1. Introduction to Options

Option is a derivative contract in which the buyer pays a sum of money to the seller or writer, and receives the right to either buy or sell an underlying asset at a fixed price either on a specific expiration date or at any time prior to the expiration date.

An option is a right, but not an obligation.

Default in options is possible only from the short to the long.

1.1 Options Types

1.1.1 Terminology

Strike Price is exercise price specified in the contract.
Premium, option premium, initial cost, initial investment, or up-front cost is paid by the buyer of option when one buys the option.
Expiration Date/Maturity Date is the date after which an option is void(无效).

1.1.2 Options Types

Based on type of right

A call option gives the holder the right to buy an asset by a certain date for a certain price.

• A long position in a call option: right to buy
• A short position in a call option: obligation to sell

A put option gives the holder the right to sell an asset by a certain date for a certain price.

• A long position in a put option: right to sell
• A short position in a put option: obligation to buy

The seller is sometimes referred to as the writer of the option.

Based on exercise date

European options can be exercised only on the expiration date itself.

American options can be exercised at any time up to the expiration date.

Price of American options ≥ \geq ≥ Price of European options due to more flexibility.

1.2 Payoff and Profit of Option

1.2.1 Intrinsic Value (Exercise Value)

Intrinsic value measures the value of an option if it can only be exercised immediately.

The maximum of zero and the amount that the option is in the money.

Call option: Max ( S − X , 0 ) \text{Max}(S-X,0) Max(S−X,0), Put option: Max ( X − S , 0 ) \text{Max}(X-S,0) Max(X−S,0)

1.2.2 Time Value (Speculative Value)

The amount by which the option premium(price) exceeds the intrinsic value.

Option premium = Intrinsic value + Time value \text{Option premium} = \text{Intrinsic value} + \text{Time value} Option premium=Intrinsic value+Time value

1.2.3 Payoff and Profit Option

Payoff from a long position in a European call option is Max ( S T − X , 0 ) \text{Max}(S_T-X,0) Max(ST​−X,0)

profit = Max ( S T − X , 0 ) − C 0 \text{profit}=\text{Max}(S_T-X,0)-C_0 profit=Max(ST​−X,0)−C0​, C 0 C_0 C0​: the premium of call option

Payoff from a long position in a European put option is Max ( X − S T , 0 ) \text{Max}(X-S_T,0) Max(X−ST​,0)

profit = Max ( X − S T , 0 ) − P 0 \text{profit}=\text{Max}(X-S_T,0)-P_0 profit=Max(X−ST​,0)−P0​, P 0 P_0 P0​: the premium of put option

1.2.4 Call Option

Profit of long = intrinsic value at expiration - premium

Breakeven underlying price = X - premium

1.2.5 Put Option

Profit of long = intrinsic value at expiration - premium

Breakeven underlying price = X + premium

	Long call	Short call	Long put	Short put
Payoff at T T T	Max ( S T − X , 0 ) \text{Max}(S_T-X,0) Max(ST​−X,0)	− Max ( S T − X , 0 ) -\text{Max}(S_T-X,0) −Max(ST​−X,0)	Max ( X − S T , 0 ) \text{Max}(X-S_T,0) Max(X−ST​,0)	− Max ( X − S T , 0 ) -\text{Max}(X-S_T,0) −Max(X−ST​,0)
Profit at T T T	Max ( S T − X , 0 ) − c 0 \text{Max}(S_T-X,0)-c_0 Max(ST​−X,0)−c0​	− Max ( S T − X , 0 ) + c 0 -\text{Max}(S_T-X,0)+c_0 −Max(ST​−X,0)+c0​	Max ( X − S T , 0 ) − p 0 \text{Max}(X-S_T,0)-p_0 Max(X−ST​,0)−p0​	− Max ( X − S T , 0 ) + p 0 -\text{Max}(X-S_T,0)+p_0 −Max(X−ST​,0)+p0​
Max profit	Unlimited	c 0 c_0 c0​	X − p 0 X-p_0 X−p0​	p 0 p_0 p0​
Max loss	c 0 c_0 c0​	Unlimited	p 0 p_0 p0​	X − p 0 X-p_0 X−p0​
Breakeven	X + c 0 X+c_0 X+c0​	X + c 0 X+c_0 X+c0​	X − p 0 X-p_0 X−p0​	X − p 0 X-p_0 X−p0​

1.2.6 Moneyness

In the money: immediate exercise would generate a positive payoff.

At the money: immediate exercise would generate no payoff.

Out of the money: immediate exercise would result in a loss.

Moneyness	Call Option	Put Option
In the money	S > X S>X S>X	S < X S<X S<X
At the money	S = X S=X S=X	S = X S=X S=X
Out of the money	S < X S<X S<X	S > X S>X S>X

2. Option Markets

2.1 Exchange-Traded Options on Stocks

2.1.1 Basics of Trading Rules on CBOE

The options on individual stocks are American-style, and include 100 shares.

CBOE also offers index options which is cash settled.

CBOE also provides exchange-trade products (ETPs) options such as options on ETFs.

2.1.2 Dividends and Stock Splits

Cash dividends do not affect the terms of stock options unless they are unusually high.

Stock splits and stock dividends do result in strike price adjustments.

• If a company announces a 3-to-1 stock split which means each share is replaced by 3 new shares, the strike price will be reduced to one third of the original price and the number of options is multiplied by three.
• Note that a 5% stock dividend means shareholders receive 1 new share for each 20 shares owned, this is identical to an 21-to stock split.

2.2 Trading&Margin Requirements

2.2.1 Trading Mechanics

Market Makers quote both bid and offer price on the option to ensure buy and sell orders be executed without any delays and add liquidity to the market.

• Bid price is the price market maker prepares to buy.
• Offer or ask price is the price market maker prepares to sell.

Like futures, exchange-traded options can be closed out by taking an offsetting position.

CBOE imposes position and exercise limits on traded options to prevent the market from being controlled by one investor.

2.2.2 Margin Requirements

The buyer fully paid for an option would be no margin requirements. However, the option seller does have potential future liabilities.

Options with maturities less than nine months must be paid in full price upfront.

Options with maturities greater than nine months can be bought on margin, but no more than 25% can be borrowed.

2.2.3 Option Clearing Corporation(OCC)

OCC performs the same function for options markets as the clearing house does for futures markets.

It guarantees that options writers willeir a record of all long and short positions.

When an investor instructs a broker to exercise an option, the broker notifies the OCC member that clears its trades. This member then places an exercise order with the OCC.

2.3 Other Types of Options

2.3.1 Over-the-Counter Market

Option characteristics in the OTC market can differ from those available on the exchanges.

E.g. strike prices, maturity dates, and the times when options can be exercised.

OTC market is large and the options often last longer than those trader on exchanges.

OTC options can also be exotic.

2.3.2 Warrants

Warrants are options issued by a corporation, and are usually call options on the corporation’s own stock.

Warrants can be used by firms to make debt issuances more attractive to investors.

Difference between warrant and option: exercising an option will not affect the number of the outstanding shares while if warrants are exercised, more shares will be issued.

The stock price after exercise:
N S + W K N + W \frac{NS+WK}{N+W} N+WNS+WK​

The value of the warrant
N S + W K N + W − K = N N + W ( S + W ) = N N + W ∗ Call  Value \frac{NS+WK}{N+W}-K=\frac{N}{N+W}(S+W)=\frac{N}{N+W}*\text{Call \;Value} N+WNS+WK​−K=N+WN​(S+W)=N+WN​∗Call Value

• S S S is the stock price, K K K is the strike price of warrant.
• N N N is the number of stocks outstanding, W W W is the number of new warrant issued.
• The total value of the equity and the warrants: N S + N W NS+NW NS+NW

The company’s stock price will decline by (dilution cost):
S − N S + W K N + W = W N + W ( S − K ) = W N + K ∗ C a l l V a l u e S-\frac{NS+WK}{N+W}=\frac{W}{N+W}(S-K)=\frac{W}{N+K}*Call \; Value S−N+WNS+WK​=N+WW​(S−K)=N+KW​∗CallValue

2.3.3 Convertible Bonds

Gives bondholder the right to exchange the bond for a specified number of common shares in issuing company.

Hybrid security with both debt and equity features.

Convertible bond = straight bond + call option on equity.

Convertible bond is similar to a warrant. When an investor chooses to convert, the company issues more stock to be exchanged for the bonds.

Warrants and convertibles are often traded on exchanges.

2.3.4 Employee Stock Options

Employee stock options are call options granted to its employees by a company.

There is usually a vesting period during which options cannot be exercised.

Employees may forfeit their options if they leave their jobs during the vesting period.

Employees are not permitted to sell their stock options to a third party.

3. Properties of Options

3.1 Six Factors

S 0 S_0 S0​: current stock price.

X X X: strike price of the option.

T T T: time to expiration of the option.

r r r: short-term risk-free interest rate over T T T.

D D D: present value of the dividend of the underlying stock.

σ \sigma σ: expected volatility of stock prices over T T T.

Summary of the effect on the price of a stock option of increasing one variable while holding all others factors constant.

Factor	European Call	European Put	Amercian Call	Amercian Put
S S S	+	-	+	-
X X X	-	+	-	+
T T T	?	?	+	+
σ \sigma σ	+	+	+	+
r r r	+	-	+	-
D D D	-	+	-	+

3.2 Early Exercise

3.2.1 Early Exercise for American Call and Put Options

Underlying stock without dividends

Call options: it is never optimal to exercise early an Amercian call on a non-dividend paying stock.

Put options: it may be optimal to exercise early an Amercian put on a non-dividend paying stock

Underlying stock with dividends ( D n D_n Dn​) paid at time t n t_n tn​

Call options: it more likely an Amercian call option will be exercised early. Early exercise will be optimal when
D n > X ( 1 − e − r ( T − t n ) ) D_n>X(1-e^{-r(T-t_n)}) Dn​>X(1−e−r(T−tn​))

Put options: it less likely that an Amercian put option will be exercised early. Early exercise will be optimal when
D n < X ( 1 − e − r ( T − t n ) ) D_n<X(1-e^{-r(T-t_n)}) Dn​<X(1−e−r(T−tn​))

3.2.2 Early Exercise for American Call Options

The call option on no-dividends paying underlying should not be exercised before maturity if interest rates are positive.

Early exercising call option would yield a profit equal to intrinsic value. However, selling the option yields a profit of the intrinsic value plus time value.

Deep-ITM call option with dividends may be early exercised. Exercising the option before an ex-dividend date would be the optimal decision.

3.2.3 Early Exercise for American Put Options

The decision to exercise an American put option without dividends is therefore a trade-off between:

• Receiving the strike price early so it can be invested to earn
  interest.
• Benefiting from the optionality in circumstances where the stock price moves above the strike price.

In general, early exercising becomes less attractive to the holder of a put option when:

• Stock price increases
• Interest rate decreases
• Time to maturity increases
• Dividends expected during the life of the option increase.

3.3 Upper&Lower Bounds of Value

Upper and lower bonds without dividend

Option Type	Max	Min
European Call	S 0 S_0 S0​	Max ( S 0 − X e − r T , 0 ) \text{Max}(S_0-Xe^{-rT},0) Max(S0​−Xe−rT,0)
American Call	S 0 S_0 S0​	Max ( S 0 − X e − r T , 0 ) \text{Max}(S_0-Xe^{-rT},0) Max(S0​−Xe−rT,0)
European Put	X e − r T Xe^{-rT} Xe−rT	Max ( X e − r T − S 0 , 0 ) \text{Max}(Xe^{-rT}-S_0,0) Max(Xe−rT−S0​,0)
Amercian Put	X X X	Max ( X − S 0 , 0 ) \text{Max}(X-S_0,0) Max(X−S0​,0)

Upper and lower bonds with dividend

Option Type	Max	Min
European Call	S 0 S_0 S0​	Max ( S 0 − PVD − X e − r T , 0 ) \text{Max}(S_0-\text{PVD}-Xe^{-rT},0) Max(S0​−PVD−Xe−rT,0)
American Call	S 0 S_0 S0​	- -
European Put	X e − r T Xe^{-rT} Xe−rT	Max ( X e − r T + PVD − S 0 , 0 ) \text{Max}(Xe^{-rT}+\text{PVD}-S_0,0) Max(Xe−rT+PVD−S0​,0)
Amercian Put	X X X	- -

3.4 Put-Call Parity

3.4.1 Definition

The relationship between the price of a European call option and that of a European put option with the same strike price and time to maturity.

Portfolio A: a European call option c 0 c_0 c0​ and an amount of cash equals to P V ( X ) = X e − r T PV(X)=Xe^{-rT} PV(X)=Xe−rT

Portfolio B: a European put option p 0 p_0 p0​ and one share S 0 S_0 S0​.

At maturity

p + S T = c + X e − r T p+S_T=c+Xe^{-rT} p+ST​=c+Xe−rT

Arbitrage opportunities exist when put-call parity does not hold.

3.4.2 Parity Relationship at Maturity

S T > X S_T>X ST​>X, Call will be ITM & Put will be OTM.

S T < X S_T<X ST​<X, Call will be OTM & Put will be ITM.

3.4.3 Synthetic Equivalencies

The put-call parity can be rearranged to create synthetic equivalencies. Note: the term “+” means long and “-” means short.

Synthetic call: c = S + p − X e − r T c=S+p-Xe^{-rT} c=S+p−Xe−rT

Synthetic put: p = c + X e − r T − S p=c+Xe^{-rT}-S p=c+Xe−rT−S

Synthetic stock: S = c + X e − r T − p S=c+Xe^{-rT}-p S=c+Xe−rT−p

Synthetic bond: X e − r T = p + S − c Xe^{-rT}=p+S-c Xe−rT=p+S−c

3.4.4 Put-Call Parity with Dividends

When the underlying asset provide dividends, the parity formula can be derived from portfolio.

Portfolio A: a European call option c 0 c_0 c0​ and an amount of cash equals to X e − r T + PV ( Div ) Xe^{-rT}+\text{PV}(\text{Div}) Xe−rT+PV(Div)

Portfolio B: a European put option p 0 p_0 p0​ and one share S 0 S_0 S0​.

p + S T = c + X e − r T + PV ( Div ) p+S_T=c+Xe^{-rT}+\text{PV}(\text{Div}) p+ST​=c+Xe−rT+PV(Div)

3.4.5 Put-Call Parity with American Options

The inequality relationship between the price of an American call option and that of an American put option with the same strike price and time to maturity is as follow:
S 0 − X ⩽ C 0 − P 0 ⩽ S 0 − X e − r T S_0-X\leqslant C_0-P_0 \leqslant S_0-Xe^{-rT} S0​−X⩽C0​−P0​⩽S0​−Xe−rT

4. Trading Strategies

4.1 Single Option Strategies

4.1.1 Covered Call(Long Stock + Short Call)

Short call + Long Stock

	S T < X S_T<X ST​<X	S T ≥ X S_T\geq X ST​≥X
Short Call	c c c	c − ( S T − X ) c-(S_T-X) c−(ST​−X)
Long Stock	S T − S 0 S_T-S_0 ST​−S0​	S T − S 0 S_T-S_0 ST​−S0​
Total	c + S T − S 0 c+S_T-S_0 c+ST​−S0​	c + X − S 0 c+X-S_0 c+X−S0​

4.1.2 Protective Put(Long Stock + Long Put)

Long put + Long Stock

	S T < X S_T<X ST​<X	S T ≥ X S_T\geq X ST​≥X
Long Put	( X − S T ) − p (X-S_T)-p (X−ST​)−p	− p -p −p
Long Stock	S T − S 0 S_T-S_0 ST​−S0​	S T − S 0 S_T-S_0 ST​−S0​
Total	X − S 0 − p X-S_0-p X−S0​−p	S T − S 0 − p S_T-S_0-p ST​−S0​−p

4.1.3 Principal Protected Notes(PPNs)

Principal protected note: a security created from a single option such that the investor benefits from any gain of a specified portfolio without the risk of losses.

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 preformance of the underlying assets.

Example: Long a three-year zero-coupon bond that will pay USD 10,000 in three years., and long a three-year call option on a portfolio, which is currently worth USD 10,000 with a strike price of USD 10,000.

4.2 Spread Trading Strategies

画图的时候记得strike price与option price的关系

4.2.1 Bull Call Spread

Long 1 Call 1 \text{Call}_1 Call1​ at X 1 X_1 X1​ + Short 1 Call 2 \text{Call}_2 Call2​ at X 2 X_2 X2​, and X 1 < X 2 X_1<X_2 X1​<X2​

	S T < X 1 S_T<X_1 ST​<X1​	X 1 ≤ S T < X 2 X_1\leq S_T<X_2 X1​≤ST​<X2​	S T ≥ X 2 S_T \geq X_2 ST​≥X2​
Long 1 c 1 c_1 c1​	− c 1 -c_1 −c1​	S T − X 1 − c 1 S_T-X_1-c_1 ST​−X1​−c1​	S T − X 1 − c 1 S_T-X_1-c_1 ST​−X1​−c1​
Short 1 c 2 c_2 c2​	c 2 c_2 c2​	c 2 c_2 c2​	c 2 − ( S T − X 2 ) c_2-(S_T-X_2) c2​−(ST​−X2​)
Total	c 2 − c 1 < 0 c_2-c_1<0 c2​−c1​<0	S T − X 1 + c 2 − c 1 S_T-X_1+c_2-c_1 ST​−X1​+c2​−c1​	X 2 − X 1 + c 2 − c 1 X_2-X_1+c_2-c_1 X2​−X1​+c2​−c1​

4.2.2 Bull Put Spread

Long 1 Put 1 \text{Put}_1 Put1​ at X 1 X_1 X1​ + Short 1 Put 2 \text{Put}_2 Put2​ at X 2 X_2 X2​, and X 1 < X 2 X_1<X_2 X1​<X2​

	S T < X 1 S_T<X_1 ST​<X1​	X 1 ≤ S T < X 2 X_1\leq S_T<X_2 X1​≤ST​<X2​	S T ≥ X 2 S_T \geq X_2 ST​≥X2​
Long 1 p 1 p_1 p1​	X 1 − S T − p 1 X_1-S_T-p_1 X1​−ST​−p1​	− p 1 -p_1 −p1​	− p 1 -p_1 −p1​
Short 1 p 2 p_2 p2​	p 2 − ( X 2 − S T ) p_2-(X_2-S_T) p2​−(X2​−ST​)	p 2 − ( X 2 − S T ) p_2-(X_2-S_T) p2​−(X2​−ST​)	p 2 p_2 p2​
Total	p 2 − p 1 + X 1 − X 2 p_2-p_1+X_1-X_2 p2​−p1​+X1​−X2​	p 2 − p 1 − ( X 2 − S T ) p_2-p_1-(X_2-S_T) p2​−p1​−(X2​−ST​)	p 2 − p 1 > 0 p_2-p_1>0 p2​−p1​>0

4.2.3 Bear Call Spread

Short 1 Call 1 \text{Call}_1 Call1​ at X 1 X_1 X1​ + Long 1 Call 2 \text{Call}_2 Call2​ at X 2 X_2 X2​, and X 1 < X 2 X_1<X_2 X1​<X2​

	S T < X 1 S_T<X_1 ST​<X1​	X 1 ≤ S T < X 2 X_1\leq S_T<X_2 X1​≤ST​<X2​	S T ≥ X 2 S_T \geq X_2 ST​≥X2​
Long 1 c 1 c_1 c1​	c 1 c_1 c1​	c 1 − ( S T − X 1 ) c_1-(S_T-X_1) c1​−(ST​−X1​)	c 1 − ( S T − X 1 ) c_1-(S_T-X_1) c1​−(ST​−X1​)
Short 1 c 2 c_2 c2​	− c 2 -c_2 −c2​	− c 2 -c_2 −c2​	( S T − X 2 ) − c 2 (S_T-X_2)-c_2 (ST​−X2​)−c2​
Total	c 1 − c 2 c_1-c_2 c1​−c2​	c 1 − ( S T − X 1 ) − c 2 c_1-(S_T-X_1)-c_2 c1​−(ST​−X1​)−c2​	X 1 − X 2 + c 1 − c 2 X_1-X_2+c_1-c_2 X1​−X2​+c1​−c2​

4.2.4 Bear Put Spread

Short 1 Put 1 \text{Put}_1 Put1​ at X 1 X_1 X1​ + Long 1 Put 2 \text{Put}_2 Put2​ at X 2 X_2 X2​，and X 1 < X 2 X_1<X_2 X1​<X2​

	S T < X 1 S_T<X_1 ST​<X1​	X 1 ≤ S T < X 2 X_1\leq S_T<X_2 X1​≤ST​<X2​	S T ≥ X 2 S_T \geq X_2 ST​≥X2​
Long 1 p 1 p_1 p1​	p 1 − ( X 1 − S T ) p_1-(X_1-S_T) p1​−(X1​−ST​)	p 1 p_1 p1​	p 1 p_1 p1​
Short 1 p 2 p_2 p2​	( X 2 − S T ) − p 2 (X_2-S_T)-p_2 (X2​−ST​)−p2​	( X 2 − S T ) − p 2 (X_2-S_T)-p_2 (X2​−ST​)−p2​	− p 2 -p_2 −p2​
Total	p 1 − p 2 + X 2 − X 1 p_1-p_2+X_2-X_1 p1​−p2​+X2​−X1​	p 1 − p 2 + ( X 2 − S T ) p_1-p_2+(X_2-S_T) p1​−p2​+(X2​−ST​)	p 1 − p 2 p_1-p_2 p1​−p2​

4.2.5 Box Spread

Bull call spread + Bear put spread

A box spread is a combination of a bull call spread with strike prices X 1 X_1 X1​ and X 2 X_2 X2​ and a bear put spread with the same two strike prices.

Under a no arbitrage assumption, the present value of the payoff will equal the net premium paid, which is X 2 − X 1 X_2-X_1 X2​−X1​. The value of a box spread is therefore always the present value of this payoff or ( X 2 − X 1 ) e − r T (X_2-X_1)e^{-rT} (X2​−X1​)e−rT.

If the market price of the box spread is too low, it is profitable to buy the box. If the market price of the box spread is too high, it it profitable to sell the box.

4.2.6 Butterfly Spread

with call options

Long 1 call 1 \text{call}_1 call1​ at X 1 X_1 X1​ + Short 2 call 2 \text{call}_2 call2​ at X 2 X_2 X2​ + Long 1 call 3 \text{call}_3 call3​ at X 3 X_3 X3​, X 1 < X 2 < X 3 X_1<X_2<X_3 X1​<X2​<X3​, usually X 2 = ( X 1 + X 3 ) / 2 X_2=(X_1+X_3)/2 X2​=(X1​+X3​)/2

	S T < X 1 S_T<X_1 ST​<X1​	X 1 ≤ S T < X 2 X_1\leq S_T<X_2 X1​≤ST​<X2​	X 2 ≤ S T < X 3 X_2\leq S_T<X_3 X2​≤ST​<X3​	S T ≥ X 3 S_T\geq X_3 ST​≥X3​
Long 1 c 1 c_1 c1​	− c 1 -c_1 −c1​	S T − X 1 − c 1 S_T-X_1-c_1 ST​−X1​−c1​	S T − X 1 − c 1 S_T-X_1-c_1 ST​−X1​−c1​	S T − X 1 − c 1 S_T-X_1-c_1 ST​−X1​−c1​
Short 2 c 2 c_2 c2​	2 c 2 2c_2 2c2​	2 c 2 2c_2 2c2​	2 c 2 − 2 ( S T − X 2 ) 2c_2-2(S_T-X_2) 2c2​−2(ST​−X2​)	2 c 2 − 2 ( S T − X 2 ) 2c_2-2(S_T-X_2) 2c2​−2(ST​−X2​)
Long 1 c 3 c_3 c3​	− c 3 -c_3 −c3​	− c 3 -c_3 −c3​	− c 3 -c_3 −c3​	S T − X 3 − c 3 S_T-X_3-c_3 ST​−X3​−c3​
Total	2 c 2 − c 1 − c 3 2c_2-c_1-c_3 2c2​−c1​−c3​	S T − X 1 + 2 c 2 − c 1 − c 3 S_T-X_1+2c_2-c_1-c_3 ST​−X1​+2c2​−c1​−c3​	2 X 2 − X 1 − S T + 2 c 2 − c 1 − c 3 2X_2-X_1-S_T+2c_2-c_1-c_3 2X2​−X1​−ST​+2c2​−c1​−c3​	2 X 2 − X 1 − X 3 + 2 c 2 − c 1 − c 3 2X_2-X_1-X_3+2c_2-c_1-c_3 2X2​−X1​−X3​+2c2​−c1​−c3​

with put options

Long 1 put 1 \text{put}_1 put1​ at X 1 X_1 X1​ + Short 2 put 2 \text{put}_2 put2​ at X 2 X_2 X2​ + Long 1 put 3 \text{put}_3 put3​ at X 3 X_3 X3​, X 1 < X 2 < X 3 X_1<X_2<X_3 X1​<X2​<X3​, usually X 2 = ( X 1 + X 3 ) / 2 X_2=(X_1+X_3)/2 X2​=(X1​+X3​)/2

	S T < X 1 S_T<X_1 ST​<X1​	X 1 ≤ S T < X 2 X_1\leq S_T<X_2 X1​≤ST​<X2​	X 2 ≤ S T < X 3 X_2\leq S_T<X_3 X2​≤ST​<X3​	S T ≥ X 3 S_T\geq X_3 ST​≥X3​
Long 1 p 1 p_1 p1​	X 1 − S T − p 1 X_1-S_T-p_1 X1​−ST​−p1​	− p 1 -p_1 −p1​	− p 1 -p_1 −p1​	− p 1 -p_1 −p1​
Short 2 p 2 p_2 p2​	2 p 2 − 2 ( X 1 − S T ) 2p_2-2(X_1-S_T) 2p2​−2(X1​−ST​)	2 p 2 − 2 ( X 1 − S T ) 2p_2-2(X_1-S_T) 2p2​−2(X1​−ST​)	2 p 2 2p_2 2p2​	2 p 2 2p_2 2p2​
Long 1 p 3 p_3 p3​	X 3 − S T − p 3 X_3-S_T-p_3 X3​−ST​−p3​	X 3 − S T − p 3 X_3-S_T-p_3 X3​−ST​−p3​	X 3 − S T − p 3 X_3-S_T-p_3 X3​−ST​−p3​	− p 3 -p_3 −p3​
Total	X 1 − 2 X 2 + X 3 − p 1 + 2 p 2 − p 3 X_1-2X_2+X_3 -p_1+2p_2-p_3 X1​−2X2​+X3​−p1​+2p2​−p3​	S T + X 3 − 2 X 2 − p 1 + 2 p 2 − p 3 S_T+X_3-2X_2 -p_1+2p_2-p_3 ST​+X3​−2X2​−p1​+2p2​−p3​	X 3 − S T − p 1 + 2 p 2 − p 3 X_3-S_T-p_1+2p_2-p_3 X3​−ST​−p1​+2p2​−p3​	2 p 2 − p 1 − p 3 2p_2-p_1-p_3 2p2​−p1​−p3​

4.2.7 Calendar Spread(with call/put option)

Short 1 T 1 T_1 T1​ call/put + Long 1 T 2 T_2 T2​ call/put, and T 1 < T 2 T_1<T_2 T1​<T2​

Calendar spread has a long and a short position with exercise price, but the maturity of long position is longer than that of short position.

4.3 Combined Strategies

4.3.1 Straddle

Long 1 call + long 1 put, both at X X X, wants volatility

	S T < X S_T<X ST​<X	S T ≥ X S_T\geq X ST​≥X
Long 1 c c c	− c -c −c	( S T − X ) − c (S_T-X)-c (ST​−X)−c
long1 p p p	( X − S T ) − p (X-S_T)-p (X−ST​)−p	− p -p −p
Total	X − S T − c − p X-S_T-c-p X−ST​−c−p	S T − X − c − p S_T-X-c-p ST​−X−c−p

4.3.2 Strangle

Long 1 call at X 2 X_2 X2​+ long 1 put at X 1 X_1 X1​, usually X 1 < X 2 X_1<X_2 X1​<X2​, like straddle, but cheaper

	S T < X 1 S_T<X_1 ST​<X1​	X 1 ≤ S T < X 2 X_1\leq S_T<X_2 X1​≤ST​<X2​	S T ≥ X 2 S_T \geq X_2 ST​≥X2​
Long 1 C ( X 2 ) C(X_2) C(X2​)	− c -c −c	− c -c −c	( S T − X 2 ) − c (S_T-X_2)-c (ST​−X2​)−c
Short 1 P ( X 1 ) P(X_1) P(X1​)	( X 1 − S T ) − p (X_1-S_T)-p (X1​−ST​)−p	− p -p −p	− p -p −p
Total	( X 1 − S T ) − c − p (X_1-S_T)-c-p (X1​−ST​)−c−p	− c − p -c-p −c−p	( S T − X 2 ) − c − p (S_T-X_2)-c-p (ST​−X2​)−c−p

5. Exotic Options

Definition of Exotic Options
Plain vanilla options: standara European and Amercian options on exchanges.

Exotic options: options with non-standard properties which are designed bu derivatives dealers to meet the specific needs of their clients and are ususlly traded in the OTC markets.

Transformation of American Options
Standard Amercian option has a fixed exercise price and can be exercised at any time during option’s life.

Nonstandard options make exceptions:

• Early exercise may be restricted to certain dates: Bermuda option
• Early exercise may be allowed during only part of the life of the option-lockout period: Employee stock option.
• Strike price may change during the life of an option: the warrants issued by corporations.

5.1 Single Asset Exotics

5.1.1 Package

A package is a portfolio consisting of plain vanilla options on an asset, such as bull spreads, bear spreads, butterfly spreads, calendar spreads, straddles, and strangles. Packages are sometimes regarded as exotic options because they are positions built to reflect a specific market view and risk tolerance.

For example, an investor who takes a long position in a butterfly spread is acting on his or her belief that the future asset price will be near the middle strike price. At the same time, the investor is building this position without taking on a great deal of risk. In contrast, an investor with a similar view of the market who chooses to sell a straddle or a strangle is taking on much more risk.

5.1.2 Zero-Cost Products

Any derivative product can be converted into a zero-cost product by arranging for it to be paid for in arrears.

Example: A derivative that matures at time T and has a premium equal to f f f. The derivative can be structured for the buyer to pay f ( 1 + R ) T f(1 + R)^T f(1+R)T at maturity, rather than requiring that the premium be paid upfront.

This is known as a futures-style option.

5.1.3 Forward Start Options

A forward start option is an option that will begin at a future time. It is essentially a forward on an option.

It is usually stated that the option will be at-the-money at the time it starts.

Employee stock options can be forward start options if an employer promises that they will be granted on future dates.

5.1.4 Cliquet Options

A Cliquet option is a series of forward start options with certain rules for determining the strike prices.

Example: Five one-year call options with starting now, in one-year, in two-years, in three-years and in four-years, respectively.

A simple rule for the strike prices could be that each option is initially at-the-money.

5.1.5 Chooser Options

Long position of chooser options has the right to choose whether the option is a call or put. The value of the chooser option at this point is Max ( c , p ) \text{Max}(c, p) Max(c,p).

5.1.6 Asian Options

Asian options provide a payoff dependent on an arithmetic average of the underlying asset price during the life of the option.

Asian option is less expensive than a regular option.

• Average price calls: payoff = Max ( S ave − X , 0 ) =\text{Max}(S_{\text{ave}}-X,0) =Max(Save​−X,0)
• Average price puts: payoff = Max ( X − S ave , 0 ) =\text{Max}(X-S_{\text{ave}},0) =Max(X−Save​,0)
• Average strike calls: payoff = Max ( S T − S ave , 0 ) =\text{Max}(S_T-S_{\text{ave}},0) =Max(ST​−Save​,0)
• Average strike puts: payoff = Max ( S ave − S T , 0 ) =\text{Max}(S_{\text{ave}}-S_T,0) =Max(Save​−ST​,0)

5.1.7 Lookback Options

Lookback options provide a payoff depends on the maximum or minimum asset price reached during the life of the option.

Lookback options are more expensive than regular options.

The value of a lookback option depends on how often the price of the underlying asset is observed. A lookback option increases in value as the observation frequency is increased.

• Fixed lookback calls: payoff = Max ( S max − X , 0 ) =\text{Max}(S_{\text{max}}-X,0) =Max(Smax​−X,0)
• Fixed lookback puts: payoff = Max ( X − S min , 0 ) =\text{Max}(X-S_{\text{min}},0) =Max(X−Smin​,0)
• Floating lookback calls: payoff = Max ( S T − S min , 0 ) =\text{Max}(S_T-S_{\text{min}},0) =Max(ST​−Smin​,0)
• Floating lookback puts: payoff = Max ( S max − S T , 0 ) =\text{Max}(S_{\text{max}}-S_T,0) =Max(Smax​−ST​,0)

5.1.8 Compound Options

Compound options means options of options, the first option has underlying asset of the second option, and the second option has underlying of other asset. Usually it has 2 layers. It takes similar logic as the FOF or MOM.

The advantages of compound options are that they allow for large leverage.

• Call on call: investor has the right to buy the underlying call option on the expiration date.
• Call on put: investor has the right to buy the underlying put option on the expiration date.
• Put on call: investor has the right to sell the underlying call option on the expiration date.
• Put on put: investor has the right to sell the underlying put option on the expiration date.

5.1.9 Gap Options

Gap option is a European call or put option where the price triggering a payoff is different from the price used in calculating the payoff.

Suppose the trigger price is X 2 X_2 X2​ and the price used in calculating the payoff is X 1 X_1 X1​. This means

• The payoff from a call option is S T − X 1 S_T-X_1 ST​−X1​, if S T ≥ X 2 S_T\geq X_2 ST​≥X2​
• The payoff from a put option is X 1 − S T X_1-S_T X1​−ST​, if S T ≤ X 2 S_T\leq X_2 ST​≤X2​

The payoff from a gap call can be positive, if X 1 < X 2 X_1<X_2 X1​<X2​.
The payoff from a gap call can be negative, if X 1 > X 2 X_1>X_2 X1​>X2​.

The payoff from a gap put can be negative, if X 1 < X 2 X_1<X_2 X1​<X2​.
The payoff from a gap put can be positive, if X 1 > X 2 X_1>X_2 X1​>X2​.

5.1.10 Binary Options

Binary option is a type of option where the payoff is either some fixed amount of some asset or nothing at all.

Two main types of binary option are cash-or-nothing options and asset-or-nothing option.

The cash-or-nothing option pays some fixed amount of cash if the option expires in-the-money.

The asset-or-nothing option pays the value of the underlying security.

Traditional European options can be thought of as combinations of binary options.

A regular European call option is equivalent to

• A long position in an asset-or-nothing call
• A short position in a cash-or-nothing call where the cash payoff in the cash-or-nothing call equals the strike price.

A regular European put option is equivalent to

• A long position in a cash-or-nothing put
• A short position in an asset-or-nothing put where the cash payoff in the cash-or-nothing call equals the strike price.

5.1.11 Barrier Options

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

Options that cease to exist when a barrier is reached are knock-out options, and options that come into existence when a barrier is reached are knock-in options.

In-out parity is the barrier option’s answer to put-call parity. If we combine one “in” option and one “out” barrier option with the same strikes and expirations, we get the price of a vanilla option. Note that this arugment only works for Europrean options.

Unlike other simpler options, barrier options are path-dependent. That is , the value of the option at any time depends not just on the underlying at that option, but also on the path taken by the underlying.

• Down-and-out: European option that eases to exist if the asset price moves down to the barrier level.
• Down-and-in: European option that comes into existence if the asset price moves down to the barrier level.
• Up-and-out: European option that ceases to exist if the asset price moves up to the barrier level.
• Up-and-in: European option that come to existence if the asset price moves up to the barrier level.

Barrier options are less expensive than regular options.

• For knock-out options, an increase in the volatility may lower the price.
• For knock-in options, an decrease in the volatility may lower the price.

5.2 Other Types of Exotics

5.2.1 Asset-Exchange Options

In an asset-exchange option, the holder has the right to exchange one asset for another.

An option to exchange X euros for Y Australian dollars.

An offer by Company X to acquire Company Y through the exchange of a certain number of its own shares for shares of Company Y.

5.2.2 Basket Options

A basket option is an option on a portfolio of assets.

Basket options can be appropriate hedging instruments for firms seeking to reduce costs by hedging their aggregate exposure to several assets with a single trade.

Basket options are dependent on the correlation between the returns from the assets in the basket.

It has three specifications of higher leverage, lower tax burden, and flexibility.

5.2.3 Volatility&Variance Swap

A volatility swap is a forward contract on the realized volatility of an asset during a certain period. A trader agrees to exchange a pre-specified volatility for the realized volatility at the end of the period, with both volatilities being multiplied by a certain amount of principal.

A volatility swap is appropriate for a trader who wants to take a position dependent only on volatility. While plain vanilla options provide an exposure to volatility, they also depend on the price of the underlying asset. A potential advantage of volatility swaps is that their payoffs depend solely on realized volatility.

The payoff from a variance swap is calculated analogously to the payoff from a volatility swap. As a reminder, the variance rate for an asset is the square of its volatility.

5.2.4 Static Option Replication

Two portfolios that are worth the same on some boundary (function of asset price and time) must also be worth the same at all interior points.

Some exotics are easier to hedge (e.g. Asian Exotic), while some are more difficult(e.g. Barrier Exotic).

If the boundary hasn’t been reached, static options replication hedge can be left unchanged.

If the boundary is reached, The hedge portfolio must then be unwound and a new hedge should be created.

6. Binomial Trees

6.1 One-step Binomial Trees

6.1.1 Definition of Binomial Model

Binomial model: after one period, the value of the underlying asset will either go up to S u S_u Su​ or go down S d S_d Sd​.

6.1.2 No-arbitrage Argument

Law of one price: Assets that produce identical(完全相同的) future cash flows regardless of future everts should have the same price.

We can use option and stock to construct a portfolio which has certain value after one period. If there is no arbitrage opportunity, the option price can be derived from the cost of this portfolio.

Example: The stock is currently trading at $ 50 50 50. The stock price will either go up to $ 75 75 75 or go down to $ 25 25 25 after one year. The risk-free rate is 2 % 2\% 2%. Please calculate the value of a 1-year European call option with an exercise price of $ 60 60 60 using one-step binomial tree.

We create risk-neutral portfolio, Long Δ \Delta Δ stock and short 1 1 1 call option.

The value of the portfolio after 1 year: Δ ∗ S u − C u = Δ ∗ S d − C d → Δ = 0.3 \Delta*S_u-C_u=\Delta*S_d-C_d \to \Delta=0.3 Δ∗Su​−Cu​=Δ∗Sd​−Cd​→Δ=0.3

The portfolio value at t 0 t_0 t0​: 50 × 0.3 − c 50\times0.3-c 50×0.3−c

The portfolio value at t 1 t_1 t1​: 75 × 0.3 − 15 = 7.5 75\times0.3-15=7.5 75×0.3−15=7.5

Thus, ( 50 × 0.3 − c ) × e 0.02 = 7.5 → c = 7.6485 (50\times0.3-c)\times e^{0.02}=7.5 \to c=7.6485 (50×0.3−c)×e0.02=7.5→c=7.6485

Calculata Δ \Delta Δ of a stock option: Δ × S u − C u = Δ × S d − C d \Delta \times S_u-C_u=\Delta \times S_d-C_d Δ×Su​−Cu​=Δ×Sd​−Cd​

Δ = C u − C d S u − S d \Delta=\frac{C_u-C_d}{S_u-S_d} Δ=Su​−Sd​Cu​−Cd​​

6.1.3 Risk Neutral Valuation

In a risk neutral world, all individuals are indifferent to risk. Investors requires no compensation for risk and expected return on all securities is the risk-free interest rate.

The option should be valued based on risk neutrality.

In a risk-neutral world, the expected return on all securities is the risk-free interest rate, and the discount rate is the risk-free interest rate.

u = e σ t , d = 1 u = e − σ t u=e^{\sigma\sqrt{t}}\;, d=\frac{1}{u}=e^{-\sigma\sqrt{t}} u=eσt ​,d=u1​=e−σt ​

The higher the standard deviation, the greater the dispersion between stock prices in up and down states.
π u × S 0 × u + ( 1 − π u ) × S 0 × d = S 0 e r t → π u = e r t − d u − d \pi_u\times S_0 \times u+(1-\pi_u)\times S_0 \times d=S_0e^{rt}\to \pi_u=\frac{e^{rt}-d}{u-d} πu​×S0​×u+(1−πu​)×S0​×d=S0​ert→πu​=u−dert−d​

6.1.4 Risk neutral valuation-Process

The valuation process of one-step binomial tree as follow:

• Calculate u u u and d d d, then construct the whole binomial tree.
• Calculate the payoff of the option at the maturity node.
• Compute the risk-neutral up and down probability, then calculate the expected value of option in one period.
• Use risk-free rate to discount the expected value to present.

6.2 Two-step Binomial Trees

6.2.1 Two-Step Binomial Model

The basic valuation process of a two-step European option is similar with one-step binomial model, but with more steps.

6.2.2 Two-Step Binomial Model - European

An analyst is using two-step binomial model to calculate the price of a 2-year European put option with strike price of $ 75 75 75. The continuously compounded risk-free rate is 5 % 5\% 5%. The stock pays no dividend and is trading at $ 70 70 70. The volatility of the stock price is 20 % 20\% 20%. What will be the European put option price?

Firstly, calculate u u u and d d d, then construct the whole binomial trees:
u = e σ T = e 0.02 ∗ 1 = 1.2214 , d = 1 u = e − σ T = 0.8187 u=e^{\sigma \sqrt{T}} = e^{0.02 * \sqrt{1}} =1.2214, d=\frac{1}{u}=e^{-\sigma \sqrt{T}}=0.8187 u=eσT ​=e0.02∗1 ​=1.2214,d=u1​=e−σT ​=0.8187

Secondly, calculate the payoff of the option at the maturity node:

Thirdly, compute the risk-neutral up and down probability, and then calculate the expected value in one period.
π u = e r t − d u − d = e 5 % × 1 − 0.8187 1.2214 − 0.8187 = 57.7529 % \pi_u=\frac{e^{rt}-d}{u-d}=\frac{e^{5\%\times1}-0.8187}{1.2214-0.8187}=57.7529\% πu​=u−dert−d​=1.2214−0.8187e5%×1−0.8187​=57.7529%
π d = 1 − π u = 1 − 57.7528 % = 42.2471 % \pi_d=1-\pi_u=1-57.7528\%=42.2471\% πd​=1−πu​=1−57.7528%=42.2471%

At last, use risk-free rate to discount the expected value to present.
Node 2 = ( 0 × 57.7529 % + 5 × 42.2471 % ) e − 5 % = 2.0093 \text{Node}2=(0\times 57.7529\%+5\times 42.2471\%)e^{-5\%}=2.0093 Node2=(0×57.7529%+5×42.2471%)e−5%=2.0093

Node 3 = ( 5 × 57.7529 % + 28.0811 × 42.2471 % ) e − 5 % = 14.0317 \text{Node}3=(5\times 57.7529\%+28.0811\times 42.2471\%)e^{-5\%}=14.0317 Node3=(5×57.7529%+28.0811×42.2471%)e−5%=14.0317

Node 1 = ( 2.0093 × 57.7529 % + 14.0317 × 42.2471 % ) e − 5 % = 6.7427 \text{Node}1=(2.0093\times 57.7529\%+14.0317\times 42.2471\%)e^{-5\%}=6.7427 Node1=(2.0093×57.7529%+14.0317×42.2471%)e−5%=6.7427

The European Put Option price is 6.7427 6.7427 6.7427

6.2.3 Two-Step Binomial Model - American

We need to determine if the option will be exercised at each node including Node 1.

Node 1 = ( 2.0093 × 57.7529 % + 17.691 × 42.2471 % ) e − 5 % = 8.2133 \text{Node}1=(2.0093\times 57.7529\%+17.691\times 42.2471\%)e^{-5\%}=8.2133 Node1=(2.0093×57.7529%+17.691×42.2471%)e−5%=8.2133

Node1 should also be checked. In this case, if there is an early exercise in Node 1, payoff will be 5 which is less than 8.2133. Therefore, the option will not be exercised early in Node 1 and value of this American put option is 8.2133.

6.2.4 As time periods are added

Suppose that a binomial tree with n n n steps in its life T, if n n n approaches infinity, the length of each step approaches to zero. A continuous binomial tree will be achieved and this is one of the ways that derive the Black-Scholes-Merton model.

6.2.5 Options on Other Assets

Options on stock indices with continuous dividend yield q q q

π u × S 0 × u + ( 1 − π u ) × S 0 × d = S 0 e ( r − q ) t \pi_u\times S_0 \times u+(1-\pi_u)\times S_0 \times d=S_0e^{(r-q)t} πu​×S0​×u+(1−πu​)×S0​×d=S0​e(r−q)t
π u = e ( r − q ) t − d u − d \pi_u=\frac{e^{(r-q)t}-d}{u-d} πu​=u−de(r−q)t−d​

Options on currencies with thedomestic risk-free rate R DC R_{\text{DC}} RDC​ and foreign risk-free rate R FC R_{\text{FC}} RFC​

π u = e ( r DC − r FC ) t − d u − d \pi_u=\frac{e^{(r_{\text{DC}}-r_{\text{FC}})t}-d}{u-d} πu​=u−de(rDC​−rFC​)t−d​

Option on futures
π u = 1 − d u − d \pi_u=\frac{1-d}{u-d} πu​=u−d1−d​

7. The Black-Scholes-Merton Model

7.1 Assumption

7.1.1 Stock Price Movements

The stock prices are often modeled by geometric Brownian motion (GBM).

The stock price S T S_T ST​, is log-normally distributed. A variable that is log-normally distributed has a minimum of zero and conforms to prices better than the normal distribution which would produce negative prices.

If the stock price is log-normally distributed, the continuously compounded annual return of the stock is normally distributed.

7.1.2 Historical Volatility

Square root rule: the volatility for short periods of time can be scaled to longer periods in time by the square root rule.
σ J − p e r i o d s = σ 1 − p e r i o d × J 0.5 \sigma_{J-periods}=\sigma_{1-period}\times J^{0.5} σJ−periods​=σ1−period​×J0.5

J J J number of the trading periods in a year.

7.1.3 Assumption

The option should be European-style.

There are no dividends during the life of the option.

The underlying stock price follows lognormal distribution and the rate of return of the stock follows normal distribution.

The continuously compounded risk-free rate is constant and known.

The volatility of the stock return is constant and known.

The market is frictionless.

• The short selling is permitted and continuous trading is available. The underlying asset is divisible.
• There are no transaction costs, no taxes and no regulatory constraints.
• There is no-arbitrage opportunities in the market.

7.2 Pricing European Option

7.2.1 European Option on A Non-dividend Stock

Call = S 0 N ( d 1 ) − X e − r T N ( d 2 ) \text{Call}=S_0N(d_1)-Xe^{-rT}N(d_2) Call=S0​N(d1​)−Xe−rTN(d2​)
Put = − S 0 N ( − d 1 ) + X e − r T N ( − d 2 ) \text{Put}=-S_0N(-d_1)+Xe^{-rT}N(-d_2) Put=−S0​N(−d1​)+Xe−rTN(−d2​)
d 1 = I n ( S 0 X ) + ( r f + σ 2 2 ) T σ T d_1=\frac{In(\frac{S_0}{X})+(r_f+\frac{\sigma^2}{2})T}{\sigma\sqrt{T}} d1​=σT ​In(XS0​​)+(rf​+2σ2​)T​
d 2 = I n ( S 0 X ) + ( r f − σ 2 2 ) T σ T d_2=\frac{In(\frac{S_0}{X})+(r_f-\frac{\sigma^2}{2})T}{\sigma\sqrt{T}} d2​=σT ​In(XS0​​)+(rf​−2σ2​)T​
d 2 = d 1 − σ T d_2=d_1-\sigma\sqrt{T} d2​=d1​−σT ​

• N ( X ) N(X) N(X): Standard normal cumulative distribution function.
• N ( d 1 ) N(d_1) N(d1​): Delta of call option
• N ( d 2 ) N(d_2) N(d2​): Risk-neutral probability of call exercise.

7.2.2 European Option on A Dividend-paying Stock

Discrete dividends
Call = ( S 0 − PVD ) N ( d 1 ) − X e − r T N ( d 2 ) \text{Call}=(S_0-\text{PVD})N(d_1)-Xe^{-rT}N(d_2) Call=(S0​−PVD)N(d1​)−Xe−rTN(d2​)
Put = − ( S 0 − PVD ) N ( − d 1 ) + X e − r T N ( − d 2 ) \text{Put}=-(S_0-\text{PVD})N(-d_1)+Xe^{-rT}N(-d_2) Put=−(S0​−PVD)N(−d1​)+Xe−rTN(−d2​)

Continuously compound dividends:
Call = S 0 e − q T N ( d 1 ) − X e − r T N ( d 2 ) \text{Call}=S_0e^{-qT}N(d_1)-Xe^{-rT}N(d_2) Call=S0​e−qTN(d1​)−Xe−rTN(d2​)
Put = − S 0 e − q T N ( − d 1 ) + X e − r T N ( − d 2 ) \text{Put}=-S_0e^{-qT}N(-d_1)+Xe^{-rT}N(-d_2) Put=−S0​e−qTN(−d1​)+Xe−rTN(−d2​)

Effects on options: Dividend payment will increase put values and decrease call values.

7.2.3 Implied Volatility

Volatility is the only parameter which we cannot get directly in the market. Four inputs in fives for BSM model is observable ( S S S, X X X, T T T, R R R), and if the option market price is available, the volatility can be calculated with the BSM model. The volatility of underlying asset return “implied” in the option market price.

Volatility estimation using historical data is not always representative of current market (backward-looking) while implied volatility is forward-looking.

8. The Greek Letters

Greek letters measure different aspects of risk in derivatives.

Delta: Sensitivity of derivatives to changes in the price of the underlying asset.

Gamma: Sensitivity of derivatives’ delta to changes in the price of the underlying asset.

Vega: Sensitivity of derivatives to changes in the implied volatility of the underlying asset.

Theta: Sensitivity of derivatives to the passage of time.

Rho: Sensitivity of derivatives to changes in the interest rates.

8.1 Delta

8.1.1 Introduction of Delta

Delta( Δ \Delta Δ) can be shown as the slope of the option price curve, relating the option price to the underlying asset price.

According to the BSM model,

• Long European call option Δ = N ( d 1 ) \Delta=N(d_1) Δ=N(d1​), which range from 0 to 1, and when it is at the money, Δ ≈ 0.5 \Delta \approx 0.5 Δ≈0.5
• Long European put option Δ = N ( d 1 ) − 1 \Delta=N(d_1)-1 Δ=N(d1​)−1, which range from -1 to 0, and when it is at the money, Δ ≈ − 0.5 \Delta \approx -0.5 Δ≈−0.5
  

When t → T t \to T t→T, delta is unstable.

8.1.2 Delta of Financial Instruments

	Without dividend	Continuous Dividend
Call Option	N ( d 1 ) N(d_1) N(d1​)	e − q T N ( d 1 ) e^{-qT}N(d_1) e−qTN(d1​)
Put Option	N ( d 1 ) − 1 N(d_1)-1 N(d1​)−1	e − q T [ N ( d 1 ) − 1 ] e^{-qT}[N(d_1)-1] e−qT[N(d1​)−1]
Stock	1	1
Forward	1	e − q T e^{-qT} e−qT
Futures	e r T e^{rT} erT	e ( r − q ) T e^{(r-q)T} e(r−q)T
Portfolio	∑ i = 1 n w i Δ i \sum^n_{i=1}w_i\Delta_i ∑i=1n​wi​Δi​

8.1.3 Delta Hedging

Delta neutral position: the delta of overall position is zero.

The portfolio combines the underlying assets with the options, and the portfolio value does not change with small volatility of the price of underlying assets.

Number of options needed to delta hedge:
Δ Hedged position = N call/put × Δ call/put + N stock × 1 = 0 \Delta_{\text{Hedged position}} = N_{\text{call/put}}\times\Delta_{\text{call/put}}+N_{\text{stock}}\times 1=0 ΔHedged position​=Ncall/put​×Δcall/put​+Nstock​×1=0
N call/put = − N stock Δ call/put N_{\text{call/put}}=-\frac{N_{\text{stock}}}{\Delta_{\text{call/put}}} Ncall/put​=−Δcall/put​Nstock​​

Dynamic-hedging: the delta measurement is a linear estimation and the delta-neutral position will no longer hold when the value of the underlying asset experiences large changes. In order to keep hedging, the portfolio has to be adjusted (rebalancing) periodically.

The cost of delta hedging comes from the fact that traders always buy stocks immediately after price risk and sell immediately after price fall. This is a ‘buy high, sell low’ strategy, which is almost certainly costly.

Static-hedging: the hedge is set ip initially and never rebalance it. It is also referred to be “hedge-and-forget”.

8.2 Gamma

8.2.1 Introduction of Gamma

Gamma( Γ \Gamma Γ): is the rate of change of the option’s delta with respect to the price of the underlying asset, and measures the curvature/stability of the option price.

Gamma is the same for call and put option. Long call/put option, Γ > 0 \Gamma>0 Γ>0, short call/put option, Γ < 0 \Gamma<0 Γ<0

Gamma is largest when an option is at-the-money. If the option id deep in or out of the money, gamma approaches to zero.

When time approaches to maturity, gamma gets higher.

8.2.2 Gamma Hedging

The gamma measures the stability of delta. A large gamma indicates that delta will be changing rapidly.

If gamma is higher, rebalancing more frequently is required.

If gamma is relatively small, delta changes slowly and rebalancing is required relatively infrequently.

Thus Gamma is used to correct the hedging error associated with delta-neutral position by providing added prectection against large movements in the underlying asset’s price.

For Neutral Position

• Delta Neutral Positions (the delta of overall position zero): hedge small changes in stock price.
• Delta and Gamma Neutral Positions (the gamma and delta of overall position are both zero): hedge larger changes in stock price.

Gamma of stocks or forward = 0

Create delta and gamma neutral position

• First, creating gamma-neutral position by non-linearly instruments, such as options and bonds. It is likely to change the portfolio’ delta.
  Γ hedged position = Γ portfolio + N call/put × Γ call/put = 0 \Gamma_{\text{hedged position}}=\Gamma_{\text{portfolio}}+N_{\text{call/put}}\times \Gamma_{\text{call/put}}=0 Γhedged position​=Γportfolio​+Ncall/put​×Γcall/put​=0
  N call/put = − Γ portfolio Γ call/put N_{\text{call/put}}=-\frac{\Gamma_{\text{portfolio}}}{\Gamma_{\text{call/put}}} Ncall/put​=−Γcall/put​Γportfolio​​

• Then, creating delta-neutral potion by linearly instruments, such as stocks and forwards.
  Δ hedged position = Δ portfolio + N call/put × Δ call/put + N stock/forward × 1 = 0 \Delta_{\text{hedged position}}=\Delta_{\text{portfolio}}+N_{\text{call/put}}\times \Delta_{\text{call/put}}+N_{\text{stock/forward}} \times 1=0 Δhedged position​=Δportfolio​+Ncall/put​×Δcall/put​+Nstock/forward​×1=0

N stock/forward = − ( Δ portfolio + N call/put × Δ call/put ) N_{\text{stock/forward}}=-(\Delta_{\text{portfolio}}+N_{\text{call/put}}\times \Delta_{\text{call/put}}) Nstock/forward​=−(Δportfolio​+Ncall/put​×Δcall/put​)

8.3 Vega

8.3.1 Introduction of Vega

Vega is the rate of change of the value of the option with respect to the volatility of the underlying asset.

Vega of a call is equal to the Vega of a put.

If Vega is highly positive or highly negative, the portfolio’s value is very sensitive to small changes in volatility.

Vega is greatest for options that are closely at the money. While tends to zero as the option moves deep in and out of money.

Vega of options with longer maturity is relatively larger.

Unfortunately, hedging vega risk is not as easy as hedging delta risk. The vega of a position in the underlying asset is zero. This means that trading the underlying asset does not affect the vega of a portfolio of derivatives dependent on the asset. Vega can only be adjusted by taking a position in another derivative dependent on the same asset

8.3.2 Compare Vega with Gamma

Both Gamma and Vega are positive for a long position in either a call option or put option.

Both Gamma and Vega are largest when the option is at the money, and approach zero as the option moves deep in or out of the money.

Important difference is that while Vega increases as the time to maturity increases, Gamma decreases.

8.4 Theta

Theta( θ \theta θ) is the rate of change of the value of the option with respect to the passage of time (time decay).

As time passed, most options tend to become less valuable, so theta is usually negative for an option.

There is no uncertainty about the passage of time, so it usually will not hedge against the passage of time. Despite this certainty regarding time, traders do like to monitor theta. One reason for this may be that theta and gamma are negatively related for a delta­neutral portfolio. Theta therefore contains information about gamma when delta is maintained at zero. When theta is highly negative, gamma tends to be highly positive; when theta is highly positive, gamma tends to be highly negative.

For long position, θ < 0 \theta<0 θ<0, means option lose value as time goes by. Short-term at the money option has greatest negative theta.

8.5 Rho

Rho( ρ \rho ρ) is the rate of change of the value of the option with respect to the interest rate.

The impact on option prices when interest rates change is relatively small. The influence of interest rates is generally not a major concern.

Long call ρ > 0 \rho>0 ρ>0, Long put ρ < 0 \rho <0 ρ<0

Long-term in-the-money calls and puts are more sensitive to changes in interest rates than short-term out-of-the- money options.

8.6 Summary

	Delta	Gamma	Vega	Theta	Rho
Sensitive Factor	S	Delta \text{Delta} Delta	σ \sigma σ	t	r
Long Call	+ + +	+ + +	+ + +	Usually − - −	+ + +
Short Call	− - −	− - −	− - −	Usually + + +	− - −
Long Put	− - −	+ + +	+ + +	Usually − - −	− - −
Short Put	+ + +	− - −	− - −	Usually + + +	+ + +
When t → T t \to T t→T	More Unstable	Increase	Decrease	Increase θ \theta θ

8.7. Others

8.7.1 Naked and Covered Position

If you sell a call option without owning the underlying asset, you hold a naked option.

If you sell a call option while owning the underlying asset, you have a covered position.

8.7.2 Stop-loss Strategy

Stop-loss Strategy hedges the short position of a call option by buying an underlying stock when its price rises above strike price K K K and selling it immediately when its price falls below strike price K K K.

The objective is to hold a naked position whenever the stock price is less than K K K and a covered position whenever the stock price is greater than K K K.

S < K S<K S<K → \to → naked strategy, S > K S>K S>K → \to → covered strategy

Although “superficially attractive”, the strategy becomes too expensive if the stock price crosses the strike price level many times.

8.7.3 Portfolio Insurance

A put option on portfolio can provide protection against the market declines while still keeping the potential for a gain. In addition to buying put positions within the portfolio, manager can construct by synthesizing a put position.

Using underlying assets to create reverse position which has a delta equal to the delta of the required option.

Using index futures to create reverse position which has a delta equal to the delta of the required option.

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