Manual for SOA Exam FM/CAS Exam 2.

1 1/51 Chapter 7. Derivatives for SOA Exam FM/CAS Exam 7. Derivatives Put call 2009. Miguel A. Arcones. All rights from: Arcones Manual for the SOA Exam FM/CAS Exam 2,Financial Mathematics. Fall 2009 Edition ,available 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call call parityRecall that the actions and payoffs corresponding to a call/put are:IfST<KIfK<STlong callno actionbuy the stockshort callno actionsell the stocklong putsell the stockno actionshort putbuy the stockno actionIfST<KIfK<STlong call0ST Kshort call0 (ST K)long putK ST0short put (K ST)0c 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7.

2 Derivatives Put call we have aK strike long call and aK strike short put, weare able to buy the asset at timeTforK. Hence,havingboth aK strike long call and aK strike short put isequivalent to have aK strike long forward contract into both aK strike long call and aK strike shortput is called asynthetic long , if we have aK strike short call and aK strikelong put, we are able to sell the asset at both aK strike short call and aK strike longput is equivalent to have a short forward contract into both aK strike short call and aK strike longput is called asynthetic short 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call no arbitrage cost at timeTof buying an asset using a longforward contract isF0,T.

3 The cost at timeTfor buying an assetusing aK strike long call and aK strike short put is(Call(K,T) Put(K,T))erT+ there exists no arbitrage, then:Theorem 1(Put call parity formula)(Call(K,T) Put(K,T))erT+K=F0, we use effective interest, the put call parity formula becomes:(Call(K,T) Put(K,T))(1 +i)T+K=F0, 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call ,F0,T=S0(1 +i)T. This forward price applies to assetswhich have neither cost nor benefit associated with owning the absence of arbitrage, we have the following relation betweencall and put prices:Theorem 2(Put call parity formula) For a stock which does not pay anydividends,(Call(K,T) Put(K,T))erT+K= 2009.

4 Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call the portfolio consisting of buying one share of stock andaK strike put for one share; selling aK strike call for one share;and borrowingS0 Call(K,T) +Put(K,T). At timeT, we havethe following possibilities:1. IfST<K, then the put is exercised and the call is not. Wefinish without stock and with a payoff for the put IfST>K, then the call is exercised and the put is not. Wefinish without stock and with a payoff for the call any case, the payoff of this portfolio isK. Hence,Kshould beequal to the return in an investment ofS0+Put(K,T) Call(K,T) in a zero coupon bond, (S0+Put(K,T) Call(K,T)) 2009.

5 Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call 1 The current value of XYZ stock share. XYZ stockdoes not pay any dividends. The premium of a nine month80 strike call is per share. The premium of a nine month80 strike put is per share. Find the annual effective rateof :The put call parity formula states that(Call(K,T) Put(K,T))(1 +i)T+K=S0(1 +i) ,( )(1 +i)3/4+ 80 = (1 +i) = ( ( ))(1 +i)3/4=( )(1 +i)3/4, andi= 5%.c 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call 1 The current value of XYZ stock share. XYZ stockdoes not pay any dividends. The premium of a nine month80 strike call is per share.

6 The premium of a nine month80 strike put is per share. Find the annual effective rateof :The put call parity formula states that(Call(K,T) Put(K,T))(1 +i)T+K=S0(1 +i) ,( )(1 +i)3/4+ 80 = (1 +i) = ( ( ))(1 +i)3/4=( )(1 +i)3/4, andi= 5%.c 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call 2 The current value of XYZ stock is85per share. XYZ stock doesnot pay any dividends. The premium of a six month K strike callis per share and the premium of a one year K strike putis per share. The annual effective rate of interest Find :The put call parity formula states that(Call(K,T) Put(K,T))(1 +i)T+K=S0(1 +i) , ( )( ) +K= 85( ) (85 + )( ) 2009.

7 Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call 2 The current value of XYZ stock is85per share. XYZ stock doesnot pay any dividends. The premium of a six month K strike callis per share and the premium of a one year K strike putis per share. The annual effective rate of interest Find :The put call parity formula states that(Call(K,T) Put(K,T))(1 +i)T+K=S0(1 +i) , ( )( ) +K= 85( ) (85 + )( ) 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call 3 XYZ stock does not pay any dividends. The price of a one yearforward for one share of XYZ stock is The premium of aone year55 strike put option of XYZ stock is per annual effective rate of interest Calculate the price ofa one year55 strike call option for one share of XYZ :The put call parity formula states that(Call(K,T) Put(K,T))(1 +i)T+K=F0, , (Call(55,1) )( ) + 55 = andCall(55,1) = + ( 55)( ) 1= 2009.

8 Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call 3 XYZ stock does not pay any dividends. The price of a one yearforward for one share of XYZ stock is The premium of aone year55 strike put option of XYZ stock is per annual effective rate of interest Calculate the price ofa one year55 strike call option for one share of XYZ :The put call parity formula states that(Call(K,T) Put(K,T))(1 +i)T+K=F0, , (Call(55,1) )( ) + 55 = andCall(55,1) = + ( 55)( ) 1= 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call prices of put options and call options do not satisfy the put callparity, it is possible to do (S0 Call(K,T) +Put(K,T))erT>K,we can make a profit by buying a call option, selling a putoption and shorting stock.

9 The profit of this strategy is= K+ (S0 Call(K,T) +Put(K,T)) (S0 Call(K,T) +Put(K,T))erT<K,we can do arbitrage by selling a call option, buying a putoption and buying stock. At expiration time, we get rid of thestock by satisfying the options and makeK (S0 Call(K,T) +Put(K,T)) 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call 4 XYZ stock trades at $54 per share. XYZ stock does not pay anydividends. The cost of an European call option with strike price$50 and expiration date in three months is $8 per share. The riskfree annual interest rate continuously compounded is 4%.c 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7.

10 Derivatives Put call 4 XYZ stock trades at $54 per share. XYZ stock does not pay anydividends. The cost of an European call option with strike price$50 and expiration date in three months is $8 per share. The riskfree annual interest rate continuously compounded is 4%.(i) Find the no arbitrage price of a European put option with thesame strike price and expiration 2009. Miguel A. Arcones. All rights for SOA Exam FM/CAS Exam 7. Derivatives Put call 4 XYZ stock trades at $54 per share. XYZ stock does not pay anydividends. The cost of an European call option with strike price$50 and expiration date in three months is $8 per share. The riskfree annual interest rate continuously compounded is 4%.