Question

3-step binominal tree For the case of call options, and S0 = 100, r = 0.07, q = 0.05, sigma = 0.3, T = 1 What is the Early exercise premium when K = 125? In the following, S0 is the stock price in dollars as of today, K is the strike price in dollars, r is the continuously-compounded risk-free interest (as a decimal), q is the continuous dividend yield (as a decimal), sigma is the volatility (as a decimal) and T is the time to maturity in years.

Answer 1

First we have to calculate the European Call option price with the given data.

The given excel sheet calculate the price of European call option. The formula sheet is also given below.

The Price of European Call is $13.99

ДА C D E F G H I J K L Inputs call or put? clp C 1 Params Asset price Strike price Time/step, At Volatility, o Riskfree rate, Dividend, a $100.00 $98.00 0.33 yrs 30.0% 7.0% 5.0% 1.189 0.841 1.007 0.4760 .5240 1- p 0 Node Time (yrs) 0.0 0.33 0.67 168.1380601 70.1380601 $141.40 $43.32 $118.91 $25.12 118.9109944 20.91099436 Stock Option $100.00 $13.999 $100.00 $9.72 84.09651314 $84.10 $4.52 $70.72 $0.00 59.47493384

The formula used are:

A B C D Inputs call or put? clp c =IF(C3="c"1,-1) Params 98 Asset price | 100 Strike price Time/step, At =4/12 Volatility, Riskfree rate, r 0.07 Dividend, a 0.05 0.3 u=EXP(C8*SQRT(C7)) d =EXP(-C8*SQRT(C7)) a =EXP((C9-C10)*C7) p =(F8-F7)/(F6-F7) 1-p =1-F9 Node Time (yrs) 0 =$C$7 =$C$7*2 =$C$7*3 =116*F6 =MAX(SE$3*(K13-$C$6,0) =G18*$F$6 =($F$9*K14+($F$10*K19))*EXP(-$C$9*$C$7) =E20*$F$6 =($F$9*117+($F$10*121))*EXP(-$C$9*$C$7) 1=116*F7 =MAX(SE$3*(K18-$C$6,0) = 5 Stock Option Seption =($F$9*G19+($F$10*G23))*EXP(-$C$9*$C$7 SF99-G19+($F$10“G23 =G22*F6 =($F$9*K19+($F$10*K23))*EXP(-$C$9*$C$7) =E 20*$F$7 =($F$9*121+($F$10*125))*EXP(-$C$9*$C$7) =120*F7 =MAX($E$3*(K22-$C$6,0) =G22*F7 =($F$9*K23+($F$10*K27))*EXP(-$C$9*$C$7) =124*F7 =MAX($E$3*(K26-$C$6,0)

Then we have to calculate the American Call option price with the given data.

The given excel sheet calculate the price of American call option. The formula sheet is also given below.

The Price of American Call is $14.015

Inputs call/put? c/p Asset Strike Time/step, At Volatility, o Rf rate, Div yield, a $100.00 Params $98.00 0.33 yrs 30.0% 7.0% 5.0% 1.007 47.6% 52.4% 1.189 0.841 Stock 0.00 - 2.0 3 168.14 118.91 84.10 59.47 $141.40 $100.00 $70.72 $118.91 $84.10 $100.00 Option 0.01 1.0 2.0 3 2 70.1380601 20.91099436 0.00 0.00 $43.40 $9.72 $0.00 $25.16 $4.52 $14.015

The formula used are:

LA B Inputs call/put? c/p =IF(C3="c",1,-1) Params Params Asset 100 Strike 98 Time/step, At =4/12 Volatility, 0.3 Rf rate, 0.07 Div yield, 0.05 a =EXP((C9-C10)*c7) p =(E6-E10)/(E9-E10) 1-0 =1-E7 u=EXP|C8*SORT(CZ)) d =EXP(-C8*SQRT(C7)) I Stock =E14*E9 =E14*E10 =E15*E10 =E16*E10 =D15*$E$9 =D15*$E$10 =D16*$E$10 =C16*$E$9 =C16*$E$10 O -C5 Option O =MAX((E14-$C$6)*$D$3,($E$7*F20+($E$8*F21))*EXP(-$C$9*$C$7)) =MAX((D15-$C$6)*$D$3,($E$ 7*E21+($E$8*E22))*EXP(-$C$9*$C$7)) =MAX((E15-$C$6)*$D$3,($E$7*F21+($E$8*F22))*EXP(-$C$9*$C$ 7)) =MAX((C16-$C$6)*$D$3,($E$ 7*D22+($E$8*D23) *EXP(-$C$9*$C$ 7)) =MAX((D16-$C$6)*$D$3,($E$ 7*E22+($E$8*E23))*EXP(-$C$9*$C$7)) =MAX((E16-$C$6)*$D$3,($E$7*F22+($E$8*F23))*EXP(-$C$9*$C$7)) =MAX($D$3*(F13-$C$6),0) =MAX($D$3*(F14-$C$6),0) =MAX($D$3*(F15-$C$6),0) =MAX($D$3*(F16-$C$6),0)

Early Exercise Premium is the extra premium we pay to get an early exercise. It is the extra premium pay for an american option over the price of Eurepean option.

Early Exercise Premium = American Call Option Price - European Call Option Price

= $14.015 - $13.99

= $0.025

Note: Give it a thumbs up if it helps! Thanks in advance!