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Fi8000 Option Valuation I Milind Shrikhande

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A Call Option A European call option gives the buyer of the option a right to purchase the underlying asset, at the exercise price on the expiration date. It is optimal to exercise the call option if the stock price exceeds the strike price: C T = Max {S T – X, 0}

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A Put Option A European put option gives the buyer of the option a right to sell the underlying asset, at the exercise price on the expiration date. It is optimal to exercise the put option if the stock price is below the strike price: P T = Max {X - S T, 0}

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The Put Call Parity If two portfolios have the same payoffs in every possible state and time in the future, their prices must be equal:

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Arbitrage – the Law of One Price If two assets have the same payoffs in every possible state in the future and their prices are not equal, there is an opportunity to make an arbitrage profit. We say that there exists an arbitrage opportunity if we identify that: There is no initial investment There is no risk of loss There is a positive probability of profit

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Arbitrage – a Technical Definition Let CF tj be the cash flow of an investment strategy at time t and state j. If the following conditions are met this strategy generates an arbitrage profit. (i) all the possible cash flows in every possible state and time are positive or zero - CF tj ≥ 0 for every t and j. (ii) at least one cash flow is strictly positive - there exists a pair ( t, j ) for which CF tj > 0.

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Arbitrage – an Example Is there an arbitrage opportunity if the following are the market prices of the assets: The price of one share of stock is $39; The price of a call option on that stock, which expires in one year and has an exercise price of $40, is $7.25; The price of a put option on that stock, which expires in one year and has an exercise price of $40, is $6.50; The annual risk free rate is 6%.

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Arbitrage – an Example In this case we must check whether the put call parity holds. Since we can see that this parity relation is violated, we will show that there is an arbitrage opportunity.

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The Construction of an Arbitrage Transaction Constructing the arbitrage strategy: 1. Move all the terms to one side of the equation so their sum will be positive; 2. For each asset, use the sign as an indicator of the appropriate investment in the asset. If the sign is negative then the cash flow at time t=0 is negative (which means that you buy the stock, bond or option). If the sign is positive reverse the position.

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Arbitrage – an Example In this case we move all terms to the LHS:

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Arbitrage – an Example Time: → t = 0 t = T Strategy: ↓ State: → S T < X = 40 S T > X = 40 Short stock +S=$39 -S T Write put +P=$6.5 -(X-S T ) 0 Buy call -C=(-$7.25)0 (S T -X) Buy bond -X/(1+rf)=(-$37.736)XX Total CF S+P-C- X/(1+rf) = > 0 -S T -(X-S T )+X = 0 -S T -(X-S T )+X = 0

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Valuation of Options ☺Quantitative Valuation ☺Binomial model (an algorithm) ☺Black-Scholes model (a formula) ☺Arbitrage Restrictions on the Values of Options ☺European Call and Put options ☺American vs. European Call option ☺American vs. European Put option ☺The Put-Call Parity ☺The option price and the exercise price ☺Option Price Convexity (options with the same expiration date but different exercise prices)

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Notation S = the price of the underlying asset (stock) C = the price of a call option P = the price of a put option X or K = the exercise or strike price T = the expiration date t = a time index

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The Value of a Call Option Assumptions: 1. A European Call option on a stock 2. The stock pays no dividends before expiration expiration 3. The stock is traded 4. A risk free bond is traded Arbitrage restrictions: Max{ S-PV(X), 0 } ≤ C EU < S

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The Value of a Call Option Max{ S-PV(X), 0 } ≤ C EU ≤ S ☺ 0 ≤ C : the owner has a right but not an obligation. ☺ S-PV(X) ≤ C : arbitrage proof. ☺ C ≤ S : you will not pay more than $S, the market price of the stock, for an option to buy that stock for $X. Buying the stock itself is always an alternative to buying the call option.

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The Value of a Call Option S S-PV(X) S C PV(X)0

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The Value of a Call Option Example: ☺ The current stock price is $83 ☺ The stock will not pay dividends in the next six months ☺ A call option on that stock is traded for $3 ☺ The exercise price is $80 ☺ The expiration of the option is in 6 months ☺ The 6 months risk free rate is 5% Is there an opportunity to make an arbitrage profit?

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The Value of a Call Option Example: ☺ Check for arbitrage opportunities ☺ Show the investment strategy and calculate the arbitrage profits ☺ Show the general proof for this case

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The Value of a Call Option Assumptions: 1. A Call option on a stock 2. The stock pays no dividends before expiration 3. The stock is traded 4. A risk free bond is traded Arbitrage restriction: C(European) = C(American) Hint: compare the payoff from immediate exercise to the lower bound of the European call option price. bound of the European call option price.

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The Value of a Put Option Assumptions: 1. A European Put option on a stock 2. The stock pays no dividends before expiration expiration 3. The stock is traded 4. A risk free bond is traded Arbitrage restrictions: Max{ PV(X)-S, 0 } ≤ P EU < PV(X)

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The Value of a Put Option Max{ PV(X)-S, 0 } ≤ P EU < PV(X) ☺ 0 ≤ P : the owner has a right but not an obligation. ☺ PV(X)-S ≤ P : arbitrage proof. ☺ P < PV(X) : the highest profit from the put option is realized when the stock price is zero. That profit is $X realized on date T, which is PV($X) today.

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The Value of a Put Option S P PV(X)0 PV(X)-S PV(X)

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The Value of a Put Option Example: ☺ The current stock price is $75 ☺ The stock will not pay dividends in the next six months ☺ A put option on that stock is traded for $1 ☺ The exercise price is $80 ☺ The expiration of the option is in 6 months ☺ The 6 months risk free rate is 5% Is there an opportunity to make an arbitrage profit?

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The Value of a Put Option Example: ☺ Check for arbitrage opportunities ☺ Show the investment strategy and calculate the arbitrage profits ☺ Show the general proof for this case

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The Value of a Put Option Assumptions: 1. A put option on a stock 2. The stock pays no dividends before expiration expiration 3. The stock is traded 4. A risk free bond is traded For a put option we always have: P(European) < P(American) Note: in this case an example is enough.

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Exercise Prices Let X 1 < X 2 be two different exercise prices; C 1, C 2 be the appropriate call option prices; and P 1, P 2 be the appropriate put option prices (we assume that all the options are European, on the same stock S that pays no dividends, with the same expiration date T). Then, C(X 1 ) ≥ C(X 2 ) and P(X 1 ) ≤ P(X 2 )

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Example Show that if there are two call options on the same stock (that pays no dividends), and both have the same expiration date but different exercise prices as follows, there is an opportunity to make an arbitrage profit. X 1 = $40 and X 2 = $50 C 1 = $3 and C 2 = $4

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Option Price Convexity Let X 1 < X 2 < X 3 be three different exercise prices, and C 1, C 2, C 3 be the appropriate call option prices, and P 1, P 2, P 3 be the appropriate put option prices (all the options are European, on the same stock S that pays no dividends, with the same expiration date T).

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Example Show that if there are three call options on the same stock (that pays no dividends), and all three have the same expiration date but different exercise prices as follows, there is an opportunity to make an arbitrage profit. X 1 = $40, X 2 = $50 and X 3 = $60 C 1 = $4.6, C 2 = $4 and C 3 = $3

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Binomial Option Pricing Model Assumptions: ☺ A single period ☺ Two dates: time t=0 and time t=1 ☺ The future (time 1) stock price has only two possible values ☺ The price can go up or down ☺ The perfect market assumptions ☺ No transactions costs, borrowing and lending at the risk free interest rate, no taxes…

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Binomial Option Pricing Model Example The stock price Assume S= $50, u= 10% and d= (-3%) S S u =S·(1+u) S d =S·(1+d) S=$50 S u =$55 S d =$48.5

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Binomial Option Pricing Model Example The call option price Assume X= $50, T= 1 year ( 1 period ) C C u = Max{Su-X,0} C d = Max{Sd-X,0} C C u = $5 = Max{55-50,0} C d = $0 = Max{ ,0}

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Binomial Option Pricing Model Example The bond price Assume r= 6% 1 (1+r) $1 $1.06

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Replicating Portfolio At time t=0, we can create a portfolio of N shares of the stock and an investment of B dollars in the risk-free bond. The payoff of the portfolio will replicate the t=1 payoffs of the call option: N·$55 + B·$1.06 = $5 N·$ B·$1.06 = $0 Obviously, this portfolio should also have the same price as the call option at t=0: N·$50 + B·$1 = C We get N=0.7692, B=( ) and the call option price is C=$

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A Different Replication The price of $1 in the “up” state: The price of $1 in the “down” state: ququ $1 $0 qdqd $1

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Replicating Portfolios Using the State Prices We can replicate the t=1 payoffs of the stock and the bond using the state prices: q u ·$55 + q d ·$48.5 = $50 q u ·$ q d ·$1.06 = $1 Obviously, once we solve for the two state prices we can price any other asset such as the call option: q u ·$5 + q d ·$0 = C We get q u =0.6531, q d = and the call option price is C=$

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Binomial Option Pricing Model Example The put option price Assume X= $50, T= 1 year ( 1 period ) P P u = Max{X-Su,0} P d = Max{X-Sd,0} P P u = $0 = Max{50-55,0} P d = $1.5 = Max{ ,0}

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Replicating Portfolios Using the State Prices We can replicate the t=1 payoffs of the stock and the bond using the state prices: q u ·$55 + q d ·$48.5 = $50 q u ·$ q d ·$1.06 = $1 But the assets are exactly the same and so are the state prices. The put option price is: q u ·$0 + q d ·$1.5 = P We get q u =0.6531, q d = and the put option price is P=$

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Two Period Example ☺ Assume that the current stock price is $50, and it can either go up 10% or down 3% in each period. ☺ The one period risk-free interest rate is 6%. ☺ What is the price of a European call option on that stock, with an exercise price of $50 and expiration in two periods?

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The Stock Price S= $50, u= 10% and d= (-3%) S=$50 S u =$55 S d =$48.5 S uu =$60.5 S ud =S du =$53.35 S dd =$47.05

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The Bond Price r= 6% (for each period) $1 $1.06 $1.1236

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The Call Option Price X= $50 and T= 2 periods C CuCu CdCd C uu =Max{ ,0}=$11.5 C ud =Max{ ,0}=$3.35 C dd =Max{ ,0}=$0

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State Prices in the Two Period Tree We can replicate the t=1 payoffs of the stock and the bond using the state prices: q u ·$55 + q d ·$48.5 = $50 q u ·$ q d ·$1.06 = $1 Note that if u, d and r are the same, our solution for the state prices will not change (regardless of the price levels of the stock and the bond): q u ·S·(1+u) + q d ·S ·(1+d) = S q u ·(1+r) t + q d ·(1+r) t = (1+r) (t-1) Therefore, we can use the same state-prices in the two period tree.

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The Call Option Price q u = and q d = C CuCu CdCd C uu =$11.5 C ud =$3.35 C dd =$0

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Two Period Example ☺ What is the price of a European put option on that stock, with an exercise price of $50 and expiration in two periods? ☺ What is the price of an American call option on that stock, with an exercise price of $50 and expiration in two periods? ☺ What is the price of an American put option on that stock, with an exercise price of $50 and expiration in two periods?

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Two Period Example ☺ European put option - use the tree or the put-call parity ☺ What is the price of an American call option - if there are no dividends … ☺ American put option – use the tree

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The European Put Option Price q u = and q d = P PuPu PdPd P uu =$0 P ud =$0 P dd =$2.955 P u = * *0 = $0 P d = * *2.955 = $ P EU = * * = $0.2490

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The American Put Option Price q u = and q d = P Am PuPu PdPd P uu =$0 P ud =$0 P dd =$2.955

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American Put Option Note that at time t=1 the option buyer will decide whether to exercise the option or keep it till expiration. If the payoff from immediate exercise is higher than the option value the optimal strategy is to exercise: If Max{ X-S u,0 } > P u (European) => Exercise

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The American Put Option Price q u = and q d = P PuPu PdPd P uu =$0 P ud =$0 P dd =$2.955 P u = Max{ * *0, } = $0 P d = Max{ * *2.955, } = = $1.5 P Am = Max{ * *1.5, } = $ > $ = P Eu

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Determinants of the Values of Call and Put Options Variable C – Call Value P – Put Value S – stock price IncreaseDecrease X – exercise price DecreaseIncrease σ – stock price volatility IncreaseIncrease T – time to expiration IncreaseIncrease r – risk-free interest rate IncreaseDecrease Div – dividend payouts DecreaseIncrease

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Practice Problems BKM Ch. 21: End of chapter - 1, 2 Example 21.1 and concept check Q #4 (pages 755-6) Practice set: 17-24, 25-35

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