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# 金融代写|金融工程代写FINANCIAL ENGINEERING代写|FIN285 Forward Contracts

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## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Forward Contracts

A forward contract is an agreement to buy or sell an asset on a fixed date in the future, called the delivery time, for a price specified in advance, called the forward price. The party to the contract who agrees to sell the asset is said to be taking a short forward position. The other party, obliged to buy the asset at delivery, is said to have a long forward position. The principal reason for entering into a forward contract is to become independent of the unknown future price of a risky asset. There are a variety of examples: a farmer wishing to fix the sale price of his crops in advance, an importer arranging to buy foreign currency at a fixed rate in the future, a fund manager who wants to sell stock for a price known in advance. A forward contract is a direct agreement between two parties. It is typically settled by physical delivery of the asset on the agreed date. As an alternative, settlement may sometimes be in cash.

Let us denote the time when the forward contract is exchanged by 0 , the delivery time by $T$, and the forward price by $F(0, T)$. The time $t$ market price of the underlying asset will be denoted by $S(t)$. No payment is made by either party at time 0 , when the forward contract is exchanged. At delivery the party with a long forward position will benefit if $F(0, T)S(T)$, then the situation will be reversed. The payoffs at delivery are $S(T)-F(0, T)$ for a long forward position and $F(0, T)-S(T)$ for a short position; see Figure 6.1.

If the contract is initiated at time $t<T$ rather than 0 , then we shall write $F(t, T)$ for the forward price, the payoff at delivery being $S(T)-F(t, T)$ for a long forward position and $F(t, T)-S(T)$ for a short position.

## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Forward Price

The No-Arbitrage Principle makes it possible to obtain formulae for the forward prices of assets of various kinds. We begin with the simplest case.

Stock Paying No Dividends. Consider a security that can be stored at no cost and brings no profit (except perhaps for capital gains arising from random price fluctuations). A typical example is a stock paying no dividends. We shall denote by $r$ the risk-free rate under continuous compounding and assume that it is constant throughout the period in question.

An alternative to taking a long forward position in stock with delivery at time $T$ and forward price $F(0, T)$ is to borrow $S(0)$ dollars to buy the stock at time 0 and keep it until time $T$. The amount $S(0) \mathrm{e}^{r T}$ to be paid to settle the loan with interest at time $T$ is a natural candidate for the forward price $F(0, T)$. The following theorem makes this intuitive argument formal.
Theorem $6.1$
For a stock paying no dividends the forward price is
$$F(0, T)=S(0) \mathrm{e}^{r T},$$
where $r$ is a constant risk-free interest rate under continuous compounding. If the contract is initiated at time $t \leq T$, then
$$F(t, T)=S(t) \mathrm{e}^{r(T-t)}$$

Proof
We shall prove formula (6.1). Suppose that $F(0, T)>S(0) \mathrm{e}^{r T}$. In this case, at time 0

• borrow the amount $S(0)$ until time $T$;
• buy one share for $S(0)$;
• take a short forward position, that is, agree to sell one share for $F(0, T)$ at time $T$.
Then, at time $T$
• sell the stock for $F(0, T)$;
• pay $S(0) \mathrm{e}^{r T}$ to clear the loan with interest.
This will bring a risk-free profit of
$$F(0, T)-S(0) \mathrm{e}^{r T}>0,$$
contrary to the No-Arbitrage Principle. Next, suppose that $F(0, T)<S(0) \mathrm{e}^{r T}$.

## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Forward Price

$$F(0, T)=S(0) \mathrm{e}^{r T},$$

$$F(t, T)=S(t) \mathrm{e}^{r(T-t)}$$

• 借入金额 $S(0)$ 直到时间 $T$;
• 购买一股 $S(0) ;$
• 做空远期头寸，即同意出售一股 $F(0, T)$ 有时 $T$.
那么，一时 $T$
• 卖出股票 $F(0, T)$ ；
• 支付 $S(0) \mathrm{e}^{r T}$ 用利息清算㑆款。
这将带来无风险的利润
$$F(0, T)-S(0) \mathrm{e}^{r T}>0$$
违背无套利原则。接下来，假设 $F(0, T)<S(0) \mathrm{e}^{r T}$.

## MATLAB代写

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