Trading with Options II1 Trading with Options session II TAEHOON KANG Department of Economics...

Trading with Options II 1

Trading with Options session II

TAEHOON KANG

Department of Economics

Keimyung University


UTILITY & RISKUTILITY & RISK


WHICH OF THE TWO CASES WOULD YOU CHOSE?

– For 10 bucks invested you either:

A. Lose 10 or Make 40

B. Lose 20 or Make 30


WHICH OF THE TWO CASES WOULD YOU CHOOSE?

– For 10 bucks invested you either:

A. Lose 10 (80%)

or Make 40 (20%)

B. Lose 20 (20%)

or Make 30 (80%)


WHICH OF THE TWO CASES WOULD YOU CHOOSE?

– For 10 bucks invested you either: (If you lose $20 you lose your job)

A. Lose 10 (80%)

or Make 40 (20%)

B. Lose 20 (20%)

or Make 30 (80%)


Are You Willing to Pay $8 to Change

Case B into:

B*. Make 10 (20%) or Make 30 (80%)

Net: Make 2 (20%) or Make 22 (80%)


Utility is composed of following strategic variables

– objectives

– views

– regrets


Objectives

Needs Desires Limitations-Protection-Benchmark-Satisfy Clients

-Upside Profit-GreaterAverage Yield-LowerFunding Costs

-Cash-Capital-Credit-Taxes-Regulatory


Views

•Spot Market

•Basis

•Volatility


Regrets

•Suffering Avoidable Losses

•Not Participating in Upside Gains


Risky Financial Environments• Foreign exchange rates - FX volatility increase since 1973

- British withdrawal from EMS in 1992

- Mexico peso crisis in Dec 1994

-The dollar touched bottom in Mar 1995 at 1.3455 DM and in Apr 1995 at 79.75 yen, now 1.7000 DM and 126 yen at May 1997.


Risky Financial Environments

-Laker Airlines Excess demand from UK travelers in late

1970 when weak dollar.

Purchased DC-10s financed by dollar.

In early 1980’s, rising $ against pound forced it into bankruptcy.


Risky Financial Environments• Interest rates -Rate uncertainty increase.

-S&L crisis(yield curve risk)

Short-term deposits and long-term fixed rate mortgages.

In 1970, upward sloping yield curve.

In 1980, yield curve inverted.

-Construction and real estate companies are highly exposed to interest rate risk.


WHAT IS RISK?

- Any Variation in an Outcome

- Both Adverse and Benign

Developments in Market Rates


WHAT IS RISK?

How Much is Risk “Worth”

It Depends!


Invest $1,000 For a Year 1. At 10% 1,100

2. 50% Chance of 20% 1,200

50% Chance of 0 1,000

3. 99% Chance of 11.2% 1,112

1% Chance of Lose All 0


THEREFORE...• Risk is neither created nor destroyed!

It is only transferred to a more manageable form or exchanged for expected return.

• Risk can easily be “transferred”, but any reduction of risk results in the reduction of expected return.


Option Features• Option enables the holder to avoid the bad returns

while to keep the good returns.

Call Holder Put Holder


Option Features

• Option separates the right from the obligation.• Asymmetry of returns.

short futures Long futuresProfit

Loss

right

obligation


Option Features

right only

obligation only

long put long call

short put short call


Option Features

• Option requires the premium.• Option buyers pay for the option at time of purchase.

No margin is required.• Option sellers receive the price of the option but must

post margin. The margin requirement is revalued daily.


Option Pricing Models

• Analytical Pricing Model

• Numerical Pricing Model

• Analytical Approximation


Analytical Pricing Models

Black-Scholes Option Pricing ModelGeneralization of the Balck-Scholes Model(Dividends, Taxes, Transaction costs, Interest rates, Short sales, Continuity of market and stock prices, Distribution assumptions, Volatility) Extensions to the Black-Scholes Model

Options on Futures (Black and Scholes)

Options on Currencies (Garman and Kohlhagen)

Compound Options (Geske)

Path-dependent options(Goldman, Sosin, and Gatto)

Lookback option, Average option, Barrier options

American style options (Roll; Geske; Whaley)


Numerical & Analytic Approximation

Numerical Pricing Models Binomial Models(Sharpe; Cox, Ross, and Rubinstein)

Finite Difference Mothodology (Schwartz; Courtadon)

Monte Carlo Simulation (Boyle)

Analytic Approximation Models (Macmillan; Giovanni-Barone-Adesi and Whaley)


Black-Scholes Option Pricing Model

Assumptions:• Continuous price process an

d lognormality• Constant volatility• No restrictions on short-selli

ng• No transaction cost or taxes• Continuous trading• No arbitrage opportunity• No dividend• Constant interest rate

Factors affecting option value:• Underlying Prices (S)• Exercise Prices(X)• Time to Expiration (t)• Volatility (2)• Risk-free interest rate (r)• Dividends(q)


Option FormulaC = S e-q(T-t) N(d1) - X e-r(T-t) N(d2)

where

d1 = [ ln(S/X) + (r-q+( 2/2)(T-t) ] / ((T-t)1/2)

d2 = d1 - ((T-t)1/2 )

First term: probability weighted estimate of asset price at expiration.

Second term: probability weighted discounted exercise price at expiration.

P = X e-r(T-t) N(-d2) - S e-q(T-t) N(-d1)


OPTION TRADING

• Spread Trading

• Volatility Trading

• Arbitrage Trading

• Cap, Floor, Collar

• Swaption

• Exotic Option


Spread Trading

• Vertical spread : buy an option and sell a similar option but with a different strike price.

• Horizontal spread: buy an option and sell a similar option but with a different maturity date.

• Diagonal spread: buy an option and sell a similar option but sell a different strike price and maturity date.


Spread Trading

Mar Apr May June Sep

790

780

770

760

750

+1 -1

Vertical+1

-1

+1

-1Diagonal

Horizontal


Vertical spread

Bull call spread Bear call spread

buy lower call sell higher call buy higher call sell lower call

Directional Trading


Horizontal spread

Buy long-dated option, Sell short-dated option

If little movement in underlying asset,both time decayed.Gain from short-dated option >Loss from long-dated option.

Using time-decay of two similaroptions, gain from the one with little time-decay.

Time-decay


Diagonal spread

Bullish, little-movement Bearish, little-movement

Sell short-dated higher callBuy long-dated lower call

Sell short-dated lower callBuy long-dated higher call

Mixture


Volatility Trading

• Straddle

• Strangle

• Long Butterfly

• Ratio Spread


Straddle

Long Straddle: buy a put option and a call option with identical factors.

Short Straddle: sell a put option and a call option with identical factors.

long put long call

short callshort put


Strangle

Long Strangle: same as long straddle but using options with different strike prices

Short Strangle: same as short straddle but using options with different strike prices

short OTM callshort OTM put

long OTM put long OTM call


ButterflyBuy an option at the lowest strike price, sell two options

at the middle strike price, and buy an option at the highest strike price.

– Variation of straddle– Lower risk, lower return

낮은 콜 1 개 매수

높은 콜 1 개 매수

중간 콜 2 개 매도


Ratio Spread

Buy an option and sell multiple of the same option at a different strike price


Put-Call ParityPortfolio A: C + XD

Portfolio B: P + S

Value at maturity:

S* < X S* > X

Portfolio A: X (S*-X)+X

Portfolio B: (X-S*)+S* S*

X S*

C + XD = P + S

C - P = S - XD = F (Long call + Short put = Long

futures)


Put-Call Parity

Buy the call and short the put and a stock.

$31 - $3 +$ 2.25 = $30.25

$30.25e0.1X0.25 = $31.02

A stock worth $31

risk-free rate 10% p.a.

$30 call at $3 (3-month)

$30 put at $2.25 (3-month)


Put-Call Parity

A: S* > X(=30)

Exercise call to buy one share for $30.

Net: $31.02 - $30.00 = $1.02

B: S* < X(=30)

Put to be exercised, so buy one share for $30.

Net: $31.02 - $30.00 = $1.02


Put-Call Parity

Uses of Put-Call Parity

C = F + P

-P = F - CA trader wants to buy a call option, but the call is illiquid.

She can buy a put at the desired strike along with futures.


Basic Call Option PricingBasic Call Option Pricing

1. Futures price =100

2. Exercise price =101

3. Volatility = 5 points per period

4. Periods to expiration = 1

5. Short-term interest rate = 15% per period.

Example: a one period call


Basic Call Option Pricing

F = 100

F = 105

F = 95

Strike=101

(C=4)

(C=0)

Today Expiration

50% chance of the futures prices move up or down



The expected value of the option Today:

0.5x4 + 0.5x0 = 2.0

Present value of the option:

2.0 / (1 + 0.15) = 1.74



1. Futures price =100

2. Exercise price =101

3. Volatility = 5 points per period

4. Periods to expiration = 2

5. Short-term interest rate = 15% per period.

Example: a two period call



Today Period1

F=110

F = 100

F = 105

F = 95

Strike=101(C=3.91)

(C=0)

F=100

F=90

Period2(Expiration)

(C=9)

(C=0)

(C=0)

(C=1.70)


The value of an option depends mainly on

-the likelihood that the option will finish in the money

-the difference between F and X if finishes in ITM

Volatility


Volatility


Volatility

Rule of Thumb

A doubling of daily price volatility will have the same effect on the price of an option as a quadrupling of the option’s time remaining to expiration.


Types of Volatility

Historical volatilitya statistical measure of the past variability of a futures price

Implied volatilitya measure of how variable option traders expect a futures price to be over

the remaining life of an option

Realized volatilitya statistical measure of the variability of a futures price after an option

position has been put on


Types of Volatility

과거 현재 미래

Historical Volatility

Implied Volatility

RealizedVolatility


Historical Volatility

Calculation

F1 = 82.05

F2 = 83.10

F3 = 82.70

R2 = (F2-F1)/F1 = (83.10 - 82.05)/82.05 = 0.0128(or 1.28%)

R3 = (F3-F2)/F2 = (82.70 - 83.10)/83.10 = -0.0048(or -0.48%)

Daily volatility = [(R22 + R3

2)]1/2 = [(0.0128)2+(-0.0048)2]1/2 = 0.0137

Annualized volatility = 0.0137x(250)1/2 = 0.2166 (or 21.66%)


Implied Volatility

InputCurrent Futures Price

Strike Price

Periods to Expiration

Short-term Interest Rate

Volatility

OutputTheoretical Option Price

Model

InputCurrent Futures Price

Strike Price

Periods to Expiration

Short-term Interest Rate

InputTheoretical Option Price

Model

Implied Volatility


Distribution of Returns

1 standard deviation above and below the current futures price is a range that contains about 68% of all returns.



Example: volatility of yen futures 10%, traded at .75.

Then after one year, there is a 68% chance that the futures price will be between .825(=.75x1.1) and .682(=.75/1.1). (picture)


• Delta

• Gamma

• Zeta (Vega, Kappa)

• Theta

Option Risk Parameters


Delta기초물의 가격이 1 포인트 변할 때옵션가치의 변화를 측정 .

행사가격이 100무위험 이자율이 8% 일 때의 콜옵션델타 .

기초물 변동성 만기 ( 단위 : 주 )가격 (%/ 년 ) 8 4 2 0106 20 .78 .86 .94 1.00

10 .94 .99 1.00 1.00104 20 .70 .77 .86 1.00

10 .84 .93 .98 1.00102 20 .61 .65 .71 1.00

10 .70 .77 .86 1.00100 20 .51 .51 .51

10 .50 .50 .5098 20 .41 .36 .30 0.00

10 .30 .23 .14 0.0096 20 .31 .23 .14 0.00

10 .15 .07 .01 0.0094 20 .22 .13 .05 0.00

10 .06 .01 .00 0.00


Delta

Long call options have positive deltas.

Long put options have negative deltas.

(Long futures have a delta of 1.0. Short futures have -1.0)

Other definition of DELTA

- the probability of the option will end up ITM.

- a hedge ratio

- the option’s futures equivalent


Delta

Because the probability of ending ITM is related to the volatility of the underlying futures price,

DELTA is related to :

- the level of volatility

- time to expiration

- the amount by which the option is in or out of the money.


Gamma

Gamma is the change in an option’s delta for a 1 point change in the price of the under-lying future.

Change Volatility Weeks to Expirationin future (%/year) 8 4 2 0

104-106 20 .04 .05 .04 0 10 .05 .03 .01 0

102-104 20 .05 .06 .08 0 10 .07 .08 .06 0

100-102 20 .05 .07 .10 0 10 .10 .14 .18 0

98-100 20 .05 .08 .11 0 10 .10 .14 .18 0

96-98 20 .05 .07 .08 0 10 .08 .08 .07 0

94-96 20 .05 .05 .05 0 10 .05 .03 .01 0

Gammas for calloption with an exercise price of 100.The cost of capital is 8%.


Gamma

A call option with a delta of 0.6 and a gamma of 0.05

will have a delta of 0.65 if futures goes up 1 point and

a delta of 0.55 if futures down 1 point.

An option’s gamma is always positive.

Gamma is added to an option’s delta as the market goes up and

is subtracted from the delta as the market goes down.


Zeta

Futures Volatility Weeks to ExpirationPrice (%/year) 8 4 2 0106 20 .120 .060 .020 .000

10 .060 .020 .000 .000104 20 .140 .080 .040 .000

10 .100 .040 .010 .000102 20 .150 .100 .060 .000

10 .140 .080 .040 .000100 20 .150 .110 .070 .000

10 .150 .110 .070 .00098 20 .150 .100 .060 .000

10 .130 .080 .040 .00096 20 .130 .080 .040 .000

10 .090 .130 .010 .00094 20 .110 .050 .020 .000

10 .040 .010 .000 .000

Zeta is the change in the value of an option for a 1 % change in volatility.

Zetas for an American calloption with an exercise price of 100.The cost of capital is 8%.


Zeta

An option with a zeta of .200 would increase in value by .200 for a 1 % change in futures volatility and

would lose .200 in value if futures volatility fell by 1% point.

Zeta is greatest for options that are at the money.

Zeta decreases as the option moves in or out of the money.

Zeta decreases as the time to expiration decreases.

Zeta for ITM and OTM options increases as the level of volatility increases.

Zeta is always positive.

(picture of effect of volatility increasing)


Zeta

Futures Volatility Weeks to ExpirationPrice (%/year) 8 6 4 2 0

104 20 5.525 5.172 4.773 4.321 4.000 10 4.295 4.186 4.083 4.010 4.000

102 20 4.125 3.8203.356 2.768 2.000 10 2.749 2.575 2.377 2.155 2.000

100 20 3.094 2.683 2.198 1.558 0.000 10 1.547 1.341 1.099 0.779 0.000

98 20 2.174 1.783 1.324 0.746 0.000 10 0.741 0.566 0.360 0.151 0.000

96 20 1.454 1.109 0.724 0.294 0.000 10 0.292 0.184 0.084 0.013 0.000

The values of an American call option with an exercise price of 100. The cost of capital is 8% per year. The values are shown in futures price points.


Theta

Futures Volatility Passage of Time in WeeksPrice (%/year) 8 to 6 6 to 4 4 to 2 2 to 0

104 20 .177 .200 .226 .161 10 .055 .052 .037 .005

102 20 .198 .232 .294 .384 10 .087 .099 .111 .078

100 20 .206 .243 .320 .779 10 .103 .121 .160 .390

98 20 .196 .230 .289 .373 10 .088 .098 .110 .076

96 20 .173 .193 .215 .147 10 .054 .050 .036 .007

Theta is also called time decay and is the change in the value of an option for a 1-day decrease in the time to expiration.


Theta

An option with a theta of .107 will lose .107 in value as the number of days to expiration decreases by 1.

Theta is greatest for ATM option.

Theta decreases as the option moves in or out of the money.

Theta increases as the time to expiration decreases.

Theta increases as the level of volatility increases.


Portfolio Risk Parameter

Risk parameters are additive.

ATM Call OTM Put

delta .50 -.35gamma .02 .01zeta .26 .15theta -.03 -.03

Long 50 Short 37 ATM Calls + OTM Puts = Portfolio

delta 25 13 38gamma 1.0 -3.7 .63zeta 13 -5.6 7.4theta -1.5 1.1 -.4


Trading Volatility - part ITrading Volatility - part I


Sources of Profit to an Option Position

The payoff to any option can be broken down into

Delta x Change in the Futures Price

+ Gamma* x Realized Volatility2

+ Zeta x Change in Implied Volatility

+ Theta x Number of Days Gone By


Straight Volatility Trade

• Option allows to trade the volatility, while futures allow only to trade the levels of the futures price.

• Get rid of the directional component

= Get rid of the position’s delta• Sell futures if the position has a positive delta

Buy futures if the position has a negative delta



Once removing direction, there remain three sources of profit to an option position.

• realized volatility• changes in implied volatility• the passage of time



• Long volatility positions will benefit

from any change in the futures price and

from any increase in implied volatility and

will lose from the passage of time.• Short volatility positions will suffer from

any change in the futures price but

will profit from any decrease in the level of implied volatility and

from the passage in time.


Net Profit from Realized Volatility

An option’s total time decay can be expressed as:

Time decay = Gamma* x Implied Volatility2

For this to be true, them implied volatility number must correspond to the time period over which time decay is calculated.



The nice relationship of the net profit from realized volatility:

Gamma* x [Realized Volatility2 - Implied Volatility2]

• The holder of a delta neutral option position will just break even if realized volatility = implied volatility.

• Long positions in a delta neutral position will more than break even if realized volatility > implied volatility.

• Short positions in a delta neutral position will more than break even if realized volatility < implied volatility.



Gamma plays a key role in determining the payoff to realized volatility.

Options that are near the money have more gamma than others.

To profit from discrepancies between realized and implied volatility, hold options whose strikes are near or around the money.


Net Profit from Changes in Implied Volatility

Zeta x Changes in Implied Volatility

The key ingredient of in implied volatility trade is the position’s zeta.

Options that are near the money have more zeta than others.

To profit from changes in implied volatility, hold options whose strikes are near or around the money.


Delta balancing

The purpose is to remove directional bias from a straight volatility trade.

How often to delta balance?

It depends on trader’s penchant for risk.


Delta balancing

0

P/L

0

A

B(delta=5)

Futures Down Futures Up

Long Volatility Position (Day1)


Delta balancing

P/L

0

A

Futures Down Futures Up

Long Volatility Position (Day2)

P/L(0)

B

CF E

D

G

P/L(+)

P/L(-)


Interplay between Realized and Implied Volatility

Value of Long DM Volatility Position After 1 Day

301 285 281 286 301287 270 265 271 287272 255 249 255 273258 240 233 240 259244 225 217 225 245

-.0110 -.0055 0 +.0055 +.0110

+2%+1%0%-1%-2%

ImpliedVolatilityChangeFrom15.8%

Futures Change from .5500

Position: Long 100 55 Sep Calls Long 100 55 Sep Puts

Short 2 Sep FuturesOriginal Cost = $255 thousand


Trading Volatility - part IITrading Volatility - part II


Volatility Cones

-maturity structure of volatility

-comparison of implied volatilities in the current lead and first deferred option contracts with

ranges of historical volatilities whose horizons correspond to the option’s different times remaining to expiration


Trade Decision

The purpose of the volatility cones is to provide a guide to

buying and selling volatility.

The first step is to decide whether to buy or sell a• straddle• strangle• butterfly

Further, establish policies to govern• frequency of delta balancing• zeta maintenance


Trade DecisionStraddles have higher gammas, zetas, and thetas than do strangle

s. They also cost more.

Strangles provide greater leverage than do straddles. Gamma, Zeta, and theta are all larger as a percentage of the cost of the strangle than they are relative to the cost of the straddle.

Short-term straddles and strangles provide relatively greater exposure to realized volatility than do long-term positions.

Long-term straddles and strangles provide relatively greater exposure to changes in implied volatility than do short-term positions.


Trade Decision

ATM option implied volatilities typically are lower than away-from-the-money implied volatilities.

ATM straddle trades at a lower measured implied volatility than does an OTM strangle.


Trade Decision

Now DM futures are trading at $.55/DM.

Suppose implied volatility for 55 DM options is 10%, for 54 and 56 DM options are 10.5%, and for 53 and 57 options are 11%.

Compare 55 straddle with 54/56 strangle(and with 53/57 strangle).

What if DM futures rise from $.55/DM to $.56/DM?

1.The 55 DM calls and puts in the straddle will trade at 10.5% from 10%.

The straddle gets unambiguous boost .

2. As the futures price rises, the puts move further OTM, and the puts will trade at 11%, or get 0.5% implied volatility, while the calls go from being OTM to being ATM and will lose 0.5% implied volatility.

Thus, the 54 puts get a boost from the smile while the 56 calls are dragged down.


Trade Decision

Recall that an option’s zeta gets smaller as it moves OTM, while gets bigger as it moves toward the money.

In the case of 54/56 strangle, as futures move from $.55 to $.56, the put is losing zeta while the call is gaining zeta.

As a result, the boost for the put is smaller than the loss on the call.

Thus, the net profit of the strangle is reduced somewhat by moving along the moneyness.

The volatility tends to favor straddles when you are buying volatility and strangles when you are selling volatility.


Delta Balancing

Approaches to delta balancing:• continuous• occasional• never


Zeta Maintenance

A trader can profit a change in implied volatility only if the option position has zeta.

Options lose their responsiveness to changes in implied volatility as they move in or out of the money.

The zeta of a volatility position should be maintained regularly as futures prices change or as the option in the position age.

This can be done either by rolling the position to strikes that are nearer the money(in the case of futures price changes) or by adding to the existing position (in the case of aging).

Zeta is largest for ATM options and for long-dated options.


EXOTIC OPTIONS


Modified condition

Some basic feature of the standard terms is varied.

- Semi-American or Bermudan option

- Digital option or Binary option : prefixed level of payoff.

All-or-nothing: should be ITM at expiry.

One-touch: ITM at any one time.

- Pay-Later or Contingent option:Premium is paid only when exercised.

- Delayed option: Right to receive later another option with the strike

price equal to the underlying asset price on that date.

- Chooser option: Right to choose later, whether the option to be call or

put. Like a straddle but cheaper.(choose one of the two).


Path-dependent option

Final payoff depends on the path taken by the underlying asset price over time.

- Average-price option or Asian option

- Average strike price option

- Lookback option

- Cliquet or Ratchet option

- Ladder option

- Shout option

- Barrier option (Knock-out, Knock-in)

down-and-in call, up-and-in put trigger price.

down-and-out call, up-in-out put barrier price.


Multi-Factor option

Final payoff depends upon the prices of two or more underlying assets.

- Rainbow or outperformance options C = max[0, (S1 ,S2 ,...,Sn ) - X ],

where S1 ,S2 ,...,Sn are n asset prices.

Maximum gain from either HangSeng, FTSE 100, CAC 40, DAX, Nikkei 225,

and S&P 500.

- Basket options

- Spread options

- Quanto options

Trading with Options II1 Trading with Options session II TAEHOON KANG Department of Economics...

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Transcript of Trading with Options II1 Trading with Options session II TAEHOON KANG Department of Economics...