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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|MATH4511 Problem Formulation

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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|Problem Formulation

The main project in this book consists in studying theoretical pricing models for those financial assets which are known as financial derivatives. Before we give the formal definition of the concept of a financial derivative we will, however, by means of a concrete example, introduce the single most important example: the European call option.

Let us thus consider the Swedish company $C \& H$, which today (denoted by $t=0)$ has signed a contract with an American counterpart ACME. The contract stipulates that $A C M E$ will deliver 1000 computer games to $C E H$ exactly six months from now (denoted by $t=T$ ). Furthermore it is stipulated that $C \& H$ will pay 1000 US dollars per game to $A C M E$ at the time of delivery (i.e. at $t=T$ ). For the sake of the argument we assume that the present spot currency rate between the Swedish krona (SEK) and the US dollar is 8.00 SEK/ $\$ \$$One of the problems with this contract from the point of view of C \& H is that it involves a considerable currency risk. Since C E B H does not know the currency rate prevailing six months from now, this means that it does not know how many SEK it will have to pay at t=T. If the currency rate at t=T is still 8.00 \mathrm{SEK} / \$$ it will have to pay $8,000,000 \mathrm{SEK}$, but if the rate rises to, say, $8.50$ it will face a cost of $8,500,000$ SEK. Thus $C \& B H$ faces the problem of how to guard itself against this currency risk, and we now list a number of natural strategies.

1. The most naive stratgey for $C E H$ is perhaps that of buying $\$ 1,000,000$today at the price of$8,000,000$SEK, and then keeping this money (in a Eurodollar account) for six months. The advantage of this procedure is of course that the currency risk is completely eliminated, but there are also some drawbacks. First of all the strategy above has the consequence of tying up a substantial amount of money for a long period of time, but an even more serious objection may be that$C \& B H$perhaps does not have access to$8,000,000$SEK today. 2. A more sophisticated arrangement, which does not require any outlays at all today, is that$C E H$goes to the forward market and buys a forward contract for$\$1,000,000$ with delivery six months from now. Such a contract may, for example, be negotiated with a commercial bank, and in the contract two things will be stipulated.
• The bank will, at $t=T$, deliver $\$ 1,000,000$to$C E H H$. •$C \mathcal{E} H$will, at$t=T$, pay for this delivery at the rate of$K$SEK$/ \$$. ## 数学代写|连续时间的期权定价理论代写Arbitrage Pricing in Continuous Time代考|The One Period Model We start with the one period version of the model. In the next section we will (easily) extend the model to an arbitrary number of periods. 2.1.1 Model Description Running time is denoted by the letter t, and by definition we have two points in time, t=0 (“today”) and t=1 (“tomorrow”). In the model we have two assets: a bond and a stock. At time t the price of a bond is denoted by B_t, and the price of one share of the stock is denoted by S_t. Thus we have two price processes B and S. The bond price process is deterministic and given by$$
\begin{aligned}
&B_0=1 \
&B_1=1+R .
\end{aligned}
$$The constant R is the spot rate for the period, and we can also interpret the existence of the bond as the existence of a bank with R as its rate of interest. The stock price process is a stochastic process, and its dynamical behaviour is described as follows.$$
\begin{aligned}
&S_0=s, \
&S_1= \begin{cases}s \cdot u, & \text { with probability } p_u . \
s \cdot d, & \text { with probability } p_d .\end{cases}
\end{aligned}
$$It is often convenient to write this as$$
\left{\begin{array}{l}
S_0=s, \
S_1=s \cdot Z,
\end{array}\right.
$$where Z is a stochastic variable defined as$$
Z= \begin{cases}u, & \text { with probability } p_u . \ d, & \text { with probability } p_d\end{cases}
$$We assume that today’s stock price s is known, as are the positive constants u, d, p_u and p_d. We assume that d<u, and we have of course p_u+p_d=1. We can illustrate the price dynamics using the tree structure in Fig. 2.1. ## 连续时间的期权定价理论代写 ## 数学代写|连续时间的期权定价理论代写ARBITRAGE PRICING IN CONTINUOUS TIME代考|PROBLEM FORMULATION 本书的主要项目是研究那些被称为金融衍生品的金融资产的理论定价模型。然而，在我们给出金融衍生品概念的正式定义之前，我们将通过一个具体的例子，介绍 一个最重要的例子：欧式看涨期权。 让我们考虑一下这家瑞典公司 C \& H, 今天denotedby \ t=0 hassignedacontractwithanAmericancounterpartACME. Thecontractstipulatesthat ACME willdeliver 1000 computergamestoCEHexactlysixmonths fromnow (denotedbyt- \mathrm{T} ). Furthermoreitisstipulatedthat \mathrm{C} \& \mathrm{A}. willpay 1000 USdollarspergametoACMEatthetimeofdelivery ( i. e. at \mathrm{t}=\mathrm{T} ). ForthesakeoftheargumentweassumethatthepresentspotcurrencyratebetweentheSwedishkrona ( SEK) andtheU Sdollaris 8.00 S E K / \ \$$

S_0=s, S_1=s \cdot Z,
$$ไ正确的。 \$$

$Z=\left{u, \quad\right.$ with probability $p_u \cdot d, \quad$ with probability $p_d$

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