Question

Finance - option pricing:

Alex is looking to price a 6-month European put option with a strike price of $29 on a share in Omni Consumer Products (OCP). The current price for an OCP share is $30. Alex has used past data and his own judgement to estimate the volatility of these shares to be 15% per annum. The risk-free continuously compounding interest rate is 5% per year. a) Construct a 3-step binomial tree showing the possible share prices over the next 6 months. Also, clearly show the corresponding probabilities for an upward and a downward movement in the share price. As a check the numbers 26.5419 and 31.8945 should appear in your tree. b) Using your binomial tree from pait a) and the information given above, calculate the corresponding 3- step tree containing the corresponding option values, and clearly state the fair price of the option according to the given information. As a check, the numbers 0.1537 and 0.7820 should appear in your tree. c) Using the binomial tree constructed in part b), estimate the time zero (t = 0) values of delta (A), theta (6), and gamma (T) ONLY. You do NOT need to calculate nu (v) and tho (2).

Answer 1

Use spreadsheet for the ease in computations. Enter values and formulas in the spreadsheet as shown in the image below.

D European put option on a non dividend paying stock 2 p= 0.5 3 Basic Parameters Binomial Parameters 4 AmEuro= TreeSize (fixed)= 5 CallPut= put dt= =B10/D4 6 AssetType= Stock u= =EXP(SIGMA*SQRT(dt)) 7 Asset Price =1/u 8 DivYld DE= =EXP((-IntRate)*dt) 9 Strike = 29 =(EXP((IntRate-DivYld)*dt)-d)/(u-d) 10 YrToExp= P= =(EXP(ExpRet-DivYld) * dt)-d)/(u-d) 11 IntRate= 0.05 BinAE= =IF(LEFT(B4,1)="A",1,0) 12 ExpRet= BinCP= =IF(LEFT(B5,1)="C",1,-1) 13 Volatility= 10.15 Up Down 14 Returns =u-1 =d-1 15 Exact Approx 16 True ExpRet =ExpRet*dt =TrueProba*D14+(1-TrueProba) *E14 17 RN ExpRet =IntRate*dt =p*D14+(1-p) *E14 18 RN StDev =Vol*SORT(dt) =SQRT(p*(D14-E17) 2+(1-2)*(E14-E17) 2) 19 20 Period (answer - a) lo 21 Stock price 22 =B22%u =C22*1 =D22%u =B22*d =C23*u I=D23*u 24 =C23*d =D24*u 25 =D24*d 26 Period (answer - b) O 27 European put 28 =MAX(BinAE BinCP*(B22-K).(p*C28+(1-2)*C29) *EXP(-IntRate* dt)) =MAX(BinAE*BinCP*(C22-K).(p*D28+(1-p)*D29) *EXP(-IntRate* dt))=MAX(BinAE*BinCP (D22-K).(*E28+(1-p)*E29) *EXP(-IntRate* dt))=MAX(0.BinCP*(E22-K)) 29 =MAX(BinAE*BinCP*(C23-K).p*D29+(1-p) *D30)*EXP(-IntRate*dt)) =MAX(BinAE*BinCP*(D23-K).(p*E29+(1-p)*E30)*EXP(-IntRate*dt)) =MAX(0.BinCP*(E23-K) =MAX(BinAE*BinCP*(D24-K).p*E30+(1-P) *E31)*EXP(-IntRate*dt)) =MAX(0.BinCP*(E24-K) 31 =MAX(0.BinCP* (E25-K)) 32 33 Answer - C 34 di =(((LN(S/K)+((IntRate+((Vol*Vol)/2))*YrToExp)/(Vol*(YrToExp 0.5)) 35 Ndi) =NORMDIST(B34,0,1,TRUE) 36 Delta = N(di) - 1 =B35-1 37 Theta 38 = ((fud - f)/(2(T/N)) =(D29-B28)/(2*(YrToExp/TreeSize)) 39 40 Gamma =(((D28-D29)/(D22-D23))-((D29-D30)/(D23-D24))/((D22-D24)/2) 23 30

The obtained result is provided below.

1 European put option on a non dividend paying stock Binomial Parameters TreeSize (fixed) dt= E put Stock 30.00 0.0% 29 0.50 5.0% 0.0% 15% DF= p= P= BinAE= BinCP= 2 3 Basic Parameters 4 AmEuro= 5 CallPut= 6 AssetType= 7 Asset Price 8 DivYld= 9 Strike = 10 YrToExp= 11 IntRate= 12 ExpRet= 13 Volatility= 14 15 16 17 18 19 20 Period (answer - a) 21 Stock price 22 23 24 0.1667 1.06315111 0.9406 0.9917 0.5530 0.4847 0 Returns Up 6.32% Exact 0.00% 0.83% 6.12% Down -5.94% Approx 0.00% 0.84% 6.09% True ExpRet RN ExpRet RN StDev 0 2 3 30.0000 31.8945 28.2180 33.9087 30.0000 26.5419 36.0501 31.8945 28.2180 24.9653 3 25 0 26 Period (answer - b) 27 European put 28 0.6043 0.1537 1.1731 29 0.0000 0.3467 2.2175 0.0000 0.0000 0.7820 4.0347 30 31 32 33 Answer - C 34 d1 35 N(di) 36 Delta = N(dl)-1 37 Theta 38 = ((fud - fy/(2(T/N)) 0.608362163 0.72852635 -0.27147365 -0.773035236 39 0.122792225 40 Gamma 41 42 fuu-fud – fud- fad Suu-Sud Sud-Sdd 43 Gamma =- 44 (Suu – Sdd)/2