Profit equations and graphs for buying and selling stock, buying and selling calls, buying and selling puts, covered calls, protective puts and conversions/reversals The effect of choosing different exercise prices The effect of closing out an option posi

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Important Concepts

¡ 

Profit equations and graphs for buying and selling

stock, buying and selling calls, buying and selling

puts, covered calls, protective puts and

conversions/reversals

¡ 

The effect of choosing different exercise prices

¡ 

The effect of closing out an option position early

versus holding to expiration

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Terminology and Notation

¡ 

Note the following standard symbols

¡  C = current call price, P = current put price

¡  S₀ = current stock price, S_T = stock price at expiration

¡  T = time to expiration

¡  X = exercise price

¡  _Π = profit from strategy

¡ 

The number of calls, puts and stock is given as

¡  N_C = number of calls

¡  N_P = number of puts

¡  N_S = number of shares of stock

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Terminology and Notation

(continued)

¡ 

These symbols imply the following:

¡  N_C,N_P, or N_S > 0 implies buying (going long)

¡  N_C, N_P, or N_S < 0 implies selling (going short)

¡ 

The Profit Equations

¡  Profit equation for calls held to expiration

¡  Π = N_C[Max(0,S_T - X) - C]

¡  For buyer of one call (N_C = 1) this implies

Π = Max(0,S_T - X) – C

¡  For seller of one call (N_C = -1) this implies

Π = -Max(0,ST - X) + C

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Terminology and Notation

(continued)

¡ 

The Profit Equations (continued)

¡  Profit equation for puts held to expiration

¡  Π = N_P[Max(0,X - S_T) - P]

¡  For buyer of one put (N_P = 1) this implies

Π = Max(0,X - S_T) - P

¡  For seller of one put (N_P = -1) this implies

Π = -Max(0,X - S_T) + P

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Terminology and Notation

(continued)

¡ 

The Profit Equations (continued)

¡  Profit equation for stock

¡  Π = N_S[S_T - S₀]

¡  For buyer of one share (N_S = 1) this implies

Π = S_T - S₀

¡  For short seller of one share (N_S = -1) this implies

Π = -S_T + S₀

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Terminology and Notation

(continued)

¡ 

Different Holding Periods

¡  Three holding periods: T₁ < T₂ < T

¡  For a given stock price at the end of the holding

period, compute the theoretical value of the option using the Black-Scholes-Merton or other appropriate model.

¡  Remaining time to expiration will be either T - T₁,

T - T₂ or T - T = 0 (we have already covered the

latter)

¡  For a position closed out at T₁, the profit will be

¡  where the closeout option price is taken from the

Black-Scholes-Merton model for a given stock

price at T₁.

7 ∏

=

N

_c

[C(S

_T

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Terminology and Notation

(continued)

¡ 

Different Holding Periods (continued)

¡  Similar calculation done for T₂

¡  For T, the profit is determined by the intrinsic value, as

already covered

¡ 

Assumptions

¡  No dividends

¡  No taxes or transaction costs

¡  We continue with the DCRB (a fictional large

high-tech company traded on NASDAQ) options. See Table 6.1.

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Stock Transactions

¡ 

Buy Stock

¡  Profit equation: _Π = N_S[S_T - S₀] given that N_S > 0

¡  See Figure 6.1 for DCRB, S₀ = $125.94

¡  Maximum profit = _∞, minimum = -S₀

¡ 

Sell Short Stock

¡  Profit equation: _Π = N_S[S_T - S₀] given that N_S < 0

¡  See Figure 6.2 for DCRB, S₀ = $125.94

¡  Maximum profit = S₀, minimum = - _∞

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Call Option Transactions

¡ 

Buy a Call

¡  Profit equation: _Π = N_C[Max(0,S_T - X) - C] given that

N_C > 0. Letting N_C = 1,

¡  Π = S_T - X - C if S_T > X

¡  Π = - C if S_T ≤ X

¡  See Figure 6.3 for DCRB June 125, C = $13.50

¡  Maximum profit = _∞, minimum = -C

¡  Breakeven stock price found by setting profit

equation to zero and solving: S_T*_{= X + C}

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Call Option Transactions

(continued)

¡ 

Buy a Call (continued)

¡  See Figure 6.4 for different exercise prices. Note

differences in maximum loss and breakeven.

¡  For different holding periods, compute profit for

range of stock prices at T₁, T₂, and T using

Black-Scholes-Merton model. See Table 6.2 and Figure 6.5.

¡  Note how time value decay affects profit for given

holding period.

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Call Option Transactions

(continued)

¡ 

Write a Call

¡  Profit equation: _Π = N_C[Max(0,S_T - X) - C] given that

N_C < 0. Letting N_C = -1,

¡  Π = -S_T + X + C if S_T > X

¡  Π = C if S_T ≤ X

¡  See Figure 6.6 for DCRB June 125, C = $13.50

¡  Maximum profit = +C, minimum = - _∞

¡  Breakeven stock price same as buying call:

S_T*_{= X + C}

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Call Option Transactions

(continued)

¡ 

Write a Call (continued)

Black-Scholes-Merton model. See Figure 6.8.

holding period.

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Put Option Transactions

¡ 

Buy a Put

¡  Profit equation_:_Π = N_P[Max(0,X - S_T) - P] given that N_P

> 0. Letting N_P = 1,

¡  Π = X - S_T - P if S_T < X

¡  Π = - P if S_T ≥ X

¡  See Figure 6.9 for DCRB June 125, P = $11.50

¡  Maximum profit = X - P, minimum = -P

equation to zero and solving: S_T*_{= X - P}

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Put Option Transactions

(continued)

¡ 

Buy a Put (continued)

holding period.

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Put Option Transactions

(continued)

¡ 

Write a Put

¡  Profit equation: _Π = N_P[Max(0,X - S_T)- P] given that N_P

< 0. Letting N_P = -1

¡  Π = -X + S_T + P if S_T < X

¡  Π = P if S_T ≥ X

¡  See Figure 6.12 for DCRB June 125, P = $11.50

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

equation to zero and solving: S_T*_{= X - P}

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Put Option Transactions

(continued)

¡ 

Write a Put (continued)

holding period.

¡ 

Figure 6.15

summarizes these payoff graphs.

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Calls and Stock: the

Covered Call

¡ 

One short call for every share owned

¡ 

Profit equation:

_Π

= N

_S

(S

_T

- S

₀

) + N

_C

[Max(0,S

_T

- X) -

C] given N

_S

> 0, N

_C

< 0, N

_S

= -N

_C

. With N

_S

= 1, N

_C

=

-1,

¡  _Π = S_T - S₀ + C if S_T _≤ X

¡  _Π = X - S₀ + C if S_T > X

¡ 

See

Figure 6.16

for DCRB June 125,

S

₀

= $125.94, C = $13.50

¡ 

Maximum profit = X - S

₀

+ C, minimum = -S

₀

+ C

¡ 

Breakeven stock price found by setting profit

equation to zero and solving: S

_T*

_{= S}

0

- C

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Calls and Stock: the

Covered Call (continued)

¡ 

See

Figure 6.17

for different exercise prices. Note

differences in maximum loss and breakeven.

¡ 

For different holding periods, compute profit for

range of stock prices at T

₁

, T

₂

, and T using

Black-Scholes-Merton model. See

Figure 6.18

.

¡ 

Note the effect of time value decay.

¡ 

Some General Considerations for Covered Calls:

¡  alleged attractiveness of the strategy

¡  misconception about picking up income

¡  rolling up to avoid exercise

¡ 

Opposite is short stock, buy call

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Puts and Stock: the

Protective Put

¡ 

One long put for every share owned

¡ 

Profit equation:

_Π

= N

_S

(S

_T

- S

₀

) + N

_P

[Max(0,X - S

_T

) -

P] given

N

_S

> 0, N

_P

> 0, N

_S

= N

_P

. With N

_S

= 1, N

_P

= 1,

¡  _Π = S_T - S₀ - P if S_T _≥ X

¡  _Π = X - S₀ - P if S_T < X

¡ 

See

Figure 6.19

for DCRB June 125, S

₀

= $125.94,

P = $11.50

¡ 

Maximum profit =

_∞

, minimum = X - S

₀

- P

¡ 

Breakeven stock price found by setting profit

equation to zero and solving: S

_T*

_{= P + S}

0

¡ 

Like insurance policy

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Puts and Stock: the

Protective Put (continued)

¡ 

See

Figure 6.20

for different exercise prices. Note

differences in maximum loss and breakeven.

¡ 

For different holding periods, compute profit for

range of stock prices at T

₁

, T

₂

, and T using

Black-Scholes-Merton model. See

Figure 6.21

.

¡ 

Note how time value decay affects profit for

given holding period.

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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.

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Synthetic Puts and Calls

(continued)

¡ 

Profit equation:

Π

= N

_C

[Max(0,S

_T

- X) - C]

+ N

_S

(S

_T

- S

₀

) given that N

_C

> 0, N

_S

< 0, N

_S

= N

_P

.

Letting N

_C

= 1, N

_S

= -1,

¡  _Π = -C - S_T + S₀ if S_T _≤ X

¡  _Π = S₀ - X - C if S_T > X

¡ 

See

Figure 6.22

for synthetic put vs. actual put.

¡ 

Table 6.3

shows payoffs from reverse conversion

(long call, short stock, short put), used when

actual put is overpriced. Like risk-free borrowing.

¡ 

Similar strategy for conversion, used when actual

call overpriced.

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