Options Strategies & Risk Management for Individuals

Operations Strategies and Risk Management for Individuals

Sachin SachdevaIndependent Risk Management ConsultantGlobal Association of Risk Professionals

October 2016

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The views expressed in the following material are the

author’s and do not necessarily represent the views of

the Global Association of Risk Professionals (GARP),

its Membership or its Management.

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Options Strategies and Risk

Management for Individuals

Terminology

Basic Positions(Covered Calls, Protective Puts and Synthetic Options)

Advanced Techniques(Spreads and Straddles)

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Terminology and Notation

• Note the following standard symbols

• C = current call price, P = current put price

• S0 = current stock price, ST = stock price at expiration

• T = time to expiration

• X = exercise price

• P = profit from strategy

• The following will represent the number of calls,

puts and stock held

• NC = number of calls

• NP = number of puts

• NS = number of shares of stock

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• These symbols imply the following:

– NC or NP or NS > 0 implies buying (going long)

– NC or NP or NS < 0 implies selling (going short)

• The Profit Equations

– Profit equation for calls held to expiration

• P = NC[Max(0,ST - X) - C]

– For buyer of one call (NC = 1) this implies

P = Max(0,ST - X) - C

– For seller of one call (NC = -1) this implies

P = -Max(0,ST - X) + C

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• The Profit Equations (continued)

– Profit equation for puts held to expiration

• P = NP[Max(0,X - ST) - P]

– For buyer of one put (NP = 1) this implies

P = Max(0,X - ST) - P

– For seller of one put (NP = -1) this implies

P = -Max(0,X - ST) + P

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• The Profit Equations (continued)

– Profit equation for stock

P = NS[ST - S0]

– For buyer of one share (NS = 1) this implies

P = ST - S0

– For short seller of one share (NS = -1) this implies

P = -ST + S0

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• Different Holding Periods

– T1<T2

• Assumptions

– No dividends

– No taxes or transaction costs

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Stock Transactions• Buy Stock

– Profit equation: P = NS[ST - S0] given that NS > 0

– Maximum profit = infinite, Maximum loss = -S0

profit

Stock Price at end of holding period

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Stock Transactions

• Sell Short Stock

– Profit equation: P = NS[ST - S0] given that NS < 0

– Maximum = S0, minimum = -infinity

profit

Stock Price at end of holding period

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Call Option Transactions

• Buy a Call

– Profit equation: P = NC[Max(0,ST - X) - C] given that NC > 0. Letting NC = 1,

P = ST - X - C if ST > X

P = - C if ST X

Maximum = infinite, minimum = -C

Breakeven stock price found by setting profit equation to zero and solving: ST

* = X + C

ST

X

Profit

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• POINT TO NOTE….

– Buy a Call with differing time horizons different times

– For a given stock price, the longer the position is held, the more

time value it losses and the lower the profit.

T

T1

T2ST

Profit

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• Write a Call

– Profit equation:

P = NC[Max(0,ST - X) - C] given that NC < 0.

• Letting NC = -1,

P = -ST + X + C if ST > X

P = C if ST X

Maximum profit = +C, minimum = -infinity

Breakeven stock price same as buying call: ST

* = X + C

Profit

STX

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Put Option Transactions

• Buy a Put

– Profit equation: P = NP[Max(0,X - ST) - P] given that NP

> 0. Letting NP = 1,

P = X - ST - P if ST < X

P = - P if ST X

Maximum profit = X - P, minimum = -P


* = X - P

Profit

ST

X

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Put Option Transactions

• Write a Put

– Profit equation: P = NP[Max(0,X - ST)- P] given that NP < 0. Letting NP = -1

P = -X + ST + P if ST < X

P = P if ST X

Maximum profit = +P, minimum = -X + P


* = X - P

Profit

STX

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Holding Single Positions

– Buying a stock, taking a call or writing a put is a bullish strategy

– Short selling a stock, writing a call or taking a put is a bearish

strategy.

– If we were now to combine more than one instrument with

offsetting values this usually leads to hedging partial risk

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Calls and Stock: the Covered Call

– Purpose: Limit possible loss from writing a call / buying a stock

– Position to hold: for every call written you must buy the stock

– Lets look at the payoff table:

ST < X ST > X

Stock ST ST

Call 0 X - ST

Payoff ST X

Profit ST+C-S0 X+C-S0

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Spot PriceX

X

C-S0

0

X+C-S0

Profit

Sell a call

Buy stock

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P = NS(ST - S0) + NC[Max(0,ST - X) - C] – given NS > 0, NC < 0, NS = -NC.

• With NS = 1, NC = -1,

P = ST - S0 + C if ST X

P = X - S0 + C if ST > X

– Maximum profit = X - S0 + C, minimum = -S0 + C

– Breakeven stock price found by setting profit equation

to zero and solving: ST* = S0 - C

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Puts and Stock: the Protective Put

• Purpose: Limit possible loss from buying a stock

– Minimum Value = X

• Position to hold: Go long on both the stock and put

• Lets look at the payoff table:

ST < X ST > X

Stock ST ST

Put X - ST 0

Payoff X ST

Profit X-S0-P ST-S0-P

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STX

Profit=Payoff - (S0+P)

0S0

X-(S0+ P)

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P = NS(ST - S0) + NP[Max(0,X - ST) - P]

– given NS > 0, NP > 0, NS = NP.

• With NS = 1, NP = 1,

P = ST - S0 - P if ST X

P = X - S0 - P if ST < X

– Maximum profit = infinite, minimum = X - S0 - P

– Breakeven stock price found by setting profit equation to zero and solving: ST

* = P + S0

– Like insurance policy

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Synthetic Puts and Calls

– Rearranging put-call parity to isolate put price

– This implies put = long call, short stock, long risk-free

bond with face value X.

– This is a synthetic put.

– In practice most synthetic puts are constructed without

risk-free bond, i.e., long call, short stock.

Tr

0cXeSCP

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Synthetic Puts and Calls


P = NC[Max(0,ST - X) - C] + NS(ST - S0)

– given that NC > 0, NS < 0, NS = NP.

• Letting NC = 1, NS = -1,

P = -C - ST + S0 if ST X

P = S0 - X - C if ST > X

– Difference between an actual Put / Call and

synthetic Put / Call is marginal

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Option Spreads: Basic Concepts

• Holding Multiple Calls or Puts with differing exercise prices or time to expiration

• A Money Spread is taking one option and writing another at the same time with different exercise prices.

– The bull spread

– The bear spread

• A time spread is where one takes and writes options at the same time with differing expiration dates.

– The calendar spread

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• Why do Investors Use Option Spreads?

– Risk reduction

– To lower the cost of a long position

– Types of spreads

• bull spread

• bear spread

• time spread is based on volatility

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• Notation

– For money spreads

• X1 < X2 < X3

• C1, C2, C3

• N1, N2, N3

– For time spreads

• T1 < T2

• C1, C2

• N1, N2

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Money Spreads

• Bull Spreads

– Purpose:

• limits up and downside potential. Good for arbitrage if one option is overpriced relative to another.

– Position:

• Buy call with strike X1, sell call with strike X2. Let N1 = 1, N2 = -1

– Requires initial outlay


P= Max(0,ST - X1) - C1 - Max(0,ST - X2) + C2

: P = -C1 + C2 if ST X1 < X2

P = ST - X1 - C1 + C2 if X1 < ST X2

P = X2 - X1 - C1 + C2 if X1 < X2 < ST

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Money Spreads

– Maximum profit = X2 - X1 - C1 + C2, Minimum = - C1 + C2

– Breakeven: ST* = X1 + C1 - C2

X1X2

ST

-C1

C2

Short a call

with X2 and T

Buy a call

with X1 and T

ProfitX2-X1+C2-C1

C2-C1

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Money Spreads

• Bear Spread using calls

– Purpose:

• Benefit from speculation of a stock price fall, but

with limited up / downward potential

– Position:

• Take a call with X2 and writing another with X1

• Note: Call bought costs less than call written

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Money Spreads (bear with calls)

X1X2

Profit

ST

Buy a call

with X2 and T

Write a call

with X1 and T

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Money Spreads• Bear Spreads using Puts

– Buy put with strike X2

– sell put with strike X1

• Requires initial outlay

– Let N1 = -1, N2 = 1


P = -Max(0,X1 - ST) + P1 + Max(0,X2 - ST) - P2

P = X2 - X1 + P1 - P2 if ST X1 < X2

P = P1 + X2 - ST - P2 if X1 < ST < X2

:P = P1 - P2 if X1 < X2 < ST

• Maximum profit = X2 - X1 + P1 - P2. Minimum = P1 - P2.

• Breakeven: ST* = X2 + P1 - P2.

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Money Spreads

• Collars

– Buy stock, buy put with strike X1, sell call with strike X2. NS = 1, NP = 1, NC = -1.


P = ST - S0 + Max(0,X1 - ST) - P1 - Max(0,ST - X2) + C2

P = X1 - S0 - P1 + C2 if ST X1 < X2 :+

P = ST - S0 - P1 + C2 if X1 < ST < X2 :~

P = X2 - S0 - P1 + C2 if X1 < X2 < ST :-

– A common type of collar is what is often referred to as a zero-cost collar. The call strike is set such that the call premium offsets the put premium so that there is no initial outlay for the options.

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Money Spreads• Collars (continued)

– The collar is a lot like a bull spread. The collar payoff exceeds the bull spread payoff by the difference between X1 and the interest on X1.

• Thus, the collar is equivalent to a bull spread plus a risk-free bond paying X1 at expiration.

• Buy C at X1 Buy S

• Sell C at X2 Buy P at X1

• Sell C at X2

• C(X1) - C(X2) = S + P(X1) – C(X2)

• C(X1) = S + P(X1)

• C(X1) = S + P(X1) - X(1+r)-T

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Money Spreads

• Butterfly Spreads

– Purpose:

• Speculate that large stock price movements are unlikely

– Position:

• Buy call with strike X1, buy call with strike X3, sell two calls with strike X2. Let N1 = 1, N2 = -2, N3 = 1.


P = Max(0,ST - X1) - C1 - 2Max(0,ST - X2) + 2C2 + Max(0,ST - X3) - C3

:P = -C1 + 2C2 - C3 if ST X1 < X2 < X3

< :P = ST - X1 - C1 + 2C2 - C3 if X1 < ST X2 < X3

> :P = -ST +2X2 - X1 - C1 + 2C2 - C3 if X1 < X2 < ST X3

:P = -X1 + 2X2 - X3 - C1 + 2C2 - C3 if X1 < X2 < X3 < ST

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Money Spreads - butterfly

X1 X3

Profit

STX2

Buy calls with

X1 and X3

Sell 2 calls with X2

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Money Spreads

• Butterfly Spreads (continued)

– Maximum profit = X2 - X1 - C1 + 2C2 - C3,

– Minimum = -C1 + 2C2 - C3

– Breakeven: – ST

* = X1 + C1 - 2C2 + C3 and

– ST* = 2X2 - X1 - C1 + 2C2 - C3

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Calendar Spreads

– Purpose:

• Benefit from volatility and time value decay.

– Position:

• Buy call with longer time to expiration, sell call with shorter time to expiration.

– Requires initial outlay

– Note how this strategy cannot be held to expiration because there are two different expirations.

– Early exercise can be problem.

– Can be constructed with puts as well.

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Calendar Spreads

Profit

X

Sell a call with

X and T1<<T2

Buy a call with

X and T2>>T1

ST

C1

-C2

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Straddles, Straps and Strips

– Straddle

– Purpose:

• You think the stock will move (but don’t know in which direction)

– Position:

• long an equal number of puts and calls


P = Max(0,ST - X) - C + Max(0,X - ST) - P (assuming Nc = 1, Np = 1)

P = ST - X - C - P if ST X

P = X - ST - C - P if ST < X

– Either call or put will be exercised (unless ST = X).

– Breakeven: ST* = X - C - P and ST

* = X + C + P

– Maximum profit: infinite, minimum = - C - P

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Profit

XST

-(P+C)

Buy a put and a call

with X and T

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• Applications of Straddles

– Based on perception of volatility greater than

priced by market

• A Short Straddle

– Unlimited loss potential

– Based on perception of volatility less than

priced by market

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• Straps

– Purpose:

• Doubles up bet on stock going up

– Position:

• two long calls and one long put.


P = 2Max(0,ST - X) - 2C + Max(0,X - ST) - P (assuming Nc = 2, Np = 1)

: P = 2ST - 2X - 2C - P if ST X

: P = X - ST - 2C - P if ST < X

– Breakeven: ST* = X + C + P/2; ST*=X-2C-P

– Maximum profit = infinite, minimum = -2C - P

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• Strips

– Purpose:

• Doubles up bet on stock going down

– Position:

• two long puts and one long call.


P = Max(0,ST - X) - C + 2Max(0,X - ST) - 2P (assuming Nc = 1, Np = 2)

: P = ST - X - C - 2P if ST X

: P = 2X - 2ST - C - 2P if ST < X

– Breakeven: ST* = X - P - C/2; ST*=X+C+2P

– Maximum profit = infinite, minimum = -2P - C

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Profit

X ST

Profit

X ST

Strip=2 puts+1 call Strap=1 put+2 calls

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Options Strategies & Risk Management for Individuals