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Chapter 7. Derivatives markets

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Chapter 7. Derivatives markets.Section 7.4. Call options.1/112Chapter 7. Derivatives markets.Manual for SOA Exam FM/CAS Exam 2.Chapter 7. Derivatives markets.Section 7.4. Call options.c2009. Miguel A. Arcones. All rights reserved.Extract from:”Arcones’ Manual for the SOA Exam FM/CAS Exam 2,Financial Mathematics. Fall 2009 Edition”,available at http://www.actexmadriver.com/c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.2/112Chapter 7. Derivatives markets. Section 7.4. Call options.Minimums and maximumsDefinition 1Given two real numbers a and b,(i) min(a, b) denotes the (minimum) smallest of the two numbers.(ii) max(a, b) denotes the (maximum) biggest of the two numbers.Example 1min(10, 5) = 5, max(10, 5) = 10, min(−1, 5) = −1,max(−1, 5) = 5, min(−2, −100) = −100, max(−2, −100) = −2.c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.3/112Chapter 7. Derivatives markets. Section 7.4. Call options.Definition 2Given real numbers a1, . . . , an,(i) min(a1, . . . , an) denotes the (minimum) smallest of thesenumbers.(ii) max(a1, . . . , an) denotes the (maximum) biggest of thesenumbers.Example 2min(−1, 5, 3, −6) = −6, max(−1, 5, 3, −6) = 5,min(−2, −100, −50) = −100 and max(−2, −100, −50) = −2.c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.4/112Chapter 7. Derivatives markets. Section 7.4. Call options.Theorem 1For each a , b, c ∈ R and each λ ≥ 0,Imin(a, b) = min(b, a).Imax(a, b) = max(b, a).Imin(min(a, b), c) = min(a, min(b, c)) = min(a, b, c).Imax(max(a, b), c) = max(a, max(b, c)) = max(a, b, c).Imin(a + c, b + c) = min(a, b) + c.Imax(a + c, b + c) = max(a, b) + c.Imin(λa, λb) = λ min(a, b).Imax(λa, λb) = λ max(a, b).Imin(−a, −b) = −max(a, b).Imax(−a, −b) = −min(a, b).c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.5/112Chapter 7. Derivatives markets. Section 7.4. Call options.Definition 3Given a real number a, |a| = a, if a ≥ 0; and |a| = −a, if a ≤ 0Example 3|23| = 23, | − 4| = 4.Theorem 2For each a , b ∈ R, min(a, b) + max(a, b) = a + b.Proof.min(a, b) and max(a, b) are a and b in some order. Hence,min(a, b) + max(a, b) = a + b.Theorem 3For each a ∈ R, |a| = max(a, 0) − min(a, 0).Proof.If a ≥ 0, then max(a, 0) = a, min(a, 0) = 0, andmax(a, 0) − min(a, 0) = a = |a|. If a ≤ 0, then max(a, 0) = 0,min(a, 0) = a, and max(a, 0) − min(a, 0) = −a = |a|.c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.6/112Chapter 7. Derivatives markets. Section 7.4. Call options.Call optionsDefinition 4A call option is a financial contract which gives the owner theright, but not the obligation, to buy a specified amount of a givenasset at a specified price during a specified period of time.The call option owner exercises the option by buying the asset atthe specified call price from the call writer. A call option isexecuted only if the call owner decides to do so. A call optionowner executes a call option only when it benefits him, i.e. whenthe specified call price is smaller than the current (market value)spot price. Since the owner of a call option can make money if theoption is exercised, call options are sold. The owner of the calloption must pay to its counterpart for holding a call option. Theprice of a call option is called its premium.c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.7/112Chapter 7. Derivatives markets. Section 7.4. Call options.Call optionsDefinition 4A call option is a financial contract which gives the owner theright, but not the obligation, to buy a specified amount of a givenasset at a specified price during a specified period of time.The call option owner exercises the option by buying the asset atthe specified call price from the call writer. A call option isexecuted only if the call owner decides to do so. A call optionowner executes a call option only when it benefits him, i.e. whenthe specified call price is smaller than the current (market value)spot price. Since the owner of a call option can make money if theoption is exercised, call options are sold. The owner of the calloption must pay to its counterpart for holding a call option. Theprice of a call option is called its premium.c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.8/112Chapter 7. Derivatives markets. Section 7.4. Call options.IThe (owner) buyer of a call option is called the option callholder. The holder of a call option is said to have a long callposition.IThe seller of a call option is called the option call writer.The writer of a call is said to have a short call position.IAssets used in call options are in commodities, currencyexchange, stock shares and stock indices.IA call option needs to specify the type and quality of theunderlying.IThe asset used in the call option is called the underlier orunderlying asset.IThe amount of the underlying asset to which the call optionapplies is called the notional amount.IThe specified price of an asset in a call option is called thestrike price, or exercise price.IA forward contract forces the buyer and seller to execute thesale. A call option is executed only if the call holder decidesto do so.c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.9/112Chapter 7. Derivatives markets. Section 7.4. Call options.IFor an European option, the exercise of the option mustoccur at a certain time (the expiration date).IFor an American option, the exercise of the option mustoccur any time by the expiration date.IFor a Bermudan option, the buyer can exercise the calloption during specified periods.Unless say otherwise, we will assume that an option is an Europeanoption. European options are simpler and easie r to study.c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.10/112Chapter 7. Derivatives markets. Section 7.4. Call options.Example 4Suppose that an investor buys a call option of 100 shares of XYZstock with a strike price of $76 per share. The exercise date is oneyear from now.(i) If the spot price at expiration is $70 per share, the call optionholder does not exercise the option. The option is worthless. Thecall option holder can buy stock in the market for a price smallerthan the call option price.(ii) If the (the market price) spot price at expiration is $80


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