Binomial Option Pricing Model Examples

Call Option – One Period Note growth rates, future expected returns are not in the option pricing models. These factors are already incorporated into the stock price and do not need to be added again. Assume the riskfree rate is 5% T is 6 months , X =$110 U = 1.25 , d = .80 and S = P = $100

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

Assume the riskfree rate is 5%

T is 6 months , X =$110

U = 1.25 , d = .80 and S = P = $100

Call Option – One Period, cont. --------------- $125 ($15) $100 --------------- $80 ($0) Recall, X=$110 P = (e rT – d)/(u-d) , where r=5%, T=.5 P = (1.025-.80)/(1.25-.80) = .225/.45 = .5 Therefore, 1-P = .5 Value of option today is f, where f F = e -rT *(pf up + (1-p)f down ) F= e -rT *(.5(15) + .5(0)) = e -rT (7.50) = $7.31

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

$100

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

Recall, X=$110

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

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

Therefore, 1-P = .5

Value of option today is f, where f

F = e -rT *(pf up + (1-p)f down )

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

Two Period Call Option Let u = 1.1,d=.9,T1-T2=T=.5,r=.05,S=$100,X=100 -------------------------------- $121 (21)Node D ------------------$110 (10) Node B $100-Node A --------------$99 (0) Node E ------------------$90 (0) Node C ---------------------------------$81 (0) Node F P = (e r(T1-T2) -d)/(u-d) = (1.025-.9)/(1.1-.9) P = .625 ; 1-P = .375 At Node C,E and F value of option is zero. STEP ONE Node B, Value of option f = e -rT (pf up + (1-p)f down ) F = e -.05*.5 (.625(21) +.375(0)) = .975*13.125 = 12.80

Let u = 1.1,d=.9,T1-T2=T=.5,r=.05,S=$100,X=100

-------------------------------- $121 (21)Node D

------------------$110 (10) Node B

$100-Node A --------------$99 (0) Node E

------------------$90 (0) Node C

---------------------------------$81 (0) Node F

P = (e r(T1-T2) -d)/(u-d) = (1.025-.9)/(1.1-.9)

P = .625 ; 1-P = .375

At Node C,E and F value of option is zero. STEP ONE

Node B, Value of option f = e -rT (pf up + (1-p)f down )

F = e -.05*.5 (.625(21) +.375(0)) = .975*13.125 = 12.80

STEP TWO Note p, r, T, u and d are unchanged Value of option at node A is F = e-.05*.5 (pf up + (1-p)f down ) F = .975(.625(12.80) +(.375)(0)) F = .975(8.00) = $7.80 What would have happened if the value of the option at Node B was $9.80?

Note p, r, T, u and d are unchanged

Value of option at node A is

F = e-.05*.5 (pf up + (1-p)f down )

F = .975(.625(12.80) +(.375)(0))

F = .975(8.00) = $7.80

What would have happened if the value of the option at Node B was $9.80?

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

Time to expiration is six months

Riskfree rate is 4%, X=50, S=P=50

U = 1.2 , d= .9

--------------- $60 (0) Node B

$50 Node A

--------------- $45 (5) Node C

P = (e rT – d)/(u-d) =(e .04*.5 -.9)/(1.2-.9)

P=.12/.3 = .40 (1-P) = .6

F = e -rT (.4(0) + .6(5)) = $3/1.02 = $2.94

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

T = 1 year between nodes

R = 4% , u = 1.3 , d=.7 , S=P=80, X=75

----------------------------- $135.20 (0) D

----------------- $104 (0) B Expires worthless

$80 A -------------------- $72.80 (2.20) E

----------------- $56 (19) C In the money $19

----------------------------- $39.20 (35.80) F

Note : Use backwardation to solve for the option value today. We know the values of the option at expiration.

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

At Node B, F = e –rT {pf up + (1-p)f down }

However, we must solve for p first.

P = (e rT – d)/(u-d) , where e rT = e .04*1 .

P = (1.041 - .7)/(1.3-.7) = .568

Therefore, 1-p = .432

F = e –rT { .568(0) + .432(2.20)} = $.91

STEP TWO Node C , T = T1-T2 = 1 year F = e -rT (pf up + (1-p)f down ),e -.04*1 = 1.041 P = (1.041-.7)/(1.3-.7) = .341/.6 = .568 Note: P is identical to P in the previous step. 1-P = .432 F = (.568(2.20) + .432(35.80)/1.041 F = (1.25 + 15.47)/1.041 = $16.06 What if this is an American option ?

Node C , T = T1-T2 = 1 year

F = e -rT (pf up + (1-p)f down ),e -.04*1 = 1.041

P = (1.041-.7)/(1.3-.7) = .341/.6 = .568

Note: P is identical to P in the previous step.

1-P = .432

F = (.568(2.20) + .432(35.80)/1.041

F = (1.25 + 15.47)/1.041 = $16.06

What if this is an American option ?

STEP THREE-PUT OPTION Note p, 1-p, u,d, and e -rT same as before. Therefore, at Node A F = e -rT (pf up + (1-p)f down ) F={.568($.91)+.432(16.06)}/1.041=$ 7.46/1.041 = $7.17 If it is a European put the value is $7.17 F = {.568($.91) + .432($19)}/1.041 = $8.38 , if it is an American put

Note p, 1-p, u,d, and e -rT same as before.

Therefore, at Node A

F = e -rT (pf up + (1-p)f down )

F={.568($.91)+.432(16.06)}/1.041=$ 7.46/1.041 = $7.17

If it is a European put the value is $7.17

F = {.568($.91) + .432($19)}/1.041 = $8.38 , if it is an American put

Call Options on Stock Indices Assume the stock index pays a dividend rate of q (a steady stream of dividends) The index value is $14,000, X=14,500, T=3 months, r=4%, q=2%,u=1.1 and d=.9 ----------------- $15,400 (900) B $14,000 A ------------------$12,600 (0) C P = {e (.04-.02)*.25 - .9}/(1.1-.9) = .525 1-P = .475 F = e -rT {pf up + (1-p)f down } F = e -.04*.25 {.525*900 + .475(0)} = 472.50/1.01 = $467.80

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

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

----------------- $15,400 (900) B

$14,000 A

------------------$12,600 (0) C

P = {e (.04-.02)*.25 - .9}/(1.1-.9) = .525

1-P = .475

F = e -rT {pf up + (1-p)f down }

F = e -.04*.25 {.525*900 + .475(0)} = 472.50/1.01 = $467.80

Important Note It is common practice to use the following U = e vt1/2 , where v represents volatility or standard deviation and t1/2 is the square root of T. And D = 1/U

It is common practice to use the following

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

And D = 1/U

Call Option on Canadian Currency Assume the Canadian dollar is .95 to the US dollar. That is one Canadian dollar buys 95 cents American. Assume the volatility or std. dev of the exchange rate is 10%, the riskfree rate in the US is r= 4%, the riskfree Canadian rate is r c =6%. The time to expiration is 3 months. P = (a-d)/(u-d) , where a = e (r-r c )*T U = e .10*(.25)1/2 = e.05 = 1.051271, round to 1.05 Note (.25)1/2 is the square root of .25 = .5 D = 1/U = 1/1.05 = .952

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

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

P = (a-d)/(u-d) , where a = e (r-r c )*T

U = e .10*(.25)1/2 = e.05 = 1.051271, round to 1.05

Note (.25)1/2 is the square root of .25 = .5

D = 1/U = 1/1.05 = .952

Currency Call Option, cont. For the value of a = e (.04-.06)*.25 = .98 P = (a-d)/(u-p) = (.98-.952)/(1.051-.952) = .028/.099 = .283 ; 1-P = .717 ------------------------- .998 (.038) B .95 A ------------------------- .904 (0) C F = e -.04*.25 {.283(.038) + .717(0)} = .0106

For the value of a = e (.04-.06)*.25 = .98

P = (a-d)/(u-p) = (.98-.952)/(1.051-.952) =

.028/.099 = .283 ; 1-P = .717

------------------------- .998 (.038) B

.95 A

------------------------- .904 (0) C

F = e -.04*.25 {.283(.038) + .717(0)} = .0106

Call Option on Futures Contracts It costs nothing to take a position in the futures markets, therefore in a riskfree world the futures price should have a zero expected growth rate. We will come back to this concept later. Therefore, e -rT = 1 , that is a = 1 and P = (1-d)/(u-d) Assume an asset has a volatility of .4, r=.05 and this is a 9 month call option.

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

Therefore, e -rT = 1 , that is a = 1 and

P = (1-d)/(u-d)

Assume an asset has a volatility of .4, r=.05 and this is a 9 month call option.

Binomial Call Option on Oil T = 9/12 = .75 U = e (.4{.75}1/2) = e .4(.866) =1.414, with {.75}1/2 is the square root of .75 and .4 represents the volatility of the asset. D = 1/U = .707 P = (1-d)/(u-d) = .293/.707 = .414 1-P = .586 F = e -rT {pf up + (1-p)f down } F= e -.05(.75) {.414(28.12) + .586(0)} = $11.21

T = 9/12 = .75

U = e (.4{.75}1/2) = e .4(.866) =1.414, with {.75}1/2 is the square root of .75 and .4 represents the volatility of the asset.

D = 1/U = .707

P = (1-d)/(u-d) = .293/.707 = .414

1-P = .586

F = e -rT {pf up + (1-p)f down }

F= e -.05(.75) {.414(28.12) + .586(0)} = $11.21

Graph a Condor Similar to a butterfly spread Buy call at 70, buy call at 85 Write a call at 75, write a call at 80 Flat top not pointed like a butterfly spread. What is the payoff difference between a butterfly spread and a condor ?

Similar to a butterfly spread

Buy call at 70, buy call at 85

Write a call at 75, write a call at 80

Flat top not pointed like a butterfly spread.

What is the payoff difference between a butterfly spread and a condor ?