1 lecture option spreads and stock index options version 1/9/2001 financial engineering: derivatives...

© K. Cuthbertson and D. Nitzsche 1

LECTURE

Option Spreads and Stock Index Options

Version 1/9/2001

FINANCIAL ENGINEERING:DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001)

K. Cuthbertson and D. Nitzsche

© K. Cuthbertson and D. Nitzsche

Financial Engineering: Basic Payoffs

Spread Trades

Bull Spread /Bear Spread

Straddle

Strangle

Butterfly

Horizontal Spread

Covered Positions

Stock Index Options

Topics


Financial Engineering: Basic Payoffs


Combine options to obtain the payoff structures you prefer

This usually involves a mixture of speculation (since you can make and lose money) and ‘psuedo-hedging’, since these synthetic positions limit possible outcomes.

Financial Engineering


Figure 10.1 : Payoff for calls

A. Buy (long) call

B. Write (short) call

-1

+1

0

0


Figure 10.2 : Payoff for puts

-1

-1

0

A. Buy (long) put

B. Write (short) put 0


Figure 10.3 : Payoff for futures

+1

Payout profile on futures and on the underlying stocks are the same.Futures profits occur at same time as profits from the options.Futures do not require any “up front“ costs at t = 0.

+1 -1

-1

A. Buy (long) futures B. Sell (short) futures


Figure 10.4 : Synthetic long call

Long Futures

plus

Long Put

equals

Long Call

+1

-10

0+1

+1


Figure 10.5 : Synthetic short put

Long Futures

plus

Short Call

equals

Short Put

+1

+10

0-1

+1


Figure 10.6 : Synthetic long futures

Short Put

plus

Long Call

equals

Long Futures

+1

+1

0

0+1

+1


Spread Trades


Spread Trades

•Vertical (or money) spread Use either calls or puts with the same expiration date but with different strike prices (K)

•StraddleThis is a special type of spread, uses both calls and puts

•Horizontal (or time or calendar) spread Options, same strike price (K) but different maturity dates

•Diagonal spreadUses options with different maturities and different strikes


Figure 10.7 : Bull spread with calls

Long Call (at K1)

plus

Short Call (at K2 > K1)

equals

Call Bull Spread

+10

+1

Profit

Share Price

K1

5

-3

K1=102

K2=110

SBE=105

00 -1

K2

+1

0

0Gamble on stock price rise and offset cost with sale of call

Favourable time decay


Payoff: Long call (K1) + short call (K2) = Bull Spread: { 0, +1, +1} + {0, 0, -1} = {0, +1, 0 }

C1 = 5 K1 = 102 C2 = 2 K2 = 110

= Max(0, ST-K1) –C1 – Max(0, ST-K2) + C2

= C2 - C1 if ST K1 K2

= ST - K1 + (C2 - C1) if K1 < ST K2

= (ST - K1 - C1) + (K2 - ST + C2) if ST > K1 > K2

= K2 - K1 + (C2 - C1)

SBE = K1 + (C1 – C2) = 102 + 3 = 105

Figure 10.7 : Bull spread with calls


Figure 10.8 : Bear spread with calls

Short Call (at K1)

plus

Long Call (at K2 > K1)

equals

Bear Spread

-10-1

Profit

Share Price

K1

0

0

+1

K2

-1

0

0

Gamble on stock price fall and offset cost with sale of call


Fig 10.9 : Payoff, volatility strategies

a) Long (Buy) Straddle

Profit

0

-8 ST94 110

+1-1

c) Short (Sell) Butterfly

Profit

0ST

00

b) Long (Buy) Strangle

Profit

0ST

+1-1

d) Short (Sell) Condor

Profit

0ST

00

+1-1 +1-1

0

0


Figure 10.10 : Long (buy) StraddleLong Call

plus

Long Put

equals

Long Straddle

0+1

-10

+1-1

Profit

0

SBE = 94 SBE = 110

K = 102

-8

8

8.

Relatively slow time when decay when there is a long time to maturity but rather vicious time decay in the last month.


Figure 10.10 : Long (buy) Straddle

K = 102 P = 3 C = 5 C + P = 8

profit long straddle (figure 10.10) :

[10.4] = Max (0, ST – K) - C + Max (0, K – ST) – P

for ST > K

= ST - K – (C + P) = K + (C + P) = 102 + 8 = 110

for ST < K

= K - ST – (C + P) = K - (C + P) = 102 - 8 = 94


Short (sell) Straddle: LeesonLong Call

plus

Long Put

equals

Long Straddle

0+1

-10

+1-1

Profit

0

SBE = 94 SBE = 110

K = 102

-8

8

8.

Relatively slow time when decay when there is a long time to maturity but rather vicious time decay in the last month.


Figure 10.11 : Long (buy) strangle

Long Put

plus

Long Call

equals

Long Strangle

0-1

+10

+1-1

Profit

0

0

0

0

.

Kc

Kp


Figure 10.12 : Short butterfly

Short butterfly requires:

sale of 2 'outer-strike price’ call options (K1, K3)

purchase of 2 ‘inner-strike price’ call options (K2)

Short butterfly is a ‘bet’ on a large change in price ofthe underlying in either direction (e.g. result of reference to the competition authorities)

Cost of the ‘bet’ is offset by ‘truncating’ the payoff by selling some options


Figure 10.12 : (a.) Short butterfly

-10

0+1

-1

0

K1

K2

-1

+1

0+1

0

+1

0 0 0-1

Sell Call at K1

plus

Buy 2 Calls at K2

plus

Sell 1 Call at K3


Figure 10.12 : (a.) Short butterfly (Cont.)

Profit

0

-40Stock Price

0010

+1-1 equalsShort Butterfly-1 +1


Figure 10.12 : (b.) Long butterfly

Profit

0

-10Stock Price

00

40

-1+1

+1 -1


Horizontal Spread

• Options, same strike price (K) but different maturity dates

e.g.•buying a long dated option (360-day) and selling a short dated option (180-day)~ both are at-the-money

In a relatively static market (ie. S0 = K) this spread willmake money from time decay, but will loose moneyif the stock price moves substantially (figure 10.13).


Figure 10.13 : Horizontal spread

Profit at expiry

0

Horizontal spread : a long position in a 1-year option and a short position in a 180 day option, both at-the-money

Profit profile 30 days before expiry of short dated option

Profit

S0 = K Stock price


Covered Positions

.


Figure 10.14 : Covered call

ST24

Profit

0

$4

Short Call

21 28

$3

25

K = 25

26

Long Stock

Covered call =

long stock + short call


Figure 10.15 : Protective put strategy

ST24

Profit

0

-$4

Long Put

22

-$5

2520 29

Long stock

Note : P = 5, K = 25, S0 = 24

Protective put =

long stock + long put

Long stock+ Long Put = protective put (ie. Call payoff)


Stock Index Options

.


PayoffST = 200 (index points)K = 210 (index points)long put receives :

$ Payoff = z (K-ST) = $100 (210-200) = $1,000

Quotes:PremiaC = 5 (index points) One call contract costs $550 (= $100 x 5), hence:

Invoice Price of one S&P100 Call contract = C z

Payoff and Price Quotes


15 March 1997

Value of stock portfolio, TVS = £10m

Portfolio beta, = 1.5

Current FTSE100 index, S0 = 4450

June 4450 Put Quote ,P = 103

Value of one index point = £10

Delta of June-4450 Put, p = 0.5

Note that for simplicity we choose K = 4450, the same as the initial stock price, S0

Static Hedge :

Np = = = 337 contracts

Cost of one June put contract = (103) (£10) = £1,035

Total cost 337 put contracts = 337 (£1,035) = £348,795

Percentage cost = (348,795/£10m)100 = 3.48%

Dynamic Hedge :

Np =

Note: Initial percentage Cost of Puts = 6.97%

Static and Dynamic Hedge (FTSE100)

contractsS

TVS

p

6745.0

3371

)10(£

)(


End of Slides

1 lecture option spreads and stock index options version 1/9/2001 financial engineering: derivatives...

Documents

nitzsche slide

short futures slide

s t c

calls long

nitzsche figure

bull spread

synthetic long futures

equals long futures