Options Valuation

using Binomial and

Black-Scholes models

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Session Agenda

Options Valuation

• Brief Introduction to Options

• Complications in Valuing Options

• Binomial Method of Valuing Options

• Replicating Call Option

• Replicating Put Option

• Risk Neutral Valuation

• Change in future stock price

• Generalizing Binomial Method

• Black Scholes Model

• Limitations of Black Scholes Model

• Summary

Expect around 4-5 questions in the exam from today’s lecture

• Options are contracts that give its buyer the right to buy or sell a particular asset

– In future

– At a pre-decided price (i.e. exercise or strike price)

– Without any obligations

• The seller of the option collects a payment (Option Premium) from the buyer for providing the

option

• Types of options:

– Call or Put Options

• Call Option: gives option holder the right to buy the asset at an agreed price

• Put Options: gives option holder the right to sell the asset at an agreed price

– European or American Options

• European options are those that can only be exercised on expiration.

• American options may be exercised on any trading day on or before expiration

• Positions:

• Long position: An option buyer is said to be in a long position

• Short Position: An option writer (or seller) is said to be in a short position

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What are Options?

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Complications in Valuing Options

• Standard approach for valuing any asset:

– Figure out expected cash flows and

– Discount them at risk adjusted cost of capital reflecting the opportunity cost of capital

• Complications that arise in valuing Options

– Impossible to quantify risks associated with the Option cash flows

– Risks associated with the Options change every time there is change in the price of the underlying

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Valuation Method 1: Binomial Method

• Binomial method entails

– Assuming the price of the underlying can take only two values in any given interval of time

– Determining Option pay-offs at these prices

– Valuing the option in one of the following ways:

• Replicating the same pay-offs in a package consisting of assets that can be valued

• Alternatively, determining probability of each pay-off to arrive at a certainty equivalent expected cash-

flow and discounting it to the present value at the risk-free rate (Risk Neutral Method)

S0

Su

Su2

Sud

Sd

Sd2

IV1 = Max[(Su2-X), 0]

IV2

IV3

p

1 - p

1 - p

p

p

1 - p

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Question

• Consider a six-month European call and put option on non-dividend paying stock with identical

exercise prices of Rs 85. This option is at the money. The short-term, risk-free interest rate was a bit

less than 4 percent per year, or about 2 percent for six months. The stock either falls to Rs 63.75 or

rises to Rs113.33 after six months. Determine their pay-offs at expiration:

• Solution: The pay-offs are as follows

Stock price=Rs63.75 Stock price=Rs113.33

1 Call option Rs0 Rs28.33

1 Put option Rs21.25 Rs0

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Question - Replicating Call Option

• Determine the value of the call option in previous question by replicating the call option .

• Solution: Lets look at the pay-offs from a package consisting of 0.5714 stocks and borrowing a

principal of Rs35.71 from the bank. The total amount to be repaid is Rs36.42 (including interest)

• The pay-offs are exactly the same as in the previous example for the call option. It follows that the

value of the call today should be equal to the value of 0.5714 shares less Present Value of Rs 36.42

• Thus, value of Call = (0.5714*85)- PV(36.42)= Rs 12.86

Stock price=Rs63.75 Stock price=Rs113.33

0.5714 Shares Rs36.42 Rs64.75

Repayment of loan + interest -36.42 -36.42

Total Payoff 0 Rs28.33

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Replicating Call Option

• Two questions remain, how did we determine the the number of stocks i.e. 0.5714 and how did we

determine the amount to be borrowed?

– The number of shares to be held is give by the option delta, given by:

• The amount to be borrowed is equal to the present value of the difference between the pay-offs from

the option and pay offs from the delta shares, i.e. 0.5714 share. In our example:

• The amount to be borrowed equals Present Value or PV of 36.43

Stock Price

Scenario 1

63.75

Scenario 2

113.33

Option value

Value of ∆Stock

0

36.43

28.33

64.76

Payoff from Option 0 -28.33

Portfolio Value 36.43 36.43

5714.075.6333.113

033.28

pricessharepossibleofSpread

priceoptionofPossibleSpread

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Replicating Put Option

• The Pay-offs from a put option can be replicated by selling delta share and setting aside a sum of

money in a risk-free investment

• In our example, the delta for the put option is given by:

• The amount to be placed in risk-free investment is PV(48.57). Calculated as shown below:

4286.075.6333.113

25.210

pricessharepossibleofSpread

priceoptionofPossibleSpread

Scenario 1 Scenario 2

Stock Price 63.75 113.33

Option value 21.25 0

Value of ∆Stock 27.32 48.57

Payoff from Option 21.25 0

Portfolio Value 48.57 48.57

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Replicating Put Option

• The put can be replicated as shown below:

• The value of put therefore is,

• Value of put = -0.4286 shares + PV( Rs48.57) (safe loan)

= - (0.4286 * 85) + 47.62= Rs11.19

Stock price=Rs63.75 Stock price=Rs113.33

Sale of 0.4284 Shares -Rs27.32 -Rs48.57

Repayment of loan + interest 48.57 48.57

Total Payoff 21.25 Rs0

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Risk Neutral Method

• The assumption is that investors are indifferent to risk.

– Step 1: Determine the probabilities associated with the different pay-offs

– Step 2: Determine expected cash flow under the assumption that investors are indifferent to risk

– Step 3: Discount the expected cash flow at the risk-free rate to arrive at the present value

• In our example, since the risk-free rate for six months is 2%, and investors are indifferent to risk, it

follows that:

– Expected return= [probability of rise * 33.33] + [(1- probability of rise) * (-25)] = 2.0 percent

– Therefore the probability of rise, p, = 0.463 or 46.3%

• Expected future value of the call option after six months is given by

– [Probability of rise * 28.33] + [(1- probability of rise) * 0]

= (0.463* 28.33) + (0.537 * 0)

= Rs 13.16

• The value today therefore is PV(13.16) = Rs 12.86

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Change in future stock price

• The following formula that relates the up and down changes to the standard deviation of stock returns:

– 1 + upside change = u = eσ√h

– 1 + downside change = d = 1/ u

• Where, e = base of natural logarithms = 2.718

• σ = standard deviation of (continuously compounded) stock returns

• h = time interval as fraction of year

• In our example, standard deviation of stock returns, σ = 40.69%, h = 0.5

– u= e 0.4069√0.5 = 1.3333, => upside change = 33%

– d= 1/u = 1/1.3333 = 0.75, => downside change = 25%

• Thus stock price takes the following two values

– Rs85x1.3333 = Rs113.33

– Rs85x0.75 = 63.75

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Generalizing the Binomial Method

• One step Binomial Method is simplistic

– Assumes just two values for the asset price is possible in the future

• More realism can be added by shortening the time intervals so that the calculations can allow for

greater number of values for the asset price at expiration.

– In our example if we allowed the stock to take values at the end of three months, we would have three

values at the end of six months:

• To work out the equivalent upside and downside changes when we divide the period into two three-

month intervals (h = 0.25), we use the same formula:

– 1 + upside change (3 months interval) = u = e 0.4069√0.25 = 1.226,=> upside change = 22.6%

– 1 + downside change = d = 1/ u = 1/1.226 = 0.816, => downside change = 18.6%

• We get the following tree

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Months

6

Months -18.6%

56.6

-18.6%

69.36

85

85

104.21

127.76

+22.6 or -18.6% +22.6

+22.6

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Generalizing the Binomial Method

• If the time intervals could be made extremely small, we would be able to account for a large number of

changes in the share price

• With the help of computer programs available today the binomial method can be used with very small

time intervals

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Question

Currently, shares of ABC Corp. trade at USD 100. The monthly risk neutral probability of the price

increasing by USD 10 is 30%, and the probability of the price decreasing by USD 10 is 70%.What are

the mean and standard deviation of the price after 2 months if price changes on consecutive months

are independent? (FRM 2010 Sample Paper)

Solution:

Develop a 2 step tree.

Mean = 9% (120) + 42% (100) + 49% (80) = 92

Variance = 9% (120 – 92)2 + 42% (100 – 92)2 + 49% (80 – 92)2 = 168

Thus, standard deviation = 12.96

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Black and Scholes Model

• Black and Scholes formula allows for infinitesimally small intervals as well as the need to revise

leverage for European options on Non Dividend paying stocks

• The formula is

– Value of call option = [delta * share price] – [bank loan]

– Where,

• Log is the natural log with base e

– N (d) = cumulative normal probability density function

– EX = exercise price option; PV(EX) is calculated by discounting at the risk- free interest rate rf

– t = number of periods to exercise date

– P =present price of stock

– σ = standard deviation per period of (continuously compounded) rate of return on stock

• Value of Put = [N (-d2) * PV (EX)] - [N (-d1) * P]

)](*)2([]*)1([ EXPVdNPdN

tdd

t

t

EXPVpd

12

2

)](/log[1

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Question: Black and Scholes Model

• Calculation of the value of call option

Price of stock now (P) 85

Exercise price (EX) 85

Standard deviation of continuously compounded annual returns (σ) 0.4069

Year to maturity (t ) 0.5

Risk-free interest rate per annum, rf 4%

log [P/PV (EX)] 0.02

log [P/PV (EX)]/σ√t 0.07

σ√t/2 0.14

d1 = log [P/PV (EX)]/σ√t+ σ√t/2 0.2134

d2 = d1 - σ√t -0.0743

N(d1) - Can be calculated by using NORMSDIST(d1) in excel 0.5845

N(d2) - Can be calculated by using NORMSDIST(d2) in excel 0.4704

PV(EX) = 85*e-4%/2

83.3169

Value of Call 10.49

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Question: Black and Scholes Model

• For European Options on dividend paying stocks, the present value of expected dividends during the

life of the option needs to be reduced from the present price of the stock:

Without dividend With dividend

Price of stock now 85 85

Present Value of Dividend 0 1.99

Price of stock adjusted for dividend (P) 85 83.01

Exercise price (EX) 85 85

Standard daviation of continuously compounded annual returns (σ) 0.4069 0.4069

Year to maturity (t ) 0.5 0.5

Risk-free interest rate per annum, rf 4% 4%

log [P/PV (EX)] 0.02 -0.004

log [P/PV (EX)]/σ√t 0.07 -0.01

σ√t/2 0.14 0.14

d1 = log [P/PV (EX)]/σ√t+ σ√t/2 0.2134 0.1309

d2 = d1 - σ√t -0.0743 -0.1568

N(d1) 0.5845 0.5521

N(d2) 0.4704 0.4377

PV(EX) = 85*e-4%/2

83.316 83.316

Value of Call 10.49 9.36

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Limitations of Black and Scholes Model

• Limitations:

– The model does not allow for early exercise

– Not suitable for valuing American Options that can be exercised any time during their life

– The stepwise binomial method is superior for valuing American Options, particularly American Puts and

American Calls on stocks that pay dividends

– Not suitable for valuing warrants as warrants are long term options and it is quite likely that the underlying

stock will pay dividends during the life of the warrant

– Also, when exercised warrants increase the total number of shares which adds another level of

complication in valuing warrants using Black and Scholes formula

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Summary

• Complications arise in valuing options because its impossible to quantify risks associated with options

• Options can be valued using the binomial method

– Replicating options

– Risk neutral method

• European options on non dividend paying stocks can be valued using the Black Scholes method

• Option Delta is defined as:

pricessharepossibleofSpread

priceoptionofPossibleSpread

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• Replicating a call option

• Construct a package containing

– Buy delta stocks and

– Borrow a sum of money which is equal to the difference between the pay-offs from the option and pay offs

from the delta shares

• This package has the same pay-off as a call option

• The value of the package is the value of the call option

• Replicating a put option

• Construct a package containing

– Sell delta stocks and

– Deposit a sum of money which is equal to the difference between the pay-offs from the option and pay offs

from the delta shares

• This package has the same pay-off as of a put option

• The value of the package is the value of the put option

Summary (Cont...)

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• Risk Neutral Method

• Determine the probability of upside and downside changes in stock price

• Assume investors are risk neutral

• Discount the future expected pay-off at the riskfree rate to derive the option value

Summary (Cont...)

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• Black Scholes Model

• Assumes log normal distribution of stock prices

• Provides a model for valuing European options on non dividend paying stocks:

– Value of call option = [delta * share price] – [bank loan]

– Where,

• Log is the natural log with base e

– N (d) = cumulative normal probability density function

– EX = exercise price option; PV(EX) is calculated by discounting at the risk- free interest rate rf

– t = number of periods to exercise date

– P =present price of stock

– σ = standard deviation per period of (continuously compounded) rate of return on stock

• Value of Put = [N (-d2) * PV (EX)] - [N (-d1) * P]

)](*)2([]*)1([ EXPVdNPdN

tσd1d2

2

tσ

tσ

X)]log[p/PV(Ed1

Summary (Cont...)

Other Webinars

Here are the links for the blogs of the other recent webinars on our website to

help you with CFA/FRM preparation

Linear regression analysis (11/04/2013)

Blog: http://www.edupristine.com/blog/demystifying-linear-regression-analysis-

for-frm-level-1-exam/

Understanding Income statement (12/04/2013)

Blog: http://www.edupristine.com/blog/cfa-tutorial-understanding-income-

statement-from-cfa-perspective/

Hedging strategies using futures (13/04/2013)

Blog: http://www.edupristine.com/blog/frm-tutorial-hedging-strategies-using-

futures-for-frm-level-1-exam/

You can find many more blogs on our website: www.edupristine.com/blog

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