9 - Options Basics

8/12/2019 9 - Options Basics

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OPTIONS


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Derivatives

When price of one security is derived from another

security it is called as Derivative

Such value can be derived from Stocks, Bonds,

Commodities, Currency Exchange Rates, Interest Ratesetc.

Common Example

Futures

Options

Swaps


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Options

What is an Option?

A type of contract between two investors where one grants the

other right (not the obligation) to buy or sell a specific asset in the

future at a predetermined price

The option buyer is buying the right to buy or sell the

underlying asset at some future date

The option seller is selling the right to buy or sell the

underlying asset at some future date


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Participants in Derivatives

Derivatives are used for

Hedging

Speculation/Investment

Arbitrage


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Participants in Derivatives - Hedging

If someone bears an economic risk and uses the

derivative market to reduce the risk, the person is a

hedger

Hedging is a prudent business practice; today a

prudent manager has an obligation to understand

and apply risk management techniques including the

use of derivatives


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Participants in Derivatives - Speculation

A person or firm who accepts the risk the hedger

does not want to take is a speculator

Speculators believe the potential return outweighs the

risk The primary purpose of derivatives markets is not

speculation. Rather, they permit or enable the transfer

of risk between market participants as they desire


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Participants in Derivatives - Arbitrage

Arbitrage is the existence of a riskless profit

Arbitrage opportunities are quickly exploited and

eliminated in efficient markets

Arbitrage then contributes to the efficiency of markets


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Participants in Derivatives - Arbitrage

Persons actively engaged in seeking out minor pricing

discrepancies are called arbitrageurs

Arbitrageurs keep prices in the marketplace efficient

An efficient market is one in which securities are priced inaccordance with their perceived level of risk and their

potential return

The pricing of options incorporates this concept of

arbitrage


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Types of Options

Call Option

Put Option

Call Option Put Option

Option Buyer

(Long)

Buys the right to buy the

underlying asset at the

Strike Price

Buys the right to sell the

underlying asset at the

Strike Price

Option Seller

(Short)

Has the obligation to sell

the underlying asset to theoption holder at the Strike

Price

Has the obligation to buy

the underlying asset fromthe option holder at the

Strike Price


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Options Agreement Specification

Type of Option Call / Put

Underlying Security (S) e.g. RELIANCE

Time to Expiry (t) Time at/before which option will

be valid. After passage of this option agreement willexpire

Strike Price (X) Reference price using which payoff

for option will be calculated


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Call Option Payoffs at Expiry

For Call Option Buyers (Long)

Call Option Payoff = max(0, S-X)

Net Call Option Payoff = max(0, S-X) Option Price


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Put Option Payoffs at Expiry

For Put Option Buyers (Long)

Put Option Payoff = max(0, X-S)

Net Put Option Payoff = max(0, X-S) Option Price


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Fair Value of a Choice

E.g. We are tossing a fair coin (equal probability of

head/tail). You will get Rs. 10 if its head and Rs. 0 if it is

tails. How much are you willing to pay for this game?

Probability 0.5 0.5

Payoff 10 0

Fair Price = (0.5*10) + (0.5*0) = 5

Pay


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BINOMIAL OPTIONPRICING MODEL (BOPM)


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Binomial Pricing Model

If we know stock prices at expiry then we can calculate fair

price of the option

We start with a single period.

Then, we stitch single period together to form Multi-period

Binomial Option Pricing Tree


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Binomial Pricing Model Assumptions

There are two (and only two) possible prices for the underlying asset onthe next date. The underlying price will either: Increase by a factor of u% (an uptick)

Decrease by a factor of d% (a downtick)

The uncertainty is that we do not know which of the two prices will berealized.

No dividends.

The one-period interest rate, r, is constant over the life of the option (r%per period).

Markets are perfect (no commissions, bid-ask spreads, taxes, pricepressure, etc.)


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Stock Pricing Process

Time T is the expiration day of a call option. Time T-1 is one

period prior to expiration.

Suppose that ST-1 = 800, u = 25% and d = -10%. What

are ST,u and ST,d?

ST,d = (1+d)ST-1

ST,u = (1+u)ST-1

ST-1

800

ST,u = ______

ST,d

= ______


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Stock Pricing Process

Time T is the expiration day of a call option. Time T-1 is one

period prior to expiration.

Suppose that ST-1 = 800, u = 25% and d = -10%. What

are ST,u and ST,d?

ST,d = (1+d)ST-1

ST,u = (1+u)ST-1

ST-1

800

ST,u = _1000_

ST,d

= _720__


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Option Pricing Process (Call)

Suppose that K = 900. What are CT,u and CT,d?

CT,d

= max(0, ST,d

-K) = max(0,(1+d)ST-1

-K)

CT,u = max(0, ST,u-K) = max(0,(1+u)ST-1-K)

CT-1

CT-1

CT,u = _ __

CT,d = _____


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Option Pricing Process (Call)

Suppose that K = 900. What are CT,u and CT,d?

CT,d

= max(0, ST,d

-K) = max(0,(1+d)ST-1

-K)

CT,u = max(0, ST,u-K) = max(0,(1+u)ST-1-K)

CT-1

CT-1

CT,u = _100__

CT,d = __0___

How to find price of this option?


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Equivalent Portfolio

Assume that we have another portfolio consisting of

shares and Rs. B as borrowed cash.

We also assume that r is risk free interest rate from

period T-1 to T.

(1+d)ST-1 + (1+r)B = ST,d + (1+r)B

(1+u)ST-1 + (1+r)B = ST,u + (1+r)B

ST-1+B

Portfolio

at time T-1Portfolio

at expiry


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If at expiry the portfolio value is equal to call option

payoff, then we can find out call value today

(1+d)ST-1 + (1+r)B = ST,d + (1+r)B

(1+u)ST-1 + (1+r)B = ST,u + (1+r)B

ST-1+B

Portfolio

at time T-1Portfolio

at expiry


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Set the payoff of equivalent portfolio to option

payoff

Solve for these two equations with two unknowns and ST-1

(1+u)ST-1 + (1+r)B = CT,u

(1+d)ST-1 + (1+r)B = CT,d

Known values

u = 20%

d = -10%r = 10%

CT,u = 100CT,d = 0


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If two assets offer the same payoffs at time T, then they

must be priced the same at time T-1.

Here, we have set the problem up so that the equivalentportfolio offers the same payoffs as the call.

Hence the calls value at time T-1 must equal the $ amount

invested in the equivalent portfolio.

CT-1 = ST-1 + B


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Equivalent Portfolio Solution

r)d)(1(u

d)C(1u)C(1B

SS

CC

d)S(u

CC

uT,dT,

dT,uT,

dT,uT,

1T

dT,uT,

+

++=

=

=

CT-1 = ST-1 + B


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Equivalent Portfolio Solution Simplification

du

rup)(1and

du

drp

where,

r)(1

p)C(1pCC

or,

r)(1

Cdu

ruC

du

dr

C

dT,uT,

1T

dT,uT,

1T

=

=

+

+

=

+

+

=

r)(1

p)C(1pCC du

+

+=In general,


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Interpreting p

p is the probability of an uptick in a risk-neutral world.

In a risk-neutral world, all assets (including the stock andthe option) will be priced to provide the same riskless rate

of return, r.

In our example, if p is the probability of an uptick then

ST-1 = [(0.57143)(1000) + (1- 0.57143)(720)]/1.1= 800

That is, the stock is priced to provide the same riskless

rate of return as the call option

Then why do people trade derivatives?


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Interpreting

Delta, , is the riskless hedge ratio; 0 < c < 1.

Delta, , is the number of shares needed to hedge onecall. I.e., if you are long one call, you can hedge your riskby selling shares of stock.

Therefore, the number of calls to hedge one share is 1/.I.e., if you own 100 shares of stock, then sell 1/ calls tohedge your position. Equivalently, buy shares of stockand write one call.

In continuous time, = C/S = the change in the valueof a call caused by a (small) change in the price of theunderlying asset.

Then why do people trade derivatives?


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Two Period Binomial Model

ST,dd = (1+d)2ST-2

ST,uu = (1+u)2ST-2

ST-1,u = (1+u)ST-2ST,ud = (1+u)(1+d)ST-2

ST-1,d = (1+d)ST-2

ST-2

CT,dd = max[0,(1+d)2ST-2 - K]

CT,uu = max[0,(1+u)2ST-2 - K]

CT-1,u CT,ud = max[0,(1+u)(1+d)ST-2 - K]

CT-1,d

CT-2


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Two Period Binomial Model Example

ST,dd = 648

ST,uu = 1250

ST-1,u = 1000

ST-1,d = 720

ST-2 = 800

CT,dd = 0

CT,uu = _______

CT-1,u = ____CT,ud = 50

CT-1,d = 20.41CT-2

ST,ud = 900


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Two Period Binomial Model Equivalent Portfolio

= 1B = -809.524

= 0.1984B = -122.449

= 0.6074B = -397.057

T-2 T-1

Note that as S rises, also rises. As S declines, so does .

Note that the equivalent portfolio is self financing. This means that the

cost of any purchase of shares (due to a rise in ) is accompanied by anequivalent increase in required borrowing (B becomes more negative).

Any sale of shares (due to a decline in ) is accompanied by an

equivalent decrease in required borrowing (B becomes less negative).


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n Period Binomial Model

In general, the n-period model is:

= ++

+=

n

aj

nTjnjjnjn K]Sd)(1u)[(1p)(1pj

n

r)(1

1C

Where a in the summation is the minimum number of up-ticks so that thecall finishes in-the-money.


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Suppose the Number of Periods Approaches

Infinity

S

T

In the limit, that is, as N gets large, and if u and d are consistentwith generating a lognormal distribution for ST, then the BOPMconverges to the Black-Scholes Option Pricing Model


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Binomial Put Pricing

ST,u = (1+u)ST-1

ST-1

ST,d = (1+d)ST-1

PT,u = max(0,K-ST,u) = max(0,K-(1+u)ST-1)

PT-1

PT,d = max(0,K-ST,d) = max(0,K-(1+d)ST-1)

(1+u)ST-1 + (1+r)B = ST,u + (1+r)B = PT,u

ST-1+B

(1+d)ST-1 + (1+r)B = ST,d + (1+r)B = PT,d


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PT-1 = ST-1 + B

du

dudu

SS

PP

d)S(u

PP

=

=

r)d)(1(u

d)P(1u)P(1B ud

+++

=

Where:

-1 < p < 0

A put is can be replicated by selling shares of stock short, andlending $B. and B change as time passes and as S changes.Thus, the equivalent portfolio must be adjusted as time passes.

B > 0


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r)(1

p)P(1pPP du

+

+=

durup)(1and

dudrp

=

=

Where:


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Simple vs. Continuous Compounding

Calculate yearly 10% simple interest rate on 50

Interest = 50 * (1+10%) = 55

What if, interest is paid twice half yearly

Interest = 50 * (1 + 5%) * (1+ 5%) = 55.125

What if, interest is paid four times in a year

Interest = 50 * (1 + 2.5%)4 = 55.19064

What if, interest is calculated on continuous basis Interest = 50 * e10%*1 = 55.25855


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Simple vs. Continuous Compounding

We mostly use continuous compounding in Finance.

Formula for continuous compounding is

P * erT = F

where, P Present Value

e Constant

r Continuous Interest Rate

T Time

F Future Value

How to calculate Present Value from Future Value?


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European vs. American Option

European Option

can only be exercised on the expiry date

Indian stock exchanges options are european options

There is a simple equation with a closed-form solutionthat can be used to evaluate option price easily

American Option

can be exercised at any time up to and including

expiry date They are mostly traded OTC

No standardized approach to price american option

We will only talk about European Options


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Bounds for Option Price Upper Bounds for Call

Call option gives the holder right to buy one share

of a stock for a certain price.

No matter what happens, the option can never be

worth more than the stock. Hence, C S0

What if C > S0?


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Bounds for Option Price Upper Bounds for Call

Call option gives the holder right to buy one share

of a stock for a certain price.

No matter what happens, the option can never be

worth more than the stock. Hence, C S0

What if C > S0?e.g. S

0= 40, C=50, K=20

At time 0, you sell the call at 50 and buy stock at 40. You

are left with 10

At expiry, 1) if call is in the money then option will be

exercised

2) if call is out of the money then call is

worthless. Sell stock and keep money.


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Bounds for Option Price Upper Bounds for Put

At expiry, the put option can not be worth more

than K. (Why? Stock price can minimum go to 0)

So, put option can not be worth more than present

value of K Hence, P Ke-rT

What if P > Ke-rT ?


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Bounds for Option Price Upper Bounds for Put

At expiry, the put option can not be worth more

than K. (Why? Stock price can minimum go to 0)

So, put option can not be worth more than present

value of K Hence, P Ke-rT

What if P > Ke-rT ?

At time 0, Sell the put and invest P at risk free interest rateAt expiry, maximum value for Put will be K and invested

money will be greater than K


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Bounds for Option Price Lower Bounds for Call

Lower bound for a call option is

C max(S0 - Ke-rT, 0)

Verify if breach of this condition results in an arbitrage

e.g. S0 = 20, K = 18, r = 10%, T = 1, C = 3


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Bounds for Option Price Lower Bounds for Put

Lower bound for a put option is

P max(Ke-rT - S0, 0)

Verify if breach of this condition results in an arbitrage

e.g. S0 = 37, K = 40, r = 5%, T = 0.5, P = 1


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Put-Call Parity Payoff at time T

Put-Call Parity is the relationship between prices of

Call and Put options that have same strike price and

maturity.

Consider 2 portfolios as below,

Portfolio A Portfolio B

One European Call Option

Zero coupon bond thatprovides a payoff of K at time T

One European Put Option

One share of Stock


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Put-Call Parity Payoff at maturity

Portfolio A Portfolio B

At time 0

One European Call Option Zero coupon bond thatprovides a payoff of K at time T

One European Put Option One share of Stock

At Expiry

If ST > K

Call Option = ST - K Zero Coupon Bond = K Total = ST

Put Option = 0 One Share = ST Total = ST

If ST < K

Call Option = 0

Zero Coupon Bond = K Total = K

Put Option = K - ST

One Share = ST Total = K


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Put-Call Parity Payoff at maturity


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Put-Call Parity

If two portfolios have same payoff at expiry then

the price of two portfolio should be same

Hence put-call parity,

C + Ke-rT = P + S0


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Options Moneyness

Option Premium is the amount of money option

buyer pays to the option writer to buy the option

contract

Option premium is split into two components Intrinsic Value

Time Value


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Options Moneyness Intrinsic Value

Intrinsic Value option is calculated as if option would

have exercised today

It is defined as the difference between the options

strike price (X) and stock current price (CP) For Call, Intrinsic Value = max(0, CP X)

For Put, Intrinsic Value = max(0, X CP)

Intrinsic value depends on moneyness of the option Intrinsic value can not be negative


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Options Moneyness

The concept of moneyness describes whether an

option is in-, out-, at-, or in-the-money by examining

the position of strike vs. existing market price of the

option's underlying security. In terms of moneyness, option can be classified as

In the Money (ITM)

Deep In the Money

Out of the Money (OTM)

At the Money (ATM)


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Options Moneyness - ITM

Any option that has positive intrinsic value is said to

be In the Money (ITM)


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Options Moneyness Deep ITM

These options represent a larger spread between the

strike and market price of an underlying security.

They trade at or near to their actual intrinsic values

This is because options with a significant amount ofintrinsic value built in have a very low chance of

expiring worthless.

Thus investing in the Deep ITM option is similar to

investing in the underlying asset, except the optionholder will have the benefits of lower capital outlay,

limited risk and leverage.


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Options Moneyness OTM

When intrinsic value of an option is 0, it is called as

Out of the Money (OTM) option


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Options Moneyness ATM

An option is said to be At the Money (ATM) when

the strike price of a call or put is equal to the

underlying asset.

Practically, the strike price

which is closest to the

underlying price is said to be

ATM strike


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Call Option Price

Variation of Price of call option with the stock price.

Curve moves in the direction of the arrows when

there is an increase in the interest rate, time to

maturity or stock price volatility.


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BLACK SCHOLES OPTIONPRICING MODEL (BSOPM)


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Continuous Time Option Pricing Model

Assumptions of Black Scholes Option Pricing ModelNo transaction cost

No taxes

Unrestricted short-selling of stock, with full use of short-sale proceeds

Shares are infinitely divisible

Constant riskless interest rate for borrowing/lending

No Dividends European Option

Continuous Trading


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Derivation of Black Scholes Pricing

Specify a process that the stock price will follow (i.e.,all possible paths)

Construct a riskless portfolio of:

Long Call

Short D Shares

As delta changes (because time passes and/or Schanges), one must maintain this risk-free portfolio

over time This is accomplished by purchasing or selling the

appropriate number of shares.


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Lognormal Distribution

Black Scholes Option Pricing Model assumes that thestock prices follow Geometric Brownian Motion

In turn, this implies that the distribution of the returns ofthe stock, at any future date, will be lognormally

distributed Lognormal returns are realistic for two reasons If returns are lognormally distributed, then the lowest

possible return in any period is -100%

lognormal returns distributions are positively skewed, that

is skewed to the right. Thus stock returns distribution would have a minimum return

of -100% and a maximum return can go beyond 100%


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Lognormal Distribution Price Changes

E S S e

S S e e

T

T

T

T T

( )

( ) ( )

=

=

0

0

2 2 2 1

var


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Black Scholes Option Formula For Call

( ) ( )2rT

1 dNKedSNC

=

where N(di) = the cumulative standard normal distribution function, evaluated at di,

and:

T

/2)T(rln(S/K)d

2

1

++=

Tdd 12 =

N(-di) = 1-N(di)


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Black Scholes Option Formula Example

S = 92

K = 95

T = 50 days (50/365 year = 0.137 year)

r = 7% (per annum)

= 35% (per annum)

What is the value of the call?



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Calculation

Calculate the PV of the Strike Price:

Ke-rT = 95e(-0.07)(50/365) = (95)(0.9905) = 94.093.

Calculate d1 and d2:

0.1370.35

.1370.1225/2)0(0.07ln(92/95)

T

/2)T(rln(S/K)d

2

1

++=

++=

0.10890.12955

0.017980.03209

701)(0.35)(0.3

0.01798)ln(0.96842=

+=

+=

d2 = d1 T.5 = -0.1089 (0.35)(0.137).5 = -0.2385



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Calculation

Calculating N(d1) and N(d2) Standard Normal Probability Tables

Excel Function NORMSDIST

N(-0.1089) = 0.4566; N(-0.2385) = 0.4058.



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Calculation

Solving for Call ValueC = S N(d1) Ke

-rT N(d2)

= (92)(0.4566) (94.0934)(0.4058)

= 3.8307.

Applying Put-Call Parity, the put price is:

P = C S + Ke-rT = 3.8307 92 + 94.0934

= 5.92.


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Black Scholes Option Formula For Put

( ) ( )12rT d-SN-d-NKeP =

where N(di) = the cumulative standard normal distribution function, evaluated at di,

and:

T

/2)T(rln(S/K)d

2

1

++=

Tdd 12 =

N(-di) = 1-N(di)


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Binomial and Black Scholes Model

Note the analogous structures of the BOPM and theBSOPM:

C = S + B (0<


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Binomial and Black Scholes Model

The BOPM actually becomes the BSOPM as the number ofperiods approaches , and the length of each period

approaches 0.

In addition, there is a relationship between u and d, and , so

that the stock will follow a Geometric Brownian Motion. If youcarve T years into n periods, then:

n

T

0.50.5q

equalmustq,uptick,anofyprobabilittheand,1ed

1eu

T/n

T/n

+=

=

=

Bl k S h l M d l V l l


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Black Scholes Model - Volatility

Volatility is the key to pricing options.

Believing that an option is undervalued is tantamount

to believing that the volatility of the rate of return onthe stock will be less than what the market believes

and vice versa.

We will be learning more about volatility later

Th k


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Thanks

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9 - Options Basics

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