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**Dr. Hassan Mounir El-Sady**

Chapter 5 Option Pricing Dr. Hassan Mounir El-Sady Dr. Hassan Mounir El-Sady

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**Outline Introduction Arbitrage and option pricing**

Intuition into Black-Scholes

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Introduction Option pricing developments are among the most important in the field of finance during the last 30 years The backbone of option pricing is the Black-Scholes model

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**Introduction to Arbitrage & Option Pricing**

Finance is sometimes called “the study of arbitrage” Arbitrage is the existence of a riskless profit Finance theory does not say that arbitrage will never appear Arbitrage opportunities will be short-lived

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Free Lunches The apparent mispricing may be so small that it is not worth the effort E.g., pennies on the sidewalk Arbitrage opportunities may be out of reach because of an impediment E.g., trade restrictions Modern option pricing techniques are based on arbitrage principles In a well-functioning marketplace, equivalent assets should sell for the same price (law of one price) A University Example A few years ago, a bookstore at a university was having a sale and offered a particular book title for $ Another bookstore at the same university had a buy-back offer for the same book for $10.50.

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**The Theory of Put/Call Parity**

Introduction Covered call and short put Covered call and long put No arbitrage relationships Variable definitions The put/call parity relationship

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Introduction For a given underlying asset, the following factors form an interrelated complex: Call price Put price Stock price and Interest rate

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**Covered Call and Short Put**

The profit/loss diagram for a covered call and for a short put are essentially equal Covered call Short put

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**Covered Call and Long Put**

A riskless position results if you combine a covered call and a long put Covered call Long put Riskless position + =

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**Covered Call and Long Put**

Riskless investments should earn the riskless rate of interest If an investor can own a stock, write a call, and buy a put and make a profit, arbitrage is present

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**No Arbitrage Relationships**

The covered call and long put position has the following characteristics: Write the call (C) Cash inflow, Buy the put (P) Cash outflows, Get a bank loan with (S) principal Cash inflow, Immediately buy the stock for (S) Cash outflows, Pay interest on the bank loan (S*r) Cash outflows, The bank loan (S) comes in now, but the interest on the bank loan is paid in the future

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**No Arbitrage Relationships (cont’d)**

If there is no arbitrage, then:

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**No Arbitrage Relationships (cont’d)**

If there is no arbitrage, then: The call premium should exceed the put premium by about the riskless rate of interest The difference will be greater as: The stock price increases Interest rates increase The time to expiration increases

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**Variable Definitions C = Call premium P = Put premium**

S0 = Current stock price S1 = Stock price at option expiration X = Option striking price r = riskless interest rate t = time until option expiration

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**The Put/Call Parity Relationship**

We now know how the call prices, put prices, the stock price, and the riskless interest rate are related:

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**The Put/Call Parity Relationship (cont’d)**

Equilibrium Stock Price Example You have the following information: Call price = $3.5 Put price = $1 Striking price = $75 Riskless interest rate = 5% Time until option expiration = 32 days If there are no arbitrage opportunities, what is the equilibrium stock price?

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**The Put/Call Parity Relationship (cont’d)**

Equilibrium Stock Price Example (cont’d) Using the put/call parity relationship to solve for the stock price:

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**The Put/Call Parity Relationship (cont’d)**

To understand why the law of one price must hold, consider the following information: C = $4.75 P = $3 S = $50 X = $50 r = 6.00% t = 6 months

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**The Put/Call Parity Relationship (cont’d)**

Based on the provided information, the put value should be: P = $ $50 + $50/(1.06)0.5 = $4.44 Or: C = $ $50 - $50/(1.06)0.5 = $3.31 The actual call price ($4.75) is too high or the put price ($3) is too low

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**The Put/Call Parity Relationship (cont’d)**

To exploit the arbitrage, arbitrageurs would: Write 1 $4.75 Buy 1 $3 Buy a share of stock at $50 Borrow $48.56 at 6.00% for 6 months These actions result in a profit of $0.31 at option expiration irrespective of the stock price at option expiration

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**The Put/Call Parity Relationship (cont’d)**

Stock Price at Option Expiration $0 $50 $100 Cash in or out from Call 4.75 (45.25) Cash in or out from Put 47.00 (3.00) Cash in or out from Loan (1.44) Cash in or out from Stock (50.00) 0.00 50.00 Total $0.31

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**The Binomial Option Pricing Model**

Assume the following: U.S. government securities yield 10% next year Stock XYZ currently sells for $75 per share There are no transaction costs or taxes There are two possible stock prices in one year Dr. Hassan Mounir El-Sady

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**The Binomial Option Pricing Model (cont’d)**

Possible states of the world: $100 $75 $50 Today One Year Later

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**The Binomial Option Pricing Model (cont’d)**

A call option on XYZ stock is available that gives its owner the right to purchase XYZ stock in one year for $75 If the stock price is $100, the option will be worth $25 If the stock price is $50, the option will be worth $0 What should be the price of this option? We can construct a portfolio of stock and options such that the portfolio has the same value regardless of the stock price after one year Buy the stock and write N call options

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**The Binomial Option Pricing Model (cont’d)**

Possible portfolio values: $100 - $25N $75 – (N)($C) $50 Today One Year Later

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**The Binomial Option Pricing Model (cont’d)**

We can solve for N such that the portfolio value in one year must be $50:

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**The Binomial Option Pricing Model (cont’d)**

If we buy one share of stock today and write two calls, we know the portfolio will be worth $50 in one year The future value is known and riskless and must earn the riskless rate of interest (10%) The portfolio must be worth = 50/(1+10%) = $45.45 today

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**The Binomial Option Pricing Model (cont’d)**

Assuming no arbitrage exists: The option must sell for $14.77!

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**The Binomial Option Pricing Model (cont’d)**

The option value is independent of the probabilities associated with the future stock price The price of an option is independent of the expected return on the stock

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**Put Pricing in the Presence of Call Options**

In an arbitrage-free world, the put option cannot also sell for $14.77; If it did, an smart arbitrageur would: Buy a 75 call Write a 75 put Sell the stock short Invest = 75/(1+10%) = $68.18 in T-bills These actions result in a cash flow of $6.82 today and a cash flow of $0 at option expiration

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**Put Pricing in the Presence of Call Options**

Activity Cash Flow Today Portfolio Value at Option Expiration Price = $100 Price = $50 Buy 75 call -$14.77 $25 Write 75 put +14.77 -$25 Sell stock short +75.00 -100 -50 Invest $68.18 in T-bills -68.18 75.00 Total $6.82 $0.00

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**Binomial Put Pricing Priced similarly to calls**

You can combine puts with stock so that the future value of the portfolio is known Assume a value of $100

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**Binomial Put Pricing (cont’d)**

Possible portfolio values: $100 $75 $50 + N($75 - $50) Today One Year Later

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**Binomial Put Pricing (cont’d)**

A portfolio composed of one share of stock and two puts will grow risklessly to $100 after one year

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**Binomial Pricing With Asymmetric Branches**

The size of the up movement does not have to be equal to the size of the decline E.g., the stock will either rise by $25 or fall by $15 The logic remains the same: First, determine the number of options Second, solve for the option price

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The Effect of Time More time until expiration means a higher option value

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**The Effect of Volatility**

Higher volatility means a higher option price for both call and put options Explains why options on Internet stocks have a higher premium than those for retail firms

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**Multiperiod Binomial Option Pricing**

In reality, prices change in the marketplace minute by minute and option values change accordingly The logic of binomial pricing can be easily extended to a multiperiod setting using the recursive methods of solving for the option value

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**Option Pricing With Continuous Compounding**

Continuous compounding is an assumption of the Black-Scholes model Using continuous compounding to revalue the call option from the previous example:

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**Risk Neutrality and Implied Branch Probabilities**

Risk neutrality is an assumption of the Black-Scholes model For binomial pricing, this implies that the option premium contains an implied probability of the stock rising

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**Risk Neutrality and Implied Branch Probabilities (cont’d)**

Assume the following: An investor is risk-neutral He can invest funds risk free over one year at a continuously compounded rate of 10% The stock either rises by 33.33% or falls 33.33% in one year After one year, one dollar will be worth $1.00 x e.10 = $ for an effective annual return of 10.52% A risk-neutral investor would be indifferent between investing in the riskless rate and investing in the stock if it also had an expected return of 10.52% We can determine the branch probabilities that make the stock have a return of 10.52%

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**Risk Neutrality and Implied Branch Probabilities (cont’d)**

Define the following: U = 1 + % increase if the stock goes up D = 1 – % decrease if the stock goes down Pup = probability that the stock goes up Pdown = probability that the stock goes down ert = continuously compounded interest rate factor

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**Risk Neutrality and Implied Branch Probabilities (cont’d)**

The average stock return is the weighted average of the two possible price movements:

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**Risk Neutrality and Implied Branch Probabilities (cont’d)**

If the stock goes up, the call will have an intrinsic value of $100 - $75 = $25 If the stock goes down, the call will be worthless. The expected value of the call in one year is: Discounted back to today, the value of the call today is:

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**Extension to Two Periods**

Assume two periods, each one year long, with the stock either rising or falling by 33.33% in each period What is the equilibrium value of a two-year European call shown on the next slide?

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**Extension to Two Periods (cont’d)**

$ (UU) $100 (U) $66.67 (UD = DU) $75 $50 (D) $33.33 (DD) Today One Year Later Two Years Later

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**Extension to Two Periods (cont’d)**

The option only winds up in the money when the stock advances twice (UU) There is a 65.78% probability that the call is worth $58.33 and a 34.22% probability that the call is worthless

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**Extension to Two Periods (cont’d)**

There is a 65.78% probability that the call is worth $34.72 in one year and a 34.22% probability that the call is worthless in one year The expected value of the call in one year is:

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**Extension to Two Periods (cont’d)**

$58.33 (UU) $34.72 $0 (UD = DU) $20.66 $0 $0 (DD) Today One Year Later Two Years Later

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**Recombining Binomial Trees**

If trees are recombining, this means that the up-down path and the down-up path both lead to the same point, but not necessarily the starting point To return to the initial price, the size of the up jump must be the reciprocal of the size of the down jump

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**Multiperiod Binomial Put Pricing**

To solve for the value of a put using binomial logic, just change the terminal intrinsic values and work backward just as with call pricing The branch probabilities do not change

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**Exploiting Arbitrage Arbitrage Example**

Binomial pricing results in a call price of $28.11 and a put price of $2.23. The interest rate is 10%, the stock price is $75, and the striking price of the call and the put is $60. The expiration date is in two years. What actions could an arbitrageur take to make a riskless profit if the call is actually selling for $29.00?

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**Exploiting Arbitrage (cont’d)**

Arbitrage Example (cont’d) Since the call is overvalued, and arbitrageur would want to write the call, buy the put, buy the stock, and borrow the present value of the striking price, resulting in the following cash flow today: Write 1 call $29.00 Buy 1 put ($2.23) Buy 1 share ($75.00) Borrow $60e-(.10)(2) $49.12 Arbitrage Profit $0.89 The value of the portfolio in two years will be worthless, regardless of the path the stock takes over the two-year period.

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**European Put Pricing and Time Value**

With a European put, the longer the option’s life, the longer you must wait to see sales proceeds More time means greater potential dispersion in underlying asset values, and this pushes up the put value A European put’s value with respect to time until expiration is indeterminate Often, an out-of-the-money put will increase in value with more time Often, an in-the-money put decreases in value for more distant expirations

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