale measure

In the case of a dividend-free asset the equivalent martingale measure $P_{B}$ made the discounted price process of the asset into a martingale with the crucial consequence being that the discounted price process of a self-financing trading strategy $\phi$ is a $P_{B^{-}}$local martingale (and a martingale under suitable, not very restrictive conditions on the coefficients $K_{t}, H_{t}$ of $\phi$ ). Thus an appropriate notion for an equivalent martingale measure in the case of a dividend paying asset is as an equivalent probability with respect to which the discounted price processes of self-financing trading strategies taking positions in this asset are local martingales.

Because of the equality $V_{t}^{B}(\phi)=V_{t}^{B}(\psi)$, where $\phi$ and $\psi$ are as above, this will be true of an equivalent martingale measure for the dividend-free reduction $S^{1}$ of $S$. Since the price process $S_{t}^{1}$ satisfies
$$d S_{t}^{1}=S_{t}^{1}\left((\mu(t)+q(t)) d t+\sigma(t) d W_{t}\right)$$
this equivalent martingale measure $P_{B}$ replaces the yield $\mu(t)+q(t)$ of the dividendfree reduction $S_{t}^{1}$ of $S_{t}$ with the risk-free yield $r(t)$. Equivalently, $P_{B}$ replaces the yield $\mu(t)$ of $S_{t}$ with the difference $r(t)-q(t)$.
With this notion of equivalent martingale measure $P_{B}$ the forward price $F$ of $S$ for delivery at time $T$ can again be written as
$$F_{t}=F_{t}^{1}=E_{P_{B}}\left[S_{T}^{1} \mid \mathcal{F}{t}\right]=E{P_{B}}\left[S_{T} \mid \mathcal{F}{t}\right],$$ exactly as in the dividend-free case (recall $S{T}^{1}=S_{T}$ ).
Put call parity. Since European puts and calls on $S$ have the same price processes as the corresponding puts and calls on $S^{1}$, the put call parity assumes the form $S_{t}^{1}+P_{t}-C_{t}=K e^{-R(t, T)}$, that is,
$$C_{t}-P_{t}=S_{t} e^{-Q(t, T)}-K e^{-R(t, T)}$$

微积分作业代写calclulus代考|Derivatives of securities paying dividends

2.c 支付股息的证券衍生品。到目前为止，我们假设股票小号不支付股息。这在定义自筹资金交易策略时很重要，而这反过来又是套利论点的基础。在考虑获得无风险利润的机会时，必须考虑到资产持有人支付股息的股息流。

dD吨=q(吨)小号吨d吨.

d五吨(φ)=到吨d乙吨+H吨d小号吨+q(吨)H吨小号吨d吨

d小号吨1=一种吨d小号吨+小号吨d一种吨=一种吨d小号吨+q(吨)一种吨小号吨d吨 =一种吨小号吨[μ(吨)d吨+σ(吨)d在吨]+q(吨)一种吨小号吨d吨 =小号吨1[(μ(吨)+q(吨))d吨+σ(吨)d在吨].

d五吨(ψ)=d五吨(φ)=到吨d乙吨+H吨(d小号吨+q(吨)小号吨d吨) =到吨d乙吨+(H吨/一种吨)d小号吨1.

微积分作业代写calclulus代考|Equivalent martingale measure

d小号吨1=小号吨1((μ(吨)+q(吨))d吨+σ(吨)d在吨)

$$F_{t}=F_{t}^{1}=E_{P_{B}}\left[S_{T}^{1} \mid \mathcal{F} {t}\ right]=E {P_{B}}\left[S_{T} \mid \mathcal{F} {t}\right],$$ 与无股息情况完全相同（回忆 $S {T}^{ 1}=S_{T}).磷你吨C一种一世一世p一种r一世吨和.小号一世nC和和你r○p和一种np你吨s一种ndC一种一世一世s○n小号H一种v和吨H和s一种米和pr一世C和pr○C和ss和s一种s吨H和C○rr和sp○nd一世nGp你吨s一种ndC一种一世一世s○nS^{1},吨H和p你吨C一种一世一世p一种r一世吨和一种ss你米和s吨H和F○r米S_{t}^{1}+P_{t}-C_{t}=K e^{-R(t, T)},吨H一种吨一世s,C吨−磷吨=小号吨和−问(吨,吨)−到和−R(吨,吨)$