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OPTIONS

• Call Option

• Put Option

• Option premium

• Exercise (striking) price

• Expiration date

• In, out-of, at-the-money options

• American vs European Options

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Option Valuation• Valuation of a call option at Expiration =

max{P-X, 0}

P

Vc

X

Valuation of a put option at expiration:max{X - P, 0}

P

Vp

X

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Option Valuation (Cont’d)

Binominal Call Pricing (one period)70 40%

P0 = 5045 -10%70 - 50 =20

V0 = ?0

70 - 45 25 5Hedge Ratio = = =

20 - 0 20 4HR: number of calls sold for each stock boughtBuy 1 shr of stock, sell 1.25 callsIf P1=$45, portfolio value = $45If P1=$70, portfolio value = 70 - 20(1.25)=45Return = 45/(50-1.25Vc)-1 = 0.10Vc = $7.27

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Option Valuation (Cont’d

Binominal Call Pricing (two periods)P2=98.00V2=48.00

P1=70.00V1=24.55

P2=63.00V2=13.00

P0=50.00V0=11.60 P2=63.00

V2=13.00P1=45.00V1=4.73

P2=40.50V2=0

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Option Valuation (Cont’d

At T=1, If P1 = $70.00HR = (98.00 - 63.00)/(48.00 - 13.00) = 1Buy 1 stock, sell 1 call

If P2 = 98.00 Port. Value = 98 - 48 = 50 P2 = 63.00 Port. Value = 63 - 13 = 501+Return = 50/(70 - V1) = 1.1V1 = $24.55

At T=1, If P1 = $40.50HR = (63.00 - 40.50)/(13.00) = 1.73Buy 1 stock, sell 1.73 call

If P2 = 63.00 Port. Value = 63 - 1.73x13 = 40.50 P2 = 40.50 Port. Value = 40.50 - 0 = 40.501+Return = 40.50/(40.50 - 1.73V1) = 1.1V1 = $4.73

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Option Valuation (Cont’d

At T=0

HR = (70.00 - 45.00) / (24.55 - 4.73)= 1.26

Buy 1 stock, sell 1.26 call

If P1 = 70.00 Port. value = 70 - 1.26x24.55 =39.07 P1 = 45.00 Port. Value = 45 - 1.26x4.73 = 39.07

Return = 39.07 / (50 - 1.26V0) = 1.1V0 = $11.60

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Black and Scholes OPM

)()( 210 dNe

XdNPV

rtC

d1 and d2 are deviations from the expectedvalue of a unit normal distribution. N(d) isthe probability of getting a value below d.

tdd

t

tRXP

df

12

20

1

])2/1([ln

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Black and Scholes Eg.

P0= $50.00 X = $50.00 Rf =10% =0.60

d1 ={ ln(50/50) + [0.10+ (1/2)0.602 ]1} / 0.60

= 0.28 / 0.60 = 0.4667

d2 = 0.4667 - 0.60 = -0.1333

N(0.4667) = 0.6796 N(-0.1333) = 0.4470

Vc = 50 (0.6796) - 50 e-0.10 (0.4470) = $13.76

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Put-Call Parity

Buy a share at P, sell a call, buy a put at thesame exercise price (X) as call.

Value of Portfolio ifP<X P>X

Stock P Pcall 0 X-Pput X-P 0 Portfolio X X

Therefore the value of the portfolio today mustbe equal to the PV of X:

P + Vp -VC = X/(1 +Rf)

or Vp = Vc + X/(1 +Rf) - P

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Option Investment Strategies

Writing covered calls - buy stock, write cals

Synthetic long: Buy call, sell put

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Option Investment StrategiesStraddle: simultaneously buying puts and callswith the same X and t on the same underlyingasset

Long Straddle

Short Straddle

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Option’s Delta, Gamma, and Theta

Delta: Rate of change in position value in response to a change in the value of the underlying asset.

Gamma: Rate of change in delta in response to change in the value of the underlying asset.

Theta: Change in position value as time to expiration gets closer (other things being thesame)

delta zero; gamma +

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Portfolio Insurance

Investing in a portfolio of stocks and a putoption on the portfolio simultaneously.

The problem is when you cannot find a putoption on your portfolio.

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Portfolio Insurance Cont’d

Alternatively one can combine stock portfolio with the risk free asset to have the same portfolio insurance, using OPM:

N(d1) = slope of the call option value. It gives thefall in position value for a decline of $1 in stockvalue.

For portfolio insurance, invest 1 -N(d1) in t-bills,and N(d1) in the risky portfolio.

Potential problem