Lecture7_BinomialOptionPricing

# In j38 we find about p05177 1405 1 p 04823 1729 whats

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Unformatted text preview: n options – backward from step 9 (1) It’s easy to compute the payoff in steps 9, but how It’s about step 8? In J38, we find about p=0.5177 \$14.05 1-p= 0.4823 \$17.29 \$? What’s the price of a claim in this node that pays What’s \$14.05 with p and \$17.29 with 1-p in step 9? \$14.05 Since we know the probability now and it’s European Since option, we can easily compute the claim’s price = (\$14.05*0.5177+\$17.29*0.4823)*e-r δt = 15.54 (\$14.05*0.5177+\$17.29*0.4823)* 15-10 Pricing European options – backward from step 9 (2) As a result, we get that the following: As - Put option with strike = 50, it’s price = 2.25 Put - Call option with strike = 50, it’s price = 3.47 Call Check the result with put-call parity Check -qT c – p = S e -qT – K e -rT Left-hand side: c – p = 1.22 1.22 -qT Right-hand side: S e -qT – K e -rT Right-hand = 50*EXP(-3%*0.5) – 50*EXP(-8%*0.5) = 1.22 50*EXP(-3%*0.5) 50*EXP(-8%*0.5) So, we confirm that our results with the put-call parity So, 15-11 Exercise 1 Now, compute the European price of put and call Now, based on all the same set-up, but change the step number to 15 steps, see the “Exercise 1” tab number As shown in the “Exercise 1 - ans” tab: - Call price = 3.44 Call - Put price = 2.22 Put 15-12 Compare European option pricing methods (1) Comparing the binomial pricing and BS pricing (more in Comparing future lectures) future According to BS pricing According c = S e −qT N (d1 ) − K e − rT N ( d 2 ) p = K e −rT N ( −d 2 ) − S e −qT N (−d1 ) ln( S / K ) + ( r − q + σ 2 / 2)T where d1 = σT ln( S / K ) + ( r − q − σ 2 / 2)T d 2 = d1 − σ T = σT Check the “BSpricing” tab for BS-implied prices, c = 3.39 and p = 2.18 and 15-13 Compare European option pricing methods (2) Summary: Option Binomial 2 steps Binomial 9 steps Binomial 15 steps BS model Euro Call 3.08 3.47 3.44 3.39 Euro Put 1.86 2.25 2.22 2.18 We know that the BS model is the continuous-time solution, We continuous-time and our binomial model is the discrete-time approximation and Our summary supports that, with more discrete-time steps, Our we are able to approximate BS model results 15-14 VBA for CRR binomial trees – European options 15-15 VBA for European option pricing See the function “binomial_euro” in “VBA_Euro” See tab tab It gives the price 2.25, consistent with our trees Let’s use a “smaller” tree (2 steps) for illustration Call or Put (c or p) Stock price (S) Strike price (K) Maturity ( T ) riskfree Rate ( r ) Dividend payout Rate ( q ) Volatilty (sigma) Number of steps (n) p 100 100 1 5.0% 0.0% 20.0% 2 Step (dt) u d p emrdt 0.5000 1.1519 0.8681 0.5539 0.9753 15-16 Stock price in the tree 100 115.19 86.81 Put option price in the tree 4.66...
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## This document was uploaded on 03/03/2014.

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