(1.) Using the binomial model, what are the ending values of the stock price? What are the payoffs of the call option? Current stock price, P = Risk-free rate, PRF = Strike price, X = Up factor for stock price, u = Down factor for stock price, d = Years to expiration, t = $30.00 5% $24.00 1.34 0.62 0.50 Binomial Payoffs Strike price: X = Current stock price: p = Up factor for stock price: u = Down factor for stock price: d = Up option payoff: C, = MAX[0,P(u)-X] = Down option payoff: Ca =MAX[0,P(d)-X] = $24.00 $30.00 1.34 0.62 Ending "up" stock price = P (u) = Option payoff: C = MAX[O,P(u)-X] = Current stock price P= $30 Ending "down" stock price = P (d) = Option payoff: Ca =MAX[0,P(d)-X] = (2.) Suppose you write 1 call option and buy N, shares of stock. How many shares must you buy to create a portfolio with a riskless payoff (which is called a hedge portfolio)? What is the payoff of the portfolio? We can form a portfolio by writing 1 call option and purchasing N, shares of stock. We want to choose Ng such that the payoff of the portfolio if the stock price goes up is the same as if the stock price goes down. This is a hedge portfolio because it has a riskless payoff. No = Cu-Cd P(u - d) The Hedge Portfolio with Riskless Payoffs Strike price: X = Current stock price: P = Up factor for stock price: u = Down factor for stock price: d = Up option payoff: C = MAX[0,P(u)-X] = Down option payoff: Co =MAX[0,P(d)-X] = Number of shares of stock in portfolio: N, = (Co-Ca) / P(u-d) $24.00 $30.000 1.34 0.621 = P, current stock price $30 Stock price = P (u) = Portfolio's stock payoff: = P(u)(N) Subtract option's payoff: C, Portfolio's net payoff = P(u)N, -C, = = Stock price = P (d) = Portoflio's stock payoff: = P(d)(N) Subtract option's payoff: Co = Portoflio's net payoff = P(d)N, -ca (3.) What is the present value of the hedge portfolio's riskless payoff? What is the value of the call option? The present value of the riskless payoff disounted at the risk-free rate (we assume daily compounding) is: N= I/YR = PMT = FV = PV = 182.5 0.0137% 0 $0.00 Using the PV function. = = Alternatively, use the present value equation: Pv of payoff = Payoff (1 + PRF/365)3 365"(t) The current value of the hedge portolio is the the stock value (N, x P) less the call value (Vc). But the hedge portfolio has a riskless payoff, so the hedge portfolio's value must also be equal to the present value of the riskless payoff disounted at the risk-free rate (we assume daily compounding). With a little algebra, we get: Vc = N, (P) - Present value of riskless payoff Vca (4.) What is a replicating portfolio? What is arbitrage? If you borrow an amount equal to the present value of the riskless payoff and buy N, shares of stock, the payoffs of this portfolio replicate the payoffs of the call option. NE Amount borrowed = PV of riskless payoff = Repayment of riskless payoff = Payoff if stock is up: Stock price = Value of stock in portfolio = Less repayment of borrowing = Net payoff of portfolio = Payoff if stock is down: Stock price = Value of stock in portfolio = Less repayment of borrowing = Net payoff of portfolio = Notice that these are the same payoffs of the option. BLACK-SCHOLES OPTION PRICING MODEL e. In 1973, Fischer Black and Myron Scholes developed the Black-Scholes Option Pricing Model (OPM). e. (1.) What assumptions underlie the OPM? In deriving this option pricing model, Black and Scholes made the following assumptions: 1. The stock underlying the call option provides no dividends or other distributions during the life of the option. 2. There are no transaction costs for buying or selling either the stock or the option. 3. The short-term, risk-free interest rate is known and is constant during the life of the option. 4. Any purchaser of a security may borrow any fraction of the purchase price at the short-term, risk-free interest rate. 5. Short selling is permitted, and the short seller will receive immediately the full cash proceeds of today's price for a security sold short. 6. The call option can be exercised only on its expiration date. 7. Trading in all securities takes place continuously, and the stock price moves randomly. e. (2.) Write out the three equations that constitute the model. The derivation of the Black-Scholes model rests on the concept of a riskless hedge. By buying shares of a stock and simultaneously selling call options on that stock, an investor can create a risk-free investment position, where gains on the stock are exactly offset by losses on the option. Ultimately, the Black-Scholes model utilizes these three formulas: V = Note: r is the risk-free rate. d = d2 = In these equations, V is the value of the option. P is the current price of the stock. N(d) is the area beneath the standard normal distribution corresponding to (dy). X is the strike price. PRF is the risk-free rate. t is the time to maturity. N(dz) is the area beneath the standard normal distribution corresponding to (d). O, or sigma, is the volatility of the stock price, as measured by the standard deviation. e. (3.) What is the value of the following call option according to the OPM? Looking at these equations we see that you must first solve d, and before you can proceed to value the option. This model is widely used by options traders and is generally considered to be the standard for option pricing. Many hand-held calculators and computer programs have this formula permanently stored in. We now use Excel to write a "program", if you will, for the Black-Scholes pricing model in Excel. First, we will lay out the input data given to us in the setup of the problem. Key Inputs: Key output: Vc = TRF t (in years) o $27 $25 6% 0.5 0.49 Now, we will use the formula from above to solve for dy. (d) = Having solved for dy, we will now use this value to find dz. (dz) = At this point, we have all of the necessary inputs for solving for the value of the call option. We will use the formula for V from above to find the value. The only complication arises when entering N(d1) and N(D2). Remember, these are the areas under the standard normal distribution. Luckily, Excel is equipped with a function that can determine cumulative probabilities of the normal distribution. This function is located in the list of statistical functions, as "NORMSDIST". For both N(D1) and N(d2), we will follow the same procedure of using this function in the value formula. Function Arguments ? NORMSDIST Z D254 ER = 0.481946784 = 0.685078125 This function is available for compatibility with Excel 2007 and earlier. Returns the standard normal cumulative distribution (has a mean of zero and a standard deviation of one) Z is the value for which you want the distribution. Formula result = 0.685078125 Help on this function OK Cancel Using the NORMSDIST function: (d) = (d) = Using the Black-Scholes formula and the cumulative distributions, we can solve for the option value. f. What impact does each of the following call option parameters have on the value of a call option? (1.) Current stock price (2.) Strike price (3.) Option's term to maturity (4.) Risk-free rate (5.) Variability of the stock price Let us now turn our attention to determining how sensitive the call option value is to the five factors of the Black-Scholes OPM. We will set up data tables for each factor determining the call value if the specified input is changed plus or minus 15% and 30%. Change the inputs below to see the impact on the option's price (X=25 for all cases). Let us now turn our attention to determining how sensitive the call option value is to the five factors of the Black-Scholes OPM. We will set up data tables for each factor determining the call value if the specified input is changed plus or minus 15% and 30%. Change the inputs below to see the impact on the option's price (X=25 for all cases). t = 0.5 F = 6% o? = 0.11 Data Option Pricing: Sensitivity Analysis Exercise Option Value Price $25.00 $20.00 Price of the stock $0 $5 $10 $15 $20 $25 $30 $35 $40 $45 Strike Price $28 $28 $28 $28 $28 $28 $28 $28 $28 $28 $15.00 Option Price $10.00 $5.00 $0.00 $0 $10 $20Stock Bilce $40 $50 $60 Exercise value -Option price g. What is put-call parity? (1.) Using the binomial model, what are the ending values of the stock price? What are the payoffs of the call option? Current stock price, P = Risk-free rate, PRF = Strike price, X = Up factor for stock price, u = Down factor for stock price, d = Years to expiration, t = $30.00 5% $24.00 1.34 0.62 0.50 Binomial Payoffs Strike price: X = Current stock price: p = Up factor for stock price: u = Down factor for stock price: d = Up option payoff: C, = MAX[0,P(u)-X] = Down option payoff: Ca =MAX[0,P(d)-X] = $24.00 $30.00 1.34 0.62 Ending "up" stock price = P (u) = Option payoff: C = MAX[O,P(u)-X] = Current stock price P= $30 Ending "down" stock price = P (d) = Option payoff: Ca =MAX[0,P(d)-X] = (2.) Suppose you write 1 call option and buy N, shares of stock. How many shares must you buy to create a portfolio with a riskless payoff (which is called a hedge portfolio)? What is the payoff of the portfolio? We can form a portfolio by writing 1 call option and purchasing N, shares of stock. We want to choose Ng such that the payoff of the portfolio if the stock price goes up is the same as if the stock price goes down. This is a hedge portfolio because it has a riskless payoff. No = Cu-Cd P(u - d) The Hedge Portfolio with Riskless Payoffs Strike price: X = Current stock price: P = Up factor for stock price: u = Down factor for stock price: d = Up option payoff: C = MAX[0,P(u)-X] = Down option payoff: Co =MAX[0,P(d)-X] = Number of shares of stock in portfolio: N, = (Co-Ca) / P(u-d) $24.00 $30.000 1.34 0.621 = P, current stock price $30 Stock price = P (u) = Portfolio's stock payoff: = P(u)(N) Subtract option's payoff: C, Portfolio's net payoff = P(u)N, -C, = = Stock price = P (d) = Portoflio's stock payoff: = P(d)(N) Subtract option's payoff: Co = Portoflio's net payoff = P(d)N, -ca (3.) What is the present value of the hedge portfolio's riskless payoff? What is the value of the call option? The present value of the riskless payoff disounted at the risk-free rate (we assume daily compounding) is: N= I/YR = PMT = FV = PV = 182.5 0.0137% 0 $0.00 Using the PV function. = = Alternatively, use the present value equation: Pv of payoff = Payoff (1 + PRF/365)3 365"(t) The current value of the hedge portolio is the the stock value (N, x P) less the call value (Vc). But the hedge portfolio has a riskless payoff, so the hedge portfolio's value must also be equal to the present value of the riskless payoff disounted at the risk-free rate (we assume daily compounding). With a little algebra, we get: Vc = N, (P) - Present value of riskless payoff Vca (4.) What is a replicating portfolio? What is arbitrage? If you borrow an amount equal to the present value of the riskless payoff and buy N, shares of stock, the payoffs of this portfolio replicate the payoffs of the call option. NE Amount borrowed = PV of riskless payoff = Repayment of riskless payoff = Payoff if stock is up: Stock price = Value of stock in portfolio = Less repayment of borrowing = Net payoff of portfolio = Payoff if stock is down: Stock price = Value of stock in portfolio = Less repayment of borrowing = Net payoff of portfolio = Notice that these are the same payoffs of the option. BLACK-SCHOLES OPTION PRICING MODEL e. In 1973, Fischer Black and Myron Scholes developed the Black-Scholes Option Pricing Model (OPM). e. (1.) What assumptions underlie the OPM? In deriving this option pricing model, Black and Scholes made the following assumptions: 1. The stock underlying the call option provides no dividends or other distributions during the life of the option. 2. There are no transaction costs for buying or selling either the stock or the option. 3. The short-term, risk-free interest rate is known and is constant during the life of the option. 4. Any purchaser of a security may borrow any fraction of the purchase price at the short-term, risk-free interest rate. 5. Short selling is permitted, and the short seller will receive immediately the full cash proceeds of today's price for a security sold short. 6. The call option can be exercised only on its expiration date. 7. Trading in all securities takes place continuously, and the stock price moves randomly. e. (2.) Write out the three equations that constitute the model. The derivation of the Black-Scholes model rests on the concept of a riskless hedge. By buying shares of a stock and simultaneously selling call options on that stock, an investor can create a risk-free investment position, where gains on the stock are exactly offset by losses on the option. Ultimately, the Black-Scholes model utilizes these three formulas: V = Note: r is the risk-free rate. d = d2 = In these equations, V is the value of the option. P is the current price of the stock. N(d) is the area beneath the standard normal distribution corresponding to (dy). X is the strike price. PRF is the risk-free rate. t is the time to maturity. N(dz) is the area beneath the standard normal distribution corresponding to (d). O, or sigma, is the volatility of the stock price, as measured by the standard deviation. e. (3.) What is the value of the following call option according to the OPM? Looking at these equations we see that you must first solve d, and before you can proceed to value the option. This model is widely used by options traders and is generally considered to be the standard for option pricing. Many hand-held calculators and computer programs have this formula permanently stored in. We now use Excel to write a "program", if you will, for the Black-Scholes pricing model in Excel. First, we will lay out the input data given to us in the setup of the problem. Key Inputs: Key output: Vc = TRF t (in years) o $27 $25 6% 0.5 0.49 Now, we will use the formula from above to solve for dy. (d) = Having solved for dy, we will now use this value to find dz. (dz) = At this point, we have all of the necessary inputs for solving for the value of the call option. We will use the formula for V from above to find the value. The only complication arises when entering N(d1) and N(D2). Remember, these are the areas under the standard normal distribution. Luckily, Excel is equipped with a function that can determine cumulative probabilities of the normal distribution. This function is located in the list of statistical functions, as "NORMSDIST". For both N(D1) and N(d2), we will follow the same procedure of using this function in the value formula. Function Arguments ? NORMSDIST Z D254 ER = 0.481946784 = 0.685078125 This function is available for compatibility with Excel 2007 and earlier. Returns the standard normal cumulative distribution (has a mean of zero and a standard deviation of one) Z is the value for which you want the distribution. Formula result = 0.685078125 Help on this function OK Cancel Using the NORMSDIST function: (d) = (d) = Using the Black-Scholes formula and the cumulative distributions, we can solve for the option value. f. What impact does each of the following call option parameters have on the value of a call option? (1.) Current stock price (2.) Strike price (3.) Option's term to maturity (4.) Risk-free rate (5.) Variability of the stock price Let us now turn our attention to determining how sensitive the call option value is to the five factors of the Black-Scholes OPM. We will set up data tables for each factor determining the call value if the specified input is changed plus or minus 15% and 30%. Change the inputs below to see the impact on the option's price (X=25 for all cases). Let us now turn our attention to determining how sensitive the call option value is to the five factors of the Black-Scholes OPM. We will set up data tables for each factor determining the call value if the specified input is changed plus or minus 15% and 30%. Change the inputs below to see the impact on the option's price (X=25 for all cases). t = 0.5 F = 6% o? = 0.11 Data Option Pricing: Sensitivity Analysis Exercise Option Value Price $25.00 $20.00 Price of the stock $0 $5 $10 $15 $20 $25 $30 $35 $40 $45 Strike Price $28 $28 $28 $28 $28 $28 $28 $28 $28 $28 $15.00 Option Price $10.00 $5.00 $0.00 $0 $10 $20Stock Bilce $40 $50 $60 Exercise value -Option price g. What is put-call parity