What is an Option?

What is an Option?

Definitions: An option is an agreement between two parties that gives the purchaser of the optionthe right, but not the obligation, to buy or sell a specific quantity of an asset at a specifiedprice during a designated time period.

A Call is the right to purchase the underlying asset

A Put is the right to sell the asset

The Strike Price is the pre-specified price at which the option holder can buy (in the case of a call) or sell (in the case of a put) the asset to the option seller

The Expiration date is the date (and time) after which the option expires

The Notional Amount is the quantity of the underlying asset that the option buyer has theright to buy or sell under the terms of the option contract

The Premium is the price of the option contract (the amount paid by the buyer to the seller).The option buyer’s maximum “downside” (possible loss) is the amount of the premium. Optionpremia are quoted as percentage of notional amount.

To Exercise means to invoke the right to buy or sell the underlying asset under the termsof the option contract

The Seller (or Writer) of the option receives a payment (the Option Premium) that then obligates him to sell (in the case of a call) or buy (in the case of a put) the asset.

WEMBA 2000 Real Options 1

Payoff Diagram for a Long Call Option

Optionpayoff

S: Price of Underlying Asset

at expiration

K

Strike Price = KPrice of Underlying Asset = S

Profit/Loss Analysis

At expiration, there are two possible outcomes:

(i) S >=K. Exercise the call and purchase the asset for K. Asset has market value S Payoff = S - K(ii) S <K. Option expires worthless. Payoff = 0

General Formula for call payoffLong call payoff = Max (0, S - K)


Optionpayoff

K

Strike Price = KPrice of Underlying Asset = S

Profit/Loss Analysis

At expiration, there are two possible outcomes:

(i) S <= K Exercise the put and sell the asset for K. Asset has market value S Payoff = K - S(ii) S >K. Option expires worthless. Payoff = 0

General Formula put payoffLong put payoff = Max (0, K - S)

Payoff Diagram for a Long Put Option



at expiration

Payoff Diagrams for Short Options Positions

Optionpayoff


at expiration

K

Short Call PositionStrike Price = K

Optionpayoff

K

Short Put PositionStrike Price = K

Notes:(i) The short position payoff diagrams are mirror images of the long positions.(ii) The above payoff charts do not include the cost of buying (or income from selling) the option.

Short call payoff = Min (0, K-S) Short put payoff = Min (0, S-K)

Question: Which is potentially riskier, a long option position or a short option position?

WEMBA 2000 Real Options


at expiration


Payoff Diagrams for Short Options Positions


Payoff Diagrams for some Option Combinations

PositionProfit/Loss


at expiration

K

Profit/Lossfrom stock

Profit/loss from short call

net profit/loss

"Covered Call" or "Buy-Write"

PositionProfit/Loss


at expiration

"Call Spread"

Profit/loss from short call

Profit/loss from long call

net profit/loss

Note: The above profit/loss charts include the cost of buying (or income from selling) the option

Factors that Influence Option Prices

The six variablesthat affect option prices:

1. Current (spot) price on the underlying security

2. Strike price

3. Time to expiration

4. Implied (expected) volatility on the underlying security

5. The riskfree rate over the time period of the option

6. Any dividends or other cashflows that will be paid or received on the underlying asset during the life of the option


Valuation of Options: Put-Call Parity


We construct two portfolios and show they always have the same payoffs, hence they must cost the same amount.

Portfolio 1: Buy 1 share of the stock today for price S0 and borrow an amount PV(X) = X e-rT

How much will this portfolio be worth at time T ?

Cashflow CashflowPosition Time = 0 Time = T

Buy Stock -S0 ST

Borrow PV(K) -K

Net: Portfolio 1 PV(K) - S0 ST - K

Portfoliopayoff

at time T

ST

K

Payoff from borrowing

Payoff from borrowing

Payoff from stock

net payoff

-K

S

Portfolio 2: Buy 1 call option and sell 1 put option with the same maturity date T and the same strike price K.How much will this portfolio be worth at time T ?

Cashflow Cashflow: Time = TPosition Time = 0 ST < K ST > K

Buy Call - c 0 ST - K

Sell Put p - (K - ST ) 0

Net: Portfolio 2 p - c ST - K ST - K


Portfoliopayoff

at time T

STK

Payoff on short put

Payoff onlong call

net payoff

-K



Payoff from Portfolio 1 and Portfolio 2 is the same, regardless of level of ST , hence costof both portfolios (cashflows at time T = 0 ) must be the same.

Hence: S0 - PV(K) = c - p Put-Call Parity

Rearranging: c = p + S0 - PV(K) (1)

Put-Call parity: a worked example

Stock is selling for $100. A call option with strike price $90 and maturity 3 months hasa price of $12. A put option with strike price $90 and maturity 3 months has a price of $2. The risk-free rate is 5%.

Question: Is there an arbitrage? Test Put-Call parity:

Right-hand side of (1): p + S0 - PV(K) = 2 + 100 - 90 e -0.05*0.25

= 13.12Left-hand side of (1): c = 12 13.12 !

Market Price of c is too low relative to the other three.Buy the call, and Sell the "replicating portfolio".


Cashflow Cashflow: Time = TPosition Time = 0 ST < 90 ST > 90

Buy Call - 12 0 ST - 90

Sell Put 2 ST - 90 0

Sell stock 100 - ST - ST

Lend money -90 e 0.05*0.25 90 90

Net Payoff 1.12 0 0

Valuation of Options: Put-Call Parity Example

Result: arbitrage profit of 1.12 today, regardless of the value of the stock price!


Valuation of Options: Black-Scholes Formula for Calls and Puts

S = Current stock price

K = Strike price on the option

T = Time to maturity of the option in years (e.g. 5 months = 5/12 = 0.417)

r = Riskfree rate of interest

= Expected ("Implied") volatility (standard deviation) of the underlying stock over the life of the option

Black-Scholes Call Price c = S N( d1 ) - X e -rT N( d2 ) (2)

where: d1 = ln (S/k) + (r + 2 / 2) T

T

d2 = d1 - T

N(d ) = cumulative standard normal probability of value less than d

Black-Scholes Put Price p = X e -rT N( - d2 ) - S N( - d1 ) (3)


Valuation of Options: Black-Scholes Formula for Calls and Puts

Example: Options on Compaq stock

On Dec 20, Compaq stock closed at $76.753 month riskfree rate: 5.5% (e.g. yield on 3 month T-bill)Estimated volatility: 41%

What are the values of 3 month call and put options with Strike = $75 ?

Black-Scholes formula inputs and calculations:

Observed inputs: Option contract inputs: Estimated input (the future level of volatility is not observable)

S = 76.75 K = 75 = 41%r = 5.5% T = 0.25

d1 = [ ln (76.75/75) + (0.055 + 0.412 / 2) 0.25 ]

0.41 0.25 = 0.2821

d2 = d1 - T = 0.0771

N(d1) = 0.6111 [obtained from Excel "normsdist" function]N(d2) = 0.5307 [obtained from Excel "normsdist" function]

c = 7.638 [from equation (2) ]p = 4.864 [from equation (3) ]


Binomial Pricing Method 1: Creating a replicating portfolio


Bluejay Corp share price is $20. Possible price at the end of three months: either $22 or $18.Value a call option on Bluejay with strike 21, expiration 3 months. Riskfree rate = 2% over 3 months.

20

22

18

c

22-21 = 1

0

Share Price Option Value {Reminder: the value of the call at expiration is Max[0, S - K]}

(a) Create a portfolio: purchase one share of the stock, and borrow money at the riskfree rateHINT: Choose amount to borrow so that the portfolio outcome is zero in one scenario

20-PV(18)=2.35

22-18=4

18-18=0

Portfolio: Buy 1 share & borrow PV(18)

Compare the payoff betweenthe call option and the portfolio.How many call options do we need to buy to make the payoffs identical?

(i) (ii)

(iii)


4*c

4

0

2.35

4

0

Portfolio: Buy 1 share & borrow PV(18)

Option Value (4 calls)

(b) Calculate number of call options to buy so that the payoff from the calls matches the portfolio payoff in all scenarios. Hence the call price must equal the value of the portfolio (Law of One Price).

4 * c = 2.35 c = 0.59

Call premium (price)

How many shares of stock to buy to replicate the payoff from one call? 4 calls replicate payoffs from 1 share, hence 1 call is replicated by 0.25 shares. The fraction of shares needed to replicate 1 call is called the delta () or hedge ratio.

= 0.25

delta ()

(iia)=(ii)*4

(iii)

Binomial Pricing Method 1: Creating a replicating portfolio

equal

How do we create areplicating portfolio

for puts?

20

22

18

24.2

19.8

16.2

Binomial Pricing Method 1: Extending to two time-steps

Share Price Tree Call Option Tree

[24.2 - 21] = 3.2

0

0

cu

cd

c

Methodology:

Step 1: Calculate u and cu , the delta and call value at the upper intermediate node

Step 2: Calculate d and cd , the delta and call value at the lower intermediate node (note: u and d will be different)

Step 3: Calculate and c, the delta and the call price today


Bluejay Corp share price is currently $20. Possible price moves in each period: either up by 10%or down by 10%. Period length: 3months. Value a call option on Bluejay with strike 21, expiration 6 months. Riskfree rate = 2% over each 3 month period.

20

22

18

24.2

19.8

16.2


Share Price Tree


Call Option Tree

3.2

0

0

cu

cd

c

22-PV(19.8)=2.59

24.2 - 19.8= 4.4

19.8 - 19.8=0

Replicating Portfolio to calculate cu

(a) Purchase 1 share and borrowmoney so that the portfolio payoff

is zero in one scenario

3.2*(1/u)= 4.4

0

(1/u)cu

Step 1: Calculating cu and u

Match replicating portfolio payoffs at ending nodes

(b) Purchase the appropriate number of calls so that the payoff at each terminal nodematches the payoffs from the portfolio.

u = 3.2/4.4 = 0.727cu = u * 2.59 = 1.88

equal



Step 2: Calculating cd and d

0

0

cd

Call payoff in either scenario is zero.Hence cd = 0, replicating portfolio = 0.By implication, d = 0

Step 3: Calculating c and

20-PV(18)=2.35

22 - 18= 4

18 - 18=0

Replicating Portfolio to calculate c

(a) Purchase 1 share and borrowmoney so that the portfolio payoff

is zero in one scenario(note: this is identical to the 1-step tree)

0

(1/)*c

Match replicating portfolio payoffs at ending nodes

cu = 1.88*(1/)=4

(b) Purchase the number of calls necessaryso that the payoff at each terminal nodematches the payoffs from the portfolio.

= 1.88/4 = 0.47c = * 2.35 = 1.10

equal


Bluejay Corp share price is currently $20. Possible price at the end of three months: either $22 or $18.Value a call option on Bluejay with strike 21, expiration 3 months. Riskfree rate = 2% over 3 months.

20

22

18

c

22-21 = 1

0

Share Price Option Value

Create a riskless portfolio: sell 1 call, buy d shares (where d is a fraction of a share)

- c + 20

22 - 1

18

Question: how can we make this portfolio riskless?

{Reminder: the value of the callat expiration is Max[0, S - K]}

Riskless Portfolio

Binomial Pricing Method 2: Creating a riskless portfolio


Riskless Portfolio

-c + 20

22 - 1

18

For the portfolio to be riskless, the two outcomes must have identical values.

HINT: Choose so that: 22 - 1 = 18

= 0.25

Portfolio Terminal value = 4.5 (in either scenario)

Portfolio Present value = 4.5/(1.02) (discount at riskfree rate) = 4.41

Hence: 4.41 = -c + 20

c = 0.59

Portfolio "delta"

Call premium (price)

Note: this is the same call price and delta that we obtained using method 1.



1

0

Option Value

0.59

q

1-q

Call price = 0.59 = [1 * q + 0 * (1 - q)]/1.02

q = 0.6

What does the value q represent?

It does not represent the probability that the stock price will move up or down! It is sometimes referred to as the “risk-neutral” probability that the stock price will move up or down.

22

18

Stock Price

20

q

1-q

Stock price = 20 = [22 * q + 18 * (1 - q)]/1.02

q = 0.6


Binomial Pricing Method 2: Generalization


S

Su

Sd

c

cu

cd

*Share Price Option Value Portfolio

S-c

Su - cu

Sd - cd

For portfolio to be riskless, choose so that Su - cu = Sd - cd

hence = cu - cd Su - Sd

Now the riskless terminal value, discounted at the riskless rate rf , should equal the portfolio cost:

Su - cu = S - c (1 + rf )

Substitute for from (1): c = q cu + (1-q)cd (1 + rf )

where q = (1 + rf) - d (u - d)

(1)

(2)

(3)

+ =


S

Su

Sd

Su2

Sud

Sd2

c

cu

cd

cuu

cud

cdd

S = Stock price todayu = proportional change in S on an up-moved = proportional change in S on a down-moverf = riskfree ratec = call price todaycu = call value after one up-movecd = call value after one down-movecuu , cud , cdd = terminal call valuesK = strike on the call

cuu = max[0, Su2 - K]cud = max[0, Sud - K]cdd = max[0, Sd2 - K]

q = (1+rf) - d (u - d)cu = [p cuu + (1-p)cud ] (1+rf )cd = …..c = …..

Binomial Pricing Method 2: Generalization over two time-steps

Example: compare with 2-step example using Method 1

S = 20, u=1.1, d = 0.9, rf = 2%cuu = 3.2; cud = cdd = 0q = [(1.02)-0.9]/(1.02) = 0.6cu = [0.6 * 3.2 + 0.4 * 0]/1.02 = 1.88cd = 0c = [0.6 * 1.88 + 0.4 * 0]/1.02 = 1.10

compare these results with those from Method 1

Valuation of Options: Binomial Pricing Method


We can evaluate a call option either by creating a replicating portfolio of the underlying stock and borrowing, or by creating a riskless portfolio of the call and the underlying stockThe two methods yield identical results

What have we shown?

The delta or hedge ratio: the fraction of the underlying stock that we need to purchase relativeto selling a single call option to obtain a riskless portfolioThe risk-neutral probability of an upmove or downmove in the underlying stock

What other information do we obtain from these methods?

The Black-Scholes formula effectively represents the binomial tree model over manyhundreds or thousands of periods

Binomial Tree methodology: Option price = delta * share price - bank loanBlack Scholes formula: Option price = N(d1 )* S - N(d2)* PV(K)

What are the underlying assumptions of these methods?

That we can freely buy and sell the underlying stock without transactions costsThat we can borrow or lend money at the riskless rate of interest

What are the limitations of these methods?

They become very complex over a large number of steps (although computers can help)

What is the connection between these methods and the Black-Scholes formula?

Valuation of Options: Call and Put Price Sensitivities


As each input to the option pricing model varies, the call and put prices respond by increasing or decreasing as follows:

Increase In: Call Price Put Price Why? [to be discussed in class]

S

X

T

r

Debt and Equity as Options

Suppose a firm has debt with a face value of $1MM outstanding that matures at the endof the year. What is the value of debt and equity at the end of the year?

Firm Value (V) Payoff to shareholders Payoff to debtholders

0.3 MM 0 0.3 MM 0.6 MM 0 0.6 MM 0.9 MM 0 0.9 MM 1.2 MM 0.2 MM 1.0 MM 1.5 MM 0.5 MM 1.0 MM

Payoffs

Firm Value V0 $1 MM

EquityholdersBondholders

Payoff to Equityholders = max [0, V - $1MM] equivalent to a call option, K=$1MM

Payoff to Bondholders = V - max [0, V - $1MM] equivalent to the total value of the firm less a call option, K=$1MM


Documents

What is an Option?