Stock Profit Patterns

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1 Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce 50 costs. Moreover, an actve future contract market for Farsta Shppng s avalable Buy Stock Sol Short Stock 1

2 Futures an Forwars Proft Patterns Sell Forwar contract(kort Buy Forwar contract (lang

3 Smple Optons Proft Patterns Buy call opton Sell call opton Buy put opton 3

4 Formulas Buy Sol Short Buy Sell Stock Stock call opton call opton B1-$B$3 $B$3-B1 (MAA(0,(B1-$D$3-$D$4 (-MAA(0,(B1-$D$3$D$4 Buy Sell Sell Buy put opton put opton Forwar contraforwar contr (MAA(0,($D$3-B1-$D$5 (-MAA(0,($D$3-B1$D$5 $D$6-B1 -$D$6B1 4

5 Propostons (arbtrage Restrctons for Optons No. 1 Assume that the present value of a rsk less securty pays off at tme T: e rt Proposton 1: Lower boun: C max [,0] rt S e 0 Example: Assume 80, S 0 83, T0.5 an r 10% C max [ ] e, Proof: (we bul on the conseraton of the CASH FLOW from a partcular strategy at Tme 0 at Tme T Buy one share of the stock Exercse f proftable Borrow the PV of the Opton Repay borrowe Funs Wrte a call on the stock 5

6 Propostons (arbtrage Restrctons for Optons No. 1 Toay Acton Buy the Stock Borrow PV( Wrte a call on the stock Total CF Cash Flow -S 0 e -rt C -S 0 e -rt C at Tme T S T < S T > S T S T (S T - S T -<0 0 A contract that has only non-postve pay-offs n the future must have a postve ntal cash flow: C rt S0 e > 0 or C > S0 The value of a call can n no case be less than zero: e rt C max [,0] rt S e 0 6

7 Propostons (arbtrage Restrctons for Optons No. Proposton : Early Exercse: Conser exercse t < T; then S t > 0. However, Proposton 1 says that the value of a call s at least r( T t Snce, S t e S t e r( T t > S t The opton holer s better off sellng the opton n the market than exercsng t. 7

8 Propostons (arbtrage Restrctons for Optons No. 3 P Strategy: max Acton Cash Flow Short the Stock S 0 Len PV( Toay [ 0, e S ] rt -e -rt Wrte a put on the stock P Total CF S 0 -e -rt P 0 at Tme T S T < S T > -S T -S T -(- S T S T <0 A contract that has only non-postve pay-offs n the future must have a postve ntal cash flow: P rt rt S0 e > 0 or P > e S 0 The value of a put can n no case be less than zero: P max [ 0, e S ] rt 0 8

9 Propostons (arbtrage Restrctons for Optons No. 4 C e rt P S 0 Toay Acton Buy the Call on the stock Buy a bon PV( Wrte a put on the stock Short on share of the stock Total CF Cash Flow -C -e -rt P S 0 -C -e -rt P S 0 at Tme T S T < S T > 0 S T - -(-S T 0 -S T -S T 0 0 A contract that has only zero pay-offs n the future must have a zero ntal cash flow: rt C e P S0 0 rt C e P S0 9

10 Propostons (Call Opton Prce Convexty No. 5 Exercse Prce 1 3 Opton Prce C 1 C C 3 1 Toay 3 then C C 1 3 < Proof: at Tme T Acton Cash Flow S T < 1 1 <S T < <S T < 3 S T > 3 Buy a Call on the stock -C 1 0 S T - S 1 T - 1 S T - 1 Buy a Call on the stock -C S T - 3 Wrte calls on the stock C 0 0 -(S T - -(S T - Total CF C -C 1 -C 3 0 S T - 1 >0-1 -S T 0 3 -S T >0 C1 C3 C C1 C3 < 0 C < C 10

11 Propostons (Put Opton Prce Convexty No then P < P 1 P 3 11

12 Propostons (Dven Payng Stock No. 7 We assume that stock pays ven (D at t < T. C max Toay Tme t Acton Cash Flow [,0] rt rt S De e 0 at Tme T S T < S T > Buy the stock -S 0 D S T S T Borrow PV (D De -rt -D Borrow PV ( e -rt - - Wrte Call C 0 -(S T - Total CF De -rt e -rt C-S 0 0 S T -<0 0 D e e C S > 0 0 rt rt Ths result C>0 > C max [,0] rt rt S De e 0 1

13 The Bnomal Opton Prcng Moel Fn the combnaton of the bons an stocks that exactly replcates (tracks the call optons payoff. 13

14 The Bnomal Opton Prcng Moel Two equatons an two unknowns: 55 A 1.06 B A 1.06 B 0 The secon equaton B A: The frst equaton: 55A * A A B B

15 15 Styrng og Fonsmeglng The Bnomal Opton Prcng Moel: State Prces Thnk about the market etermnng a prce q u for NOK 1 n the up state of the worl an a prce q for NOK 1 n the own state of the worl. Hence, for the stock 1 (1 (1 (1 (1 q u q S S q u S q u u An for the bon 1 (1 1 ( q q u Solves to gve: ( (1 ( (1 u u q u q u

16 16 Styrng og Fonsmeglng The Bnomal Opton Prcng Moel: State Prces Hence The put-call party gves: [ ] [ ],0 (1,0 1 ( S max q u S max q C u S 0 e C P T Consequently: [ ] [ ],0 (1 (1,0 (1 (1 n n u n n u u S max q q n P u S max q q n C

17 Prcng Amercan Optons Usng the Bnomal Prcng Moel The Bnomal Prcng Moel can prce Amercan Optons. Example: S50, 50, an 6%. q u 10% an q 3% Stock Prce Amercan Put payoffs Date 0 Date 1 Date Date 0 Date 1 Date {Max( ,0} 55?? ?? 0.00 {Max( ,0} 48.5?? {Max( ,0} 17

18 Prcng Amercan Optons Usng the Bnomal Prcng Moel At ate 1, the holer of an Amercan put can choose whether to hol the put or to exercse t. Hence, Put value Value of put f exercse at ate max qu Put payoff at uu q Put payoff at u state u max( Su max qu Put payoff at uu q Put payoff at u A smlar functon hols for the put value n state at ate. At ate 0, a smlar value functon recurs. 18

19 Prcng Amercan Optons Usng the Bnomal Prcng Moel The value tree for the Amercan put: Max(max( S (1 u,0, q putpayoff q putpayoff u uu u Max(max( S (1 u,0, q putpayoff q putpayoff u u Max(max( S,0, q putvalue q putvalue u u 19

20 The Lognormal Dstrbuton How o stock prces look lke? 0

21 Stock Prces: 5 Propertes (Black & Scholes 1. Wggly Lnes. Lnes that are contnuous, wth no obvous jumps 3. Lnes that are always postve an never cross zero, no matter how low they get 4. That at a gven pont n tme, the average over all plausble lnes s greater than the ntal prce of the stock. The farther out we go, the hgher ths average becomes. 5. That the stanar evaton over all plausble lnes s greater the farther out we go. 1

22 Lognormal Prce Dstrbutons an Geometrc Dffusons Let us enote by S t the prce at tme t of a share of stock. The lognormal strbuton assumes that the natural logarthm og one plus the return from holng a share of stock between tme t an t t s normally strbute wth mean µ an stanar evaton σ. Let us enote the uncertan return over the nterval t wth ~. Hence, S t t S t e [ ~ r t ] t In the lognormal strbuton the rate of return over a short tme pero t s normally strbute wth mean µ t an varance σ t. Hence, St S t [ t Z t ] t exp µ σ where Z s a stanar normal varable (mean0, varance1. ~ r t r t

23 Lognormal Prce Smulaton; VBA Proceure Sub prcepathsmulaton( Range("starttme" Tme Applcaton.screenupatng False N Range("runs".Value mean Range("mean" sgma Range("sgma" elta_t 1 / ( * N ReDm prce(0 To * N As Double prce(0 Range("ntal_prce" For Inex 1 To N start: Statc ran1, ran, S1, S, 1, ran1 * Rn - 1 ran * Rn - 1 S1 ran1 ^ ran ^ If S1 > 1 Then GoTo start S Sqr(- * Log(S1 / S1 1 ran1 * S ran * S prce( * Inex - 1 prce( * Inex - * Exp(mean * elta_t _ sgma * Sqr(elta_t * 1 prce( * Inex prce( * Inex - 1 * Exp(mean * elta_t _ sgma * Sqr(elta_t * Next Inex For Inex 0 To * N Range("output".Cells(Inex 1, prce(inex Next Inex Range("stoptme" Tme Range("elapse" Range("stoptme" - Range("starttme" Range("elapse".NumberFormat "hh:mm:ss" En Sub 3

24 Calculatng Parameters for Lognormal Dstrbuton (Stock Prces 1. Mean S t t E ln( E S σ t. Varance St Var ln( S t lnearty n tme. Estmaton: µ [ µ t Z t ] Var [ µ t σ Z t ] σ t t St Mean ln( S t t t t, σ St Var ln( S t t 4

25 Calculatng Parameters for Lognormal Dstrbuton (Stock Prces Return: Mean Annual Returns n * Mean Peroc Return where n s the number of peros n one year (aly > n 5 Stanar evaton: σ Annual Returns Sqrt(n * σ Peroc Return where n s the number of peros n one year (aly > n Sqrt(5 5

26 The Black & Scholes Moel; Call Opton. C S N rt ( e N( 1 where 1 ln( S / ( r σ σ T / T 1 σ T Where C enotes the prce of a call, S s the prce of the unerlyng nstrument, s the exercse prce of the call, T s the tme to exercse an r s the rsk free nterest rate. 6

27 where P The Black & Scholes Moel; Put Opton. Applyng the put-call party relaton a put wth the same exercse ate T, exercse prce : P C S e -rt. Hence, 1 S ln( S 1 rt N( 1 e N( / σ ( r σ σ T T / T Where P enotes the prce of a put, S s the prce of the unerlyng nstrument, s the exercse prce of the call, T s the tme to exercse an r s the rsk free nterest rate. 7

28 The Black & Scholes Moel; VBA-Functons. Call Opton: Put Opton: Functon CallOpton(Stock, Exercse, Tme, Interest, sgma CallOpton Stock * Applcaton.NormSDst(One(Stock, Exercse, _ Tme, Interest, sgma - Exercse * Exp(-Tme * Interest * _ Applcaton.NormSDst(One(Stock, Exercse, Tme, Interest, sgma _ - sgma * Sqr(Tme En Functon Functon PutOpton(Stock, Exercse, Tme, Interest, sgma PutOpton CallOpton(Stock, Exercse, Tme, Interest, sgma _ Exercse * Exp(-Interest * Tme - Stock En Functon 8

29 The Black & Scholes Moel; VBA-Imple Volatlty. Imple Volatlty: Functon CallVolatlty(Stock, Exercse, Tme, Interest, Target Hgh Low 0 Do Whle (Hgh - Low > If CallOpton(Stock, Exercse, Tme, Interest, (Hgh Low / > _ Target Then Hgh (Hgh Low / Else: Low (Hgh Low / En If Loop CallVolatlty (Hgh Low / En Functon 9

30 The Bang of the Bucks wth Optons. Suppose that you are convnce that a gven stock wll go up n a very short pero of tme. You want to buy calls on the stock that have a maxmum bang for the buck - that s, you want the percentage proft on your opton nvestment to be maxmal. Apply the B&S formula: Buy calls wth shortest possble maturty Buy calls that are most hghly out of the money (.e. wth the hghest exercse prce possble Implement! 30

31 Valung Optons on More Complex Assets The Forwar Prce Verson of the B&S moel: One we know how to calculate the forwar prce of an unerlyng asset, t s possble to etermne the B&S value for a European call on the unerlyng asset. Remember: sf S 0 (1 where s F s the forwar prce of an unerlyng asset. Hence, r T C e rt [ s N( N( F 1 ] 1 ln( S / ( r σ σ T / T 1 σ T 31

32 Valung Optons on More Complex Assets The U.S. Dollar rsk-free rate s assume to be 6% per year an all σ s are assume to be 5% per year. E want to apply the forwar prce verson of B&S to value European call optons one year from now: a 1 DM at a strke prce of $0.50. The Spot Exchange rate s $0.40/DM an the one year rsk free rate s 8% Germany. b A 30-year bon wth an 8% sem-annual coupon at a strke prce of $100. The bon s currently sellng at a full prce of $10 (nclue accrue nterest per $100 of face value an has two scheule coupons of $4 before opton expraton, to be pa sx months an one year form now. c The S&P500 contract wth a strke prce of $850. The S&P500 has a current prce of $800 an a present value of next year s vens of $0. A barrel of Ol wth a strke prce of $0. A barrel of ol currently sells for $18. The present value of next year storage costs s $1 an the present value of the convenence yel havng a barrel of ol avalable over the next year (e.g., f there are long gas lnes ue to an ol embargo an eplorable government polcy s $1. 3

33 Forwar prces: a b c Valung Optons on More Complex Assets ($ /1.08 $0.396 $ $ $4 $ ($ 800 $ $86.80 ($ 18 $1 $ $19.08 Forwar Prce verson of B&S moel: $0.396 N( $0.5 N( ln(0.396 / 0.5 C 1 where, $ N( $100 N( ln( / $86.80 N( $850 N( ln(86.80 / 850 C 1 where, $19.08 N( $0 N( ln(19.08/ 0 C 1 where, a 1 >C$0.01 b C 1 where, >C$9.39 c >C$68. >C$

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