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.