# 金融代写|衍生品代写Derivatives代考|IE2042

## 金融代写|衍生品代写Derivatives代考|Bachelier model

One of the features of the Black Scholes Merton (BSM) and Black models is they assume that the underlying prices are lognormally distributed. From a price perspective, this means the models do not allow for the possibility that prices could turn negative.

For conventional financial assets such as shares, bonds, and foreign exchange this is a reasonable assumption; when a company goes bust, their shares will stop trading rather than turn negative.

As ever, commodities are different. Negative prices have been experienced in power and gas (see chapters 7 and 8 ) and calendar spread trades can flip between positive and negative values. The issue of negative prices and index values came more to the fore in 2020 as a result of activity in the shipping market (e.g. Baltic Capesize Index) and in April 2020 for the first time, crude oil (see see chapter 6). This brought into question the use of models such as Black 1976 for valuing and risk managing options on futures.

An alternative to this model is the Bachelier model, which dates from around $1900 .$ Some of the key characteristics of the Bachelier model are:

• It assumes that the underlying price follows a normal rather than lognormal distribution and so allows for the possibility of negative prices.
• Volatility is expressed as a monetary value rather than as a proportional percentage value. For example, assume an asset is trading at a price of USD $10.00$ with an implied volatility of $10 \%$ per day. This would suggest that the asset is expected to trade with an approximate range of prices between USD 9.00-11.00. If the price drops to USD $5.00$, but the implied volatility remains the same, then the approximate range of prices would be $+/-50$ cents. Using the same set of prices, a Bachelier model would express volatility as a dollar value, say, USD 1.00. So, at USD $10.00$ the range of prices would be similar. But if prices fell to USD $5.00$ then under a Bachelier framework, the range of prices remains at $+/-$ USD 1.00. At very low prices (e.g. USD 1.00), but with volatility expressed as a dollar value (e.g. USD $20.00$ ), it means that the model implies a range of values from -USD $19.00$ to +USD 21.00.
• Since the model allows for negative prices, OTM puts will be valued relatively higher using a Bachelier model rather than a BSM framework. Since the premium reflects the expected payout the Bachelier model reflects, this pushes the price relatively higher than the BSM framework. For example, using a simple Bachelier model a three-month put option with a zero strike, an underlying price of zero, and a dollar volatility of USD $40.00$ returns a premium of USD 7.98. A Black 1976 model returns a premium value of USD $0.00$.

## 金融代写|衍生品代写Derivatives代考|Put-call parity: the theory

Although pricing models provide one linkage between the underlying price, the forward rate and the option premium, the concept of put-call parity is an alternative representation. Tompkins (1994) provides a detailed analysis of the concept.

Put-call parity is a concept that attempts to link options with their underlying assets such that arbitrage opportunities could be identified. The conditions of put-call parity will hold if the strike, maturity, and amount are the same.

Although put-call parity varies according to the underlying asset, in its simplest form it can be represented by the following expression:
$$C-P=F-E$$
Where
$\mathrm{C}=$ Price of a call option
$\mathrm{P}=$ Price of a put option
$\mathrm{F}=$ Forward price of the underlying asset
$\mathrm{E}=$ Strike rate for the option
Strictly speaking the right hand side of the expression would need to be present valued as the strike and the forward price relate to a future time period, whereas the call and put premiums are expressed in present value terms.

Although put-call parity was designed to identify price discrepancies between markets, it is possible to adapt the formula so it can be used from a strategy perspective. In this case the expression can be written as:
$$\mathrm{C}-\mathrm{P}=\mathrm{F}$$
Where $\mathrm{F}$ is redefined as a position in the underlying, in this case a forward position. Each of the symbols in the expression can be annotated with either a ‘ $+$ ‘or a ‘-‘ to indicate either a buying or selling position, respectively. So:
$$+\mathrm{C}-\mathrm{P}=+\mathrm{F}$$
That is, buying a call and selling a put is equivalent to a long position in the underlying. Although this may seem a very dry concept, the relationship is frequently used in financial engineering to create innovative structures.

## 金融代写|衍生品代写Derivatives代考|Bachelier model

Black Scholes Merton (BSM) 和 Black 模型的特征之一是它们假设基础价格呈对数正态分布。从价格的角度来看，这意味着模型不允许价格转为负数。

• 它假设基础价格遵循正态分布而不是对数正态分布，因此允许负价格的可能性。
• 波动率表示为货币价值，而不是比例百分比值。例如，假设一项资产以美元的价格交易10.00隐含波动率10%每天。这表明该资产预计将在 9.00-11.00 美元之间的大致价格范围内交易。如果价格跌至美元5.00，但隐含波动率保持不变，则价格的大致范围为+/−50美分。使用同一组价格，Bachelier 模型将波动性表示为美元价值，例如 1.00 美元。所以，在美元10.00价格范围将是相似的。但如果价格跌至美元5.00然后在 Bachelier 框架下，价格范围保持在+/−1.00 美元。以非常低的价格（例如 1.00 美元），但波动性以美元价值表示（例如美元20.00)，这意味着该模型暗示了从 -USD 开始的一系列值19.00至 +21.00 美元。
• 由于该模型允许负价格，使用 Bachelier 模型而不是 BSM 框架，OTM 看跌期权的价值将相对较高。由于溢价反映了 Bachelier 模型所反映的预期支出，这推动价格相对高于 BSM 框架。例如，使用一个简单的 Bachelier 模型，一个 3 个月的看跌期权，行使价为零，基础价格为零，美元波动率为 USD40.00返回 7.98 美元的溢价。1976 年黑色模型返回美元的溢价0.00.

## 金融代写|衍生品代写Derivatives代考|Put-call parity: the theory

$$C-P=F-E$$

$\mathrm{C}=$ 看涨期权的价格
$\mathrm{P}=$ 看跌期权的价格
$\mathrm{F}=$ 标的资产的远期价格
$\mathrm{E}=$ 期权的执行价格

$$\mathrm{C}-\mathrm{P}=\mathrm{F}$$

$$+\mathrm{C}-\mathrm{P}=+\mathrm{F}$$

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