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# Derivatives Binomial Option Pricing Model Examples

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• Very explcit example of calculating call and put option

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### Transcript

• 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