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2. Lattice Methods 2.1 One-step binomial tree model (Hull, Chap. 11, page 241) Math6911 S08, HM Zhu Outline 1. No-Arbitrage Evaluation 2. Its relationship to “risk-neutral valuation”. 2 Math6911, S08, HM ZHU A Simple Binomial Model A stock price is currently $20 After 3 months it will be either $22 or $18 A 3-month European call option on the stock with K=$21 Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0 3 months 3 Math6911, S08, HM ZHU 1 No-Arbitrage Evaluation Consider the risklessPortfolio: long ∆ shares short 1 call option Portfolio Value 22∆ –1 18∆ Portfolio is riskless when 22∆ –1 = 18∆ i.e. ∆ = 0.25 4 Math6911, S08, HM ZHU Valuing the Portfolio and the Option (Risk-Free Rate r = is 12%) 1. What is the value of the portfolio at the expiry of the option? 2. What is the value of the portfolio today? 3. What is the value of the option today? 5 Math6911, S08, HM ZHU General One-Step Binomial Tree (No-arbitrage evaluation) A derivative lasts for time T and is dependent on a stock S0 u (u > 1) vu S0 v (d < 1) S0d vd 6 Math6911, S08, HM ZHU 2 General One-Step Binomial Tree (No-arbitrage evaluation) Consider the portfolio: long ∆ shares, short 1 derivative S0 u∆ – vu S0∆– v S0d∆ – vd The portfolio is riskless when S0u ∆ – vu = S0d ∆ – vd or v − v ∆ = u d S 0 u − S 0 d 7 Math6911, S08, HM ZHU General One-Step Binomial Tree (No-arbitrage evaluation) Value of the portfolio at time T: S0u ∆ – vu Value of the portfolio today: –rT S0 ∆– v =(S0u ∆ – vu )e Hence –rT v = S0 ∆ –(S0u ∆ – vu )e 8 Math6911, S08, HM ZHU Another approach: risk-neutral evaluation Substituting for ∆, we obtain S u −rT 0 Where v = []pvu + ()1− p vd e p vu S0 rT v ed− S d p = (1 – 0 p ) ud− vd • The variables p and (1 – p ) can be interpreted as the risk- neutral probabilities of up and down movements • The value of a derivative can be considered as its expected payoff in a risk-neutral world discounted at the risk-free rate 9 Math6911, S08, HM ZHU 3 Comments and example • Note: When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant • Use risk-neutral evaluation to calculate option price in the original example (r = 12%, T = 3/12): S0u = 22 p vu = 1 S0 v (1 S0d = 18 – p ) vd = 0 10 Math6911, S08, HM ZHU Original example revisited (risk-neutral evaluation) erT −d e0.12×0.25 −0.9 p = = = 0.6523 u −d 1.1−0.9 S0u = 22 523 vu = 1 0.6 S0 v 0.3 S0d = 18 477 vd = 0 Therefore, the value of the option is v = e -0.12×3 / 12 []0.6523 ×1 + 0.3477 × 0 = 0.633 11 Math6911, S08, HM ZHU Risk-neutral evaluation • In a risk-neutral world, all investors are indifferent to risk, i.e., investors require no compensation for risk and the expected returns of all securities is the risk- free interest rate • The option price resulted from risk-neutral evaluation is correct not only in the risk-neutral world, but also in other world as well • Further reading: Cox, Ross, and Rubinstein, “Option pricing: a simplified approach”, J. of Financial Economics 7:229-264 (1979) 12 Math6911, S08, HM ZHU 4 2. Lattice Methods 2.2 Binomial Methods (Hull, Sec 11.7, Sec. 17.1) Math6911 S08, HM Zhu Outline 1. Extension to N-step binomial method 2. Determine the parameters 3. Binomial methods in option pricing 14 Math6911, S08, HM ZHU Binomial Trees • Binomial trees are frequently used to approximate the movements in the price of a stock or other asset • In each small interval of time (∆t) the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d p uS S 1 – p dS Stock movements in time ∆t 15 Math6911, S08, HM ZHU 5 Assumption 1 To model a continuous walk by a discrete walk in the following fashion : u >1 d <1 p 2 u S p : probability p uS S p1-p u d S 1-p dS 1-p d2S time 0 ∆t2 ∆t T=M ∆t M≥30 16 Math6911, S08, HM ZHU Assumption 1 (cont) The parameters ud,, and p in such a way that mean and variance of the discrete random walk coincide with those of the continuous random walk 17 Math6911, S08, HM ZHU Assumption 2: Risk-Neutral Valuation We may assume investors are risk-neutral (even though most are not) In terms of option pricing (or other derivatives), we can assume: • The expected return from all traded securities is the risk-free interest rate • Future cash flows can be valued by discounting their expected values at the risk-free interest rate The tree is designed to represent the behavior of a stock price in risk-neutral world 18 Math6911, S08, HM ZHU 6 Binomial Method: Assumption 2 Under the assumption of risk-neutral world, the value Vmtm of the put option at ∆= the expected value Vmtm+1of option at ()+1 ∆ discounted by the risk-free interest rate r mrtm−∆ +1 VEeV= ⎣⎦⎡⎤ 19 Math6911, S08, HM ZHU Tree parameters p, u, and d for non- dividend paying stock The parameters u, d, and p in such a way that mean and variance of the discrete random walk coincide with those of the continuous random walk in risk-neutral world, i.e.,: ES⎡⎤⎡⎤mm++11 S= ES mm S cb⎣⎦⎣⎦ and var⎡⎤⎡⎤ Smm++11 S= var S mm S cb⎣⎦⎣⎦ The parameters pu, , and d must give correct values for the mean and variance of stock price changes during ∆t. 20 Math6911, S08, HM ZHU Tree parameters u, d, and p ES⎡⎤⎡⎤mm++11 S= ES mm S cb⎣⎦⎣⎦ rt∆ m m ⇒=+−eS⎣⎦⎡⎤ pu()1 pdS ⇒=+−epupdrt∆ ()1(1) 21 Math6911, S08, HM ZHU 7 Tree parameters u, d, and p var⎡⎤⎡⎤ Smm++11 S=⇒ var S mm S cb⎣⎦⎣⎦ mrtt222222∆∆σ 2 rtm ∆ ⇒−=+−−()Se() e11⎣⎦⎡⎤ pupde() () S 2 ⇒=+−epupd222rt∆+σ ∆ t ()12 () In order to determine these three unknowns uniquely, we need to add another condition (somewhat arbitrary). Note: ST follows a lognormal distribution. It can be shown that 1) The expected value of ST : µ()Tt− E()SSeT = 2) The variance of ST : 2 var S=− S2 e2µσ()Tt−−⎡⎤ e () Tt 1 22 ()T ⎣⎦ Math6911, S08, HM ZHU 1 One further condition: u = d 1 One of the popular choices is u = . d It generates a symmetric tree w.r.t. the initial value S0 : ⎛⎞σσ22 ⎛⎞ σσ∆+tr⎜⎟ − ∆ t − ∆+ tr ⎜⎟ − ∆ t rt∆ ⎜⎟22 ⎜⎟ ed− ue=⎝⎠ ,= de ⎝⎠ , p= . ud− [reference: Cox, Ross and Rubinstein (1979)] Often, ignoring the terms in ∆∆tt2 and higher power of gives edrt∆ − ue=σσ∆−∆tt ,= de , p= . ud− Note: If ∆<t is too large, p(-p) or 1 0. The binomial method fails. 23 Math6911, S08, HM ZHU 1 Symmetric Tree: u = d 24 Math6911, S08, HM ZHU 8 1 Another further condition: p = 2 1 One of the popular choices is p = . 2 It generates an asymmetric tree, oriented in the direction of the drift: 221 ue=1rt∆∆+− eσσ t 1 ,=1 de rt ∆∆ −− e t 1 , p = ()()2 []Kwok, 1998; Wilmott et al, 1995 Note: If ∆t is too large, d <0. The binomial method fails. 25 Math6911, S08, HM ZHU 1 Asymmetric Tree: p = 2 26 Math6911, S08, HM ZHU 2. Lattice Methods 2.3 General Binomial Tree Models (Hull, Sec. 17.1, , page 391) Math6911 S08, HM Zhu 9 The Idea of Binomial Methods 1. Build a tree of possible values of asset prices and their probabilities, given an initial asset price 2. Once we know the value of the option at the final nodes, work backward through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate Advantages: easily deal with possibility of early exercise and with dividend payments 28 Math6911, S08, HM ZHU Binomial Method for valuing non- dividend-paying options Step 1. Build up a tree of possible asset prices for all time points 0 and their probabilities, given the initial price S0 mnmn− 0 Sn ==ud S0 for n0 ,, 1 ,m m Sn : the n-th possible value of S at m-th time-step 29 Math6911, S08, HM ZHU Build A Complete Tree of Asset Prices 4 3 S0u S0u 2 S0u 2 S0u S0u S0u S0 S0 S d S0 0 S0d 2 S0d 2 3 S0d S0d Tree structure used the relationship u = 1/d 4 S0d 30 Math6911, S08, HM ZHU 10 Step 2: Valuing an European option: Step 2. Use this tree to calculate the possible value of option at expiry. Then work back down the tree to caculate the price of the option 2.1. Value the option at expiry, i.e, time-step Mt∆ MM Put:VmaxKS,n,,Mnn==() - 0 0 1… Call: 2.2. Find the expected value of the option at a time step prior to expiry mrtmrtm⎡⎤−∆ +11 −∆⎡⎤ + m+1 VEeVnn==+−⎣⎦ e⎣⎦ pV n+1 ()1 pV n and so on, back to time-step 0. 31 Math6911, S08, HM ZHU Example 1: European 2-year Put Option S0 = 50; K = 52; r =5%; σ = 30%; T = 2 years; ∆t = 1 year The parameters imply ue == 03.
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