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  • 1/97

    Chapter 7. Derivatives markets.

    Manual for SOA Exam FM/CAS Exam 2.Chapter 7. Derivatives markets.

    Section 7.8. Spreads.

    c2009. 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/

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 2/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Spreads

    An option spread (or a vertical spread) is a combination of onlycalls or only puts, in which some options are bought and someothers are sold. By buying/selling several call/puts we can createportfolios useful for many different objectives. A ratio spread is acombination of buying m calls at one strike price and selling n callsat a different strike price.

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 3/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Speculating on the increase of an asset price. Bull spread.

    Definition 1A bull spread consists on buying a K1strike call and selling aK2strike call, both with the same expiration date T and nominalamount, where 0 < K1 < K2.

    A way to speculate on the increase of an asset price is buying theasset. This position needs a lot of investment. Another way tospeculate on the increase of an asset price is to buy a call option.A bull spread allows to speculate on increase of an asset price bymaking a limited investment.

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 4/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    The payoff for buying a K1strike call is max(ST K1, 0).The payoff for selling a K2strike call is max(ST K2, 0).

    6

    -

    max(ST K1, 0)

    STK1 K2

    6

    -@@@@@

    max(ST K2, 0)

    STK1 K2

    Figure 1: Payoff of the calls in a bull spread

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 5/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    The bull spread payoff is

    max(ST K1, 0)max(ST K2, 0)=max(ST K1, 0) + K2 K1 max(ST K1,K2 K1)=max(ST K1, 0) + K2 K1 max(ST K1, 0,K2 K1)=max(ST K1, 0) + K2 K1 max(max(ST K1, 0),K2 K1)=min(max(ST K1, 0),K2 K1)

    =

    0 if ST < K1,ST K1 if K1 ST < K2,K2 K1 if K2 ST .

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 6/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    6

    -

    max(ST K1, 0)max(ST K2, 0)

    STK1 K2

    Figure 2: Payoff of a bull spread

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 7/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    The profit of a bull spread is

    min(max(ST K1, 0),K2 K1) (Call(K1,T ) Call(K2,T ))(1 + i)T

    =

    (Call(K1,T ) Call(K2,T ))(1 + i)T if ST < K1,ST K1 (Call(K1,T ) Call(K2,T ))(1 + i)T if K1 ST < K2,K2 K1 (Call(K1,T ) Call(K2,T ))(1 + i)T if K2 ST .

    Figure 3 shows a graph of the profit of a bull spread. Notice thatthe profit is positive for values of ST large enough.

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 8/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Figure 3: Profit for a bull spread.

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 9/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    In this section, we will use the values of calls/puts in Table 1.

    Table 1: Prices of some calls and some puts.

    K 65 70 75 80 85

    Call(K ,T ) 14.31722 10.75552 7.78971 5.444947 3.680736Put(K ,T ) 1.221977 2.422184 4.218281 6.635423 9.633117

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 10/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Example 1

    (Use Table 1) Ronald buys a $70strike call and sells a $85strikecall for 100 shares of XYZ stock. Both have expiration date oneyear from now. The current price of one share of XYZ stock is$75. The risk free annual effective rate of interest is 5%. Thepremium per share of $70strike and $85strike calls are 10.75552and 3.680736 respectively.

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 11/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Example 1

    (Use Table 1) Ronald buys a $70strike call and sells a $85strikecall for 100 shares of XYZ stock. Both have expiration date oneyear from now. The current price of one share of XYZ stock is$75. The risk free annual effective rate of interest is 5%. Thepremium per share of $70strike and $85strike calls are 10.75552and 3.680736 respectively.

    (i) Find the profit at expiration as a function of the strike price.

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 12/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Example 1

    (Use Table 1) Ronald buys a $70strike call and sells a $85strikecall for 100 shares of XYZ stock. Both have expiration date oneyear from now. The current price of one share of XYZ stock is$75. The risk free annual effective rate of interest is 5%. Thepremium per share of $70strike and $85strike calls are 10.75552and 3.680736 respectively.

    (i) Find the profit at expiration as a function of the strike price.Solution: (i) Ronalds profit is

    (100) (min(max(ST 70, 0), 85 70) (10.75552 3.680736)(1.05))=(100) (min(max(ST 70, 0), 85 70) 7.428523)

    =

    (100)(7.428523) if ST < 70,(100)(ST 70 7.428523) if 70 ST < 85,(100)(85 70 7.428523) if 85 ST .

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 13/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Example 1

    (Use Table 1) Ronald buys a $70strike call and sells a $85strikecall for 100 shares of XYZ stock. Both have expiration date oneyear from now. The current price of one share of XYZ stock is$75. The risk free annual effective rate of interest is 5%. Thepremium per share of $70strike and $85strike calls are 10.75552and 3.680736 respectively.

    (ii) Make a table with Ronalds profit when the spot price at expi-ration is $65, $70, $75, $80, $85, $90.

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 14/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Example 1

    (Use Table 1) Ronald buys a $70strike call and sells a $85strikecall for 100 shares of XYZ stock. Both have expiration date oneyear from now. The current price of one share of XYZ stock is$75. The risk free annual effective rate of interest is 5%. Thepremium per share of $70strike and $85strike calls are 10.75552and 3.680736 respectively.

    (ii) Make a table with Ronalds profit when the spot price at expi-ration is $65, $70, $75, $80, $85, $90.Solution: (ii)

    Spot Price 65 70 75

    Ronalds profit 742.8523 742.8523 242.8523

    Spot Price 80 85 90

    Ronalds profit 257.1477 757.1477 757.1477

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 15/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Example 1

    (Use Table 1) Ronald buys a $70strike call and sells a $85strikecall for 100 shares of XYZ stock. Both have expiration date oneyear from now. The current price of one share of XYZ stock is$75. The risk free annual effective rate of interest is 5%. Thepremium per share of $70strike and $85strike calls are 10.75552and 3.680736 respectively.

    (iii) Draw the graph of Ronalds profit.

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 16/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Example 1

    (Use Table 1) Ronald buys a $70strike call and sells a $85strikecall for 100 shares of XYZ stock. Both have expiration date oneyear from now. The current price of one share of XYZ stock is$75. The risk free annual effective rate of interest is 5%. Thepremium per share of $70strike and $85strike calls are 10.75552and 3.680736 respectively.

    (iii) Draw the graph of Ronalds profit.Solution: (ii) The graph of the profit is Figure 3.

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 17/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Let 0 < K1 < K2. We have that

    [long K1 strike call ] + [short K1 strike put ] [buying asset for K1 at time T ],[long K2 strike call ] + [short K2 strike put ] [buying asset for K2 at time T ].

    Since the two investment strategies differ by a constant, they havethe same profit. Hence, so do the following strategies,

    [long K1 strike call ] + [short K2 strike call ],[long K1 strike put ] + [short K2 strike put ].

    In other words, buying a K1strike call and selling a K2strike callhas the same profit as buying a K1strike put and selling aK2strike put.We can form a bull spread either buying a K1strike call and sellinga K2strike call, or buying a K1strike put and selling a K2strikeput.

    c2009. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.

  • 18/97

    Chapter 7. Derivatives markets. Section 7.8. Spreads.

    Example 2

    The current price of XYZ stock is $75 per share. The effectiveannual interest rate is 5%. Elizabeth, Daniel and Catherine believethat the price of XYZ stock is going to appreciate significantly inthe next year. Each per