# 金融代写|期权理论代写Mathematical Introduction to Options代考|MATH451

## 金融代写|期权理论代写Mathematical Introduction to Options代考|ADAPTATION TO DIFFERENT MARKETS

(i) The objective of this book is to provide the reader with a grounding in option theory, which can be applied to a variety of different markets. Most readers will be interested in one specific market, and it is always easier to read material which is narrowly specific to ones own area of interest, but unfortunately this is not a practicable way to write a book. This section tries to ease the reader’s burden of adapting the material to his own specific area of interest.

In much of the forgoing, the market used to develop the theory was the equity market. This was chosen since it is the most straightforward and widely understandable for newcomers to finance theory: everyone understands what the price of one share of stock means and roughly how dividends work; a futures price or convenience yield is more arcane. Where equity failed to provide an adequate example, as in the discussions of arbitrage or futures, we have turned to other markets such as foreign exchange or commodities. At the risk of some repetition, we now summarize how the theory is adapted to other markets.
(ii) Equities: This is the easiest, since the theory has been developed largely with reference to this market. It is a very straightforward cash market, i.e. the commodity (stock) is purchased directly with physical delivery as soon as possible after purchase. In most established markets there are traded options on the most important stocks, although forwards and futures on single stocks have not yet become established.

This begs the following question: in the absence of a forward market, can we really price an option using the arbitrage arguments of Section 1.2, which were developed for the foreign exchange market with its large forward market which can be used to execute arbitrage trades? The answer is an emphatic yes; foreign exchange was merely used as a simple illustration of the no-arbitrage principle in its various forms. The notion of a forward can be used in pricing an option, even though no formal forward market exists. The arbitrage that is actually performed if an option is mispriced is not buying spot and selling forward, but extracting the option’s fair value through delta hedging.

A formal forward market is not needed to calculate the fair value of an option from the notional forward price; but the delta hedge must exist. In some markets, shorting stock is illegal or restricted to certain categories of market participant, and often stock is just not available for borrowing. This means that positive delta positions (short puts, long calls) cannot be hedged and arbitrage arguments do not apply. The “fair value” is then no more than a hypothetical construction.

The “dividend” $q$ may be different for delta hedging with long or short stock positions. If the stock is held long, $q$ will indeed be the continuous dividend yield; but if the stock is held short, $q$ will be the total cash that needs to be paid out on the short position, i.e. continuous dividend plus stock borrowing cost.

## 金融代写|期权理论代写Mathematical Introduction to Options代考|OPTIONS ON FORWARDS AND FUTURES

(i) Following the last section, we now examine what happens if the underlying security is itself a futures contract. For example, it was seen in the last section that a call option on a stock index could be dynamically hedged by buying or selling the appropriate number of stock index futures contracts; now we consider a call option on a stock index futures price rather than on the index itself.

The analysis is very similar for forward contracts and futures contracts, so these are treated together, with any divergence in behavior pointed out as we go along. Futures contracts are of course far more important in practice, since these are traded on exchanges, while active forward markets are normally interbank (especially in foreign exchange).

It is critical that the reader has a clear understanding of the concepts and notation of paragraph (iii) of the last section.
(ii) The payoff of an option on a futures or
forward contract is more abstract than for

• Options on the Underlying Stock Price: the contract is an option to buy one share of stock at a price $X$.
$$\text { Payoff }=\max \left[\left(S_\tau-X\right), 0\right]$$
• Options on the Forward Price: this is an option that at time $\tau$ we can enter a forward contract maturing at time $T$, at a forward price of $X$. The value of this forward contract at time $\tau$ will be $\left(F_{T \tau}-X\right) \mathrm{e}^{-r(\tau-T)}$.
• $$• \text { Payoff }=\max \left[\left(F_{\tau T}-X\right) \mathrm{e}^{-r(T-\tau)}, 0\right] •$$
• Options on the Futures Price: as in the last case, this is an option to enter a futures contract at time $\tau$ and price $X$; however, futures are marked to market daily so that a profit of $\Phi_{\tau T}-X$ would immediately be realized within one day of time $\tau$.
$$\text { Payoff }=\max \left[\left(\Phi_{\tau T}-X\right), 0\right]$$
iii) The forward price is given by $F_{t T}=S_t \mathrm{e}^{(r-q)(T-t)}$, and if interest rates are constant, we also have $F_{t T}=\Phi_{t T}$. We may therefore write
Volatility of $F_{t T}=$ volatility of $\Phi_{t T}=\sqrt{\operatorname{var}\left\langle\ln S_t\right\rangle}=\sigma$
In general, the volatility of the forward price equals the volatility of the spot price; the volatility of the futures price equals the volatility of the underlying stock if the interest rate is constant.

# 期权理论代考

## 金融代写|期权理论代写期权数学介绍代考|适应不同的市场

“红利”$q$可能是不同的delta对冲与多或空股票头寸。如果股票长期持有，$q$确实会有持续的股息收益;但如果股票被做空，$q$将是需要在做空头寸上支付的现金总额，即持续分红加上股票借款成本。

## 金融代写|期权理论代写期权数学介绍代考| Options ON远期和期货

(ii)期货或远期合约的期权支付比更为抽象

• 标的股票价格期权:该合同是一种以$X$价格购买一股股票的期权。
$$\text { Payoff }=\max \left[\left(S_\tau-X\right), 0\right]$$
• 远期价格期权:该期权是一种在$\tau$时间我们可以进入一份在$T$时间到期的远期合同，远期价格为$X$。这个远期合约在$\tau$时刻的值将是$\left(F_{T \tau}-X\right) \mathrm{e}^{-r(\tau-T)}$ .
• $$• \text { Payoff }=\max \left[\left(F_{\tau T}-X\right) \mathrm{e}^{-r(T-\tau)}, 0\right] •$$
• 期货价格期权:与最后一种情况一样，这是一种在时间$\tau$，价格$X$进入期货合约的期权;然而，期货按日计价，因此$\Phi_{\tau T}-X$的利润将在一天内立即实现$\tau$ .
$$\text { Payoff }=\max \left[\left(\Phi_{\tau T}-X\right), 0\right]$$
iii)远期价格由$F_{t T}=S_t \mathrm{e}^{(r-q)(T-t)}$给出，如果利率不变，我们也有$F_{t T}=\Phi_{t T}$。因此我们可以写
$F_{t T}=$的波动率$\Phi_{t T}=\sqrt{\operatorname{var}\left\langle\ln S_t\right\rangle}=\sigma$的波动率
一般来说，远期价格的波动率等于现货价格的波动率;如果利率不变，期货价格的波动率等于标的股票的波动率。

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