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Binomial Option Pricing Model Examples
Call Option – One Period <ul><li>Note growth rates, future expected returns are not in the option pricing models. These fa...
Call Option – One Period, cont. <ul><ul><li>---------------  $125 ($15) </li></ul></ul><ul><ul><li>$100 </li></ul></ul><ul...
Two Period Call Option <ul><li>Let u = 1.1,d=.9,T1-T2=T=.5,r=.05,S=$100,X=100 </li></ul><ul><li>--------------------------...
STEP TWO <ul><li>Note p, r, T, u and d are unchanged  </li></ul><ul><li>Value of option at node A is </li></ul><ul><li>F =...
Put Option – One Period <ul><li>Time to expiration is six months </li></ul><ul><li>Riskfree rate is 4%, X=50, S=P=50 </li>...
Two Period Put Option <ul><li>T = 1 year between nodes </li></ul><ul><li>R = 4% , u = 1.3 , d=.7 , S=P=80, X=75 </li></ul>...
STEP ONE <ul><li>At Node B,  F = e  –rT  {pf up  + (1-p)f down } </li></ul><ul><li>However, we must solve for p first.  </...
STEP TWO <ul><li>Node C , T = T1-T2 = 1 year </li></ul><ul><li>F = e -rT (pf up  + (1-p)f down ),e -.04*1  = 1.041 </li></...
STEP THREE-PUT OPTION <ul><li>Note p, 1-p, u,d, and e -rT  same as before. </li></ul><ul><li>Therefore, at Node A </li></u...
Call Options on Stock Indices <ul><li>Assume the stock index pays a dividend rate of q (a steady stream of dividends) </li...
Important Note <ul><li>It is common practice to use the following </li></ul><ul><li>U = e  vt1/2   , where v represents vo...
Call Option on Canadian Currency <ul><li>Assume the Canadian dollar is .95 to the US dollar. That is one Canadian dollar b...
Currency Call Option, cont.  <ul><li>For the value of a = e (.04-.06)*.25 = .98 </li></ul><ul><li>P = (a-d)/(u-p) = (.98-....
Call Option on Futures Contracts <ul><li>It costs nothing to take a position in the futures markets, therefore in a riskfr...
Binomial Call Option on Oil <ul><li>T = 9/12 = .75 </li></ul><ul><li>U = e  (.4{.75}1/2)   = e  .4(.866) =1.414, with {.75...
Graph a Condor <ul><li>Similar to a butterfly spread </li></ul><ul><li>Buy call at 70, buy call at 85 </li></ul><ul><li>Wr...
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Derivatives Binomial Option Pricing Model Examples

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Derivatives Binomial Option Pricing Model Examples

  1. 1. Binomial Option Pricing Model Examples
  2. 2. Call Option – One Period <ul><li>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. </li></ul><ul><li>Assume the riskfree rate is 5% </li></ul><ul><li>T is 6 months , X =$110 </li></ul><ul><li>U = 1.25 , d = .80 and S = P = $100 </li></ul>
  3. 3. Call Option – One Period, cont. <ul><ul><li>--------------- $125 ($15) </li></ul></ul><ul><ul><li>$100 </li></ul></ul><ul><ul><li>--------------- $80 ($0) </li></ul></ul><ul><ul><li>Recall, X=$110 </li></ul></ul><ul><ul><li>P = (e rT – d)/(u-d) , where r=5%, T=.5 </li></ul></ul><ul><ul><li>P = (1.025-.80)/(1.25-.80) = .225/.45 = .5 </li></ul></ul><ul><ul><li>Therefore, 1-P = .5 </li></ul></ul><ul><ul><li>Value of option today is f, where f </li></ul></ul><ul><ul><li>F = e -rT *(pf up + (1-p)f down ) </li></ul></ul><ul><ul><li>F= e -rT *(.5(15) + .5(0)) = e -rT (7.50) = $7.31 </li></ul></ul>
  4. 4. Two Period Call Option <ul><li>Let u = 1.1,d=.9,T1-T2=T=.5,r=.05,S=$100,X=100 </li></ul><ul><li>-------------------------------- $121 (21)Node D </li></ul><ul><li>------------------$110 (10) Node B </li></ul><ul><li>$100-Node A --------------$99 (0) Node E </li></ul><ul><li>------------------$90 (0) Node C </li></ul><ul><li>---------------------------------$81 (0) Node F </li></ul><ul><li>P = (e r(T1-T2) -d)/(u-d) = (1.025-.9)/(1.1-.9) </li></ul><ul><li>P = .625 ; 1-P = .375 </li></ul><ul><li>At Node C,E and F value of option is zero. STEP ONE </li></ul><ul><li>Node B, Value of option f = e -rT (pf up + (1-p)f down ) </li></ul><ul><li>F = e -.05*.5 (.625(21) +.375(0)) = .975*13.125 = 12.80 </li></ul>
  5. 5. STEP TWO <ul><li>Note p, r, T, u and d are unchanged </li></ul><ul><li>Value of option at node A is </li></ul><ul><li>F = e-.05*.5 (pf up + (1-p)f down ) </li></ul><ul><li>F = .975(.625(12.80) +(.375)(0)) </li></ul><ul><li>F = .975(8.00) = $7.80 </li></ul><ul><li>What would have happened if the value of the option at Node B was $9.80? </li></ul>
  6. 6. Put Option – One Period <ul><li>Time to expiration is six months </li></ul><ul><li>Riskfree rate is 4%, X=50, S=P=50 </li></ul><ul><li>U = 1.2 , d= .9 </li></ul><ul><li>--------------- $60 (0) Node B </li></ul><ul><li>$50 Node A </li></ul><ul><li>--------------- $45 (5) Node C </li></ul><ul><li>P = (e rT – d)/(u-d) =(e .04*.5 -.9)/(1.2-.9) </li></ul><ul><li>P=.12/.3 = .40 (1-P) = .6 </li></ul><ul><li>F = e -rT (.4(0) + .6(5)) = $3/1.02 = $2.94 </li></ul>
  7. 7. Two Period Put Option <ul><li>T = 1 year between nodes </li></ul><ul><li>R = 4% , u = 1.3 , d=.7 , S=P=80, X=75 </li></ul><ul><li>----------------------------- $135.20 (0) D </li></ul><ul><li>----------------- $104 (0) B Expires worthless </li></ul><ul><li>$80 A -------------------- $72.80 (2.20) E </li></ul><ul><li>----------------- $56 (19) C In the money $19 </li></ul><ul><li>----------------------------- $39.20 (35.80) F </li></ul><ul><li>Note : Use backwardation to solve for the option value today. We know the values of the option at expiration. </li></ul>
  8. 8. STEP ONE <ul><li>At Node B, F = e –rT {pf up + (1-p)f down } </li></ul><ul><li>However, we must solve for p first. </li></ul><ul><li>P = (e rT – d)/(u-d) , where e rT = e .04*1 . </li></ul><ul><li>P = (1.041 - .7)/(1.3-.7) = .568 </li></ul><ul><li>Therefore, 1-p = .432 </li></ul><ul><li>F = e –rT { .568(0) + .432(2.20)} = $.91 </li></ul>
  9. 9. STEP TWO <ul><li>Node C , T = T1-T2 = 1 year </li></ul><ul><li>F = e -rT (pf up + (1-p)f down ),e -.04*1 = 1.041 </li></ul><ul><li>P = (1.041-.7)/(1.3-.7) = .341/.6 = .568 </li></ul><ul><li>Note: P is identical to P in the previous step. </li></ul><ul><li>1-P = .432 </li></ul><ul><li>F = (.568(2.20) + .432(35.80)/1.041 </li></ul><ul><li>F = (1.25 + 15.47)/1.041 = $16.06 </li></ul><ul><li>What if this is an American option ? </li></ul>
  10. 10. STEP THREE-PUT OPTION <ul><li>Note p, 1-p, u,d, and e -rT same as before. </li></ul><ul><li>Therefore, at Node A </li></ul><ul><li>F = e -rT (pf up + (1-p)f down ) </li></ul><ul><li>F={.568($.91)+.432(16.06)}/1.041=$ 7.46/1.041 = $7.17 </li></ul><ul><li>If it is a European put the value is $7.17 </li></ul><ul><li>F = {.568($.91) + .432($19)}/1.041 = $8.38 , if it is an American put </li></ul>
  11. 11. Call Options on Stock Indices <ul><li>Assume the stock index pays a dividend rate of q (a steady stream of dividends) </li></ul><ul><li>The index value is $14,000, X=14,500, T=3 months, r=4%, q=2%,u=1.1 and d=.9 </li></ul><ul><li>----------------- $15,400 (900) B </li></ul><ul><li>$14,000 A </li></ul><ul><li>------------------$12,600 (0) C </li></ul><ul><li>P = {e (.04-.02)*.25 - .9}/(1.1-.9) = .525 </li></ul><ul><li>1-P = .475 </li></ul><ul><li>F = e -rT {pf up + (1-p)f down } </li></ul><ul><li>F = e -.04*.25 {.525*900 + .475(0)} = 472.50/1.01 = $467.80 </li></ul>
  12. 12. Important Note <ul><li>It is common practice to use the following </li></ul><ul><li>U = e vt1/2 , where v represents volatility or standard deviation and t1/2 is the square root of T. </li></ul><ul><li>And D = 1/U </li></ul>
  13. 13. Call Option on Canadian Currency <ul><li>Assume the Canadian dollar is .95 to the US dollar. That is one Canadian dollar buys 95 cents American. </li></ul><ul><li>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. </li></ul><ul><li>P = (a-d)/(u-d) , where a = e (r-r c )*T </li></ul><ul><li>U = e .10*(.25)1/2 = e.05 = 1.051271, round to 1.05 </li></ul><ul><li>Note (.25)1/2 is the square root of .25 = .5 </li></ul><ul><li>D = 1/U = 1/1.05 = .952 </li></ul>
  14. 14. Currency Call Option, cont. <ul><li>For the value of a = e (.04-.06)*.25 = .98 </li></ul><ul><li>P = (a-d)/(u-p) = (.98-.952)/(1.051-.952) = </li></ul><ul><li>.028/.099 = .283 ; 1-P = .717 </li></ul><ul><li>------------------------- .998 (.038) B </li></ul><ul><li>.95 A </li></ul><ul><li>------------------------- .904 (0) C </li></ul><ul><li>F = e -.04*.25 {.283(.038) + .717(0)} = .0106 </li></ul>
  15. 15. Call Option on Futures Contracts <ul><li>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. </li></ul><ul><li>Therefore, e -rT = 1 , that is a = 1 and </li></ul><ul><li>P = (1-d)/(u-d) </li></ul><ul><li>Assume an asset has a volatility of .4, r=.05 and this is a 9 month call option. </li></ul>
  16. 16. Binomial Call Option on Oil <ul><li>T = 9/12 = .75 </li></ul><ul><li>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. </li></ul><ul><li>D = 1/U = .707 </li></ul><ul><li>P = (1-d)/(u-d) = .293/.707 = .414 </li></ul><ul><li>1-P = .586 </li></ul><ul><li>F = e -rT {pf up + (1-p)f down } </li></ul><ul><li>F= e -.05(.75) {.414(28.12) + .586(0)} = $11.21 </li></ul>
  17. 17. Graph a Condor <ul><li>Similar to a butterfly spread </li></ul><ul><li>Buy call at 70, buy call at 85 </li></ul><ul><li>Write a call at 75, write a call at 80 </li></ul><ul><li>Flat top not pointed like a butterfly spread. </li></ul><ul><li>What is the payoff difference between a butterfly spread and a condor ? </li></ul>

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