Lecture 2: Option Theory

E xam ple

Form ula

B inom ia l m odelfor option pricing

P aram eters 'im pac t on

option values

Form ula

B lack & Scholesoption pric ing

form ula

E xam ple

Form ula

P ut-ca ll parity

P ut options

C all op tions

A rb itrageres tric tions

F indings

P opularresearchsubjec ts

O ption pric ing

How To Price Options

The critical factor when trading in The critical factor when trading in options, is determining a fair price for options, is determining a fair price for the option.the option.

Computes the value of a call optionComputes the value of a call option

The value can change only at the end of the The value can change only at the end of the period period (t+1) (t+1) and the possible maximal and and the possible maximal and minimal values are currently minimal values are currently (t) (t) known. known.

The binomial model for option pricing

The Binomial Option Pricing Formula

C p C p C ru d ( ) /1

p u ru d

1

Explanations of the parameters

C = Call option = Call option = Call value at t+1, when stock price goes = Call value at t+1, when stock price goes to max.to max. = Call value at t+1, when stock price goes = Call value at t+1, when stock price goes to min.to min.

= Riskless interest rate= Riskless interest rateu = Multiplicative upward movement in = Multiplicative upward movement in stock stock

price (S)price (S)d = Multiplicative downward movement in = Multiplicative downward movement in stock stock

price (S)price (S)

Cu

Cd

r rf ( )1

Features of the binomial model

The formula doesn’t depend on investors’ The formula doesn’t depend on investors’ attitudes towards riskattitudes towards risk

Investors can agree on the relationship Investors can agree on the relationship between between C, S and and r even if they have even if they have different expectations about the upward or different expectations about the upward or downward movement of downward movement of C

The only random variable on which the call The only random variable on which the call value depends, is the stock price itselfvalue depends, is the stock price itself

p can take values between 0 and 1 can take values between 0 and 1 (0 < (0 < p < 1). If investors were risk < 1). If investors were risk

neutral then neutral then p=q. .

A Short Example

= 10 = 10 monetary unitsmonetary units = 0 = 0 mumu

r = 1,06 ( 6 %) = 1,06 ( 6 %)u = 1,25d = 0,75

Cu

Cdrf

KK = 15 = 15 mumu (striking price) (striking price)

SS = 20 = 20 mumu= 25 = 25 mumu= 15 = 15 mumu

Su

Sd

1 25 1 061 25 0 75

1 0 62, ,, ,

,

mu85,506,1/1062,0

rCpCpC du /)1(

p u ru d

1

The Black & Scholes formula(A continuous time formula)

““It is possible to create a risk free portfolio” by It is possible to create a risk free portfolio” by owning 1 stock and writing owning 1 stock and writing hh call options on it.” call options on it.”

The most frequently used option pricing formula The most frequently used option pricing formula Originally a heat transfer equation in physicsOriginally a heat transfer equation in physics

Assumptions behind the formula:

1. 1. The stock price follows a continuous Wiener-The stock price follows a continuous Wiener- process and the future stock prices are process and the future stock prices are lognormally distributed.lognormally distributed.

2. 2. There exist no transaction costs or taxes.There exist no transaction costs or taxes.3. 3. No dividends are paid during the lifetime of the No dividends are paid during the lifetime of the

option.option.4. 4. The capital market is perfect: there exist no The capital market is perfect: there exist no

arbitrage-possibilities.arbitrage-possibilities.5. 5. The composition of the portfolio can be The composition of the portfolio can be

continuously adjusted.continuously adjusted.6. 6. The risk free interest rate is constant during the The risk free interest rate is constant during the

lifetime of the option.lifetime of the option.

Black & Scholes Option Pricing Formula

wherewhere

d d t2 1

d S K r ttf

1

20 5

(ln( / ) ( , )

)()( 21 dNeKdNSC trf

Copeland & Weston. 1988. p 276.

ParametersC = The price of the call optionC = The price of the call optionS = Stock priceS = Stock priceN = The standard normaldistributionN = The standard normaldistribution

= The continuous risk free rate of return = The continuous risk free rate of return [=ln(1+ )][=ln(1+ )]t t = The time to expiration = The time to expiration

(if 63 days, then t=63/365)(if 63 days, then t=63/365)K = Striking price K = Striking price = = The variance of the stock return The variance of the stock return

Rfrf

N(dN(d11) = the inverse hedge ratio) = the inverse hedge ratio

e.g. for each stock that is owned, e.g. for each stock that is owned, 1/N(d ) 1/N(d ) options has to be written for options has to be written for the portfolio to be risk the portfolio to be risk free.free. = the discounted value of the striking= the discounted value of the striking price. price.

N(dN(d22) = the probability for the option to be “in ) = the probability for the option to be “in the money” the money” on due date (e.g. the option on due date (e.g. the option will be exercised). will be exercised).

K e rf t

The price of an option is dependent on the following parameters:

PP CC Current stock price (S)Current stock price (S) Time to expiration (t)Time to expiration (t) Striking price (K)Striking price (K) Stock volatilityStock volatility Interest ratesInterest rates

(Cash dividends)(Cash dividends)

The Put-Call Parity:

There is a connection between the price of a There is a connection between the price of a call and a put optioncall and a put option

If the prices differ from this equation, there If the prices differ from this equation, there exist arbitrage opportunitiesexist arbitrage opportunities

P S C K rtt

0 0 0

An intuitive example Buy a share: S = 19 Buy a share: S = 19 mumu Buy a put option: P = 1 Buy a put option: P = 1 mumu (striking price K = 20 (striking price K = 20 mumu)) Buy a call option with the same striking price and Buy a call option with the same striking price and

maturity: C = 1,50 maturity: C = 1,50 mumu Deposit 18,50 Deposit 18,50 mumu at the risk free rate r = 10,95% at the risk free rate r = 10,95%

KCSP ttt

1 + 19 = MAX(19-20,0) + 20 = 20

Outcome 9 months later on expiration date

Put + Share = Call + DepositionPut + Share = Call + Deposition 3 + 173 + 17 = 0 + 20 = 0 + 20

out of the moneyout of the moneyin the moneyin the money

P S C K rt t tt 0

Input vaules: r=10,95%;t=0,75; K=18,50;C=1,50;P=1;S=19Input vaules: r=10,95%;t=0,75; K=18,50;C=1,50;P=1;S=19

P S C K rtt

0 0 0 Discounted to t(0)Discounted to t(0)Prolonged to tProlonged to t

If we take cash dividends (D) into consideration If we take cash dividends (D) into consideration the formula is slightly modified the formula is slightly modified

Further modifications required if applied on Further modifications required if applied on american optionsamerican options

S P D C K rtt

0 0 0 0

DividendsDividends

Arbitrage restrictions on call values(1) The value of a call is never less than the larger of:

zerozero S - KS - K

S C S K S Kr Dt max[ , , ]0

(2) The value of a call is never greater than: the price of its underlying stockthe price of its underlying stock

In other words:In other words:

S Kr Dt

The amount of The amount of cash depositedcash deposited

(equals to K)(equals to K)

Call price

StockStockpriceprice

Kr Dt

Cox & Rubinstein. 1985. p 131.

C = SC = S

C S Kr Dt

Cox & Rubinstein. 1985. p 136.

StrikingStriking priceprice

Call price

C = S - KS

C

Arbitrage restrictions on put valuesThe value of a put is never less than the larger of

zerozero K-SK-S

K P K S D Kr St max[0, , ]

D Kr St

The value of a put is never greater than its striking priceits striking price

In other words:In other words:The amount of The amount of cash depositedcash deposited

(equals K)(equals K)

Put price

StockStockpriceprice

K - S

Cox & Rubinstein. 1985. p 147.

K

Putprice

StrikingStrikingpricepriceCox & Rubinstein. 1985. p 148.Cox & Rubinstein. 1985. p 148.

P = KP = K

P = K - SP = K - S

PP

Popular research subjects Forecasting the future volatility Forecasting the future volatility

(Engle Robert F.: “Statistical Models for Financial Volatility”, (Engle Robert F.: “Statistical Models for Financial Volatility”, Financial Analysts’ Journal Jan/Feb 1993)Financial Analysts’ Journal Jan/Feb 1993)

Comparing the theoretical and the actual Comparing the theoretical and the actual pricing of optionspricing of options

(Kahra Hannu: “Pricing FOX Options Under Conditional (Kahra Hannu: “Pricing FOX Options Under Conditional Heteroscedasticity In Returns”, Tampere Economic Studies 1/1992)Heteroscedasticity In Returns”, Tampere Economic Studies 1/1992)

Option Valuation under Stochastic Option Valuation under Stochastic Volatility. With Mathematica Code.Volatility. With Mathematica Code.

((Alan L. Lewis, Finance Press, NewPort California, 2000Alan L. Lewis, Finance Press, NewPort California, 2000))

Option pricing, using distributions other Option pricing, using distributions other than the standard normal distribution than the standard normal distribution

(Cox John C & Steven A. Ross: “The Valuation of Options for (Cox John C & Steven A. Ross: “The Valuation of Options for Alternative Stochastic Processes”, Journal of Financial Alternative Stochastic Processes”, Journal of Financial Economics Jan-Mar 1976)Economics Jan-Mar 1976)

Findings:

Call options tend to be overpricedCall options tend to be overpriced

Put options tend to be underpricedPut options tend to be underpriced

Volatility seems to be forecastable in the short runVolatility seems to be forecastable in the short run

Only the broker makes profits in the long runOnly the broker makes profits in the long run