# 金融代写|衍生品代写Derivatives代考|AEM4210

## 金融代写|衍生品代写Derivatives代考|Volatility term structure

Recall that within the BSM option valuation framework, volatility was assumed to be a constant. In Section 2.7.5.1 it was shown that empirically volatility varied with respect to option strikes. It can also be seen that volatility varies with respect to time to maturity.
In most financial markets the so-called term structure of volatility is upward sloping (see for example Schofield, 2017). This suggests that longer-dated options will be valued using higher volatilities. One reasonable way of interpreting this shape is to suggest that uncertainty over the ‘at maturity’ outcomes of the underlying asset rise with respect to time to maturity. However, in many commodity markets volatility displays an inverse term structure, i.e. volatility decreases with respect to maturity. Consider the following option on a fictitious asset (Table 2.10). The underlying asset price is 100 and in all instances the option is struck ATM.
Two points arise from this analysis:

• For a given maturity there is a positive relationship between an option premium and the level of implied volatility.
• In all three conditions, longer-dated options will have a higher value than shorter-dated options.

The following figures illustrate the relationship for different commodities:
Recall the forward curves for crude oil in Figure 2.2. That figure illustrated that the short end of the forward curve tends to be the most volatile. Logically this would also impact the volatility term structure of options (figure $2.10$ ) and since it is inverted it suggests a wider dispersion of ‘at maturity’ values for shorter-dated options than longer-dated options. Since it is plausible to suggest that commodity prices mean revert then at a simple level, there is less expected variability for longer-dated options. This is also shown in the forward curve diagram where there is an element of ‘convergence’ for longer-dated prices. A similar argument could be used to explain the shape of the volatility term structure for natural gas (Figure 2.11). Again, gold is something of an exception. If one were to accept that it is traded more like a financial asset, then the upward sloping volatility term structure is not unexpected (Figure 2.12). The figure for copper indicates that like their observed skew profiles, term structures are not as well defined (Figure 2.13).

## 金融代写|衍生品代写Derivatives代考|Subcategories of risk

Within each of the individual risk headings it is possible to break down the risk into a series of subheadings. For example, within market risk one could consider:

m Interest rate risk – the risk that entity loses money from an adverse movement in interest rates. For example, a company may borrow money on a variable rate basis and then be faced with higher costs if interest rates subsequently rise. Equally this exposure also applies to interest earned on cash surpluses, which is often ignored.

• Inflation risk-although not always obvious, this may be an exposure for a corporate whose revenues or expenses are linked to inflation. A corporate may be faced with a hidden inflation cost if they were to offer a pension scheme to their employees which is linked to a change in inflation
• Foreign exchange rate risk – typically foreign exchange rate risk arises from two sources. Transactional FX risk arises because of a company’s day-to-day business. For example, a US importer of goods and services from abroad will have a foreign currency payable. Translational FX risk arises from expressing a foreign currency asset or liability in the company’s domestic accounting currency. With respect to commodities, since they are mostly traded and quoted in US dollars, foreign producers or consumers will also be exposed to FX risk.
• Equity risk – this can arise in several different forms from a corporate perspective. Again, one source of equity risk could be hidden in a company’s defined benefit pension scheme. Proposed share buybacks and employee share option schemes may require the company to buy equity at an unfavourable price. Investments in other publicly quoted companies gives rise to a form of equity investment risk
• Liquidity risk – liquidity risk can be thought of as the potential inability of a company to meet its short-term cash requirements. This may arise from the inability to borrow money from its bankers, or the inability to be able to liquidate assets to cover any shortfall.
• Commodity risk – simply, this is the exposure that a company will face because of a change in commodity prices. This may be either explicit or as a side product of a company’s business. For example, a gold producer will be exposed to a fall in the price of the commodity, which is a direct function of his production. A haulage company’s exposure to diesel prices is arguably a secondary exposure to their main line of business. A company that is fully integrated along a particular supply chain will arguably face offsetting price risks.

## 金融代写|衍生品代写Derivatives代考|Volatility term structure

• 对于给定的期限，期权溢价与隐含波动率水平之间存在正相关关系。
• 在所有三种情况下，较长期期权的价值都高于短期期权。

## 金融代写|衍生品代写Derivatives代考|Subcategories of risk

m 利率风险——实体因利率不利变动而蒙受损失的风险。例如，一家公司可能会以浮动利率借入资金，然后如果利率随后上升，则将面临更高的成本。同样，这种风险也适用于现金盈余所赚取的利息，这通常被忽略。

• 通胀风险——尽管并不总是显而易见的，但对于收入或支出与通胀相关的公司来说，这可能是一种风险。如果公司向员工提供与通货膨胀变化相关的养老金计划，他们可能会面临隐藏的通货膨胀成本
• 外汇风险——通常外汇风险来自两个来源。交易外汇风险源于公司的日常业务。例如，从国外进口商品和服务的美国进口商将有外币应付。换算外汇风险源于以公司的国内会计货币表示外币资产或负债。在大宗商品方面，由于大多以美元进行交易和报价，外国生产商或消费者也将面临外汇风险。
• 股权风险——从公司的角度来看，这可能以几种不同的形式出现。同样，股权风险的一个来源可能隐藏在公司的固定收益养老金计划中。提议的股票回购和员工股票期权计划可能要求公司以不利的价格购买股权。对其他上市公司的投资会产生某种形式的股权投资风险
• 流动性风险——流动性风险可以被认为是公司可能无法满足其短期现金需求。这可能是由于无法从银行家借钱，或者无法清算资产以弥补任何短缺。
• 商品风险——简单地说，这是公司因商品价格变化而面临的风险。这可能是明确的，也可能是公司业务的副产品。例如，黄金生产商将面临商品价格下跌的风险，这是其生产的直接函数。一家运输公司对柴油价格的敞口可以说是对其主营业务的二次敞口。一家完全整合在特定供应链中的公司可能会面临抵消价格风险。

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