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What is an Option? Definitions: An option is an agreement between two parties that gives the purchaser of the option the right, but not the obligation, to buy or sell a specific quantity of an asset at a specified price during a designated time period. A Call is the right to purchase the underlying asset A Put is the right to sell the asset The Strike Price is the pre-specified price at which the option holder can buy (in the case of a call) or sell (in the case of a put) the asset to the option seller The Expiration date is the date (and time) after which the option expires The Notional Amount is the quantity of the underlying asset that the option buyer has the right to buy or sell under the terms of the option contract The Premium is the price of the option contract (the amount paid by the buyer to the seller). The option buyer’s maximum “downside” (possible loss) is the amount of the premium. Option premia are quoted as percentage of notional amount. To Exercise means to invoke the right to buy or sell the underlying asset under the terms of the option contract The Seller (or Writer) of the option receives a payment (the Option Premium) that then obligates him to sell (in the case of a call) or buy (in the case of a put) the asset. WEMBA 2000Real Options1

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Payoff Diagram for a Long Call Option Option payoff S: Price of Underlying Asset at expiration K Strike Price = K Price of Underlying Asset = S Profit/Loss Analysis At expiration, there are two possible outcomes: (i) S >=K. Exercise the call and purchase the asset for K. Asset has market value S Payoff = S - K (ii) S <K. Option expires worthless. Payoff = 0 General Formula for call payoff Long call payoff = Max (0, S - K) WEMBA 2000Real Options2

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Option payoff K Strike Price = K Price of Underlying Asset = S Profit/Loss Analysis At expiration, there are two possible outcomes: (i) S <= K Exercise the put and sell the asset for K. Asset has market value S Payoff = K - S (ii) S >K. Option expires worthless. Payoff = 0 General Formula put payoff Long put payoff = Max (0, K - S) Payoff Diagram for a Long Put Option WEMBA 2000Real Options3 S: Price of Underlying Asset at expiration

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Payoff Diagrams for Short Options Positions Option payoff S: Price of Underlying Asset at expiration K Short Call Position Strike Price = K Option payoff K Short Put Position Strike Price = K Notes: (i) The short position payoff diagrams are mirror images of the long positions. (ii) The above payoff charts do not include the cost of buying (or income from selling) the option. Short call payoff = Min (0, K-S) Short put payoff = Min (0, S-K) Question: Which is potentially riskier, a long option position or a short option position? WEMBA 2000Real Options S: Price of Underlying Asset at expiration WEMBA 2000Real Options4 Payoff Diagrams for Short Options Positions

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WEMBA 2000Real Options5 Payoff Diagrams for some Option Combinations Position Profit/Loss S: Price of Underlying Asset at expiration K Profit/Loss from stock Profit/loss from short call net profit/loss "Covered Call" or "Buy-Write" Position Profit/Loss S: Price of Underlying Asset at expiration "Call Spread" Profit/loss from short call Profit/loss from long call net profit/loss Note: The above profit/loss charts include the cost of buying (or income from selling) the option

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Factors that Influence Option Prices The six variablesthat affect option prices: 1. Current (spot) price on the underlying security 2. Strike price 3. Time to expiration 4. Implied (expected) volatility on the underlying security 5. The riskfree rate over the time period of the option 6. Any dividends or other cashflows that will be paid or received on the underlying asset during the life of the option WEMBA 2000Real Options6

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Valuation of Options: Put-Call Parity WEMBA 2000Real Options7 We construct two portfolios and show they always have the same payoffs, hence they must cost the same amount. Portfolio 1: Buy 1 share of the stock today for price S 0 and borrow an amount PV(X) = X e -rT How much will this portfolio be worth at time T ? Cashflow PositionTime = 0Time = T Buy Stock -S 0 S T Borrow PV(K) -K Net: Portfolio 1PV(K) - S 0 S T - K Portfolio payoff at time T STST K Payoff from borrowing Payoff from borrowing Payoff from stock net payoff -K-K S

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Portfolio 2: Buy 1 call option and sell 1 put option with the same maturity date T and the same strike price K. How much will this portfolio be worth at time T ? Cashflow Cashflow: Time = T PositionTime = 0 S T K Buy Call - c 0 S T - K Sell Put p - (K - S T ) 0 Net: Portfolio 2p - c S T - K S T - K Valuation of Options: Put-Call Parity Portfolio payoff at time T STST K Payoff on short put Payoff on long call net payoff -K-K WEMBA 2000Real Options8

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Valuation of Options: Put-Call Parity Payoff from Portfolio 1 and Portfolio 2 is the same, regardless of level of S T, hence cost of both portfolios (cashflows at time T = 0 ) must be the same. Hence: S 0 - PV(K) = c - pPut-Call Parity Rearranging:c = p + S 0 - PV(K)(1) Put-Call parity: a worked example Stock is selling for $100. A call option with strike price $90 and maturity 3 months has a price of $12. A put option with strike price $90 and maturity 3 months has a price of $2. The risk-free rate is 5%. Question: Is there an arbitrage? Test Put-Call parity: Right-hand side of (1): p + S 0 - PV(K) = 2 + 100 - 90 e -0.05*0.25 = 13.12 Left-hand side of (1): c = 12 13.12 ! Market Price of c is too low relative to the other three. Buy the call, and Sell the "replicating portfolio". WEMBA 2000Real Options9

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Cashflow Cashflow: Time = T PositionTime = 0 S T 90 Buy Call - 12 0 S T - 90 Sell Put 2 S T - 90 0 Sell stock 100 - S T - S T Lend money-90 e 0.05*0.25 90 90 Net Payoff 1.12 0 0 Valuation of Options: Put-Call Parity Example Result: arbitrage profit of 1.12 today, regardless of the value of the stock price! WEMBA 2000Real Options10

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Valuation of Options: Black-Scholes Formula for Calls and Puts S = Current stock price K = Strike price on the option T = Time to maturity of the option in years (e.g. 5 months = 5/12 = 0.417) r = Riskfree rate of interest = Expected ("Implied") volatility (standard deviation) of the underlying stock over the life of the option Black-Scholes Call Price c = S N( d 1 ) - X e -rT N( d 2 )(2) where: d 1 = ln (S/k) + (r + 2 / 2) T T d 2 = d 1 - T N(d ) = cumulative standard normal probability of value less than d Black-Scholes Put Price p = X e -rT N( - d 2 ) - S N( - d 1 )(3) WEMBA 2000Real Options11

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Valuation of Options: Black-Scholes Formula for Calls and Puts Example: Options on Compaq stock On Dec 20, Compaq stock closed at $76.75 3 month riskfree rate: 5.5% (e.g. yield on 3 month T-bill) Estimated volatility: 41% What are the values of 3 month call and put options with Strike = $75 ? Black-Scholes formula inputs and calculations: Observed inputs:Option contract inputs: Estimated input (the future level of volatility is not observable) S = 76.75K = 75 = 41% r = 5.5%T = 0.25 d 1 = [ ln (76.75/75) + (0.055 + 0.41 2 / 2) 0.25 ] 0.41 0.25 = 0.2821 d 2 = d 1 - T = 0.0771 N(d 1 ) = 0.6111[obtained from Excel "normsdist" function] N(d 2 ) = 0.5307[obtained from Excel "normsdist" function] c = 7.638[from equation (2) ] p = 4.864[from equation (3) ] WEMBA 2000Real Options12

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Binomial Pricing Method 1: Creating a replicating portfolio WEMBA 2000Real Options13 Bluejay Corp share price is $20. Possible price at the end of three months: either $22 or $18. Value a call option on Bluejay with strike 21, expiration 3 months. Riskfree rate = 2% over 3 months. 20 22 18 c 22-21 = 1 0 Share PriceOption Value {Reminder: the value of the call at expiration is Max[0, S - K]} (a) Create a portfolio: purchase one share of the stock, and borrow money at the riskfree rate HINT: Choose amount to borrow so that the portfolio outcome is zero in one scenario 20- PV(18) =2.35 22-18=4 18-18=0 Portfolio: Buy 1 share & borrow PV(18) Compare the payoff between the call option and the portfolio. How many call options do we need to buy to make the payoffs identical? (i) (ii) (iii)

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WEMBA 2000Real Options14 4*c 4 0 2.35 4 0 Portfolio: Buy 1 share & borrow PV(18) Option Value (4 calls) (b) Calculate number of call options to buy so that the payoff from the calls matches the portfolio payoff in all scenarios. Hence the call price must equal the value of the portfolio (Law of One Price). 4 * c = 2.35 c = 0.59 Call premium (price) How many shares of stock to buy to replicate the payoff from one call? 4 calls replicate payoffs from 1 share, hence 1 call is replicated by 0.25 shares. The fraction of shares needed to replicate 1 call is called the delta ( ) or hedge ratio. = 0.25 delta ( ) (iia) =(ii)*4 (iii) Binomial Pricing Method 1: Creating a replicating portfolio equal How do we create a replicating portfolio for puts?

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20 22 18 24.2 19.8 16.2 Binomial Pricing Method 1: Extending to two time-steps Share Price Tree Call Option Tree [24.2 - 21] = 3.2 0 0 cucu cdcd c Methodology: Step 1: Calculate u and c u, the delta and call value at the upper intermediate node Step 2: Calculate d and c d, the delta and call value at the lower intermediate node (note: u and d will be different) Step 3: Calculate and c, the delta and the call price today WEMBA 2000Real Options15 Bluejay Corp share price is currently $20. Possible price moves in each period: either up by 10% or down by 10%. Period length: 3months. Value a call option on Bluejay with strike 21, expiration 6 months. Riskfree rate = 2% over each 3 month period.

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20 22 18 24.2 19.8 16.2 Binomial Pricing Method 1: Extending to two time-steps Share Price Tree WEMBA 2000Real Options16 Call Option Tree 3.2 0 0 cucu cdcd c 22- PV(19.8) =2.59 24.2 - 19.8 = 4.4 19.8 - 19.8 =0 Replicating Portfolio to calculate c u (a) Purchase 1 share and borrow money so that the portfolio payoff is zero in one scenario 3.2*(1/ u ) = 4.4 0 (1/ u)c u Step 1: Calculating c u and u Match replicating portfolio payoffs at ending nodes (b) Purchase the appropriate number of calls so that the payoff at each terminal node matches the payoffs from the portfolio. u = 3.2/4.4 = 0.727 c u = u * 2.59 = 1.88 equal

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Binomial Pricing Method 1: Extending to two time-steps WEMBA 2000Real Options17 Step 2: Calculating c d and d 0 0 cdcd Call payoff in either scenario is zero. Hence c d = 0, replicating portfolio = 0. By implication, d = 0 Step 3: Calculating c and 20- PV(18) =2.35 22 - 18 = 4 18 - 18 =0 Replicating Portfolio to calculate c (a) Purchase 1 share and borrow money so that the portfolio payoff is zero in one scenario (note: this is identical to the 1-step tree) 0 (1/ )*c Match replicating portfolio payoffs at ending nodes c u = 1.88*(1/ )=4 (b) Purchase the number of calls necessary so that the payoff at each terminal node matches the payoffs from the portfolio. = 1.88/4 = 0.47 c = * 2.35 = 1.10 equal

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WEMBA 2000Real Options18 Bluejay Corp share price is currently $20. Possible price at the end of three months: either $22 or $18. Value a call option on Bluejay with strike 21, expiration 3 months. Riskfree rate = 2% over 3 months. 20 22 18 c 22-21 = 1 0 Share PriceOption Value Create a riskless portfolio: sell 1 call, buy d shares (where d is a fraction of a share) - c + 20 22 - 1 18 Question: how can we make this portfolio riskless? {Reminder: the value of the call at expiration is Max[0, S - K]} Riskless Portfolio Binomial Pricing Method 2: Creating a riskless portfolio

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WEMBA 2000Real Options19 Riskless Portfolio -c + 20 22 - 1 18 For the portfolio to be riskless, the two outcomes must have identical values. HINT: Choose so that: 22 - 1 = 18 = 0.25 Portfolio Terminal value = 4.5 (in either scenario) Portfolio Present value = 4.5/(1.02) (discount at riskfree rate) = 4.41 Hence: 4.41 = -c + 20 c = 0.59 Portfolio "delta" Call premium (price) Note: this is the same call price and delta that we obtained using method 1. Binomial Pricing Method 2: Creating a riskless portfolio

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WEMBA 2000Real Options20 1 0 Option Value 0.59 q 1-q Call price = 0.59 = [1 * q + 0 * (1 - q)]/1.02 q = 0.6 What does the value q represent? It does not represent the probability that the stock price will move up or down! It is sometimes referred to as the “risk-neutral” probability that the stock price will move up or down. 22 18 Stock Price 20 q 1-q Stock price = 20 = [22 * q + 18 * (1 - q)]/1.02 q = 0.6 Binomial Pricing Method 2: Creating a riskless portfolio

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Binomial Pricing Method 2: Generalization WEMBA 2000Real Options21 SS Su Sd c cucu cdcd *Share PriceOption Value Portfolio S-c Su - c u Sd - c d For portfolio to be riskless, choose so that Su - c u = Sd - c d hence = c u - c d Su - Sd Now the riskless terminal value, discounted at the riskless rate r f, should equal the portfolio cost: Su - c u = S - c (1 + r f ) Substitute for from (1):c = q c u + (1-q)c d (1 + r f ) where q = (1 + r f ) - d (u - d) (1) (2) (3) + =

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WEMBA 2000Real Options22 S Su Sd Su 2 Sud Sd 2 c cucu cdcd c uu c ud c dd S = Stock price today u = proportional change in S on an up-move d = proportional change in S on a down-move r f = riskfree rate c = call price today c u = call value after one up-move c d = call value after one down-move c uu, c ud, c dd = terminal call values K = strike on the call c uu = max[0, Su 2 - K] c ud = max[0, Sud - K] c dd = max[0, Sd 2 - K] q = (1+r f ) - d (u - d) c u = [p c uu + (1-p)c ud ] (1+r f ) c d = ….. c = ….. Binomial Pricing Method 2: Generalization over two time-steps Example: compare with 2-step example using Method 1 S = 20, u=1.1, d = 0.9, r f = 2% c uu = 3.2; c ud = c dd = 0 q = [(1.02)-0.9]/(1.02) = 0.6 c u = [0.6 * 3.2 + 0.4 * 0]/1.02 = 1.88 c d = 0 c = [0.6 * 1.88 + 0.4 * 0]/1.02 = 1.10 compare these results with those from Method 1

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Valuation of Options: Binomial Pricing Method WEMBA 2000Real Options23 We can evaluate a call option either by creating a replicating portfolio of the underlying stock and borrowing, or by creating a riskless portfolio of the call and the underlying stock The two methods yield identical results What have we shown? The delta or hedge ratio: the fraction of the underlying stock that we need to purchase relative to selling a single call option to obtain a riskless portfolio The risk-neutral probability of an upmove or downmove in the underlying stock What other information do we obtain from these methods? The Black-Scholes formula effectively represents the binomial tree model over many hundreds or thousands of periods Binomial Tree methodology: Option price = delta * share price - bank loan Black Scholes formula:Option price = N(d 1 )* S - N(d 2 )* PV(K) What are the underlying assumptions of these methods? That we can freely buy and sell the underlying stock without transactions costs That we can borrow or lend money at the riskless rate of interest What are the limitations of these methods? They become very complex over a large number of steps (although computers can help) What is the connection between these methods and the Black-Scholes formula?

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Valuation of Options: Call and Put Price Sensitivities WEMBA 2000Real Options24 As each input to the option pricing model varies, the call and put prices respond by increasing or decreasing as follows: Increase In: Call Price Put PriceWhy? [to be discussed in class] S X T r

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Debt and Equity as Options Suppose a firm has debt with a face value of $1MM outstanding that matures at the end of the year. What is the value of debt and equity at the end of the year? Firm Value (V)Payoff to shareholdersPayoff to debtholders 0.3 MM 00.3 MM 0.6 MM 0 0.6 MM 0.9 MM 0 0.9 MM 1.2 MM 0.2 MM1.0 MM 1.5 MM 0.5 MM1.0 MM Payoffs Firm Value V 0 $1 MM Equityholders Bondholders Payoff to Equityholders = max [0, V - $1MM] equivalent to a call option, K=$1MM Payoff to Bondholders = V - max [0, V - $1MM] equivalent to the total value of the firm less a call option, K=$1MM WEMBA 2000Real Options25

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