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FIN 4021 Fall 2015 Homework Assignment #3 Due: Friday, December 4th Submit one copy of your solutions per group. You may email this to me anytime on the due date. Include the names of all your group members and your group's number. Give your file a title that is very easy for me to identify. I suggest you use something like FIN_4021_HW_3_Group_XX. 1. Black-Scholes-Merton Option Pricing: Use EXCEL to answer this question. Suppose that Syzsmanski & Co. (SYZ) stock is trading at $69.81. The annualized standard deviation of daily returns for SYZ is 71.65%. The T-Bill rate is 4.95%. a. Call Price without Dividends: Assume that SYZ pays no dividends. What is the BSM price of a call with an exercise price of $70 that matures in 146 days (assume that a year is 252 days)? Note: Use the NORMSDIST function in EXCEL to find N(d1) and N(d2). b. Call Price with Dividends: Assume that SYZ pays dividends at annual rate of 1%. Now, what is the BSM price of a call with an exercise price of $70 that matures in 146 days (assume that a year is 252 days)? c. Put Price without Dividends: Assume that SYZ pays no dividends. What is the BSM price of a put with an exercise price of $70 that matures in 146 days (assume that a year is 252 days)? d. Put Price with Dividends: Assume that SYZ pays dividends at annual rate of 1%. Now, what is the BSM price of a put with an exercise price of $70 that matures in 146 days (assume that a year is 252 days)? 2. Using EXCEL to compute the BSM model 'Greeks': Suppose that Ono & Co. (ONO) stock does not pay dividends. The annualized standard deviation of daily returns for ONO is 20%. The T-Bill rate is 5%. In this problem we will analyze puts and calls with a strike price of $100 and 6 months to maturity. a. Delta: In the BMS model, the sensitivity of an option's price to a change in the underlying stock price is called Delta. The Delta for puts and calls is given by: = ( 1 ) and = (1 ) On the same graph, plot the Put and Call Deltas for ONO for stock for prices ranging from $75 to $125. Note 1: You can use the NORMSDIST function in EXCEL to find N(d1) and N(-d1). Note 2: The Call Delta ranges from 0 to 1. The Put Delta ranges from -1 to 0. b. Gamma: As in the binomial model, delta in the BSM model changes with the value of the underlying asset. The sensitivity of delta to changes in the underlying stock price is called Gamma. The Gamma for puts and calls is given by: 1 = = 1 2 (1 )2 2 Plot the Call Gamma for ONO for stock for prices ranging from $75 to $125. Note: Gamma is always positive and is maximized near the money. c. Theta: Theta measures the impact of time on option values. For reasons you will see in this calculation, it often called time decay. The Theta for puts and calls is given by: = = 1 2 2 1 2 2 (1 )2 2 (1 )2 2 ( 2 ) + (2 ) On the same graph, plot the Call and Put Theta for ONO for stock prices ranging from $75 to $125. Note: the Put Theta will plot a constant distance above the Call Theta. Also, the Call Theta will always be negative (time decay always reduces call values); but, the Put Theta should be positive for low stock prices (deep in-the-money). At these low prices, the passage of time actually increases the value of the put because we are coming closer to receiving the strike price (in this range the time value of money dominates the insurance value of the option). d. Vega: The BSM model assumes the volatility of the underlying asset is constant. In reality, of course, this isn't true. Traders compute the sensitivity of BSM model prices to changes in volatility. The result is called Vega. The Vega for puts and calls is given by: = = 2 (1 )2 2 Plot the Call Vega for ONO for stock prices ranging from $75 to $125. Note: Vega is always positive and is maximized near the money. 3. Implied Volatility: Suppose the S&P 500 (SPX) is trading at 1,512.84. SPX has a dividend yield of 1.62%. The T-Bill rate is 4.62%. We will evaluate options with a maturity of 61 days (out of 252 trading days). Consider the following observed market prices for puts and calls: Exercise Price Call Price Put Price 1,450 1,500 1,520 1,540 1,600 $105.60 $21.70 $63.40 $32.40 $45.50 $39.50 $40.00 $42.00 $11.50 $76.50 Compute the implied volatility for each option. Plot the implied volatility for each option on the same graph where the x-axis is exercise price and the y-axis is volatility. Connect the put volatility estimates (so that they trace out a smile, smirk, etc.). Do the same for the call volatility estimates. Note: the put and call volatility estimates will not generally be the same even at the same strike price. Solution strategy: Compute the difference between the BSM model price and the observed market price for each exercise price. Use a dummy value of to compute the initial BSM model price. You can use the EXCEL SOLVER tool to find the implied volatility by trial & error. Instruct SOLVER to set the difference between the BSM model price and the observed market price to zero (within a very small distance) by changing the volatility