Binomial Option Binomial Option Pricing ModelPricing Model

ExamplesExamples

Call Option – One PeriodCall Option – One Period

Note growth rates, future expected Note growth rates, future expected returns are not in the option pricing returns are not in the option pricing models. These factors are already models. These factors are already incorporated into the stock price and incorporated into the stock price and do not need to be added again.do not need to be added again.

Assume the riskfree rate is 5%Assume the riskfree rate is 5% T is 6 months , X =$110T is 6 months , X =$110 U = 1.25 , d = .80 and S = P = $100U = 1.25 , d = .80 and S = P = $100

Call Option – One Period, Call Option – One Period, cont.cont.

--------------- $125 ($15)--------------- $125 ($15)

$100$100

--------------- $80 ($0)--------------- $80 ($0)

Recall, X=$110Recall, X=$110

P = (eP = (erTrT – d)/(u-d) , where r=5%, T=.5 – d)/(u-d) , where r=5%, T=.5

P = (1.025-.80)/(1.25-.80) = .225/.45 = .5P = (1.025-.80)/(1.25-.80) = .225/.45 = .5

Therefore, 1-P = .5Therefore, 1-P = .5

Value of option today is f, where fValue of option today is f, where f

F = eF = e-rT-rT *(pf *(pfupup + (1-p)f + (1-p)fdowndown) )

F= eF= e-rT-rT*(.5(15) + .5(0)) = e*(.5(15) + .5(0)) = e-rT-rT(7.50) = $7.31(7.50) = $7.31

Two Period Call OptionTwo Period Call Option Let u = 1.1,d=.9,T1-T2=T=.5,r=.05,S=$100,X=100Let u = 1.1,d=.9,T1-T2=T=.5,r=.05,S=$100,X=100 -------------------------------- $121 (21)Node D-------------------------------- $121 (21)Node D ------------------$110 (10) Node B------------------$110 (10) Node B $100-Node A --------------$99 (0) Node E$100-Node A --------------$99 (0) Node E ------------------$90 (0) Node C------------------$90 (0) Node C ---------------------------------$81 (0) Node F---------------------------------$81 (0) Node F P = (eP = (er(T1-T2)r(T1-T2)-d)/(u-d) = (1.025-.9)/(1.1-.9)-d)/(u-d) = (1.025-.9)/(1.1-.9) P = .625 ; 1-P = .375P = .625 ; 1-P = .375 At Node C,E and F value of option is zero. At Node C,E and F value of option is zero. STEP STEP

ONEONE Node B, Value of option f = eNode B, Value of option f = e-rT-rT(pf(pfupup + (1-p)f + (1-p)fdowndown)) F = eF = e-.05*.5-.05*.5(.625(21) +.375(0)) = .975*13.125 = 12.80(.625(21) +.375(0)) = .975*13.125 = 12.80

STEP TWOSTEP TWO

Note p, r, T, u and d are unchanged Note p, r, T, u and d are unchanged Value of option at node A isValue of option at node A is F = F = e-.05*.5e-.05*.5(pf(pfupup + (1-p)f + (1-p)fdowndown)) F = .975(.625(12.80) +(.375)(0))F = .975(.625(12.80) +(.375)(0)) F = .975(8.00) = $7.80F = .975(8.00) = $7.80 What would have happened if the What would have happened if the

value of the option at Node B was value of the option at Node B was $9.80?$9.80?

Put Option – One PeriodPut Option – One Period Time to expiration is six monthsTime to expiration is six months Riskfree rate is 4%, X=50, S=P=50Riskfree rate is 4%, X=50, S=P=50 U = 1.2 , d= .9U = 1.2 , d= .9 --------------- $60 (0) Node B--------------- $60 (0) Node B $50 Node A$50 Node A --------------- $45 (5) Node C--------------- $45 (5) Node C P = (eP = (erTrT – d)/(u-d) =(e – d)/(u-d) =(e.04*.5.04*.5 -.9)/(1.2-.9) -.9)/(1.2-.9) P=.12/.3 = .40 (1-P) = .6P=.12/.3 = .40 (1-P) = .6 F = eF = e-rT-rT (.4(0) + .6(5)) = $3/1.02 = $2.94 (.4(0) + .6(5)) = $3/1.02 = $2.94

Two Period Put OptionTwo Period Put Option T = 1 year between nodesT = 1 year between nodes R = 4% , u = 1.3 , d=.7 , S=P=80, X=75R = 4% , u = 1.3 , d=.7 , S=P=80, X=75 ----------------------------- $135.20 (0) D----------------------------- $135.20 (0) D ----------------- $104 (0) B Expires worthless----------------- $104 (0) B Expires worthless $80 A -------------------- $72.80 (2.20) E$80 A -------------------- $72.80 (2.20) E ----------------- $56 (19) C In the money $19----------------- $56 (19) C In the money $19 ----------------------------- $39.20 (35.80) F----------------------------- $39.20 (35.80) F Note : Use backwardation to solve for the Note : Use backwardation to solve for the

option value today. We know the values of option value today. We know the values of the option at expiration.the option at expiration.

STEP ONESTEP ONE

At Node B, F = e At Node B, F = e –rT–rT {pf {pfupup + (1-p)f + (1-p)fdowndown}} However, we must solve for p first. However, we must solve for p first. P = (eP = (erTrT – d)/(u-d) , where e – d)/(u-d) , where erTrT = e = e .04*1.04*1 . . P = (1.041 - .7)/(1.3-.7) = .568P = (1.041 - .7)/(1.3-.7) = .568 Therefore, 1-p = .432Therefore, 1-p = .432 F = e F = e –rT–rT { .568(0) + .432(2.20)} = $.91 { .568(0) + .432(2.20)} = $.91

STEP TWOSTEP TWO

Node C , T = T1-T2 = 1 yearNode C , T = T1-T2 = 1 year F = eF = e-rT-rT(pf(pfupup + (1-p)f + (1-p)fdowndown),e),e-.04*1-.04*1 = 1.041 = 1.041 P = (1.041-.7)/(1.3-.7) = .341/.6 = .568P = (1.041-.7)/(1.3-.7) = .341/.6 = .568 Note: P is identical to P in the previous Note: P is identical to P in the previous

step.step. 1-P = .432 1-P = .432 F = (.568(2.20) + .432(35.80)/1.041 F = (.568(2.20) + .432(35.80)/1.041 F = (1.25 + 15.47)/1.041 = $16.06 F = (1.25 + 15.47)/1.041 = $16.06 What if this is an American option ?What if this is an American option ?

STEP THREE-PUT STEP THREE-PUT OPTIONOPTION

Note p, 1-p, u,d, and eNote p, 1-p, u,d, and e-rT-rT same as before. same as before. Therefore, at Node ATherefore, at Node A F = eF = e-rT-rT (pf (pfupup + (1-p)f + (1-p)fdowndown)) F={.568($.91)+.432(16.06)}/1.041=$ F={.568($.91)+.432(16.06)}/1.041=$

7.46/1.041 = $7.17 7.46/1.041 = $7.17 If it is a European put the value is $7.17If it is a European put the value is $7.17 F = {.568($.91) + .432($19)}/1.041 = F = {.568($.91) + .432($19)}/1.041 =

$8.38 , if it is an American put$8.38 , if it is an American put

Call Options on Stock Call Options on Stock IndicesIndices

Assume the stock index pays a dividend rate of q Assume the stock index pays a dividend rate of q (a steady stream of dividends)(a steady stream of dividends)

The index value is $14,000, X=14,500, T=3 The index value is $14,000, X=14,500, T=3 months, r=4%, q=2%,u=1.1 and d=.9months, r=4%, q=2%,u=1.1 and d=.9

----------------- $15,400 (900) B ----------------- $15,400 (900) B $14,000 A$14,000 A ------------------$12,600 (0) C------------------$12,600 (0) C P = {e P = {e (.04-.02)*.25(.04-.02)*.25 - .9}/(1.1-.9) = .525 - .9}/(1.1-.9) = .525 1-P = .475 1-P = .475 F = eF = e-rT-rT{pf{pfupup + (1-p)f + (1-p)fdowndown} } F = eF = e-.04*.25-.04*.25 {.525*900 + .475(0)} = 472.50/1.01 = {.525*900 + .475(0)} = 472.50/1.01 =

$467.80$467.80

Important NoteImportant Note

It is common practice to use the It is common practice to use the followingfollowing

U = e U = e vt1/2vt1/2 , where v represents , where v represents volatility or standard deviation and volatility or standard deviation and t1/2 is the square root of T. t1/2 is the square root of T.

And D = 1/UAnd D = 1/U

Call Option on Canadian Call Option on Canadian CurrencyCurrency

Assume the Canadian dollar is .95 to the US Assume the Canadian dollar is .95 to the US dollar. That is one Canadian dollar buys 95 dollar. That is one Canadian dollar buys 95 cents American.cents American.

Assume the volatility or std. dev of the Assume the volatility or std. dev of the exchange rate is 10%, the riskfree rate in the exchange rate is 10%, the riskfree rate in the US is r= 4%, the riskfree Canadian rate is US is r= 4%, the riskfree Canadian rate is rrcc=6%. The time to expiration is 3 months.=6%. The time to expiration is 3 months.

P = (a-d)/(u-d) , where a = e (r-rP = (a-d)/(u-d) , where a = e (r-rcc)*T)*T U = e U = e .10*(.25)1/2.10*(.25)1/2 = e.05 = 1.051271, round to 1.05 = e.05 = 1.051271, round to 1.05 Note (.25)1/2 is the square root of .25 = .5Note (.25)1/2 is the square root of .25 = .5 D = 1/U = 1/1.05 = .952 D = 1/U = 1/1.05 = .952

Currency Call Option, Currency Call Option, cont. cont.

For the value of a = e (.04-.06)*.25 = .98For the value of a = e (.04-.06)*.25 = .98 P = (a-d)/(u-p) = (.98-.952)/(1.051-.952) P = (a-d)/(u-p) = (.98-.952)/(1.051-.952)

= = .028/.099 = .283 ; 1-P = .717.028/.099 = .283 ; 1-P = .717 ------------------------- .998 (.038) B------------------------- .998 (.038) B .95 A.95 A ------------------------- .904 (0) C------------------------- .904 (0) C F = eF = e-.04*.25-.04*.25 {.283(.038) + .717(0)} {.283(.038) + .717(0)}

= .0106= .0106

Call Option on Futures Call Option on Futures ContractsContracts

It costs nothing to take a position in It costs nothing to take a position in the futures markets, therefore in a the futures markets, therefore in a riskfree world the futures price should riskfree world the futures price should have a zero expected growth rate. We have a zero expected growth rate. We will come back to this concept later.will come back to this concept later.

Therefore, eTherefore, e-rT-rT = 1 , that is a = 1 and = 1 , that is a = 1 and P = (1-d)/(u-d) P = (1-d)/(u-d) Assume an asset has a volatility of .4, Assume an asset has a volatility of .4,

r=.05 and this is a 9 month call option.r=.05 and this is a 9 month call option.

Binomial Call Option on Binomial Call Option on OilOil

T = 9/12 = .75T = 9/12 = .75 U = e U = e (.4{.75}1/2)(.4{.75}1/2) = e = e .4(.866).4(.866)=1.414, with =1.414, with

{.75}1/2 is the square root of .75 and .4 {.75}1/2 is the square root of .75 and .4 represents the volatility of the asset. represents the volatility of the asset.

D = 1/U = .707D = 1/U = .707 P = (1-d)/(u-d) = .293/.707 = .414P = (1-d)/(u-d) = .293/.707 = .414 1-P = .5861-P = .586 F = eF = e-rT-rT {pf {pfupup + (1-p)f + (1-p)fdowndown} } F= e F= e -.05(.75)-.05(.75){.414(28.12) + .586(0)} = $11.21 {.414(28.12) + .586(0)} = $11.21

Graph a CondorGraph a Condor

Similar to a butterfly spreadSimilar to a butterfly spread Buy call at 70, buy call at 85Buy call at 70, buy call at 85 Write a call at 75, write a call at 80Write a call at 75, write a call at 80 Flat top not pointed like a butterfly Flat top not pointed like a butterfly

spread.spread. What is the payoff difference What is the payoff difference

between a butterfly spread and a between a butterfly spread and a condor ?condor ?