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# 金融代写|金融工程代写FINANCIAL ENGINEERING代写|CMSE11471 Time Value of Options

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## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Time Value of Options

The following convenient terminology is often used. We say that at time $t$ a call option with strike price $X$ is

• in the money if $S(t)>X$,
• at the money if $S(t)=X$,
• out of the money if $S(t)<X$.
Similarly, for a put option we say that it is
• in the money if $S(t)<X$,
• at the money if $S(t)=X$,
• out of the money if $S(t)>X$.
Also convenient, though less precise, are the terms deep in the money and deep out of the money, which mean that the difference between the two sides in the respective inequalities is considerable.

An American option in the money will bring a positive payoff if exercised immediately, whereas an option out of the money will not. We use the same terms for European options, though their meaning is different: Even if the option is currently in the money, it may no longer be so on the exercise date, when the payoff may well turn out to be zero. A European option in the money is no more than a promising asset.

## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Option Pricing

Suppose that for any contingent claim $D(T)$ there exists a replication strategy, that is, an admissible strategy $x(t), y(t)$ with final value $V(T)=D(T)$. Then the price $D(0)$ of the contingent claim at time 0 must be equal to that of the replicating strategy, $V(0)=D(0)$.
Proof
The proof is just a modification of that of Proposition 1.3. If $D(0)>V(0)$, then we write the derivative security and take a long position in the strategy. Our obligation will be covered by the strategy, the difference $D(0)-V(0)$ being our arbitrage profit. If $D(0)<V(0)$, then we take the opposite positions, with $V(0)-D(0)$ the resulting arbitrage profit.

Replication also solves the problem of hedging the position of the option writer. If the cash received for the option is invested in the replicating strategy, then all the risk involved in writing the option will be eliminated.

In this chapter we shall gradually develop such pricing methods for options, starting with a comprehensive analysis of the one-step binomial model, which will then be extended to a multi-step model. Finally, the Black-Scholes formula in continuous time will be introduced.

## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Time Value of Options

• 如果在钱小号(吨)>X,
• 钱如果小号(吨)=X,
• 如果没钱小号(吨)<X.
同样，对于看跌期权，我们说它是
• 如果在钱小号(吨)<X,
• 钱如果小号(吨)=X,
• 如果没钱小号(吨)>X.
同样方便但不太精确的还有钱深和钱外这两个术语，这意味着两个方面在各自的不等式上的差异是相当大的。

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