Posted on Categories:Arbitrage Pricing in Continuous Time, 数学代写, 连续时间的期权定价理论

# 数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|MATH4511 Portfolios and Arbitrage

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## 数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|Portfolios and Arbitrage

We will study the behaviour of various portfolios on the $(B, S)$ market, and to this end we define a portfolio as a vector $h=(x, y)$. The interpretation is that $x$ is the number of bonds we hold in our portfolio, whereas $y$ is the number of units of the stock held by us. Note that it is quite acceptable for $x$ and $y$ to be positive as well as negative. If, for example, $x=3$, this means that we have bought three bonds at time $t=0$. If on the other hand $y=-2$, this means that we have sold two shares of the stock at time $t=0$. In financial jargon we have a long position in the bond and a short position in the stock. It is an important assumption of the model that short positions are allowed.
Assumption 2.1.1 We assume the following institutional facts.

• Short positions, as well as fractional holdings, are allowed. In mathematical terms this means that every $h \in R^2$ is an allowed portfolio.
• There is no bid-ask spread, i.e. the selling price is equal to the buying price of all assets.
• There are no transactions costs of trading.
• The market is completely liquid, i.e. it is always possible to buy and/or sell unlimited quantities on the market. In particular it is possible to borrow unlimited amounts from the bank (by selling bonds short).

## 数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|Contingent Claims

Let us now assume that the market in the preceding section is arbitrage free. We go on to study pricing problems for contingent claims.

Definition 2.7 A contingent claim (financial derivative) is any stochastic variable $X$ of the form $X=\Phi(Z)$, where $Z$ is the stochastic variable driving the stock price process above.

We interpret a given claim $X$ as a contract which pays $X$ SEK to the holder of the contract at time $t=1$. See Fig. 2.2, where the value of the claim at each node is given within the corresponding box. The function $\Phi$ is called the contract function. A typical example would be a European call option on the stock with strike price $K$. For this option to be interesting we assume that $s dK$ then we use the option, pay $K$ to get the stock and then sell the stock on the market for $s u$, thus making a net profit of $s u-K$. If $S_1<K$ then the option is obviously worthless. In this example we thus have
$$X= \begin{cases}s u-K, & \text { if } Z=u, \ 0, & \text { if } Z=d,\end{cases}$$
and the contract function is given by
\begin{aligned} &\Phi(u)=s u-K, \ &\Phi(d)=0 . \end{aligned}

## 数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代 考|Contingent Claims

$$X={s u-K, \quad \text { if } Z=u, 0, \quad \text { if } Z=d,$$

$$\Phi(u)=s u-K, \quad \Phi(d)=0 .$$

## MATLAB代写

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