Lecture7_BinomialOptionPricing

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online