Problem1 Consider pricing put options on a stock.We use the binomial lattice tec
ID: 1170489 • Letter: P
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Problem1 Consider pricing put options on a stock.We use the binomial lattice technique to price options. The current price of the stock is $20. The strike price is $21 and the option maturity is 6 months. The risk-free interest rate is 12% per annum (compounded continuously)We shall take the length of each time step of the lattice ?t equal to!months. Also assume that the up and down parameters of the stock price lattice u and d are given by I.1 and 0.9 respectively (a) Draw the binomial lattice for the stock price. (b) Compute the risk-neutral probability q c) Calculate the European option price. (d) Calculate the American option price.Explanation / Answer
(a) Put Option Strike Price = K = $ 21, Current Price of Asset = $ 20, Risk-Free Interest rate = 12 % per annum, Option Maturity = 6 months, Time Period Length = 2 months, u = 1.1 and d = 0.9
(b) u = 1.1 and d = 0.9, risk-free rate = 12 % per annum and length of each period = 2 months
Risk Neutral Probability (of up movement) = q = [EXP{0.12 x (2/12)} - 0.9] / [1.1 - 0.9] = 0.601
(c) Payoff at Node 7 = $ 0
Payoff at Node 8 = $ 0
Payoff at Node 9 = $ 3.18
Payoff at Node 10 = $ 6.42
Present Value(PV) of Payoff at Node 4 = 0.601 x 0 + 0.399 x 0 / EXP[0.12 x (2/12)] = $ 0
PV of Payoff at Node 5 = 0.601 x 0 + 0.399 x 3.18 / EXP[0.12 x (2/12)] = $ 1.244
PV of Payoff at Node 6 = 0.601 x 3.18 + 0.399 x 6.42 / EXP[0.12 x (2/12)] = $ 4.384
PV of Payoff at Node 2 = 0.601 x 0 + 0.399 x 1.244 / EXP[0.12 x (2/12)] = $ 1.02
PV of Payoff at Node 3 = 0.601 x 1.244 + 0.399 x 4.384 / EXP[0.12 x (2/12)] = $ 2.447
PV of Payoff at Node 1 = Put Option Price = 1.02 x 0.601 + 2.447 x 0.399 / EXP[0.12 x (2/12)] = $ 1.558
(d) American option differs from European option in the fact that the former can be exercised before its maturity. Hence, the pricing for an American Option would involve comparing the option's intrinsic value at each node to the PV of payoff at that particular node. The greater of the two quantity would be selected for calculation of expected payoff and then discounted period wise to arrive at the put option's price.
NOTE: IV denotes intrinsic option value and PVP denotes the present value of payoff.
Payoff at Node 7 = $ 0
Payoff at Node 8 = $ 0
Payoff at Node 9 = $ 3.18
Payoff at Node 10 = $ 6.42
Present Value(PV) of Payoff at Node 4 = 0.601 x 0 + 0.399 x 0 / EXP[0.12 x (2/12)] = $ 0
PV of Payoff at Node 5 = 0.601 x 0 + 0.399 x 3.18 / EXP[0.12 x (2/12)] = $ 1.244
PV of Payoff at Node 6 = 0.601 x 3.18 + 0.399 x 6.42 / EXP[0.12 x (2/12)] = $ 4.384
PV of Payoff at Node 2 = 0.601 x 0 + 1.244 x 0.399 / EXP[0.12 x (2/12)] = $ 0.486
PV of Payoff at Node 3 =0.601 x 1.244 + 0.399 x 4.8 / EXP[0.12 x (2/12)] = $ 2.61
PV of Payoff at Node 1 = American Put Option Price = 0.601 x 0.486 + 0.399 x 3 / EXP[0.12 x (2/12)] = $ 1.4596
t = 0 months t = 2 months t = 4 months t = 6 months $ 26.62 (node 7) $ 24.2 (node 4) $ 22 (node 2) $ 21.78 (node 8) $ 20 (node 1) $ 19.8 (node 5) $ 18 (node 3) $ 17.82 (node 9) $ 16.2 (node 6) $ 14.58 (node 10)Related Questions
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