FRM-Option

Properties of options

Call options

Put options

Put-call Parity

\[ P+S=C+PV(K) \]

• European option -- same underlying, strike, maturity

• Div adjustment

  • discrete: \(P+S-D=C+Ke^{-rt}\)
  • discrete rate q: \(P+Se^{-qt}=C+Ke^{-rt}\)

• annual compound: \(P+S=C+\frac{K}{(1+R_f)^t}\)

• American option: no exact relationship

Portfolio construction

\[ P+S=C+Ke^{-rt} \\ P=C+Ke^{-rt}-S \]

long put = long call, long zero-compound bond, short stock

Arbitrary

\[ P+S (buy)<C+Ke^{-rt}(sell) \]

###Pricing

Simple Strategies

option + underlying

Covered Call

Protective Put

Principal Protected Notes(PPN)

• A PPN is structured as a zero-coupon bond and an option with a payoff that is linked to an underlying asset, index, or benchmark.
• It guarantees a minimum return equal to the investor's initial investment (the principal amount), regardless of the performance of the underlying assets.

Spread Strategies

Bull spread

outlook is bullish, buy low sell high ~ 低买高卖，行权价格

Bear Spread

outlook is bearish, buy high, sell low

Box spread

• A box spread is a combination of a bull call spread with strike prices \(K_1\) and \(K_2\) and a bear put spread with the same two strike prices.
• The payoff from a box spread is always \(K_2 – K_1\) .
• Box spread = Bull spread + Bear spread

Butterfly Spread

long option \(k_1\), long option \(k_3\), short option \(k_2\), \(k_2=\frac{k_1+K_3}{2}\)

• Expects low volatility

• Capped risk

Calender Spread

long long-term, short short-term

Combination Spread

Call+Put

Straddle and Strangle

Strangle is cheaper than straddle——购买高行权价格的看涨期权，购买低行权价格的看跌期权，期权费更加便宜。

Exotic Option

Gap Option

• Call gap option
  • \(S_T>K_2\), \(Payoff=S_T-K_1\)
• Put option
  • \(S_T<K_2\), \(Payoff=K_1-S_T\)

Forward Start option

A forward start option is an advance purchase of a put or call option that will become active at some specified future time. It is essentially a forward on an option

Compound option

• Options on options

• A call on a call, a put on a call, a call on a put, and a put on a put

• If both options are exercised, the total premium will be more than the premium on a single option

Chooser option

After a specified period of time, the holder can choose whether the option is a call or a put.

Barrier Option

• Payoffs and existence depend on whether the underlying’s asset price reaches a certain barrier level over the life of the option.

• A knock-out option ceases to exist when the underlying asset price reaches a certain barrier while a knock-in option comes into existence only when the underlying asset price reaches a barrier.

==In-out parity==

down-and-out call + doan-and-in call = call option

Binary options

• ==Cash-or-Nothing== : Pays some fixed amount of cash if the option expires in-the-money.
• ==Asset-or-Nothing== : Pays the value of the underlying security.
• A regular European call option is equivalent to a long position in an asset-or-nothing call and a short position in a cash-or-nothing call.
• A regular European put option is equivalent to a long position in a cash-or-nothing put and a short position in an asset-or-nothing put.

Call option = long asset or nothing call + short cash or nothing call

Put option = long cash or nothing put + short asset or nothing put

Lookback options

• Payoffs depend on maximum or minimum price of the underlying asset
• With floating strike and with fixed strike

Asian options

• Payoff dependon arithmetic average of the underlying asset price
• Average price option and average strike option.

==Path-dependence==

Volatility and Variance Swap

Volatility Swap

• Exchanging of volatility based on a national principal

• Payments base on pre-specified volatility and realized volatility.

Variance Swap

• Exchanging pre-specified fixed variance rate for realized variance rate

Static options Replication

This technique involves searching for a portfolio of actively traded options (regular options) that approximately replicates the exotic option. Shorting this position provides the hedge.

下半年更加努力吧~

Daily English Reading 06-10

As George Floyd is laid to rest, his legacy reverberates across the globe

https://edition.cnn.com/2020/06/10/politics/george-floyd-legacy/index.html

Missing Boris

https://www.bilibili.com/video/BV1fZ4y1x7SP

Email Writing

English greetings

• Formal
  • Dear Lina,
  • Dear Sir/Madam,
  • Dear Mr Hill,
• Informal
  • Hello,
  • Hi Ryan,
• Business Email
  • I hope you're well.
  • How are you?

Formal: Dear + name

Neutral: Hello/Hi + name

Informal: Hi/Hey, or no greeting

Explaining why you are writing

Formal Opening sentences

• I'm writing regarding
• I wanted to follow up on
• I would like to ask about
• I'm concerned about sth
• I need to bring sth to your attention

Informal opening sentences

• Do you know when the database issues will be fixed?
• Let's confirm our plans for this month
• How will the new budget affect out department

Adding details to your email

• 2 to 3 sentences
• as short as possible

change topic

• There's one more thing I'd like to discuss with you
• Regarding
• I'd also like to ask you about

Adding a call to action to your email

• please by tomorrow at the latest
• As a matter of urgency, you need to
• could you please
• I would like you to

Less direct

• Can I ask you to
• This is just to keep you updated
• This doesn't require any immediate response, but please keep an eye on the situation

Adding a sign-off to your email

• Formal
  • Regards,
  • Best Wishes,
  • Kind Regards,
• Informal
  • Cheers,
  • Take care,

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Microeconomic Review

• Resevation price: a person's maximum willingness to pay for sth
• Utility is simply a way of describing preferences.
• A Utility Function is a way of assigning a number to every possible consumption bundle such that more-preferred bundles get assigned larger numbers than less-preferred bundles
• Ordinal Utility: The magnitude of the utility function is only important to rank the different consumption bundles; the size of the utility difference between any two consumption bundles doesn't matter.
• Monotonic Transformation is a way of transforming one set of numbers into another set of numbers in a way that preserves the order of the numbers.
• a monotonic transformation of a utility function is a utility function that represents the same preferences as the original utility function.
• Cardinal Utility: attach a significance to the magnitude of utility.
• A utility function is simply a way to represent or summarize a preference ordering. The numerical magnitudes of utility levels have no intrinsic meaning.
• \(MRS = -\frac{MU_1}{MU_2}\)

Macroeconomics Review

• One way to view GDP is as the total income of everyone in the economy; another way is as the total expenditure on the economy’s output of goods and services
• National Income Accounting, the system used to measure GDP and many related statistics.
• A stock is a quantity measured at a given point in time, whereas a flow is a quantity measured per unit of time.
• The Treatment of Inventories
  • spoil: not alter GDP
  • put into inventory: GDP rises (regarded as the owner bought the inventory)
• GDP includes only the value of final goods.
• Economists call the value of goods and services measured at current prices Nominal GDP. Notice that nominal GDP can increase either because prices rise or because quantities rise.
• Real GDP, which is the value of goods and services measured using a constant set of prices. That is, real GDP shows what would have happened to expenditure on output if quantities had changed but prices had not.
• The GDP deflator is the ratio of nominal GDP to real GDP
  • GDP Deflator = \(\frac{\text{Nominal GDP}}{\text{Real GDP}}\)
  • The GDP deflator reflects what’s happening to the overall level of prices in the economy.
• GDP is the sum of consumption, investment, government purchases, and net export
  • The general rule is that the economy’s investment does not include purchases that merely reallocate existing assets among different individuals.
• Whereas GDP measures the total income produced domestically, GNP measures the total income earned by nationals (residents of a nation).
• To obtain net national product (NNP), we subtract from GNP the depreciation of capital—the amount of the economy’s stock of plants, equipment, and residential structures that wears out during the year
  • NNP = GNP - depreciation
  • National Income = NNP - Statistical Discrepancy.
  • Disposable Personal Income = Personal Income - Personal Taxes.

READING 12: TOPICS IN DEMAND AND SUPPLY ANALYSIS

12.1 Elasticity

12.a: Calculate and interpret price, income, and cross-price elasticities of demand and describe factors that affect each measure

Own-Price Elasticity of Demand

Own-price elasticity is a measure of the responsiveness of the quantity demanded to a change in price [negative downward-sloping]

Two Extreme Cases