I need a step by step solutions for the 10 problems in the excel file. The correct answers are provided I only need the solutions steps. Subject is Finance (options valuation)

On March 2, a Treasury bill expiring on April 20 had a bid discount of 5.80, and an ask discount of 5.86. What is the best estimate of the risk-free rate as given in the text? Answer: 6.11 % The stock price was 113.25. The risk-free rates were 7.30 percent (November), 7.50 percent (December) and 7.62 percent (January). The times to expiration were 0.0384 (November), 0.1342 (December), and 0.211 (January). Assume no dividends unless indicated. Puts Calls Strike Nov Dec Jan Nov Dec Jan 105 8.4 10 11.5 5.3 1.3 2 110 4.4 7.1 8.3 0.9 2.5 3.8 115 1.5 3.9 5.3 2.8 4.8 4.8 What is the intrinsic value of the December 115 put? Answer: 1.75 Consider a binomial world in which the current stock price of 80 can either go up by 10 percent or down by 8 percent. The risk-free rate is 4 percent. Assume a one-period world. Answer questions 12 through 15 about a call with an exercise price of 80. What is the theoretical value of the call? Answer: 5.15 A stock priced at 50 can go up or down by 10 percent over two periods. The risk-free rate is 4 percent. Which of the following is the correct price of an American put with an exercise price of 55? Answer: 5.00 The following information is given about options on the stock of a certain company S0 = 23 X = 20 rc = 0.09 T = 0.5 2 = 0.15 What value does the Black-Scholes-Merton model predict for the call? (Due to differences in rounding your calculations may be slightly different. \"none of the above\" should be selected only if your answer is different by more than 10 cents.) Answer: 4.73 The following information is given about options on the stock of a certain company S0 = 23 X = 20 rc = 0.09 T = 0.5 2 = 0.15 If we now assume that the stock pays a dividend at a known constant rate of 3.5 percent, what stock price should we use in the model? (Due to differences in rounding your calculations may be slightly different. \"none of the above\" should be selected only if your answer is different by more than 10 cents.) Answer: 22.60 Consider a stock priced at $30 with a standard deviation of 0.3. The risk-free rate is 0.05. There are put and call options available at exercise prices of 30 and a time to expiration of six months. The calls are priced at $2.89 and the puts cost $2.15. There are no dividends on the stock and the options are European. Assume that all transactions consist of 100 shares or one contract (100 options). What is the breakeven stock price at expiration $27? Answer: $27.11 The following prices are available for call and put options on a stock priced at $50. The risk-free rate is 6 percent and the volatility is 0.35. The March options have 90 days remaining and the June options have 180 days remaining. The Black-Scholes model was used to obtain the prices. Strike 45 50 55 Calls March June 6.84 8.41 3.82 5.58 1.89 3.54 Puts March June 1.18 2.09 3.08 4.13 6.08 6.93 Assume that each transaction consists of one contract (for 100 shares) unless otherwise indicated. What is the profit if the stock price at expiration is $47? Answer = -102 The following prices are available for call and put options on a stock priced at $50. The risk-free rate is 6 percent and the volatility is 0.35. The March options have 90 days remaining and the June options have 180 days remaining. The Black-Scholes model was used to obtain the prices. Strike 45 50 55 Calls March June 6.84 8.41 3.82 5.58 1.89 3.54 Puts March June 1.18 2.09 3.08 4.13 6.08 6.93 Assume that each transaction consists of one contract (for 100 shares) unless otherwise indicated. What are the two breakeven stock prices at expiration using the June 50 options. Answer = $59.71 and $40.29 A futures contract covers 5000 pounds with a minimum price change of $0.01 is sold for $31.60 per pound. If the initial margin is $2,525 and the maintenance margin is $1,000, at what price would there be a margin call? Answer: 31.91 Here, work through the steps to value a call with a one-period binomial model: S $125 x $125 T = 1 year 1 u 1.25 d 0.80 Su $156 Sd $100 r (annual) 5.650% a. What are Cu and Cd Cu Cd $31.25 $0.00 b. What is the hedge ratio? Hedge ratio is ratio between the change in an option's theoretical value and the change in price of the underlying stock at a given point in time. In this case it is 0.556 Hedge 0.556 c. What does the hedge ratio mean in terms of number of shares and calls? Hedge ratio of 0.556 means buying 55.6% of short positions sahres as call. d. What is the value of p (the risk neutral probability of the stock price increasing)? P 0.57 e. What is the value of 1-p (the risk-neutral probability of the stock price decreasing)? 1-P 0.43 f. What is the value of the call? Call $16.86 g. What would "u" be for the single 1 year period if the standard deviation of the stock were 25% per year (T=1) t stadard deviation u 1 25% 1.28 Consider a two-period (total of one year), two-state world. Let the current stock price be 48 and the (cc) risk-free rate be 4.879%. Each period the stock price can go up by 10% or down by 10%. A call option expiring at the end of the second 6 month period has an exercise price of $43. Stock Price $ 48.00 Exercise price Given Data 43.00 risk-free rate $ 4.88% Up a. What are the stock prices at each point? Su Suu Sud Sd Sdd 52.8 58.08 47.52 43.2 38.88 b. What is the value of p (the probability of the stock price increasing)? P 1-P 0.74395 0.25605 c. What are the final period call prices? Cuu Cud Cdd $ $ 15.08 4.52 0.00 d. What are the 1st period call prices? Cu Cd 11.800372 3.2062224 e. What is the value of the call? Call $9.15 f) What is the initial hedge ratio? Hedge 0.8952 10% Down 10% Consider the following binomial option pricing problem involving an American call. This call has two (one year each) periods to go before expiring. Its stock price is $32, and its exercise price is $27. The risk free rate is .05, stock's standard deviation of returns is 25%. The stock pays a dividend at the end of the first period at the rate of 8%. Find the value of the American call. Given Data S 32 X 27 Rf 5% SD 25% T 1 D 8% UP 1.28 DWN 0.78 a. What are the stock prices at each point? Su (pre dividend) Su (ex-dividend) Suu Sud Sd (pre dividend) Sd (ex-dividend) Sdd 41.09 37.80 48.54 29.44 24.92 22.93 17.86 b. What is the value of p (the probability of the stock price increasing)? P 1-P 0.54 0.46 c. What are the final period call prices? Cuu Cud Cdd 21.54 2.44 0 d. What are the 1st period call prices? Cu Cd 12.09 1.25 e. What is the value of the American call at time 0? C $7.75 14.09 Use the Black-Scholes-Merton model to calculate the stock price, if: The current stock price is 174, the exercise price is 174, the risk free interest rate (cc) is 5.71%, the stock's standard deviation is 21% and the option has 135 days (365 day year) to maturity. Given Data S X t (year) Rf SD 174 174 0.37 5.71% 21.00% DTM 135 A. What is d1 What is d2 What is N(d1) What is N(d2) What is the price of the call based on BSM? What is the hedge ratio? A PV of X SD*t^.5 d1 d2 N(d1) N(d2) Hedge Ratio N(d2)*PV(EX) 170.36 0.13 0.2292 0.1015 0.5907 0.5404 0.06 92.07 Price of Call Based on BSM 10.70 B. If the actual price of the call were $9.75, what strategy should we take? For consistency of analysis, assume that you will purchase or sell 1000 calls (10 contracts).This is where put-call parity comes in, but only use calls and stocks for this one. B The actual price of the call were $9.75 0.95 591 Short sell 591 shares and buy 1,000 calls. C. Given your strategy in part b, what happens to the value of the calls if the price of the stock changes by $1 C If the price of stock: Increased $1: the value will increase by 5.71% Decreased $1 : the value will decrease 5.71% DY 365 Use the Excel spreadsheet BSMbin9e.xls and determine the value of thecall option and a put option on a stock currently priced at 100, where the risk free rate is 5% (annually compounded), the exercise price is 100, the volatility is 30%, the option expires in one year, and there are no dividends on the stock.Use the Binomial part of this model: Let the number of binomial periods be 25.Verify that put-call parity exists. Hints on the homework:Double check the drop-down box to be sure you are using the right discounting method (discrete).Once you have the Binomial solutions for the call and put values, use put-call parity, or PCP, (from Chance CH 3) to determine the PCP price for a call. Show your work by copying the BSM model output here. Show your PCP calculations, I want to see each of the factors used as your inputs. BSM model output here: d1 d2 N(d1) N(d2) PV of divs PV of strike S - PV divs 0.312634 0.012634 0.622721 0.50504 0 95.12294 100 European Call Put Price 14.17301 9.411101 Delta (D) 0.622721 -0.377279 Gamma (G) 0.012664 0.012664 Theta (Q) -8.045481 -3.398799 Vega 37.99147 37.99147 Rho 48.09905 -47.13904 European Call Put Price 14.28359146 9.5216867 Delta (D) 0.625936919 -0.3740631 Gamma (G) 0.012270592 0.01227059 Theta (Q) -7.98503388 -3.3383398 Price Delta (D) Gamma (G) Theta (Q) American Call Put 14.28359 10.0123593 0.625937 -0.403814 0.012271 0.01396244 -7.985034 -3.92198575 Givien Data S 100 X 100 r 5% Stock 100 X 100 SD P+S 109.41 30% x 0.95238 T 1 C + PV(X) Difference 109.41 0.00 Not opportunity for arbriage since different is 0 D 0% P 25 Homework 2 - Derivatives Osama Alfaraj Date 17-Jul 21-Aug 16-Oct Rate 0.0516 0.055 0.0588 The stock is priced @ 165.13 Calls Strike 155 160 165 170 Jul 10.5 6 2.7 0.8 A) July B) October C) August Aug 11.8 8.1 5.2 3.2 160 155 170 Oct 14.00 11.1 8.1 6 B 5.13 0.87 0.9985 5.37 Oct 2.75 4.5 6.7 9 There is no arbitage opportinints since all peices imimate the boundary A Intrinsic value Value Time Lower bound Jul 0.2 0.75 2.35 5.8 Puts Aug 1.25 2.75 4.7 7.5 Intrinsic value Value Time Lower bound 10.13 3.87 0.9842 12.585205 C Intrinsic value Value Time Lower bound 0 3.2 0.9933 0.00 Rate 17-Jul 21-Aug 16-Oct Rate 0.0516 0.055 0.0588 The stock is priced@ 165.13 Calls Strike 155 160 165 170 Jul 10.5 6 2.7 0.8 A) July B) October C) August Aug 11.8 8.1 5.2 3.2 Puts Oct 14.00 11.1 8.1 6 165 160 170 Aug 1.25 2.75 4.7 7.5 Oct 2.75 4.5 6.7 9 There is no arbitage opportinints since all peices imimate the boundary A Intrinsic value Time value Lower bound: Jul 0.2 0.75 2.35 5.8 B 0 2.35 0 Intrinsic value Time value Lower bound: C 0 2.75 0 Intrinsic value Time value 4.87 4.13 Rate 17-Jul 21-Aug 16-Oct Rate 0.0516 0.055 0.0588 Strike 155 160 165 170 Jul 10.5 6 2.7 0.8 A) July B) August C) October The stock is priced@ Calls Aug 11.8 8.1 5.2 3.2 Puts Oct 14.00 11.1 8.1 6 Jul 0.2 0.75 2.35 5.8 Aug 1.25 2.75 4.7 7.5 Oct 2.75 4.5 6.7 9 155 160 170 A P + So C+X Diffrence 165.13 B 165.33 165.2652 0.064845 P + So C+X Diffrence C 167.88 167.024017673 0.855982327 P + So C+X Diffrence 174.13 173.3072 0.822805 Rate 17-Jul 21-Aug 16-Oct Strike 155 160 165 170 Rate 0.0516 0.055 0.0588 Jul 10.5 6 2.7 0.8 The stock is priced@ Calls Aug 11.8 8.1 5.2 3.2 165.13 Puts Oct 14.00 11.1 8.1 6 Jul 0.2 0.75 2.35 5.8 Aug 1.25 2.75 4.7 7.5 Oct 2.75 4.5 6.7 9 Difference in premiums of calls should not more than the difference in X price, except if it was Eurpean call. Augest 155, 160 October 160,165 Difference in Premiums 3.7 3 Difference in Excersise Price 5 5 No valioation for both cases here. Rate 17-Jul 21-Aug 16-Oct Strike 155 160 165 170 Rate 0.0516 0.055 0.0588 Jul 10.5 6 2.7 0.8 The stock is priced@ Calls Aug 11.8 8.1 5.2 3.2 165.13 Puts Oct 14.00 11.1 8.1 6 Jul 0.2 0.75 2.35 5.8 Aug 1.25 2.75 4.7 7.5 Oct 2.75 4.5 6.7 9 Difference in premiums of calls should not more than the difference in X price, except if it was Eurpean call. Augest 155, 160 October 160,170 Difference in Premiums 1.5 4.5 Difference in Excersise Price 5 10 No valioation for both cases here. 1 A short position in stock can be protected by holding a call option. Determine the profit equations for this postion and identify the breakeven stock price at expiration and maximum and minimum profits. Profit from strategy (excel)= Max(0, ST - X) - C - (ST - S0)) When Stock price becomes less than or equal to Excersize the Profit from strategy = - C - ST + S0 If Stock price becomes greater than Excersize then Profit from strategy = ST - X - C - ST + S0 = S0 - X - C The breakeven value is when of stock price becomes less than or equal Excerize price which will be= - Price of call - Breackeven price at expiration + Stock price rice today = Zero So the Breackeven price at expiration will be = Stock proce today - Price of call The maximum profit if the stock price becomes zero will be = - Price of call + Stock price today The minimum profit if the stock price becomes greater than the excersize will be = Stock price today - Excersize - Price of call and it going to be loss because the price of the call should be more than the stock price today - Excersize 2 A short stock can be protected by selling a put. Determine the profit equations for this postion, and identify the breakeven stock price at expiration and maximum and minimum profits. The profit from the strategy (excel) = -Max(0, X - ST) + P - (ST - S0) If Stock price becomes less than Excersize then Profit from strategy = - X + ST + P - ST + S0 = P + S0 - X When Stock price becomes greater than or equal to Excersize the Profit from strategy = P - ST + S0 The breakeven value is when of stock price becomes greater then Excerize price which will be= Price of put- Breackeven price at expiration + Stock price rice today = Zero So solving Breackeven price at expiration = Put price + Stock price today The maximum profit if the stock price becomes less than the excersize will be = Put price + Stock price today - Excersize Minimum profit is unimited loss since stock price can go up unlimited 3 Explain the advantages and disadvantages to a covered call writer of closing out the position prior to expiration One advantage of closing out a covered call position before epiration is giving the stock price shorter time to move which minmize the risk. One disadvantage is that the call value becomes more costly to repurchase which minimize the gain. The following option prices were observed for a stock on July 6 of a particular year. Use this information for the following problems. Unless otherwise indicated, ignore dividends on the stock. The stock is priced a 165.13 The expirations and associated risk free rates for each term are: Date 17-Jul 21-Aug 16-Oct Strike 160 165 170 Rate 5.16% 5.50% 5.88% Calls Jul Aug 6.00 2.70 0.80 Puts Jul Oct 8.10 5.20 3.20 11.10 8.10 6.00 Aug Oct 0.75 2.35 5.80 2.75 4.70 7.50 4.50 6.70 9.00 In the following problems, determine the profits for possible stock prices of 150, 155, 160, 165, 170, 175, and 180. Answer any other questions as requested. Note the Excel spreadsheet available via the course website (to a link on the publisher/author website): Stratlyz9e.xls will be useful here for obtaining graphs as requeted, but it does not allow you to calculate the p 4 Buy one August 165 call contract. Hold it until the options expire. Determine the profits and graph the results. Then identify the breakeven stockStratlyz8e.xls to generate the graphs maximum lossthe horizontal axis will be labeled as the \"Stock Price at Expiration\". Here it has been changed to \"Stock Price at End of Holding Period\". Note: If you use price at expiration. What is the in this chapter, on this transaction. Call Breakeven Maxumum loss St 165 170.20 (520.00) Value at expiration 150 0 155 0 160 0 165 0 170 5 175 10 180 15 980.00 480.00 P (520.00) (520.00) (520.00) (520.00) (20.00) 480.00 980.00 150 155 160 165 (520.00) (520.00) (520.00) (20.00) 170 (520.00) ST Repeat the above problem, but close the postion on August 1. Use the spreadsheet to find the profits for the possible stock prices on 5 August 1. Generate a graph and use it to identify the approximate breakeven stock price. Stock Price 165.13 Value at expiration P Excersize 165 St (471.30) SD 30% 150 0.487 (396.13) 155 1.2387 2.6434 (255.66) 8/1/2015 8/21/2015 160 (33.26) Time 5.48% 165 4.8674 273.68 Risk-free rate 5.50% 170 7.9368 653.56 Call 5.20 175 11.7356 1,086.80 Breakeven 170.33 180 16.068 Max Loss 520.00 1,086.80 653.56 273.68 175 180 150 155 (33.26) 160 165 170 175 180 (255.66) (471.30) (396.13) 6 Buy one October 165 put contract. Hold it until the options expire. Determine the profits and graph the results. Identify the breakeven stock price at expiration. What are the maximum and possible gain and loss on this transaction? Buy Put price Breakeven Max loss Maxumum profit 165 6.70 158.30 (670.00) 15,830.00 Value at expiration P 150 15 155 10 160 5 165 0 170 0 175 0 180 0 St 830.00 830.00 330.00 (170.00) (670.00) (670.00) (670.00) (670.00) 330.00 150 155 160 165 170 175 180 (670.00) (170.00) (670.00) (670.00) (670.00) 7 Construct a bear money spread using October 165 and 170 calls. Hold the postion until the options expire Determine the profits and graph the results. Identify the breakeven stock price at expiration and the maximum and minimum profits. Dicuss any special 165 170 8.10 6.00 Excersize Call Breakeven Max loss Max profit October:165 150 0 155 0 160 0 165 0 170 5 175 10 180 15 St 167.10 (290.00) 210.00 210.00 210.00 210.00 October:170 0.00 0.00 0.00 0.00 0.00 5.00 10.00 Profit 210.00 210.00 210.00 210.00 (290.00) (290.00) (290.00) 210.00 Short call could be used as strategy when the stock price go up 150 155 160 165 170 175 180 (290.00) (290.00) (290.00) 8 Construct a long straddle using October 165 options. Hold until the options expire. Determine the profits and graph the results. Identify the breakeven stock prices at expiration and the minimum profit. October 165 Breakeven 179.80 Put 6.70 Max loss (1,480.00) 20.00 Call 8.10 Max profit Unlimited 150 155 160 165 St Put:170 150 15.00 155 10.00 160 0.00 165 0.00 170 0.00 175 0.00 180 0.00 Call:165 0 0 0 0 5 10 15 Profit 20.00 (480.00) (980.00) (1,480.00) (980.00) (480.00) 20.00 (480.00) Item Value at expiration Profit Maximum profit Maximum loss Breakeven Bear Spread With Puts max(0,St-X1)-max(0,St-X2) max(0,X2-St))-max(0,X1-St) Vt-C1+C2 X2-X1-C1+C2 C1-C2 Vt-P2+P1 X2-X1-P2-P1 P2-P1 X1+C1-C2 X2-P2+P1 175 (480.00) Profit (980.00) (980.00) (1,480.00) Stock Price at Expiration 9 Complete the following table with the correct formula related to various spread strategies Bull Spread With Calls 170 Butterfly Spread With Calls Butterfly Spread With Calls max(0,St-X1)-2max(0,StX2)+max(0,St,X3) max(0,St-X1)-2max(0,StX2)+max(0,St,X3) Vt - C1+ 2C2 - C3 X2-X1-C1+2C2-C3 C1-2C2+C3 St = X1+C1- 2C2+C3 & = 2X2-X1-C1+2C2C3 Vt - C1+ 2C2 - C3 X2-X1-C1+2C2-C3 C1-2C2+C3 St = X1+C1- 2C2+C3 & St = 2X2-X1-C1+2C2-C3 20.00 180 profits for several user-specified asset prices. It permits you to specify one asset price and a maximum an dminimum. Uset Stratlyz9e.xls to prodict the graph for a range of prices from 150 to 180 but determine the profits for the pric for the prices of 150, 160....180 by hand for positions held at expiration. For positions closed prior to expiration, use the spreadsheet BSMbin9e.xls to determine the option price when the postion is closed; then calculate the profit by he profit by hand. Unit 1: Principles of Options Pricing Introduction Different types of derivatives Presuppositions for financial markets, risk preferences, risk-return tradeoff, and market efficiency Theoretical fair value Arbitrage, storage, and delivery The role of derivative markets Criticisms of derivatives Ethics Definitions and examples of call and put options Institutional characteristics of options markets Options available for trading Placing an options order The clearinghouse Accessing option price quotations Transaction costs Regulation of options markets Margins and taxes in option transactions Derivative Markets and Instruments Business risk vs. financial risk Derivatives A derivative is a financial instrument whose return is derived from the return on another instrument. Derivatives play a key role in transferring risks in the economy By using various types of derivatives, businesses can help reduce their uncertainties. Size of the OTC derivatives market at year-end 2010 $601 trillion notional principal GDP is only $15 trillion See Figure 1.1 and Figure 1.2 Real vs. financial assets (Return to text slide) (Return to text slide) The OTC Market Prior to 2008 Largely unregulated Banks acted as market makers quoting bids and offers Master agreements usually defined how transactions between two parties would be handled But some transactions were handled by central counterparties (CCPs). A CCP stands between the two sides to a transaction in the same way that an exchange does The OTC Market Since 2008 OTC market has become regulated. Objectives: Reduce systemic risk Increase transparency In the U.S and some other countries, standardized OTC products must be traded on swap execution facilities (SEFs) which are similar to exchanges CCPs (Central counterparties) must be used for standardized transactions between dealers in most countries All trades must be reported to a central registry Derivative Markets and Instruments Derivative Markets Over-the-counter and exchange traded Exchange traded derivatives volume in 2010 was over 22 billion contracts on at least 78 derivatives exchanges, according to Futures Industry magazine (a leading source of derivatives industry information Derivatives trade all over the world See Table 1.1 for the top ten derivatives exchanges (Return to text slide) Derivative Markets and Instruments Options Definition: a contract between two parties that gives one party, the buyer, the right to buy or sell something from or to the other party, the seller, at a later date at a price agreed upon today Option terminology price/premium call/put exchange-listed vs. over-the-counter options Derivative Markets and Instruments (continued) Forward Contracts Definition: a contract between two parties for one party to buy something from the other at a later date at a price agreed upon today Exclusively over-the-counter The difference between and option and a forward contract is that in a forward, both the buyer and seller are obligated to complete the transaction (or close their position prior to expiration) where for options, the buyer can decide if exercising his option is to his advantage or not. The forward price for a contract is the delivery price that would be applicable to the contract if were negotiated today (i.e., it is the delivery price that would make the contract worth exactly zero) The forward price may be different for contracts of different maturities (as shown by the table) Derivative Markets and Instruments (continued) Futures Contracts Definition: a contract between two parties for one party to buy something from the other at a later date at a price agreed upon today; subject to a daily settlement of gains and losses and guaranteed against the risk that either party might default Exclusively traded on a futures exchange Agreement to buy or sell an asset for a certain price at a certain time Similar to forward contract Whereas a forward contract is traded OTC, a futures contract is traded on an exchange Standardized High value - often $1mm or more Derivative Markets and Instruments (continued) Options on Futures (also known as commodity options or futures options) Definition: a contract between two parties giving one party the right to buy or sell a futures contract from the other at a later date at a price agreed upon today Exclusively traded on a futures exchange Derivative Markets and Instruments (continued) Swaps and Other Derivatives Definition of a swap: a contract in which two parties agree to exchange a series of cash flows Exclusively over-the-counter Other types of derivatives include swaptions and hybrids. Their creation is a process called financial engineering. The Underlying Asset Called the underlying A derivative derives its value from the underlying. Some Important Concepts in Financial and Derivative Markets Presuppositions - rule of law, property rights, culture of trust Risk Preference Risk aversion vs. risk neutrality Risk premium Short Selling Repurchase agreements (repos) Return and Risk Risk defined The risk-return tradeoff (see Figure 1.3) (Return to text slide) Some Important Concepts in Financial and Derivative Markets (continued) Market Efficiency and Theoretical Fair Value Efficient market defined: A market in which the price of an asset equals its true economic value. An efficient market is a consequence of rational and knowledgeable investor behavior In derivatives, traders can take short positions to offset the risk of long positions. The concept of theoretical fair value The true economic value Fundamental Linkages Between Spot and Derivative Markets Arbitrage and the Law of One Price Arbitrage defined: A type of profit-seeking transaction where the same good trades at two prices. Example: See Figure 1.4 The concept of states of the world The Law of One Price The Storage Mechanism: Spreading Consumption across Time Delivery and Settlement (Return to text slide) The Role of Derivative Markets Risk Management Hedging vs. speculation Setting risk to an acceptable level Example: Southwest Airlines Price Discovery Operational Advantages Transaction costs Liquidity Ease of short selling Market Efficiency Derivatives and Ethics Codes of ethics and standards of professional conduct are vital components of the derivatives profession Examples CFA Institute Professional Risk Managers International Association Global Association of Risk Professionals Criticisms of Derivatives Speculation Comparison to gambling Misuses of Derivatives High leverage Inappropriate use Derivatives and Your Career Financial management in a business Small businesses ownership Investment management Public service Option Terminology and Examples Option terminology price/premium call vs. put exercise price/strike price/striking price expiration date Everyday examples of options rain check discount coupon airline ticket with cancellation right right to drop a course Development of Options Markets Early origins Put and Call Brokers and Dealers Association Chicago Board Options Exchange, 1973 Resurgence of over-the-counter market Call Options Current example Objective of a call buyer Moneyness concepts In-the-money Out-of-the-money At-the-money Put Options Current example Objective of a put buyer Moneyness concepts In-the-money Out-of-the-money At-the-money Options Trading Activity In 2010, exchange-traded option volume (number of contracts) approximately 11.1 billion contracts (Futures Industry magazine) In 2013, over-the-counter option volume approximately $67 trillion notional principal and $2 trillion market value (Bank of International Settlements) OTC options notional amount outstanding fell dramatically during the Financial Crisis of 2008 (see Figure 2.1) OTC options market value outstanding rose sharply during initial phase of Financial Crisis of 2008 (see Figure 2.2) (Return to text slide) (Return to text slide) Over-the-Counter Options Market Worldwide Credit risk Customized terms Private transactions Unregulated Options on stocks and stock indices, bonds, interest rates, commodities, swaps & currencies Organized Options Trading The concept of an options exchange Listing Requirements Contract Size Exercise Prices Expiration Dates Position and Exercise Limits Option Traders Liquidity Providers Provide bid and ask prices to facilitate trading Scalpers, position traders, spreaders Lead market makers, designated primary market makers Floor Broker - acts as agent for customers Order Book Official Limit orders Electronic order processing Options Traders (continued) Other Option Trading Systems Specialists Registered options traders Electronic trading systems Off-Floor Option Traders Option brokers Proprietary options traders Mechanics of Trading Placing an Opening Order Types of orders Role of the Clearinghouse Options Clearing Corporation (OCC) Clearing firms See Figure 2.3 Margin (see Appendix 2.A) Open interest (Return to text slide) Mechanics of Trading (continued) Placing an Offsetting Order In the exchange-listed options market In the over-the-counter options market Exercising an Option European vs. American style Assignment Cash settlement Problems Delayed information Non-synchronized prices Types of Options Stock Options Index Options Currency Options Other Types of Options interest rate options options attached to bonds exotic options warrants, callable bonds, convertible bonds non-traded executive options Real Options Specification of ExchangeTraded Options Expiration date \"Cycles\" are January, February or March January cycle months are January, April, July & October February cycle months are February, May, Aug & Nov March cycle months are March, June, Sept and Dec Strike price - typically spaced $2.50, $5 or $10 apart European or American Call or Put (option class) Transaction Costs in Option Trading Floor Trading and Clearing Fees Commissions Bid-Ask Spread Other Transaction Costs The Regulation of Options Markets Federal regulation Industry regulation Over-the-counter market regulation The issue of which agency has regulatory responsibility has occasionally arisen. Appendix 2.A: Margin Requirements Definitions Margin Initial margin Maintenance margin Margin Requirements on Stock Transactions Margin Requirements on Option Purchases Margin Requirements on the Uncovered Sale of Options Margin Requirements on Covered Calls Appendix 2.A: Margin Requirements Return on Investment in Stock and Return on Stock Investment with Dividend Initial investment = $100; 2 Scenarios - Down 50% and Up 100% Appendix 2.A: Margin Requirements Return on Portfolio of 2 Assets Portfolio variance is equal to variance of risky asset If trading on margin, money is borrowed by issuing the risk-free asset as a loan. Weights of the portfolio can be greater than the portfolio size reflecting additional leverage Return and variance of the portfolio would increase with increased leverage Appendix 2.A: Margin Requirements Short selling involves borrowing a security and then selling it at a later date A profit is generated as the security's price declines and the return is similar to a loan where the payback is less than the loan amount Potential loss is unlimited Dividends are paid out by the short seller and erodes the potential return over time Appendix 2.A: Margin Requirements Margin trading is defined by two notions Initial Margin - Percentage of investor's portfolio represented by his own equity on the day in which a trade is done. Maintenance Margin - Percentage of investor's portfolio represented by his own equity on the days after the initial trade is executed. If borrow $40 and use $60 of capital to buy $100 stock (30% margin req) If stock falls to $70, the margin is If stock falls to $50, the margin is The amount to post is Appendix 2.A: Margin Requirements For short selling, the amount borrowed reflects the collateral for the investor to repurchase the stock If borrow $100 stock and initial req = 50% If the stock goes to $110, margin is If the stock goes to $130, margin is The amount to post is Appendix 2.B: Taxation of Option Transactions Taxation of Long Call Transactions Taxation of Short Call Transactions Taxation of Long Put Transactions Taxation of Short Put Transactions Taxation of Non-Equity Options Wash and Constructive Sales Chapter 3: Principles of Option Pricing Important Concepts in Chapter 3 Role of arbitrage in pricing options Minimum value, maximum value, value at expiration and lower bound of an option price Effect of exercise price, time to expiration, risk-free rate and volatility on an option price Difference between prices of European and American options Put-call parity Basic Notation and Terminology Symbols S0 (stock price) X (exercise price) T (time to expiration = (days until expiration)/365) r (see below) ST (stock price at expiration) C(S0,T,X), P(S0,T,X) Basic Notation and Terminology (continued) Computation of risk-free rate (r) Date: May 14. Option expiration: May 21 T-bill bid discount = 4.45, ask discount = 4.37 Average T-bill discount = (4.45+4.37)/2 = 4.41 T-bill price = 100 - 4.41(7/360) = 99.91425 T-bill yield = (100/99.91425)(365/7) - 1 = 0.0457 So 4.57 % is risk-free rate for options expiring May 21 Other risk-free rates: 4.56 (June 18), 4.63 (July 16) See Table 3.1 for prices of DCRB options Stocks and Strike Prices For a call, the future payoff is based on the difference between the current stock price and the exercise price The higher the stock price, the greater the potential difference between the stock and exercise prices Similarly, the lower the exercise price, the greater the profit potential For puts, the relationship is the opposite The lower the stock price, the greater the potential difference between strike and stock prices Likewise, higher the strike price, the greater the difference in strike and stock prices Options Pricing Assumptions There are no transactions costs All trading profits (net of trading losses) are subject to the same tax rate Borrowing and lending are possible at the risk-free rate of interest The risk free rate is the risk free rate of interest for an investment maturing at time T (stated as an annual rate) The risk free rate is assumed to be > 0 Put and call prices are options to buy one share of stock Principles of Call Option Pricing Minimum Value of a Call C(S0,T,X) 0 (for any call) For American calls: Ca(S0,T,X) Max(0,S0 - X) Concept of intrinsic value: Max(0,S0 - X) Proof of intrinsic value rule for DCRB calls Concept of time value See Table 3.2 for time values of DCRB calls See Figure 3.1 for minimum values of calls Principles of Call Option Pricing (continued) Maximum Value of a Call C(S0,T,X) S0 See Figure 3.2, which adds this to Figure 3.1 Value of a Call at Expiration C(ST,0,X) = Max(0,ST - X) For American and European options See Figure 3.3 Upper Bounds on Call Option Prices Both American and European call options give the right to buy a share of stock (at the strike price X) The call can never be worth more than the stock itself CA S0 and CE S0 If these relationships weren't true, there would be arbitrage opportunities If the call price were more than the stock price, investors would buy the stock and sell the call - The market prices assets such that this type of arbitrage doesn't happen. Principles of Call Option Pricing (continued) Effect of Time to Expiration Two American calls differing only by time to expiration, T1 and T2 where T1 S0 - X prior to expiration so Ca(S0,T,X) Max(0,S0 - X(1+r)-T) Look at Table 3.6 for lower bounds of DCRB calls If there are no dividends on the stock, an American call will never be exercised early. It will always be better to sell the call in the market. Intuition Principles of Call Option Pricing (continued) Early Exercise of American Calls on Dividend-Paying Stocks If a stock pays a dividend, it is possible that an American call will be exercised as close as possible to the ex-dividend date. (For a currency, the foreign interest can induce early exercise.) Intuition Effect of Interest Rates Effect of Stock Volatility Principles of Put Option Pricing Minimum Value of a Put P(S0,T,X) 0 (for any put) For American puts: Pa(S0,T,X) Max(0,X - S0) Concept of intrinsic value: Max(0,X - S0) Proof of intrinsic value rule for DCRB puts See Figure 3.6 for minimum values of puts Concept of time value See Table 3.7 for time values of DCRB puts Principles of Put Option Pricing (continued) Maximum Value of a Put Pe(S0,T,X) X(1+r)-T Pa(S0,T,X) X Intuition See Figure 3.7, which adds this to Figure 3.6 Value of a Put at Expiration P(ST,0,X) = Max(0,X - ST) Proof/intuition For American and European options See Figure 3.8 Principles of Put Option Pricing (continued) Effect of Time to Expiration Two American puts differing only by time to expiration, T1 and T2 where T1 X T T 0 ST X X X Total Portfolio C Call option Zero-coupon bond Portfolio A X ST Put Option X ST 0 Share ST ST Total X ST Principles of Put-Call Parity Value of Portfolios at Expiration At expiration, if the stock price is less than the strike price, the call option is worth nothing, and the zero coupon bond is now worth X for a total portfolio value of X In portfolio C, the put option is in the money and pays off X-St when the stock price is less than X and the share price is worth St, combined the portfolio is worth X, the same as Portfolio A Put-Call Parity Result Portfolios providing the same payoffs at maturity must be valued the same today. Both portfolios payoff the max of the share price or the strike price at maturity. Put-call parity for European calls is then a portfolio having a call plus a risk free bond worth the present value of the strike price is equal to a portfolio containing a put plus a share of the underlying stock CE + X(1+r)-T = PE + S0 From this put-call parity equation, the value a put can be derived from the value of a call of on the same stock, same strike price and time to maturity Principles of Put-Call Parity Put-Call Parity Form portfolios A and B where the options are European. See Table 3.11. The portfolios have the same outcomes at the options' expiration. Thus, it must be true that S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T This is called put-call parity. It is important to see the alternative ways the equation can be arranged and their interpretations. Principles of Put-Call Parity Put-Call parity for American options can be stated only as inequalities: N Ca (S , T, X) X D j (1 r) ' 0 t j j1 S0 Pa (S'0 , T, X) Ca (S'0 , T, X) X(1 r) T See Table 3.12 for put-call parity for DCRB options See Figure 3.11 for linkages between underlying asset, risk-free bond, call, and put through put-call parity. Principles of Put-Call Parity The Effect of Interest Rates The Effect of Stock Volatility Summary See Table 3.13. Appendix 3: The Dynamics of Option Boundary Conditions: A Learning Exercise Principles of Options Arbitrage If from put-call parity it is determined that the market price of calls or puts is too high based on the market prices of the other instruments, an investor can create synthetic instruments to take advantage of any mispricing Say the a call is overpriced in the market, using the basics of arbitrage, where the overpriced asset should be sold and the underpriced assets purchased, the investor could sell a call, go short or issue a bond, buy a put and the underlying stock and achieve a risk free profit. In this example we have a stock priced at $31 having calls of a given expiration selling at $3 and a risk free rate of 10%. IF the calls and puts have a strike price of $30 and puts are priced at $2.25, is there an opportunity for arbitrage? We start by calculating the value of each side of the equation. The call premium plus the PV of the exercise price is equal to $32.29 but the sum of the put premium and the stock price is equal to $33.25 CE + X(1+r)-T= PE + S0 3 + 30(1+.10)-.25 = 2.25 + 31 $32.29 $33.25 Because the two sides of the parity equation are not equal, there is an arbitrage opportunity Principles of Options Arbitrage We can see that the put plus stock are overpriced relative to the call plus the PV of the strike price The call + PV of X = $32.29 and the put + stock = $33.25 So, the put+stock is overpriced An investor would then go long on the underpriced portfolio, which is the call plus a bond and go short on the put and the underlying stock. When we are looking at the value at time 0, here we are looking at cash flows. We pay the call price and we buy the bond and indicate this by a negative cash flow. Shorting the put and the underlying stock result in cash inflows. The initial gain on a per share basis is 96 cents. Value at Time 0 Payoff Value at Expiration 30 Transaction > 30 Long call -3.00 0 ( - 30) Buy bond -29.29 +30 +30 Short put +2.25 -(30 - ) 0 Short underlying +31.00 - Total +0.96 0 Factor in effect of options: Risk Free Profit = $0.96 (Return to text slide) (Return to text slide 5) (Return to text slide 7) (Return to text slide) (Return to text slide) (Return to text slide) (Return to text slide) (Return to text slide 8) (Return to text slide 9) (Return to text slide) (Return to text slide) (Return to text slide) (Return to text slide) (Return to text slide) (Return to text slide 13) (Return to text slide 15) (Return to text slide) (Return to text slide) (Return to text slide) (Return to text slide 16) (Return to text slide 17) (Return to text slide) (Return to text slide) (Return to text slide) (Return to text slide) (Return to text slide) (Return to text slide) (Return to text slide) UNIT 3: OPTION PRICING MODELS: THE BINOMIAL MODEL 1 IMPORTANT CONCEPTS IN CHAPTER 4 The concept of an option pricing model The one- and two-period binomial option pricing models Explanation of the establishment and maintenance of a risk-free hedge Illustration of how early exercise can be captured The extension of the binomial model to any number of time periods Alternative specifications of the binomial model Ch. 4: 2 IMPORTANT CONCEPTS IN CHAPTER 4 Definition of a model A simplified representation of reality that uses certain inputs to produce an output or result Definition of an option pricing model A mathematical formula that uses the factors that determine an option's price as inputs to produce the theoretical fair value of an option. Ch. 4: 3 ONE-PERIOD BINOMIAL MODEL Conditions and assumptions One period, two outcomes (states) S = current stock price u = 1 + return if stock goes up d = 1 + return if stock goes down r = risk-free rate Value of European call at expiration one period later Cu = Max(0,Su - X) or Cd = Max(0,Sd - X) See Figure 4.1 Ch. 4: 4 (Return to text slide) Ch. 4: 5 ONE-PERIOD BINOMIAL MODEL (CONTINUED) Important point: d X. Ch. 5: 76 A NOBEL FORMULA (CONTINUED) nCharacteristics of the Black-Scholes-Merton Formula (continued) u Interpretation of the Formula (continued) F The first term of the formula is the expected value of the stock price given that it exceeds the exercise price times the probability of the stock price exceeding the exercise price, discounted to the present. F The second term is the expected value of the payment of the exercise price at expiration. Ch. 5: 77 A NOBEL FORMULA (CONTINUED) Characteristics of the Black-Scholes-Merton Formula (continued) The Black-Scholes-Merton Formula and the Lower Bound of a European Call Recall from Chapter 3 that the lower bound would be The Black-Scholes-Merton formula always exceeds this value as seen by letting S0 be very high and then let it approach zero. Max(0, S0 Xe rcT ) Ch. 5: 78 A NOBEL FORMULA (CONTINUED) Characteristics of the Black-Scholes-Merton Formula (continued) The Formula When T = 0 At expiration, the formula must converge to the intrinsic value. It does but requires taking limits since otherwise it would be division by zero. Must consider the separate cases of ST X and ST