Options. Pricing. Binomial models. Black-Scholes model. Greeks

Transcription

1 Options. Priing. Binomial moels. Blak-Sholes moel. Greeks 1. Binomial moel,. Blak-Sholes moel, assmptions, moifiations (iviens, rreny options, options on ftres 3. Implie volatility 4. Sensitivity measres The binomial moel has the avantage of allowing to prie Amerian options. This is a isrete time moel. Senarios are shown on a binomial tree. The proess of valing an option is often alle risk-netral valation. The BSM moel is a ontinos time moel se to prie only Eropean options. Simple Binomial Priing Moel [ox, Ross an Rbinstein] Serity S S S all Option Max (, S - E Max (, S - E Figre. 1. Simple Binomial Priing Moel. all Option (1 q + (1 - q (1 + r T ( q S (1 + r T - S S - S

2 Derivatives on Finanial Market Two-Perio Binomial Priing Moel Serity S S S S S all Option S Figre. Two-Perio Binomial Priing Moel. all Option q + (1 q rt S rt S q S S S rt S q S S q [ q + (1 q ] + (1 q [ q + (1 q ] rt

3 Derivatives on Finanial Market Blak-Sholes option priing moel Aoring to the Blak-Sholes moel, the vale of a all option is given as rt (1 SN(1 Ee N( where ln(s/e + (r + σs /T 1 σ T s 1 σs T σs is the variane of the asset s retrns N(x is the mlative probability for a nit normal variable allate at a vale of x. Assmptions 1 Retrns for the nerlying asset are log normally istribte an inepenent over time. onstant variane 3 onstant interest rate 4 No iviens 5 No early exerise Table 1. Option Priing. Eqity an Debt as Options Option Priing Eqity an Debt premim eqity S spot prie maret vale of assets E exerise prie book vale of ebt S- ebt T matrity matrity of ebt * R B risk-free rate risk-free rate σ s volatility of retrns volatility of ROA Sensitivity Measres Table. Sensitivity Measres Miernik Notaja all Pt Delta N( 1 1 p S Gamma Theta Rho Vega γ S n( Sσ 1 γ p Sσn(1 rt θ θ ree N( T T ρ rt ρ TEe N( r υ σ T υ S Tn(1 θ ρ p p γ γ θ ρ + ree TEe υ υ p rt rt where: e / n( π 3

4 Derivatives on Finanial Market Problem 1. Binomial Priing Moel The rrent serity prie is $1. The exerise prie on the option is $11. It will either go p to $15 or own to $9. The riskless rate of interest is 5%. Matrity is 36 ays, T 1. (a allate the prie of the all option, the hege ratio, probabilities of the p an own movements sing ox, Ross an Rbinstein moel. ompare the reslt with the prie allate sing BSM moel. allate the present vale of the ening payoff. (b allate the weights for the repliating strategy, the ening payoff of the all, option an the prie of the all option. The bon prie is $1. Soltion (a Serity prie Exerise The payoff of the The hege ratio: prie all option (S S h -1,5 ( 9 11 The ening payoff B S + h 9, B S + h 9, q + (1 q,5 9,5 (1 + rt The option premim sing Blak-Sholes moel: BS 1,3 The present vale of the ening payoff : B B rt 85,7 B S + h 85,7 (b S(1 + rt S q S S w w a b S S w as + w S - S ( S S P b P 1 h 66,7% -6,% 4 w as + w w S + w a b b P P 9,5 4

5 Derivatives on Finanial Market Problem. Binomial Priing Moel. Mlti-perio Binomial Moel. Dration an onvexity T 1, 1, 1, 1, 1, f,%,4%,8% 4,% 5,% z,%,1998%,3995%,7973% 3,341% z+shift,%,1998%,3995%,7973% 3,341% σ 1% 1% 9% 1% 1% 5,% E 1, 1, 1, OAS,8% Reqire allate the prie of a bon with a all option. allate the effetive ration an onvexity. 5, S 1, 5, r 11,977% S 97,143 5, r 8,885% 5, S 98,5 S 1, 5, r 4,8133% 5, r 9,916% S 1, S 98,345 r 3,851% 5, r 6,7673% 5, S 1,14 S 99,78 S 1, r,8% 5, r 4,15% 5, r 8,591% S 1, S 99,351 r 3,% 5, r 5,6856% 5, S 1, S 1, r 3,6% 5, r 6,97% S 1, r 4,8% 5, S 1, r 5,8% ena opji 15-1,14,86 V- V+ D V y V- + V+ V V y (, ,14 -,5% 1,799 53,764,5% 11,549 5

6 Derivatives on Finanial Market Problem 3. Blak-Sholes Moel With the following parameteres S spot prie 1, E exerise prie 11, Time to expiration (nmber of ays 36 Risk free interest rate 5% σ volatility 36,1% (a allate the prie of the all option sing the Blak-Sholes moel. (b allate the prie of the pt option sing pt-all parity. Soltion (a S N ( rt Ee N ( 1 1 ln( S / E + ( r + σ / T σ s T s σ T 1 s T,9863 1,511 -,376 N(1,54 N(,379 e^(-rt,95 Ee^(-rT 14,769 1,33 (b The prie of the pt option rt [ ( 1] [ ( 1] rt 1 P S + Ee S N Ee N 17,4 6

7 Derivatives on Finanial Market Problem 4. Implie. Volatility S spot prie 1 E exerise prie 1 Time to expiration (nmber of ays 45 Risk free interest rate 7% all prie 4 Reqire allate the implie volatility sing sensitivity ananlysis. Soltion Expete volatility all Premim 1% 3,8 11% 3,38 1% 3,49 13% 3,61 14% 3,7 15% 3,84 16% 3,97 < above this vale 17% 4,9 18% 4, 19% 4,35 % 4,47 1% 4,6 % 4,74 3% 4,87 4% 5, 5% 5,13 7

8 Derivatives on Finanial Market Problem 5. Sensitivity Measres The all option has the following parameters S spot prie 11 E exerise prie 1 Time to expiration (nmber of ays 3 Risk free interest rate 5% σ volatility % (a allte sensitivity measres for the all option, pt option, overe all protete pt, strale, bll all sprea, bear all sprea. (b Show the sensitivity of sensitivity measres on prie hanges 8-1. Soltion (a Delta Gamma Theta Rho Vega all Option 1,96,1 -,,8,3 Pt Option -,4,1 -,1,,3 overe all 3,4 -,1, -,8,3 Protetive pt 4,96,1 -,1,,3 Strale 5,9,3 -,3,7,5 Bll all Sprea 6,14 -,3,,1 -,6 Bear all Sprea 7 -,4,1 -,1,, (b all option Spot Delta Gamma Theta Rho Vega Pt option Prie 8,,,,, 85,,,,, 9,4, -,1,, 95,1,5 -,3,,8 1,54,7 -,4,4,11 15,83,4 -,4,7,8 11,96,1 -,,8,3 115,99, -,,8,1 1 1,, -,1,8, Spot Delta Gamma Theta Rho Vega Prie 8-1,,,1 -,8, 85-1,,,1 -,8, 9 -,96,,1 -,8, 95 -,79,5 -, -,7,8 1 -,46,7 -,3 -,4, ,17,4 -, -,,8 11 -,4,1 -,1,, ,1,,,,1 1,,,,, 8

9 Derivatives on Finanial Market Delta Delta 1,5 1,,5, 8 -, , Pt all -1,5 Spot Prie all option Sensitivity Measres,14,1,1,8,6,4,, -, ,4 -,6 Spot prie Gamma Theta Rho Vega Pt option,15 Sensitivity Measres,1,5, ,5 Gamma Theta Rho Vega -,1 Spot Prie 9