Finance 350: Problem Set 8 Alternative Solutions


 Milo Gordon
 2 years ago
 Views:
Transcription
1 Finance 35: Problem Set 8 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. All payoff diagrams include the initial cost of the security, however they do not take into account the time value of this cost by computing the future value. Each payoff diagram is drawn with the future price of the security on the horizontal axis and the future payoff to the position on the vertical axes. Strictly speaking, when computing the future payoff to any position, we should calculate the future value of the money paid (or received) today when factoring this into the net position. This is not done here because the dollar impact is small and it adds little to the problem. I. Formulas This section contains the formulas you might need for this homework set: 1. Payoff to a long position in a call option: P ayoff = max {S T K, } (1) where S T is the price of the underlying security at expiration of the contract (time T ) and K is the strike price. 2. Payoff to a short position in a call option: P ayoff = max {S T K, } = min {K S T, } (2) 3. Payoff to a long position in a put option: P ayoff = max {K S T, } (3) 1
2 4. Payoff to a short position in a put option: 5. PutCall Parity (PCP): P ayoff = max {K S T, } = min {S T K, } (4) C = P + Se dt Ke rt, (5) where C is the price of a call option, P is the price of an otherwise identical put option, S is the price of the underlying security, d is the dividend yield, T is the time to maturity (in years), K is the strike price and r is the riskfree rate. II. Problems 1. 1.a The payoff at maturity (net of the initial cost of the option) to the call buyer is given by max[, S T K] F V (C), (6) where S T is the price of the underlying stock at maturity, K is the strike price and F V (C) is the future value of the call premium (price), C. The net payoff at maturity (net of the initial cost of the option) to the buyer of the put option at is, mathematically, max[, K S T ] F V (P ), (7) where F V (P ) is the future value of the put premium, P. Graphically, these payoffs are depicted in Figures 1 and 2 below. 1.b The payoff at maturity (net of the initial cost of the option) to the call seller is given by (max[, S T K] F V (C)) = min(, K S T ) + F V (C), (8) where S T is the price of the underlying stock at maturity, K is the strike price and F V (C) is the future value of the call premium (price), C. The payoff at maturity (net of the initial cost of the option) to the put buyer is given by (max[, K S T ] F V (P )) = min(, S T K) + F V (P ), (9) where F V (P ) is the future value of the put premium, P. Graphically, these payoffs are depicted in Figures 3 and 4 below. 2
3 Figure 1: Net Payoff to a Long Call Position with $5 Strike Price sition with $5 Strike Figure 2: Net Payoff to a Long Put Po Price c All of these graphs were done in Excel. The names of each position are provided upon request. You need not know them. 3
4 Figure 3: Net Payoff to a Short Position Figure 4: Net Payoff to a Short Position in a Call Option with $5 Strike Price in a Put Option with $5 Strike Price Figure 1: (Protective Put): Buy one share and a put with strike price = $ Buy Share Buy Put(5) Combined 4
5 Figure 2: (Bear Cylinder): Buy a put with strike = $5 and write (i.e. sell) a call with strike = $ Buy Put(5) Write Call(7) Combined Figure 3: (Strangle I): Buy a call with strike = $7 and buy a put with strike = $ Buy Call(7) Buy Put(5) Combined 5
6 Figure 4: (Strangle II): Buy a call with strike = $5 and buy a put with strike = $ Buy Call(5) Buy Put(7) Combined
7 Figure 5: Short one share and buy a call with strike = $ Short Share Buy Call(7) Combined
8 Figure 6: (Butterfly Spread I): Buy call with strike = $5, buy call with strike = $7 and sell 2 calls with strike =$ Buy Call(5) Buy Call(7) Combined Sell 2xCall(6) 8
9 Figure 7: (Butterfly Spread II): Buy put with strike = $5, buy put with strike = $7 and sell 2 puts with strike =$ Buy Put(5) Buy Put(7) Combined Sell 2xPut(6) 1.d We will compute the price of a call implied by putcall parity (equation (5)), using the price of a put. We could have just as easily computed the price of a put implied by putcall parity, using the price of the call. It makes no difference. To avoid a redundant calculation, the time to maturity in years for all options is 71/365 =.195. Table (1) below computes the relevant comparisons. The first column is simply the different strike prices. The second column repeats the market value of each call from the table on the problem set. The third column reports the price implied by putcall parity. That is, the price of the call option using equation (5). The fourth column reports the difference between the market price and the implied price. All units are in dollars, so the dollar sign is excluded. Clearly, there are a number of discrepancies between the market values and the putcall parity implied value. However, these discrepancies are likely due to market frictions such as: 1. The closing times for the options on the exchange might be different from those of the stock 9
10 Table 1: Call Prices Implied from PutCall Parity vs. Actual Market Prices Market Call Implied PCP Price Strike (K) Price (P + S Ke rt ) Difference PutCall parity is for European options. These are American options. 3. The difference for the options at 5 are less than the transaction costs for the arbitrage. 4. Putcall parity relation above applies to nondividend paying options only. (we assumed d=) Thus, if 3Com was paying a dividend, we would have to account for this by using a nonzero value for d (which we would have to estimate). 1.e The price of portfolio iii from part 1.c is: Call(7) + P ut(5) = = (1) The price of portfolio iv from part 1.c is: Call(5) + P ut(7) = = (11) Clearly, for the set of prices examined here, portfolio iv is more expensive. However, we must show that this is always the case, at any point in time and for any maturity. Intuitively, this result must be true for the following reason. Portfolio iv s assets (the 5 call and 7 put) are more likely to finish inthemoney than portfolio iii s (7 call and 5 put). A call is more likely to finish inthemoney, the lower the strike price, and a put is more likely to finish inthemoney the higher the strike price. So you are more likely to make money with portfolio iv than portfolio iii. Hence, the higher cost. We will now prove this result more formally. We want to show that the cost of portfolio iv, Call(5) + Put(7), is always greater than the cost of 1
11 portfolio iii, Call(7) + Put(5). Mathematically, we need to show: which is equivalent to showing Call(5) + P ut(7) > Call(7) + P ut(5), Call(5) + P ut(7) Call(7) P ut(5) >. (12) Putcall parity allows us to write the value of the call options in terms of put options, the stock and cash: Call(5) = P ut(5) + Se dt 5e rt (13) Call(7) = P ut(7) + Se dt 7e rt, (14) We now plug in the results from equations (13) and (14), into equation (12). This yields, [P ut(5)+se dt 5e rt ]+P ut(7) [P ut(7)+se dt 7e rt ] P ut(5) > Simplification yields, 7e rt 5e rt >, which is true for all values of r and T. The result is proved. 1.f Since portfolios vi & vii have the same payoff, they should have the same price. However, we see that the cost of portfolio vi is: Call(5) + Call(7) 2 Call(6) = 2 (if you want to think of this in terms of cash flows, just reverse the signs on each position). The cost of portfolio vii is: P ut(5) + P ut(7) 2 P ut(6) = We can show that these two portfolios should be equal by using putcall parity in an identical manner as used above in part 1.e. 11
12 1.g First, take a look at portfolio i. This portfolio is hedged in the sense that the downside risk associated with owning the share is hedged by buying the put. As the share price falls, we lose money on the share we own. However, we make money from the put option as the price falls and this offsets the loss from owning the share. Now examine portfolio v. When we shortsell a share, we lose money as the price increases above the price at which we purchased the share. By buying the call, however, we make money as the share price increases above the strike. This offsets losses in our shortsell position. 1.h If we believe that the volatility of the underlying asset price is going to be high, what we are saying is that we expect large swings in the stock price. That is, the stock price may move up or down, but when it moves, it will move a lot! In this scenario, we want to be holding portfolio iii since this pays off when the stock price moves far from its current position of $5.75. If we believe volatility will be falling, we are saying that we do not expect the stock price to move far from its current position. In this case, we want to be holding portfolio vi, since it pays off when the price is near its current position. 1.i Exactly the same logic applies. Greater volatility yields a desire to hold portfolio iv. Less volatility yields a desire to hold portfolio vii. 1.j Putcall parity begins with equation (5): C = P + S Ke rt, (15) ignoring dividends. We can rearrange this equation to obtain C P = S Ke rt. Thinking in terms of cash flows, the lefthand side of the equation represents a cash inflow of C (sell a call) and a cash outflow of P (buy a put). The cash flows on the righthand side represent a cash inflow of S (buy the stock) and 12
13 the cash outflow of Ke rt (lend cash). Plugging in values for S, K, r and T yields, C P = e.5 71/365 = This is a cash outflow, or cost, of $8.669, which differs from the market value, equal to = Again, rearranging the original putcall parity equation (5) yields, P C = Ke rt S (ignoring the dividend yield). The left hand side corresponds to cash inflow of P (sell a put) and a cash outflow of C (buy a call). The right hand side corresponds to a cash inflow of Ke rt (borrowing cash) and a cash outflow of S (buy the asset). Plugging in the numbers yields: P C = 65e.5 71/ = This is a cash inflow (or a negative cost). Note, this too differs from the market price of the portfolio: = 15.5 (likely due to market frictions, such as those mentioned in part 1.d. 1.k See Excel Solutions 1.l See Excel Solutions 2. 2.a Our exposure (amount at risk) is 5 million SFR in 4 days. The American firms concern is that the exchange rate ($/SFR) will fall, meaning that each SFR can buy fewer $US. To hedge we will want to sell the 5 million SFR forward (i.e. in the future) by selling futures contracts on the SFR. Since our exposure is 5 million SFR and each futures contract is for.125 million SFR, we need 5 /.125 = 4 contracts to hedge all of our exposure. 13
14 2.b Scenario 1 assumes that the spot exchange rate, 4 days hence, is.65 $/SFR. This implies that we can sell the 5 million SFR s we receive from the sale of goods for 65 cents per SFR, netting us 3.25 million $US. However, our cash balance from the short futures position is ( ) * 5 =.265 million $US, offsetting any loss from the transaction in the spot market. Our aggregate gain is = million $US. Scenario 2 assumes that the exchange rate will be.7 $/SFR in March. The company receives.7 * 5 = 3.5 million $US from sale of goods. Our futures position pays the company ( ) * 5 =.15 million $US, for an aggregate position of = million $US. Scenario 3 assumes the exchange rate will be.75 $/SFR, implying that the company receives.75 * 5 = 3.75 million $US from the sale of goods (after exchanging the SFR s at the spot rate). The futures position suffers a loss of ( ) * 5 = million $US. The aggregate position is: = million $US. Note, in all scenarios the aggregate position is unaffected by the spot exchange rate in March. This is the perfect hedge. 2.c Since we want to sell Swiss Francs in the future, we want to buy put options, which gives us the option to sell the underlying asset in the future. Since each contract is for.625 million SFR s, we need to sell 5/.625 = 8 contracts. Table (2) through (4) below looks at the effect of entering into 3 types of put contracts differentiated by their strike price, when the spot exchange rate in March is.65,.7 and.75 $/SFR. Table 2: Scenario 1: Spot Exchange Rate in March =.65 $/SFR Buy 8 contracts: Exchange rate =.65 Future Spot exchange rate Put Option Strike price Premium per SFR (Cost per unit) Total premium (cost per 8 contracts) 53,5 98, 221, Value when exercised (K > S T ) 225, 3, 45, Value from sale of 5m SFR 3,25, 3,25, 3,25, Net payoff (disregard discounting) 3,421,5 3,452, 3,479, 14
15 Table 3: Scenario 2: Spot Exchange Rate in March =.7 $/SFR Buy 8 contracts: Exchange rate =.7 Future Spot exchange rate Put Option Strike price Premium per SFR (Cost per unit) Total premium (cost per 8 contracts) 53,5 98, 221, Value when exercised (K > S T )  5, 2, Value from sale of 5m SFR 3,5, 3,5, 3,5, Net payoff (disregard discounting) 3,446,5 3,452, 3,479, 15
16 Table 4: Scenario 3: Spot Exchange Rate in March =.75 $/SFR Buy 8 contracts: Exchange rate =.75 Future Spot exchange rate Put Option Strike price Premium per SFR (Cost per unit) Total premium (cost per 8 contracts) 53,5 98, 221, Value when exercised (K > S T ) Value from sale of 5m SFR 3,75, 3,75, 3,75, Net payoff (disregard discounting) 3,696,5 3,652, 3,529, The conclusions to be made from the tables are as follows. Options do not offer a perfect hedge in the sense that we do not know for certain what our future payoff will be. They do offer a floor, or minimum amount, along with unlimited upside. Of course, the price of maintaining this upside is the cost of the option. Their is also a tradeoff between the higher strike price (more likely to finish inthemoney and the higher premium). The hedge transaction itself has a zero NPV. 2.d The futures hedge locks in an exchange rate. We know for certain what our future payoff will be, regardless of what happens to the underlying spot exchange rate. The option hedge is more uncertain. While we are guaranteed a minimum payoff, we do need a bit of upside movement in the exchange rate to recoup the initial cost of the option contracts (there is no cost to entering futures contracts, beyond transaction costs). Beyond recovering the initial cost of the option contracts, the upside gains are unlimited for the option hedge. Assuming there is no private information, there is little reason to hedge using the options when the futures contracts accomplish this goal at a cheaper price and with no uncertainty. 2.e The CEO is incorrect. Assuming the company has another buyer offering an amount in $US that the firm would otherwise get from the sale to the Swiss firm at.695 $/SFR, the firm should exit the sale and take the cash from their futures position. Agreeing to the sale at that point would force the firm to take a loss on the sale of goods. Something they need not do under the exit clause. 16
17 2.f The company breaks even at an exchange rate of.695 $/SFR. That is, they are indifferent between selling at.695 and not selling at all. So, the exit clause acts in the exact same manner as the put option. Therefore, it should have the same value as the put, or $53,5 (see above). This is an example of a real option. 3. The parameters of the problem are: 1. Price of call option (C) = $ Price of put option (P) = $.4 3. Strike (K) = $45 4. Current Stock Price (S) = $ Riskfree interest rate (r) = 5% 6. Time to expiration, in years (T) = 1/12 =.833 Putcall parity is violated since C = P + S Ke rt = e = 2.19, which is less than the market price of $3.1. We can use the putcall parity violation to tell us what arbitrage strategy to undertake. The intuition is as follows. We know the market price of the call is too high so that C > P +S Ke rt. Now let s rearrange the equation so that we get an expression that is greater than zero. By doing this, and interpreting the dollar amounts as cash flows, we are saying that the cash flows today will be greater than zero, implying an arbitrage since we know the cash flows in the future will be the same. A little algebra reveals C P S + Ke rt >. 17
18 Table 5: Arbitrage Table At Maturity Position Today S t < K S t > K Sell Call 3.1 (S T 45) Buy Put S T Buy Stock S T S T Borrow Cash Net Cash Flows.91 Think of this equation in terms of the cash flows at time (i.e. today). To receive a positive cash flow of C today, we must sell the call option. But, this makes sense since the call option price in the market is too high. The negative cash flows of P and S correspond to buying the put and buying the stock. The positive cash flow of Ke rt reflects borrowed cash. The arbitrage table is presented in (5). These solutions are produced by Michael R. Roberts. Thanks go to Jen Rother for her excellent assistance, and to an anonymous TA. Any remaining errors are mine. 18
CHAPTER 20 Understanding Options
CHAPTER 20 Understanding Options Answers to Practice Questions 1. a. The put places a floor on value of investment, i.e., less risky than buying stock. The risk reduction comes at the cost of the option
More informationFinance 350: Problem Set 6 Alternative Solutions
Finance 350: Problem Set 6 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas
More informationOptions Markets: Introduction
Options Markets: Introduction Chapter 20 Option Contracts call option = contract that gives the holder the right to purchase an asset at a specified price, on or before a certain date put option = contract
More informationExample 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).
Chapter 4 PutCall Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.
More information2. Exercising the option  buying or selling asset by using option. 3. Strike (or exercise) price  price at which asset may be bought or sold
Chapter 21 : Options1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. BlackScholes
More informationLecture 5: Put  Call Parity
Lecture 5: Put  Call Parity Reading: J.C.Hull, Chapter 9 Reminder: basic assumptions 1. There are no arbitrage opportunities, i.e. no party can get a riskless profit. 2. Borrowing and lending are possible
More informationFactors Affecting Option Prices
Factors Affecting Option Prices 1. The current stock price S 0. 2. The option strike price K. 3. The time to expiration T. 4. The volatility of the stock price σ. 5. The riskfree interest rate r. 6. The
More informationTrading Strategies Involving Options. Chapter 11
Trading Strategies Involving Options Chapter 11 1 Strategies to be Considered A riskfree bond and an option to create a principalprotected note A stock and an option Two or more options of the same type
More informationProperties of Stock Options. Chapter 10
Properties of Stock Options Chapter 10 1 Notation c : European call option price C : American Call option price p : European put option price P : American Put option price S 0 : Stock price today K : Strike
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationSession X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London
Session X: Options: Hedging, Insurance and Trading Strategies Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Option
More information9 Basics of options, including trading strategies
ECG590I Asset Pricing. Lecture 9: Basics of options, including trading strategies 1 9 Basics of options, including trading strategies Option: The option of buying (call) or selling (put) an asset. European
More informationCHAPTER 20: OPTIONS MARKETS: INTRODUCTION
CHAPTER 20: OPTIONS MARKETS: INTRODUCTION 1. Cost Profit Call option, X = 95 12.20 10 2.20 Put option, X = 95 1.65 0 1.65 Call option, X = 105 4.70 0 4.70 Put option, X = 105 4.40 0 4.40 Call option, X
More informationEconomics 1723: Capital Markets Lecture 20
Economics 1723: Capital Markets Lecture 20 John Y. Campbell Ec1723 November 14, 2013 John Y. Campbell (Ec1723) Lecture 20 November 14, 2013 1 / 43 Key questions What is a CDS? What information do we get
More informationFinance 436 Futures and Options Review Notes for Final Exam. Chapter 9
Finance 436 Futures and Options Review Notes for Final Exam Chapter 9 1. Options: call options vs. put options, American options vs. European options 2. Characteristics: option premium, option type, underlying
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the
More information11 Option. Payoffs and Option Strategies. Answers to Questions and Problems
11 Option Payoffs and Option Strategies Answers to Questions and Problems 1. Consider a call option with an exercise price of $80 and a cost of $5. Graph the profits and losses at expiration for various
More informationChapter 11 Properties of Stock Options. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull
Chapter 11 Properties of Stock Options 1 Notation c: European call option price p: European put option price S 0 : Stock price today K: Strike price T: Life of option σ: Volatility of stock price C: American
More informationChapter 5 Financial Forwards and Futures
Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment
More informationCHAPTER 22: FUTURES MARKETS
CHAPTER 22: FUTURES MARKETS 1. a. The closing price for the spot index was 1329.78. The dollar value of stocks is thus $250 1329.78 = $332,445. The closing futures price for the March contract was 1364.00,
More informationChapter 3: Commodity Forwards and Futures
Chapter 3: Commodity Forwards and Futures In the previous chapter we study financial forward and futures contracts and we concluded that are all alike. Each commodity forward, however, has some unique
More informationCHAPTER 22: FUTURES MARKETS
CHAPTER 22: FUTURES MARKETS PROBLEM SETS 1. There is little hedging or speculative demand for cement futures, since cement prices are fairly stable and predictable. The trading activity necessary to support
More informationFigure S9.1 Profit from long position in Problem 9.9
Problem 9.9 Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances
More informationChapter 21: Options and Corporate Finance
Chapter 21: Options and Corporate Finance 21.1 a. An option is a contract which gives its owner the right to buy or sell an underlying asset at a fixed price on or before a given date. b. Exercise is the
More informationChapter 2 An Introduction to Forwards and Options
Chapter 2 An Introduction to Forwards and Options Question 2.1. The payoff diagram of the stock is just a graph of the stock price as a function of the stock price: In order to obtain the profit diagram
More informationChapter 15. Option on Stock Indices and Currencies. Promble Promble Promble Promble Finger 15.1 Tree for Promble 15.
Chapter 15 Option on Stock Indices and Currencies Promble 15.1. When the index goes down to 7, the value of the portfolio can be expected to be 1*(7/8)=$8.75 million.(this assumes that the dividend yield
More informationOptions Pricing. This is sometimes referred to as the intrinsic value of the option.
Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the PutCall Parity Relationship. I. Preliminary Material Recall the payoff
More informationCHAPTER 21: OPTION VALUATION
CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given
More informationCHAPTER 20: OPTIONS MARKETS: INTRODUCTION
CHAPTER 20: OPTIONS MARKETS: INTRODUCTION PROBLEM SETS 1. Options provide numerous opportunities to modify the risk profile of a portfolio. The simplest example of an option strategy that increases risk
More informationOther observable variables as arguments besides S.
Valuation of options before expiration Consider European options with time t until expiration. Value now of receiving c T at expiration? (Value now of receiving p T at expiration?) Have candidate model
More informationWeek 1: Futures, Forwards and Options derivative three Hedge: Speculation: Futures Contract: buy or sell
Week 1: Futures, Forwards and Options  A derivative is a financial instrument which has a value which is determined by the price of something else (or an underlying instrument) E.g. energy like coal/electricity
More informationAmerican Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options
American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus
More informationFUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing
FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing I. PutCall Parity II. OnePeriod Binomial Option Pricing III. Adding Periods to the Binomial Model IV. BlackScholes
More informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 081 Institutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationPutCall Parity. chris bemis
PutCall Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008. Options
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options the socalled plain vanilla options. We consider the payoffs to these
More informationOne Period Binomial Model
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing
More informationEXP 481  Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0
EXP 481  Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Blackcholes model PutCall Parity Implied Volatility Options: Definitions A call option gives the buyer the
More informationChapter 2 Introduction to Option Management
Chapter 2 Introduction to Option Management The prize must be worth the toil when one stakes one s life on fortune s dice. Dolon to Hector, Euripides (Rhesus, 182) In this chapter we discuss basic concepts
More informationProtective Put Strategy Profits
Chapter Part Options and Corporate Finance: Basic Concepts Combinations of Options Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options
More informationOPTIONS MARKETS AND VALUATIONS (CHAPTERS 16 & 17)
OPTIONS MARKETS AND VALUATIONS (CHAPTERS 16 & 17) WHAT ARE OPTIONS? Derivative securities whose values are derived from the values of the underlying securities. Stock options quotations from WSJ. A call
More informationArbitrage spreads. Arbitrage spreads refer to standard option strategies like vanilla spreads to
Arbitrage spreads Arbitrage spreads refer to standard option strategies like vanilla spreads to lock up some arbitrage in case of mispricing of options. Although arbitrage used to exist in the early days
More informationLecture 12. Options Strategies
Lecture 12. Options Strategies Introduction to Options Strategies Options, Futures, Derivatives 10/15/07 back to start 1 Solutions Problem 6:23: Assume that a bank can borrow or lend money at the same
More informationSession IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics
Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock
More informationFIN 411  Investments Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices
FIN 411  Investments Option Pricing imple arbitrage relations s to call options Blackcholes model PutCall Parity Implied Volatility Options: Definitions A call option gives the buyer the right, but
More informationOption Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration
CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value  profit that could be made if the option was immediately exercised Call: stock price  exercise price Put:
More informationHedging with Futures and Options: Supplementary Material. Global Financial Management
Hedging with Futures and Options: Supplementary Material Global Financial Management Fuqua School of Business Duke University 1 Hedging Stock Market Risk: S&P500 Futures Contract A futures contract on
More informationLecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the
More informationFundamentals of Futures and Options (a summary)
Fundamentals of Futures and Options (a summary) Roger G. Clarke, Harindra de Silva, CFA, and Steven Thorley, CFA Published 2013 by the Research Foundation of CFA Institute Summary prepared by Roger G.
More informationIntroduction to Options
Introduction to Options By: Peter Findley and Sreesha Vaman Investment Analysis Group What Is An Option? One contract is the right to buy or sell 100 shares The price of the option depends on the price
More informationCHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT PROBLEM SETS 1. In formulating a hedge position, a stock s beta and a bond s duration are used similarly to determine the expected percentage gain or loss
More informationLecture 3: Put Options and DistributionFree Results
OPTIONS and FUTURES Lecture 3: Put Options and DistributionFree Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distributionfree results? option
More information1. Assume that a (European) call option exists on this stock having on exercise price of $155.
MØA 155 PROBLEM SET: Binomial Option Pricing Exercise 1. Call option [4] A stock s current price is $16, and there are two possible prices that may occur next period: $15 or $175. The interest rate on
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationLOCKING IN TREASURY RATES WITH TREASURY LOCKS
LOCKING IN TREASURY RATES WITH TREASURY LOCKS Interestrate sensitive financial decisions often involve a waiting period before they can be implemented. This delay exposes institutions to the risk that
More informationK 1 < K 2 = P (K 1 ) P (K 2 ) (6) This holds for both American and European Options.
Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated.
More informationChapter 20 Understanding Options
Chapter 20 Understanding Options Multiple Choice Questions 1. Firms regularly use the following to reduce risk: (I) Currency options (II) Interestrate options (III) Commodity options D) I, II, and III
More informationOPTIONS ON SHORTTERM INTEREST RATE FUTURES*
OPTIONS ON SHORTTERM INTEREST RATE FUTURES* Anatoli Kuprianov Options are contracts that give their owners the right, but not the obligation, to buy or sell a specified item at a set price on or before
More informationCHAPTER 13 Suggested answers and solutions to the end of chapter questions and problems
CHAPTER 13 Suggested answers and solutions to the end of chapter questions and problems Questions 1. How would you define transaction exposure? How is it different from economic exposure? Transaction exposure
More informationOption Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9)
Option Properties Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 9) Liuren Wu (Baruch) Option Properties Options Markets 1 / 17 Notation c: European call option price.
More informationChapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.
Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of riskadjusted discount rate. Part D Introduction to derivatives. Forwards
More informationName Graph Description Payoff Profit Comments. commodity at some point in the future at a prespecified. commodity at some point
Name Graph Description Payoff Profit Comments Long Commitment to purchase commodity at some point in the future at a prespecified price S T  F S T F No premium Asset price contingency: Always Maximum
More informationFX, Derivatives and DCM workshop I. Introduction to Options
Introduction to Options What is a Currency Option Contract? A financial agreement giving the buyer the right (but not the obligation) to buy/sell a specified amount of currency at a specified rate on a
More information1 The BlackScholes Formula
1 The BlackScholes Formula In 1973 Fischer Black and Myron Scholes published a formula  the BlackScholes formula  for computing the theoretical price of a European call option on a stock. Their paper,
More informationADVANCED COTTON FUTURES AND OPTIONS STRATEGIES
ADVANCED COTTON FUTURES AND OPTIONS STRATEGIES Blake K. Bennett Extension Economist/Management Texas Cooperative Extension, The Texas A&M University System INTRODUCTION Cotton producers have used futures
More informationCHAPTER 3. Arbitrage and Financial Decision Making. Chapter Synopsis
CHAPTER 3 Arbitrage and Financial Decision Making Chapter Synopsis 3.1 Valuing Decisions When considering an investment opportunity, a financial manager must systematically compare the costs and benefits
More informationCHAPTER 8 SUGGESTED ANSWERS TO CHAPTER 8 QUESTIONS
INSTRUCTOR S MANUAL: MULTINATIONAL FINANCIAL MANAGEMENT, 9 TH ED. CHAPTER 8 SUGGESTED ANSWERS TO CHAPTER 8 QUESTIONS. On April, the spot price of the British pound was $.86 and the price of the June futures
More informationOverview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies
Options and Derivatives Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Overview Option basics and option strategies Noarbitrage bounds on option prices Binomial option pricing BlackScholesMerton
More informationCHAPTER 20. Financial Options. Chapter Synopsis
CHAPTER 20 Financial Options Chapter Synopsis 20.1 Option Basics A financial option gives its owner the right, but not the obligation, to buy or sell a financial asset at a fixed price on or until a specified
More informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More informationJournal of Business & Economics Research Volume 2, Number 8
LongTerm Capital Gains Tax Strategies: Correlated Protective Put Strategy John R. Aulerich, (EMail: aulericj@uww.edu), University of WisconsinWhitewater James Molloy, University of WisconsinWhitewater
More informationUnderstanding Volatility and Making It Work for You. The Options Industry Council
Understanding Volatility and Making It Work for You December 16, 2013 Joe Burgoyne, OIC www.optionseducation.org 2 The Options Industry Council Options involve risks and are not suitable for everyone.
More informationLecture 11: RiskNeutral Valuation Steven Skiena. skiena
Lecture 11: RiskNeutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena RiskNeutral Probabilities We can
More informationModelFree Boundaries of Option Time Value and Early Exercise Premium
ModelFree Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 331246552 Phone: 3052841885 Fax: 3052844800
More informationOptions. Moty Katzman. September 19, 2014
Options Moty Katzman September 19, 2014 What are options? Options are contracts conferring certain rights regarding the buying or selling of assets. A European call option gives the owner the right to
More informationCHAPTER 21: OPTION VALUATION
CHAPTER 21: OPTION VALUATION PROBLEM SETS 1. The value of a put option also increases with the volatility of the stock. We see this from the putcall parity theorem as follows: P = C S + PV(X) + PV(Dividends)
More informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE QUESTIONS Financial Economics
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Financial Economics June 2014 changes Questions 130 are from the prior version of this document. They have been edited to conform
More informationConvenient Conventions
C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. YuhDauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff
More informationOptions: Valuation and (No) Arbitrage
Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial
More informationOptions (1) Class 19 Financial Management, 15.414
Options (1) Class 19 Financial Management, 15.414 Today Options Risk management: Why, how, and what? Option payoffs Reading Brealey and Myers, Chapter 2, 21 Sally Jameson 2 Types of questions Your company,
More informationChapter 15 OPTIONS ON MONEY MARKET FUTURES
Page 218 The information in this chapter was last updated in 1993. Since the money market evolves very rapidly, recent developments may have superseded some of the content of this chapter. Chapter 15 OPTIONS
More informationCaput Derivatives: October 30, 2003
Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor
More informationContents. iii. MFE/3F Study Manual 9th edition Copyright 2011 ASM
Contents 1 PutCall Parity 1 1.1 Review of derivative instruments................................................ 1 1.1.1 Forwards............................................................ 1 1.1.2 Call
More informationHedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging
Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in
More informationWHAT ARE OPTIONS OPTIONS TRADING
OPTIONS TRADING WHAT ARE OPTIONS Options are openly traded contracts that give the buyer a right to a futures position at a specific price within a specified time period Designed as more of a protective
More informationCall and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options
Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder
More informationExpected payoff = 1 2 0 + 1 20 = 10.
Chapter 2 Options 1 European Call Options To consolidate our concept on European call options, let us consider how one can calculate the price of an option under very simple assumptions. Recall that the
More informationSection 4, PutCall Parity
HCMSAF1033B, Financial Economics, 6/30/10, age 49 Section 4, utcall arity The value of an otherwise similar call and put option on the same asset are related. Adam and Eve Example: XYZ stock pays no
More informationb. June expiration: 9523 = 95 + 23/32 % = 95.71875% or.9571875.9571875 X $100,000 = $95,718.75.
ANSWERS FOR FINANCIAL RISK MANAGEMENT A. 24 Value of Tbond Futures Contracts a. March expiration: The settle price is stated as a percentage of the face value of the bond with the final "27" being read
More information2. How is a fund manager motivated to behave with this type of renumeration package?
MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] In the discussions of some of the models in this course, we relied on the following type of argument: If two investment strategies have the same payoff
More informationFin 3710 Investment Analysis Professor Rui Yao CHAPTER 14: OPTIONS MARKETS
HW 6 Fin 3710 Investment Analysis Professor Rui Yao CHAPTER 14: OPTIONS MARKETS 4. Cost Payoff Profit Call option, X = 85 3.82 5.00 +1.18 Put option, X = 85 0.15 0.000.15 Call option, X = 90 0.40 0.000.40
More informationFINANCIAL ECONOMICS OPTION PRICING
OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.
More informationECMC49F Options Practice Questions Suggested Solution Date: Nov 14, 2005
ECMC49F Options Practice Questions Suggested Solution Date: Nov 14, 2005 Options: General [1] Define the following terms associated with options: a. Option An option is a contract which gives the holder
More informationUnderlying (S) The asset, which the option buyer has the right to buy or sell. Notation: S or S t = S(t)
INTRODUCTION TO OPTIONS Readings: Hull, Chapters 8, 9, and 10 Part I. Options Basics Options Lexicon Options Payoffs (Payoff diagrams) Calls and Puts as two halves of a forward contract: the PutCallForward
More informationIntroduction, Forwards and Futures
Introduction, Forwards and Futures Liuren Wu Zicklin School of Business, Baruch College Fall, 2007 (Hull chapters: 1,2,3,5) Liuren Wu Introduction, Forwards & Futures Option Pricing, Fall, 2007 1 / 35
More informationFutures Price d,f $ 0.65 = (1.05) (1.04)
24 e. Currency Futures In a currency futures contract, you enter into a contract to buy a foreign currency at a price fixed today. To see how spot and futures currency prices are related, note that holding
More informationCHAPTER 8: TRADING STRATEGES INVOLVING OPTIONS
CHAPTER 8: TRADING STRATEGES INVOLVING OPTIONS Unless otherwise stated the options we consider are all European. Toward the end of this chapter, we will argue that if European options were available with
More informationOption Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values
Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X
More informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationManual for SOA Exam FM/CAS Exam 2.
Manual for SOA Exam FM/CAS Exam 2. Chapter 7. Derivatives markets. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall
More information