11. option pricing

Option PricingOption Pricing
..

Stochastic ProcessStochastic Process
A variable whose value changes overA variable whose value changes over
time in an uncertain way is said totime in an uncertain way is said to
follow a stochastic process.follow a stochastic process.
Stochastic processes can be “discreteStochastic processes can be “discrete
time” or “continuous time” and alsotime” or “continuous time” and also
“discrete variable” or “continuous“discrete variable” or “continuous
variable”variable”

Markov ProcessMarkov Process
It is a particular type of stochastic processIt is a particular type of stochastic process
where only the present value of a variable iswhere only the present value of a variable is
relevant for predicting the future. The pastrelevant for predicting the future. The past
history of the variable and the way in which thehistory of the variable and the way in which the
present value has emerged from the past arepresent value has emerged from the past are
irrelevant.irrelevant.
It is consistence with the weak form of marketIt is consistence with the weak form of market
efficiency and means that while statisticalefficiency and means that while statistical
properties of the stock prices may be useful inproperties of the stock prices may be useful in
determining the characteristics of the stochasticdetermining the characteristics of the stochastic
process followed by the stock price but theprocess followed by the stock price but the
particular path followed in the past is irrelevant.particular path followed in the past is irrelevant.

Wiener ProcessWiener Process
It is a particular type of Markov StochasticIt is a particular type of Markov Stochastic
Process and has been used in physics toProcess and has been used in physics to
describe the motion of a particular subjected todescribe the motion of a particular subjected to
a large number of small molecular shocks and isa large number of small molecular shocks and is
sometimes referred to as Brownian Motion. Itsometimes referred to as Brownian Motion. It
has two properties;has two properties;
1.1. Small change is equal to root of change inSmall change is equal to root of change in
time multiplied by a random variable following atime multiplied by a random variable following a
standardized normal distribution.standardized normal distribution.
2.2. For two different short intervals, the smallFor two different short intervals, the small
changes are independent.changes are independent.

Lognormal DistributionLognormal Distribution
It is the variable whose logarithmIt is the variable whose logarithm
values are normally distributed. Wevalues are normally distributed. We
need to convert lognormalneed to convert lognormal
distributions (stochastic stock pricedistributions (stochastic stock price
changes) to normal distributions sochanges) to normal distributions so
that one could undertake analysisthat one could undertake analysis
using confidence limits, hypothesisusing confidence limits, hypothesis
testing etc.testing etc.

Option Pricing ModelsOption Pricing Models
DCF criterion cannot be used since risk ofDCF criterion cannot be used since risk of
anan
option is virtually indeterminate and henceoption is virtually indeterminate and hence
the discount rate is impossible to bethe discount rate is impossible to be
estimated. The two popular models are:estimated. The two popular models are:
 The Binomial ModelThe Binomial Model
 Black – Scholes ModelBlack – Scholes Model

The Binomial ModelThe Binomial Model
The model assumes,The model assumes,
 The price of asset can only go up orThe price of asset can only go up or
go down in fixed amounts ingo down in fixed amounts in
discrete time.discrete time.
 There is no arbitrage between theThere is no arbitrage between the
option and the replicating portfoliooption and the replicating portfolio
composed of underlying asset andcomposed of underlying asset and
risk-less asset.risk-less asset.

 Current stock price = SCurrent stock price = S
 Next Year values = uS or dSNext Year values = uS or dS
 B amount can be borrowed at ‘r’.B amount can be borrowed at ‘r’.
Interest factor is (1+r) = RInterest factor is (1+r) = R
 d < R < u (no risk free arbitraged < R < u (no risk free arbitrage
possible)possible)
 E is the exercise priceE is the exercise price

Depending on the change in stockDepending on the change in stock
value, option value will bevalue, option value will be
Cu = Max (uS – E, 0)Cu = Max (uS – E, 0)
Cd = Max (dS – E, 0)Cd = Max (dS – E, 0)

..
S
Su
Sd
Su2
Sud
Sd2

We now set a portfolio of ∆ shares and B amountWe now set a portfolio of ∆ shares and B amount
ofof
debt such that its payoff is equal to that of calldebt such that its payoff is equal to that of call
option after 1 year. Then,option after 1 year. Then,
Cu = ∆uS + RB……………Cu = ∆uS + RB…………… (1)(1)
Cd = ∆dS + RB…………….Cd = ∆dS + RB……………. (2)(2)
Solving these equations,Solving these equations,
(Cu – Cd)(Cu – Cd)
∆∆ == ; and; and
S(u-d)S(u-d)
(uCd – dCu)(uCd – dCu)
B =B =
(u – d)R(u – d)R
Hence C = ∆S + B, since portfolio has same payoff asHence C = ∆S + B, since portfolio has same payoff as

IllustrationIllustration
A stock is currently selling for Rs.40.A stock is currently selling for Rs.40.
The call option on the stockThe call option on the stock
exercisable a year from now at aexercisable a year from now at a
strikestrike
price of Rs.45 is currently selling atprice of Rs.45 is currently selling at
Rs.8. The risk-free rate is 10%. TheRs.8. The risk-free rate is 10%. The
stock can either rise or fall after astock can either rise or fall after a
year.year.
It can fall by 20%. By whatIt can fall by 20%. By what
percentagepercentage

Black-Scholes Model as the LimitBlack-Scholes Model as the Limit
of the Binomial Modelof the Binomial Model
The Binomial Model converges to the Black-The Binomial Model converges to the Black-
Scholes model as the number of timeScholes model as the number of time
periods increases.periods increases.

Black-Scholes Model: The originBlack-Scholes Model: The origin
 1820s – Scottish scientist Robert Brown1820s – Scottish scientist Robert Brown
observed motion of suspended particles inobserved motion of suspended particles in
water.water.
 Early 19Early 19thth
century – Albert Einstein usedcentury – Albert Einstein used
Brownian motion to explain movements ofBrownian motion to explain movements of
molecules, many research papers.molecules, many research papers.
 1900 – French scholar, Louis Bachelier wrote1900 – French scholar, Louis Bachelier wrote
dissertation on option pricing and developed adissertation on option pricing and developed a
model strikingly similar to BSM.model strikingly similar to BSM.
 1951 – Japanese mathematician Kiyoshi Ito1951 – Japanese mathematician Kiyoshi Ito
developed Ito’s Lemma that was used in optiondeveloped Ito’s Lemma that was used in option
pricing.pricing.

Black-Scholes Model: The originBlack-Scholes Model: The origin
 Fischer Black and Myron Scholes worked inFischer Black and Myron Scholes worked in
Finance Faculty at MIT Published paper in 1973.Finance Faculty at MIT Published paper in 1973.
They were later joined by Robert Merton.They were later joined by Robert Merton.
 Fischer left academia in 1983, died in 1995 atFischer left academia in 1983, died in 1995 at
57.57.
 1997 – Scholes and Merton got Nobel Prize1997 – Scholes and Merton got Nobel Prize

Black-Scholes ModelBlack-Scholes Model
Fischer Black and Myron Scholes, The Journal of Political Economy, 1973Fischer Black and Myron Scholes, The Journal of Political Economy, 1973
Assumptions:Assumptions:
 The underlying stock pays no dividends.The underlying stock pays no dividends.
 It is a European option.It is a European option.
 The stock price is continuous and is distributedThe stock price is continuous and is distributed
lognormally.lognormally.
 There are no transaction costs and taxes.There are no transaction costs and taxes.
 No restrictions or penalty on short sellingNo restrictions or penalty on short selling
 The risk free rate is known and is constant overThe risk free rate is known and is constant over
the life of the option.the life of the option.

CC00 = S= S00 N (dN (d11 ) – E/e) – E/ertrt
N (dN (d22 ) where,) where,
CC00 = Present equilibrium value of call option= Present equilibrium value of call option
SS00 = Current stock price= Current stock price
EE = Exercise price= Exercise price
ee = Base of natural logarithm= Base of natural logarithm
rr = Continuously compounded risk free interest rate= Continuously compounded risk free interest rate
tt = length of time in years to expiration= length of time in years to expiration
N (*)N (*) = Cumulative probability distribution function of a= Cumulative probability distribution function of a
standardized normal distributionstandardized normal distribution

C = S N (dC = S N (d11 ) – K) – Kee-rt-rt
N (dN (d22 ) where,) where,
CC = Present equilibrium value of call option= Present equilibrium value of call option
SS = Current stock price= Current stock price
KK = Exercise price= Exercise price
ee = Base of natural logarithm= Base of natural logarithm
rr = Continuously compounded risk free interest rate= Continuously compounded risk free interest rate
tt = length of time in years to expiration= length of time in years to expiration
N (*)N (*) = Cumulative probability distribution function of a= Cumulative probability distribution function of a
standardized normal distributionstandardized normal distribution

llnn (S(S00 /E) + (r + ½/E) + (r + ½ σσ22
)t)t
dd11 ==
σσ √t√t
llnn (S(S00 /E) + (r - ½/E) + (r - ½ σσ22
)t)t
dd22 ==
σσ √t√t
where lwhere lnn is the natural logarithmis the natural logarithm

llnn (S/K(S/K ee-rt-rt
))
dd11 == + 0.5+ 0.5 σσ √t√t
σσ √t√t
dd22 == dd11 -- σσ √t√t
where lwhere lnn is the natural logarithmis the natural logarithm

The standard deviation of the continuouslyThe standard deviation of the continuously
compounded stock price change for acompounded stock price change for a
company is estimated to be 20% per year.company is estimated to be 20% per year.
The stock currently sells for Rs.80 and theThe stock currently sells for Rs.80 and the
effective annual interest rate is Rs.15.03%.effective annual interest rate is Rs.15.03%.
What is the value of a one year call optionWhat is the value of a one year call option
on the stock of the company if the exerciseon the stock of the company if the exercise
price is Rs.82?price is Rs.82?

The Linkage between Calls, Puts,The Linkage between Calls, Puts,
Stock, and Risk-Free BondsStock, and Risk-Free Bonds
..
Call
Stock
Put
Risk-Free
Bond
Black-Scholes
Call Option
Pricing Model
Put-Call
Parity
Black-Scholes
Put Option
Pricing Model

Put-Call Parity TheoremPut-Call Parity Theorem
Payoffs just beforePayoffs just before
expirationexpiration
If SIf S11 < E< E If SIf S11 > E> E
1.1. Buy the equity stockBuy the equity stock SS11 SS11
2.2. Buy a put optionBuy a put option E-SE-S11 00
3.3. Borrow amount equalBorrow amount equal
to exercise priceto exercise price - E- E - E- E
1+2+3=Buy a call option1+2+3=Buy a call option 00 SS11 - E- E

Using Black-Scholes ModelUsing Black-Scholes Model
1.1. Find the Standard Deviation of theFind the Standard Deviation of the
continuously compounded asset value changecontinuously compounded asset value change
and the square root of the time left toand the square root of the time left to
expirationexpiration
2.2. Calculate ratio of the current asset value toCalculate ratio of the current asset value to
the present value of the exercise pricethe present value of the exercise price
3.3. Consult the table giving %age relationshipConsult the table giving %age relationship
between the value of the Call Option and thebetween the value of the Call Option and the
stock price corresponding to the value instock price corresponding to the value in
steps 1 and 2steps 1 and 2
4.4. Value of Put Option = Value of Call Option +Value of Put Option = Value of Call Option +
PV of exercise price – Stock PricePV of exercise price – Stock Price

Find the value of a one year call option asFind the value of a one year call option as
well as a put option, if the current stockwell as a put option, if the current stock
price is Rs.120, exercise price is Rs.125 andprice is Rs.120, exercise price is Rs.125 and
the S.D. of continuously compounded pricethe S.D. of continuously compounded price
change of the stock is 30%. The effectivechange of the stock is 30%. The effective
interest rate is 15.03% so that the interestinterest rate is 15.03% so that the interest
factor is 1.1503.factor is 1.1503.

Step 1: Standard Deviation × √Time = 0.30 × √1Step 1: Standard Deviation × √Time = 0.30 × √1
= 0.30= 0.30
Step 2: The ratio of stock price to the PV ofStep 2: The ratio of stock price to the PV of
exercise price = 120 ÷ 125/1.1503exercise price = 120 ÷ 125/1.1503
= 120/108.7 = 1.10= 120/108.7 = 1.10
Step 3: Consulting the table we get 16.5% of theStep 3: Consulting the table we get 16.5% of the
stock price as the value of call option i.e.stock price as the value of call option i.e.
120×0.165 = 19.8120×0.165 = 19.8
Step 4: Value of Put OptionStep 4: Value of Put Option
= 19.8 + 108.7 – 120 = 8.5= 19.8 + 108.7 – 120 = 8.5

Variants of Black-Scholes ModelVariants of Black-Scholes Model
 To price European options on dividendTo price European options on dividend
paying options and American options onpaying options and American options on
non-dividend paying stocks (Robertnon-dividend paying stocks (Robert
Merton, 1973 and Clifford Smith, 1976).Merton, 1973 and Clifford Smith, 1976).
 American call options on dividend-payingAmerican call options on dividend-paying
stocks (Richard Roll, 1977; Robertstocks (Richard Roll, 1977; Robert
Whaley, 1981; and Richard Geske andWhaley, 1981; and Richard Geske and
Richard Roll, 1984)Richard Roll, 1984)

Variants of Black-Scholes ModelVariants of Black-Scholes Model
 When price changes are discontinuousWhen price changes are discontinuous
(Cox, Rubinstein and Ross, 1979). This(Cox, Rubinstein and Ross, 1979). This
was published in Journal of Financialwas published in Journal of Financial
Economics as “Option Pricing: AEconomics as “Option Pricing: A
Simplified Approach”. This is the BinomialSimplified Approach”. This is the Binomial
Model for Option Pricing.Model for Option Pricing.

Garman - Kohlhagen ModelGarman - Kohlhagen Model
The foreign currency option pricing modelThe foreign currency option pricing model
is equivalent to the Black-Scholes modeis equivalent to the Black-Scholes mode
except that the spot rate is discounted byexcept that the spot rate is discounted by
the foreign interest rate and appearsthe foreign interest rate and appears
instead of the Stock Price.instead of the Stock Price.

Sensitivity of Option PremiumsSensitivity of Option Premiums
 An option’s intrinsic value is the amount byAn option’s intrinsic value is the amount by
which it is in the money and the time valuewhich it is in the money and the time value
is the difference between actual premiumis the difference between actual premium
and the intrinsic value i.e. premium =and the intrinsic value i.e. premium =
intrinsic value + time value.intrinsic value + time value.
 At the money option has highest likelihoodAt the money option has highest likelihood
of gaining intrinsic value as compared toof gaining intrinsic value as compared to
that of losing. It has no value to lose butthat of losing. It has no value to lose but
50-50 chance of gaining.50-50 chance of gaining.

Delta (Delta (δδ):): Change in option price relativeChange in option price relative
to the price of underlying asset. Reverseto the price of underlying asset. Reverse
of Delta is used to calculate a hedge ratio.of Delta is used to calculate a hedge ratio.
Gamma:Gamma: The rate of change of Delta. It isThe rate of change of Delta. It is
the second derivative of option price withthe second derivative of option price with
respect to price of the asset and is alsorespect to price of the asset and is also
known as option’s curvature. High gammaknown as option’s curvature. High gamma
makes option less attractive.makes option less attractive.

Lambda:Lambda: Change in option price relative toChange in option price relative to
change in volatility. Its value lies betweenchange in volatility. Its value lies between
zero and infinity and declines as optionzero and infinity and declines as option
approaches maturityapproaches maturity
Theta:Theta: Change in option price relative toChange in option price relative to
Time to Expiration. The value of theta liesTime to Expiration. The value of theta lies
between zero and total value of option.between zero and total value of option.
Rho:Rho: Change in option value in relation toChange in option value in relation to
interest rates and varies from type to typeinterest rates and varies from type to type
of the options.of the options.

Implied volatility is obtained by finding theImplied volatility is obtained by finding the
S.D. that when used in the Black-ScholesS.D. that when used in the Black-Scholes
model makes the model price equal tomodel makes the model price equal to
market price of the option.market price of the option.
The pattern of implied volatility acrossThe pattern of implied volatility across
expirations is often called the termexpirations is often called the term
structure of volatility, and the pattern ofstructure of volatility, and the pattern of
volatility across exercise prices is oftenvolatility across exercise prices is often
called the volatility smile or skew.called the volatility smile or skew.

If on February 1, one wants to price a MarchIf on February 1, one wants to price a March
European call option of a company,European call option of a company,
WhereWhere SS == Rs.92.00Rs.92.00
KK == Rs.95.00Rs.95.00
tt == 50 days,50 days, or (50/365=0.137 years)or (50/365=0.137 years)
rr == 7.12%7.12%
σσ == 35%35%
and the company does not pay anyand the company does not pay any
dividendsdividends

Return RelativesReturn Relatives
 If P(0) is the beginning wealth andIf P(0) is the beginning wealth and
P(T), the ending wealth, the priceP(T), the ending wealth, the price
relative R(0,T) is given by P(T)/P(0).relative R(0,T) is given by P(T)/P(0).
Since P(T) is a random variable,Since P(T) is a random variable,
P(T)/P(0) is also a random variable.P(T)/P(0) is also a random variable.
 Holding period return is the effectiveHolding period return is the effective
return r(T) is related to R(0,T).return r(T) is related to R(0,T).
 Continuous holding period return rContinuous holding period return r cc (T)(T)
is related to price relative by:is related to price relative by: rrcc (T) =(T) =
ln(R(0,T)).ln(R(0,T)).
 Thus all these are random variables.Thus all these are random variables.

11. option pricing

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11. option pricing