# Contents. iii. MFE/3F Study Manual 9th edition Copyright 2011 ASM

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1 Contents 1 Put-Call Parity Review of derivative instruments Forwards Call and put options Combinations of options Put-call parity Stock put-call parity Synthetic stocks and Treasuries Synthetic options Exchange options Currency options Exercises Solutions Comparing Options Bounds for Option Prices Early exercise of American options Time to expiry Different strike prices Three inequalities Options in the money Exercises Solutions Binomial Trees Stock, One Period Risk-neutral pricing Replicating portfolio Volatility Exercises Solutions Binomial Trees General Multi-period binomial trees American options Currency options Futures Other assets Exercises Solutions Risk-Neutral Pricing Pricing with True Probabilities Risk-Neutral Pricing and Utility Exercises Solutions iii

2 iv CONTENTS 6 Binomial Trees: Miscellaneous Topics Understanding early exercise of options Lognormality and alternative trees Lognormality Alternative trees Estimating volatility Exercises Solutions Modeling Stock Prices with the Lognormal Distribution The normal and lognormal distributions The normal distribution The lognormal distribution Jensen s inequality The lognormal distribution as a model for stock prices Stocks without dividends Stocks with dividends Confidence intervals Conditional payoffs using the lognormal model Exercises Solutions Fitting Stock Prices to a Lognormal Distribution Estimating lognormal parameters Evaluating the fit graphically Exercises Solutions The Black-Scholes Formula Black-Scholes Formula for common stock options Black-Scholes formula for currency options Black-Scholes formula for options on futures Exercises Solutions The Black-Scholes Formula: Greeks Greeks Delta Gamma Vega Theta Rho Psi Greek measures for portfolios Elasticity and related concepts Elasticity Related concepts Elasticity of a portfolio What will I be tested on? Exercises Solutions

3 CONTENTS v 11 The Black-Scholes Formula: Applications and Volatility Profit diagrams before maturity Call options and bull spreads Calendar spreads Volatility Implied volatility Historical volatility Exercises Solutions Delta Hedging Overnight profit on a delta-hedged portfolio The delta-gamma-theta approximation Greeks for binomial trees Rehedging Hedging multiple Greeks What will I be tested on? Exercises Solutions Asian, Barrier, and Compound Options Asian options Barrier options Maxima and minima Compound options Compound option parity American options on stocks with one discrete dividend Bermudan options Exercises Solutions Gap, Exchange, and Other Options All-or-nothing options Gap options Definition of gap options Pricing gap options using Black-Scholes Delta hedging gap options Exchange options Other exotic options Chooser options Forward start options Exercises Solutions Monte Carlo Valuation Introduction Generating lognormal random numbers Simulating derivative instruments Control variate method Other variance reduction techniques Exercises

4 vi CONTENTS Solutions Brownian Motion Brownian motion Arithmetic Brownian motion Geometric Brownian motion Covariance Exercises Solutions Differentials Differentiating The language of Brownian motion Exercises Solutions Itô s Lemma 399 Exercises Solutions The Black-Scholes Equation 411 Exercises Solutions Sharpe Ratio Calculating and using the Sharpe ratio Risk Free Portfolios Exercises Solutions Risk-Neutral Pricing and Proportional Portfolios Risk-neutral pricing Proportional portfolio Exercises Solutions S a Valuing a forward on S a The Itô process for S a Derivation of F0,T P ) using true pricing Examples of the formula for forwards on S a Exercises Solutions Stochastic Integration Integration Quadratic variation Differential equations Exercises Solutions

5 CONTENTS vii 24 Binomial Tree Models for Interest Rates Binomial Trees The Black-Derman-Toy model Construction of a Black-Derman-Toy binomial tree Pricing forwards and caps using a BDT tree Exercises Solutions The Black Formula for Bond Options Pricing Options with the Black Formula Pricing caps with the Black formula Exercises Solutions Equilibrium Interest Rate Models: Vasicek and Cox-Ingersoll-Ross The impossible model Equilibrium models The Rendleman-Bartter model The Vasicek model The Cox-Ingersoll-Ross model Characteristics of Vasicek and Cox-Ingersoll-Ross models Risk-neutral pricing A(t, T )e B(t,T )r (t ) form B(t, T ) (Vasicek model) Sharpe ratio questions P/ t Yield on infinitely-lived bonds Delta hedging Delta-gamma approximation Exercises Solutions Practice Exams Practice Exam Practice Exam Practice Exam Practice Exam Practice Exam Practice Exam Practice Exam Practice Exam Practice Exam 9 641

6 viii CONTENTS 10 Practice Exam Practice Exam Appendices 673 A Solutions for the Practice Exams 675 Solutions for Practice Exam Solutions for Practice Exam Solutions for Practice Exam Solutions for Practice Exam Solutions for Practice Exam Solutions for Practice Exam Solutions for Practice Exam Solutions for Practice Exam Solutions for Practice Exam Solutions for Practice Exam Solutions for Practice Exam B Solutions to Old Exams 791 B.1 Solutions to SOA Exam MFE, Spring B.2 Solutions to CAS Exam 3, Spring B.3 Solutions to CAS Exam 3, Fall B.4 Solutions to Exam MFE/3F, Spring B.5 Solutions to Sample Questions C Lessons Corresponding to Questions on Released and Practice Exams 843 D Standard Normal Distribution Function Table 845

8 2 1. PUT-CALL PARITY Let us determine F t,t. As we do so, we shall introduce assumptions and terminology used throughout the course. One assumption we make throughout the course is that there is a risk-free interest paying investment, an investment which will never default. If you invest K in this investment, you are sure to receive K (1 + i ) at the end of a year, where i is some interest rate. We will always express interest as a continuously compounded rate instead of as an annual effective rate, unless we say otherwise. We will use the letter r for this rate. In Financial Mathematics, the letter δ is used for this concept, but we will be using δ for a different (although related) concept. As you learned in Financial Mathematics, 1 + i = e r. So if the risk-free effective annual return was 5%, we would say that r = ln 1.05 = rather than In real life, Treasury securities play the role of a risk-free interest paying investment. Here are the assumptions we make: 1. It is possible to borrow or lend any amount of money at the risk-free rate. 2. There are no transaction charges or taxes. 3. Arbitrage is impossible. An Arbitrage is a set of transactions which when combined have no cost, no possibility of loss and at least some possibility (if not certainty) of profit. In order to make arbitrage impossible, if there are two sets of transactions leading to the same end result, they must both have the same price, since otherwise you could enter into one set of transactions as a seller and the other set as a buyer and be assured a profit with no possibility of loss. Forwards on stock We will consider 3 possibilities for the stock: 1. The stock pays no dividends. 2. The stock pays discrete dividends. 3. The stock pays continuous dividends. Forwards on nondividend paying stock We begin with a forward on a nondividend paying stock. The forward agreement is entered at time 0 and provides for transferring the stock at time T. Let S t be the price of the stock at time t. There are two ways you can own the stock at time T : Two Methods for Owning Non-Dividend Paying Stock at Time T Method #1: Buy stock at time 0 and hold it to time T Method #2: Buy forward on stock at time 0 and hold it to time T Payment at time 0 S 0 0 Payment at time T 0 F 0,T By the principle of no arbitrage, the two ways must have the same cost. The cost at time 0 of the first way is S 0, the price of the stock at time 0. The cost of the second way is a payment of F 0,T at time T. Since you can invest at the risk-free rate, every dollar invested at time 0 becomes e r T dollars at time T. You can pay F 0,T at time T by investing F 0,T e r T at time 0. We thus have We have priced this forward agreement. F 0,T e r T = S 0 F 0,T = S 0 e r T

9 1.1. REVIEW OF DERIVATIVE INSTRUMENTS 3 Forwards on a stock with discrete dividends Now let s price a forward on a dividend paying stock. This means it pays known cash amounts at known times. There are the same two ways to own a stock at time T : Two Methods for Owning Dividend Paying Stock at Time T Method #1: Buy stock at time 0 and hold it to time T Method #2: Buy forward on stock at time 0 and hold it to time T Payment at time 0 S 0 0 Payment at time T 0 F 0,T However, if you use the first way, you will also have the stock s dividends accumulated with interest, which you won t have with the second way. Therefore, the price of the second way as of time T must be cheaper than the price of the first way as of time T by the accumulated value of the dividends. Equating the 2 costs, and letting CumValue(Div) be the accumulated value at time T of dividends from time 0 to time T, we have F 0,T = S 0 e r T CumValue(Div) (1.1) Forwards on a stock index with continuous dividends We will now consider an asset that pays dividends at a continuously compounded rate which get reinvested in the asset. In other words, rather than being paid as cash dividends at certain times, the dividends get reinvested in the asset continuously, so that the investor ends up with additional shares of the asset rather than cash dividends. The continuously compounded dividend rate will be denoted δ. This could be a model for a stock index. A stock index consists of many stocks paying dividends at various times. As a simplification, we assume these dividends are uniform. We will often model a single stock, not just a stock index, as if it had continuous reinvested dividends, using δ instead of explicit dividends. Let s price a forward on a stock index paying dividends at a rate of δ. If you buy the stock index at time 0 and it pays continuous dividends at the rate δ, you will have e δt shares of the stock index at time T. To replicate this with forwards, since the forward pays no dividends, you would need to enter a forward agreement to purchase e δt shares of the index. So here are two ways to own e δt shares of the stock index at time T : Two Methods for Owning e δt Shares of Continuous Dividend Paying Stock Index at Time T Method #1: Buy stock index at time 0 and hold it to time T Method #2: Buy e δt forwards on stock index at time 0 and hold it to time T Payment at time 0 S 0 0 Payment at time T 0 e δt F 0,T Thus to equate the two ways, you should buy e δt units of the forward at time 0. At time T, the accumulated cost of buying the stock index at time 0 is S 0 e r T, while the cost of e δt units of the forward agreement is F 0,T e δt, so we have F 0,T e δt = S 0 e r T F 0,T = S 0 e (r δ)t (1.2) We see here that δ and r work in opposite directions. We will find in general that δ tends to work like a negative interest rate.

10 4 1. PUT-CALL PARITY? Quiz For a stock, you are given: (i) It pays quarterly dividends of (ii) It has just paid a dividend. (iii) Its price is 50. (iv) The continuously compounded risk-free interest rate is 5%. Calculate the forward price for an agreement to deliver 100 shares of the stock six months from now. A forward on a commodity works much like a forward on a nondividend paying stock. If the commodity produces income (for example if you can lease it out), the income plays the role of a dividend; if the commodity requires expenses (for example, storage expenses), these expenses work like a negative dividend. Forwards on currency For forwards on currency, it is assumed that each currency has its own risk-free interest rate. The risk-free interest rate for the foreign currency plays the role of a continuously compounded dividend on a stock. For example, suppose a forward agreement at time 0 provides for the delivery of euros for dollars at time T. Let r \$ be the risk-free rate in dollars and r the risk-free rate in euros. Let x 0 be the \$/ rate at time 0. The two ways to have a euro at time T are: 1. Buy e r T euros at time 0 and let them accumulate to time T. 2. Buy a forward for 1 euro at time T, with a payment of F 0,T at time T. The first way costs x 0 e r T dollars at time 0. The second way can be funded in dollars at time 0 by putting aside e r\$t F 0,T dollars at time 0. Thus e r\$t F 0,T = x 0 e r T F 0,T = e (r\$ r )T x 0? Quiz 1-2 A yen-denominated forward agreement provides for the delivery of \$100 at the end of 3 months. The continuously compounded risk-free rate for dollars is 5%, and the continuously compounded risk-free rate for yen is 2%. The current exchange rate is 110 /\$. Calculate the forward price in yen for this agreement. A summary of the forward formulas is in Table 1.1. This table makes the easy generalization from agreements entered at time 0 to agreements entered at time t. It is also possible to have forwards on bonds, and bond interest would play the same role as stock dividends in such a forward. More typical are forward rate agreements, which guarantee an interest rate Call and put options A call option permits, but does not require, the purchaser to pay a specific amount K at time T in return for an asset. In a call option, there are two cash flows: a payment of the value of the call option is made at time 0, and a possible payment of K in return for an asset is made at time T. The payment at time 0, which is the premium of the call, is denoted by C (K, T ). 2 At time T, rather than actually paying K and receiving the asset, a monetary 1 Solutions to quizzes are at the end of each lesson, after the solutions to the exercises. 2 I am following McDonald textbook notation, which is not consistent. At this point, C and P have two arguments, but later on, they will have three arguments: S, K, and T, where S is the stock price. When we get to Black-Scholes (Lesson 9), it will have six arguments. Sometimes the arguments will be omitted, and sometimes C will be used generically for both call and put options.

11 1.1. REVIEW OF DERIVATIVE INSTRUMENTS 5 Table 1.1: Forward prices at time t for settlement at time T Underlying asset Forward price r (T t ) Non-dividend paying stock S t e Dividend paying stock S t e r (T t ) CumValue(Divs) (r δ)(t t ) Stock index S t e Currency, denominated in currency d for delivery of currency f (rd r f )(T t ) x t e Net payoff 20 Net payoff C (40, T ) 10 0 P(40, T ) (a) Payoff net of initial investment on call option S T (b) Payoff net of initial investment on put option S T Figure 1.2: Payoff net of initial investment (ignoring interest) on option with a strike price of 40 settlement of the difference is usually made. If S T is the value of the asset at time T, the payoff is max(0,s T K ). The one who sells the call option is called the writer of the call option. A put option is the counterpart to a call option. It permits, but does not require, the purchaser to sell an asset at time T for a price of K. Thus there is a payment at time 0 of the premium of the put option, and a possible receipt of K in return for the asset at time T. The payment at time 0, which is the premium of the put, is denoted P(K, T ). The monetary settlement at time T if the asset is then worth S T is max(0, K S T ). K is called the strike price. Figure 1.2 shows the net payoff of a call or a put, assuming the strike price is 40. The cost of the option is added to the payoff at the time T. Interest is ignored, although we won t ignore it when we price the option. The options we have described are European options. European options can only be exercised at time T, not before. In contrast, American options allow the option to be exercised at any time up to time T Combinations of options Various strategies involving buying or selling more than one option are possible. These are discussed in chapter 3 of the McDonald textbook, and you learn about them in Course FM/2. It is probably not necessary to know about them for this exam, since exam questions about them will define them for you. Nevertheless, here is a short summary of them. When we buy X, we are said to be long X, and when we sell X, we are said to be short X. Long and short may be used as adjectives or verbs.

15 1.1. REVIEW OF DERIVATIVE INSTRUMENTS 9 Net profit 20 Net profit S T S T (a) Net profit on collar (b) Net profit on collared stock Figure 1.5: Net profits (price accumulated at interest plus final settlement) of a collar and of a collared stock using European options, assuming strike prices of 40 and 60 and current stock price of 50 Net profit Net profit S T S T (a) Net profit on straddle (b) Net profit on strangle Figure 1.6: Net profits (price accumulated at interest plus final settlement) of a straddle and of a strangle using European options, assuming strike prices of 50 for straddle and 40 and 60 for strangle Straddles: buying two options of different kinds In a straddle, you buy a call and a put, both of them at-the-money (K = S 0 ). If they both have the same strike price, the payoff is S T S 0, growing with the absolute change in stock price, so this is a bet on volatility. You have to pay for both options. To lower the initial cost, you can buy a put with strike price K 1 and buy a call with strike price K 2 > K 1 ; then this strategy is called a strangle. Diagrams of the net profits of a straddle and a strangle are shown in Figure 1.6. We re done with review.

17 1.2. PUT-CALL PARITY Stock put-call parity For a nondividend paying stock, the forward price is F 0,T = S 0 e r T. Equation (1.3) becomes C (K, T ) P(K, T ) = S 0 K e r T The right hand side is the present value of the asset minus the present value of the strike. EXAMPLE 1A A nondividend paying stock has a price of 40. A European call option allows buying the stock for 45 at the end of 9 months. The continuously compounded risk-free rate is 5%. The premium of the call option is Determine the premium of a European put option allowing selling the stock for 45 at the end of 9 months. ANSWER: Don t forget that 5% is a continuously compounded rate. C (K, T ) P(K, T ) = S 0 K e r T 2.84 P(K, T ) = 40 45e (0.05)(0.75) 2.84 P(K, T ) = 40 45( ) = P(K, T ) = = A convenient concept is the prepaid forward. This is the same as a regular forward, except that the payment is made at the time the agreement is entered, time t = 0, rather than at time T. We use the notation Ft P,T for the value at time t of a prepaid forward settling at time T. Then F0,T P = e r T F 0,T. More generally, if the forward maturing at time T is prepaid at time t, Ft P,T = e r (T t ) F t,t, so we can translate the formulas in Table 1.1 into the ones in Table 1.2. In this table PV t,t is the present value at time t of a payment at time T. Using prepaid Table 1.2: Prepaid Forward Prices at time t for settlement at time T Underlying asset Forward price Prepaid forward price Non-dividend paying stock S t e r (T t ) S t Dividend paying stock S t e r (T t ) CumValue(Divs) S t PV t,t (Divs) Stock index S t e (r δ)(t t ) δ(t t ) S t e Currency, denominated in currency d for delivery of currency f x t e (rd r f )(T t ) r f (T t ) x t e forwards, the put-call parity formula becomes C (K, T ) P(K, T ) = F P 0,T K e r T (1.4) Using this, let s discuss a dividend paying stock. If a stock pays discrete dividends, the formula becomes C (K, T ) P(K, T ) = S 0 PV 0,T (Divs) K e r T (1.5) EXAMPLE 1B A stock s price is 45. The stock will pay a dividend of 1 after 2 months. A European put option with a strike of 42 and an expiry date of 3 months has a premium of The continuously compounded risk-free rate is 5%. Determine the premium of a European call option on the stock with the same strike and expiry.

18 12 1. PUT-CALL PARITY ANSWER: Using equation (1.5), C (K, T ) P(K, T ) = S 0 PV 0,T (Divs) K e r T PV 0,T (Divs), the present value of dividends, is the present value of the dividend of 1 discounted 2 months at 5%, or (1)e 0.05/6. C (42, 0.25) 2.71 = 45 (1)e 0.05/6 42e 0.05(0.25) = ( ) = C (42, 0.25) = = ? Quiz 1-3 A stock s price is 50. The stock will pay a dividend of 2 after 4 months. A European call option with a strike of 50 and an expiry date of 6 months has a premium of The continuously compounded risk-free rate is 4%. Determine the premium of a European put option on the stock with the same strike and expiry. Now let s consider a stock with continuous dividends at rate δ. Using prepaid forwards, put-call parity becomes C (K, T ) P(K, T ) = S 0 e δt K e r T (1.6) EXAMPLE 1C You are given: (i) A stock s price is 40. (ii) The continuously compounded risk-free rate is 8%. (iii) The stock s continuous dividend rate is 2%. A European 1-year call option with a strike of 50 costs Determine the premium for a European 1-year put option with a strike of 50. ANSWER: Using equation (1.6), C (K, T ) P(K, T ) = S 0 e δt K e r T 2.34 P(K, T ) = 40e e 0.08 = 40( ) 50( ) = P(K, T ) = = ? Quiz 1-4 You are given: (i) A stock s price is 57. (ii) The continuously compounded risk-free rate is 5%. (iii) The stock s continuous dividend rate is 3%. A European 3-month put option with a strike of 55 costs Determine the premium of a European 3-month call option with a strike of 55.

19 1.2. PUT-CALL PARITY Synthetic stocks and Treasuries Since the put-call parity equation includes terms for stock (S 0 ) and cash (K ), we can create a synthetic stock with an appropriate combination of options and lending. With continuous dividends, the formula is C (K, T ) P(K, T ) = S 0 e δt K e r T S 0 = C (K, T ) P(K, T ) + K e r T e δt (1.7) For example, suppose the risk-free rate is 5%. We want to create an investment equivalent to a stock with continuous dividend rate of 2%. We can use any strike price and any expiry; let s say 40 and 1 year. We have S 0 = C (40, 1) P(40, 1) + 40e 0.05 e 0.02 So we buy e 0.02 = call options and sell put options, and buy a Treasury for At the end of a year, the Treasury will be worth e 0.05 = An option will be exercised, so we will pay 40( ) = and get shares of the stock, which is equivalent to buying 1 share of the stock originally and reinvesting the dividends. If dividends are discrete, then they are assumed to be fixed in advance, and the formula becomes C (K, T ) P(K, T ) = S 0 PV(dividends) K e r T S 0 = C (K, T ) P(K, T ) + PV(dividends) + K e r T } {{ } amount to lend For example, suppose the risk-free rate is 5%, the stock is 40, and the period is 1 year. The dividends are 0.5 apiece at the end of 3 months and at the end of 9 months. Then their present value is 0.5e 0.05(0.25) + 0.5e 0.05(0.75) = To create a synthetic stock, we buy a call, sell a put, and lend e 0.05 = At the end of the year, we ll have 40 plus the accumulated value of the dividends. One of the options will be exercised, so the 40 will be exchanged for one share of the stock. To create a synthetic Treasury 3, we rearrange the equation as follows: C (K, T ) P(K, T ) = S 0 e δt K e r T (1.8) K e r T = S 0 e δt C (K, T ) + P(K, T ) (1.9) We buy e δt shares of the stock and a put option and sell a call option. Using K = 40, r = 0.05, δ = 0.02, and 1 year to maturity again, the total cost of this is K e r T = 40e 0.05 = At the end of the year, we sell the stock for 40 (since one option will be exercised). This is equivalent to investing in a one-year Treasury bill with maturity value 40. If dividends are discrete, then they are assumed to be fixed in advance and can be combined with the strike price as follows: K e r T + PV(dividends) = S 0 C (K, T ) + P(K, T ) (1.10) The maturity value of this Treasury is K + CumValue(dividends). For example, suppose the risk-free rate is 5%, the stock is 40, and the period is 1 year. The dividends are 0.5 apiece at the end of 3 months and at the end of 9 months. Then their present value, as calculated above, is 0.5e 0.05(0.25) + 0.5e 0.05(0.75) = and their accumulated value at the end of the year is e 0.05 = Thus if you buy a stock and a put and sell a call, both options with strike prices 40, the investment will be K e = and the maturity value will be = Creating a synthetic Treasury is called a conversion. Selling a synthetic Treasury by shorting the stock, buying a call, and selling a put, is called a reverse conversion.

20 14 1. PUT-CALL PARITY? Quiz 1-5 You wish to create a synthetic investment using options on a stock. The continuously compounded risk-free interest rate is 4%. The stock price is 43. You will use 6-month options with a strike of 45. The stock pays continuous dividends at a rate of 1%. The synthetic investment should duplicate 100 shares of the stock. Determine the amount you should invest in Treasuries Synthetic options If an option is mispriced based on put-call parity, you may want to create an arbitrage. Suppose the price of a European call based on put-call parity is C, but the price it is actually selling at is C < C. (From a different perspective, this may indicate the put is mispriced, but let s assume the price of the put is correct based on some model.) You would then buy the underpriced call option and sell a synthesized call option. Since C (S, K, t ) = Se δt K e r t + P(S, K, t ) you would sell the right hand side of this equation. You d sell e δt shares of the underlying stock, sell a European put option with strike price K and expiry t, and buy a risk-free zero-coupon bond with a price of K e r t, or in other words lend K e r t at the risk-free rate. These transactions would give you C (S, K, t ). You d pay C for the option you bought, and keep the difference Exchange options The options we ve discussed so far involve receiving/giving a stock in return for cash. We can generalize to an option to receive a stock in return for a different stock. Let S t be the value of the underlying asset, the one for which the option is written, and Q t be the price of the strike asset, the one which is paid. Forwards will now have a parameter for the asset; F t,t (Q) will mean a forward agreement to purchase asset Q (actually, the asset with price Q t ) at time T. A superscript P will indicate a prepaid forward, as before. Calls and puts will have an extra parameter too: C (S t,q t, T t ) means a call option written at time t which lets the purchaser elect to receive S T in return for Q T at time T ; in other words, to receive max(0,s T Q T ). P(S t,q t, T t ) means a put option written at time t which lets the purchaser elect to give S T in return for Q T ; in other words, to receive max(0,q T S T ). The put-call parity equation is then C (S t,q t, T t ) P(S t,q t, T t ) = F P t,t (S) F P t,t (Q) (1.11) Exchange options are sometimes given to corporate executives. They are given a call option on the company s stock against an index. If the company s stock performs better than the index, they get compensated. Notice how the definitions of calls and puts are mirror images. A call on one share of Ford with one share of General Motors as the strike asset is the same as a put on one share of General Motors with one share of Ford as the strike asset. In other words P(S t,q t, T t ) = C (Q t,s t, T t ) and we could ve written the above equation with just calls: C (S t,q t, T t ) C (Q t,s t, T t ) = F P t,t (S) F P t,t (Q)

### Contents. iii. MFE/3F Study Manual 9 th edition 10 th printing Copyright 2015 ASM

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