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Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
Derivatives  Binomial  Option  Pricing  Model  Examples
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Derivatives Binomial Option Pricing Model Examples

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  • 1. Binomial Option Pricing Model Examples
  • 2. 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
  • 3. 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
  • 4. 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
  • 5. 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?
  • 6. 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
  • 7. 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.
  • 8. 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
  • 9. 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 = $16.06
    • What if this is an American option ?
  • 10. STEP THREE-PUT OPTION
    • Note p, 1-p, u,d, and e -rT same as before.
    • Therefore, at Node A
    • F = e -rT (pf up + (1-p)f down )
    • F={.568($.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
  • 11. 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
  • 12. 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
  • 13. 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
  • 14. 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
  • 15. 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.
  • 16. 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
  • 17. 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 ?

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