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Chapter 7. Derivatives markets.

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

Extract from:Arcones Manual for the SOA Exam FM/CAS Exam 2,

Financial Mathematics. Fall 2009 Edition,available at http://www.actexmadriver.com/

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Chapter 7. Derivatives markets. Section 7.8. 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.

• 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.

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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

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Chapter 7. Derivatives markets. Section 7.8. Spreads.

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 .

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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

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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.

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Chapter 7. Derivatives markets. Section 7.8. Spreads.

Figure 3: Profit for a bull spread.

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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

• 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.

• 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.

• 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 .

• 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.

• 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

• 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.

• 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.

• 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.