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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

A spread option pays off based on the difference between the price of two underlying assets, relative to a pre-agreed strike. A call option will pay off if the spread is greater than the strike, while a put option will pay off if the spread is lower than the strike.
Call payoff $=$ MAX (Price of asset $1-$ price of asset $2-$ strike, 0 )
Put payoff $=$ MAX $($ Strike $-$ price of asset $1+$ price of asset 2,0$)$
Spread options are relatively more popular in commodities than in traditional financial assets. For example, a participant may wish to hedge or take exposure between:

• The cost of an input and the revenue earned from an output (e.g. cost of crude oil and revenue from gasoline).
• The prices between a commodity traded at two different locations.
• The price of a single commodity future for two different delivery dates.
These examples will be discussed in detail in their respective chapters. From a valuation perspective, these options require an additional valuation input and that is the correlation that exists between the two asset prices. To illustrate this, consider a spread option structured as a call, which will pay off if the spread between two asset prices increases beyond a pre-agreed strike. If the two asset prices are negatively correlated, an increase in asset price 1 will be associated with a decrease in asset price 2 . This relationship increases the probability that the option will end up more deeply in the money and as such the seller will charge a higher price.

## 金融代写|衍生品代写Derivatives代考|Average rate options

Anecdotally, average rate options (‘avros’) are the most common type of commodity option. Regular or ‘vanilla’ option payoffs will reference a single underlying asset price at maturity, with the payoff being made relative to the strike. However, an average rate option will pay out based on an average of prices covering a pre-agreed period prior to maturity.
The expiry payoffs of average rate options are:
Call option = MAX (average underlying price $-$ strike price, 0 )
Put option = MAX (strike price $-$ average underlying price, 0 )
One of the reasons for the popularity of such contracts in commodities is that it reflects the way in which physical supply contracts are structured. A refiner who agrees to buy crude oil from a producer will avoid paying a potentially high price if the terms of the deal reference market prices on a single date. Although averaging prices will dampen the effect of volatile price movements it also means that the refiner will not be able to benefit if the price of the commodity suddenly falls. A common feature of commodity derivatives is that the payoff on the instrument should match that of the commercial contract to avoid a cash flow mismatch.

One of the characteristics of an average rate option is that they will show a premium reduction over the equivalent European option; it is not really a cheap alternative to a vanilla option. As an example, the cost of a six-month call option on a crude oil future with an at-the-money strike price of USD $50.00$ and an implied volatility of $30 \%$. The cost of a vanilla option on this asset is USD 4.20/barrel, but if the terms of the contract are altered such that the final settlement price is an average of the futures prices over the last three months of the transaction, the cost of the option falls to USD $3.96$. This is because the implied volatility of the average rate option references an average price series which will be lower than a non-averaged equivalent. Another way of thinking about the problem is that since the premium of an option is a function of payoff received by the buyer, a payoff based on an average price series will always be lower than a non-averaged equivalent.

# 衍生品代考

• 投入成本和产出收入（例如原油成本和汽油收入）。
• 在两个不同地点交易的商品之间的价格。
• 两种不同交割日期的单一商品期货价格。
这些示例将在各自的章节中详细讨论。从估值的角度来看，这些期权需要额外的估值输入，这就是两种资产价格之间存在的相关性。为了说明这一点，考虑一个构造为看涨期权的价差期权，如果两种资产价格之间的价差增加超过预先约定的行使价，这将得到回报。如果两种资产价格负相关，则资产价格 1 的上升将与资产价格 2 的下降相关联。这种关系增加了期权最终在货币中更深的可能性，因此卖方将收取更高的价格。

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