The Black-Scholes-Merton (BSM) Model

FNCE30007 Derivative Securities / Lecture 4

Schedule

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Introduction to Options

Properties of Stock Options

The Binomial Model

The Black-Scholes - Merton Model

Dividends and Options on Other

Instruments The Greeks Futures Markets

Hedging with Futures and

Forwards

Forward and Futures Prices Futures Options Swaps

Outline and Readings

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Outline Normal and lognormal distributions Assumptions of the model The pricing formulas Derivation of the model Implied volatility American options Link to the Binomial model

Readings Hull, 8th ed., chapter 13, Sections 13.1 13.9 Hull, 7th ed., chapter 13, Sections 13.1 13.9

Normal and Lognormal Distributions

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Normal and Standard Random Variable

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A normal random variable has a mean of and a standard deviation of .

Remember that a normal random variable is symmetric about its mean and the total area under the curve is 1.

A standard normal variable (Z) has a mean of zero and a standard deviation of 1.

The probability of a Z b can be calculated from a standard normal table which gives the area under the standard normal curve for a range of different values of Z.

Lognormal Random Variable

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If X is a random variable, then any function of it is also a random variable. If lnX is a normal random variable, then X is a lognormal

random variable. The mean of lnX is and standard deviation is . Note that

X is greater than zero. X is not symmetric around its mean

Stock Prices and Lognormal and Normal Distr.

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The lowest stock price is zero. Thus, the lognormal distribution is appropriate for stock prices.

Stock returns can be negative as well as positive. Therefore, the normal distribution is appropriate for returns.

0 Normal distribution Lognormal distribution

0

Assumptions of the Model

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Assumptions

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The original model assumes a European call option on a non-dividend paying stock with a current price of S, strike price of K, and maturity of T years.

Stock prices are assumed to be lognormally distributed. Volatility is constant. The markets are frictionless (no taxes, no transaction

costs, no restrictions on short selling, and securities are perfectly divisible).

The continuously compounded risk-free interest rate (r) per annum is constant.

Assumptions

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Investors can borrow and lend at the same risk-free interest rate.

There are no arbitrage opportunities. Trading is continuous.

The Pricing Formulas

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The Pricing Formulas - Call

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The current value of a call option is:

N(z) is the area under the standard normal distribution. N(d2) is the probability that the option will be exercised in

a risk-neutral world.

=

+ +=

=

1 2

2

1

2 1

( ) ( )

ln( / ) ( / 2)

rTc SN d Ke N dwhere

S K r TdT

d d T

The Pricing Formulas - Call

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N(d1) is not easy to interpret but can be considered as the factor by which the present value of the contingent receipt of the stock exceeds the current stock price.

The value of the option does not depend on the expected return of the stock.

When using the standard normal table to calculate N(d1) and N(d2), simply calculate d1 and d2 to the nearest two decimal places. Do not use the interpolation technique in Hull in the exam.

The Pricing Formulas - Put

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The current value of an otherwise equivalent put option is:

We can derive the put formula using the put-call parity.

=

+ +=

=

2 1

2

1

2 1

( ) ( )

ln( / ) ( / 2)

rTp Ke N d SN d

where

S K r TdT

d d T

Do the Formulas Make Sense?

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When the stock price is very large, we expect a call to be exercised and the value of the call is expected to be S Ke-rT. When the stock price is very large, both N(d1) and N(d2)

approach 1. When the stock price is very small, we expect a call to be

worthless and the value of the call is expected to be (close to) zero. When the stock price is very small, both N(d1) and N(d2)

approach 0.

Do the Formulas Make Sense?

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When the stock price is very small, we expect a put to be exercised and the value of the put is expected to be Ke-rT S. When the stock price is very small, both N(-d1) and N(-d2)

approach 1. When the stock price is very large, we expect a put to be

worthless and the value of the put is expected to be (close to) zero. When the stock price is very large, both N(-d1) and N(-d2)

approach 0.

Example

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Consider a nine-month option where the underlying stock has a price of $100. Assume that the exercise price of the option is $110, the continuously-compounded risk-free rate is 5% per annum, and the volatility is 40% per annum. Find Black-Scholes call and put prices.

Example Continued

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S = 100, K = 110, r = 0.05, = 0.40, T = 9/12 = 0.75 + += =

= =

=

=

=

=

2

1

2

1

2

1

2

ln(100 / 110) (0.05 0.40 / 2)(0.75) 0.0063 0.010.40 0.75

0.0063 0.40 0.75 0.3401 0.34

( ) 0.5040( ) 0.3669( ) 0.4960( ) 0.6331

d

d

N dN dN dN d

Example Continued

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In the final step, plug everything into the formulas to find the prices.

= =

= =

( 0.05)(0.75)

( 0.05)(0.75)

100(0.5040) 110 (0.3669) $11.53

110 (0.6331) 100(0.4960) $17.48

c e

p e

Risk-Neutral Valuation

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Note that a security dependent on other traded securities can be valued by assuming that investors are risk neutral.

This assumption does not mean that investors are risk neutral.

The assumption means that investors risk preferences do not affect the value of an option when the value is expressed as a function of the price of the underlying instrument.

This is why the pricing formulas do not involve the stocks expected return.

Risk-Neutral Valuation

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Note that in a risk-neutral world The expected rate of return from all investments is the risk-

free interest rate. The risk-free interest rate is the appropriate discount rate that

should be applied to all expected future cash flows. Remember that to find the value of an option in a risk

neutral world Assume that the expected return is the risk-free rate. Calculate the expected payoff. Discount the expected payoff at the risk-free rate.

Derivation of the Model

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No-Arbitrage Argument

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The stock price and the option price are both affected by the same underlying source of uncertainty which is the movements in the stock price.

Over a very small period, the option price is perfectly correlated with the stock price.

When security trading is continuous and there is perfect correlation, we can form an instantaneous riskless hedge.

However, since the hedge is instantaneous, it needs to be rebalanced continuously (compare this to the binomial model).

Derivation of the Model

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The Black-Scholes-Merton model is based on the idea of instantaneous riskless hedge (compare to the binomial model).

At each instant, we form a portfolio composed of a long position in the stock and a short position in the option so that the value of the portfolio is riskless for that instant.

Since the portfolio is riskless, it must earn the risk-free rate.

Following a similar process as in the binomial case, working through the math leads to the call option pricing formula.

Implied Volatility

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Inputs to the Model

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Share price is observable. Interest rate (continuously compounded). Should be risk

free rate with maturity close to that of the option. Strike price is in the contract. Time to maturity is in the contract. Volatility is very important but you cant observe it.

Historical (in the appendix) Implied

Implied Volatility

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Given the stock price, strike price, interest rate, time to maturity, and the current market price of an option, the implied volatility of the option is the value of the standard deviation that when substituted into the Black-Scholes formulas gives a theoretical price equal to the current market price.

The implied volatility of an option indicates the markets view of the future volatility of the stock over the life of the option.

Example

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Suppose that the current value of a three-month European call option on a non-dividend paying stock is $2.10. If the current stock price is $21, the exercise price is $20 and the continuously compounded risk-free rate is 10% per annum, find the implied volatility of the option.

We need to use an iterative approach to find the implied volatility.

Alternatively, we can use Excels Solver function.

Trading vs. Calendar Days

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Volatility is much higher when the exchange is open than when it is closed.

Practitioners tend to ignore days when the exchange is closed when estimating volatility from historical data and when calculating the life of an option.

The volatility per annum from the volatility per trading day = volatility per trading day x 252 week = volatility per week x 52 month = volatility per month x 12

Trading vs. Calendar Days

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Similarly, the life of an option is usually calculated using trading days rather than calendar days.

If T is the life of an option in years then

T = Number of trading days until option maturity252

American Options

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American Calls and Puts

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The Black-Scholes-Merton formula can also be used to price an American call option on a non-dividend paying stock.

Put option prices obtained from the Black-Scholes-Merton formula do not reflect early exercise for American puts and, thus, are extremely biased. A binomial model would be necessary to get an accurate price.

The Link to the Binomial Model

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The Binomial and BSM Models

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Keeping the life of an option fixed, as n (the number of periods) increases and t approaches zero, the option price given by the binomial model converges to the option prices given by the Black-Scholes-Merton model.

One condition for the convergence is that we have

See the Excel file on LMS.

1/

t

t

u e

d e u

=

= =

Binomial Convergence to the BSM

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4.60

4.70

4.80

4.90

5.00

5.10

5.20

5.30

5.40

2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90 94 98 102 106 110 500

Call

Pric

e

Tree Steps BSM Price Binomial Price

Appendix: Historical Volatility

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Volatility

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The volatility is the standard deviation of the continuously compounded rate of return in one year.

The standard deviation of the return in time t is:

Example: If a stock price is $40 and its volatility is 40% per year what is

the standard deviation of the price change in one week? One standard deviation move in the stock price in one week is 40 x 0.0554 = $2.216.

t

5.54% or 0554.052140.0 =

Estimating Volatility from Historical Data

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This is the volatility over a recent time period. Collect daily, weekly, or monthly returns on the stock. Convert each return to its continuously compounded

equivalent by taking ln(1 + return). Calculate variance. Annualize by multiplying by 252 (daily returns trading

days), 52 (weekly returns) or 12 (monthly returns). Take the square root.

Check out the Excel file for historical volatility on the LMS.

The Black-Scholes-Merton (BSM) ModelScheduleOutline and ReadingsNormal and Lognormal DistributionsNormal and Standard Random VariableLognormal Random VariableStock Prices and Lognormal and Normal Distr.Assumptions of the ModelAssumptionsAssumptionsThe Pricing FormulasThe Pricing Formulas - CallThe Pricing Formulas - CallThe Pricing Formulas - PutDo the Formulas Make Sense?Do the Formulas Make Sense?ExampleExample ContinuedExample ContinuedRisk-Neutral ValuationRisk-Neutral ValuationDerivation of the ModelNo-Arbitrage ArgumentDerivation of the ModelImplied VolatilityInputs to the ModelImplied VolatilityExampleTrading vs. Calendar DaysTrading vs. Calendar DaysAmerican OptionsAmerican Calls and PutsThe Link to the Binomial ModelThe Binomial and BSM ModelsBinomial Convergence to the BSMAppendix: Historical VolatilityVolatilityEstimating Volatility from Historical Data