Option Pricing Using the Binomial Model

Transcription

1 Finance 400 A. Penati - G. Pennacci Option Pricing Using te Binomial Moel Te Cox-Ross-Rubinstein (CRR) tecnique is useful for valuing relatively complicate options, suc as tose aving American (early exercise) features. In tese notes we sow ow an American put option can be value. Recall tat CRR assume tat over eac perio of lengt t, stock prices follow te process us wit probability q S (1) S wit probability 1 q Te results of our earlier analysis sowe tat te assumption of an absence of arbitrage allowe us to use a risk-neutral valuation meto to erive te value of an option. In general, tis meto of valuing a erivative security can be implemente by: i) setting te expecte rate of return on all securities equal to te risk-free rate ii) iscounting te expecte value of future casflows generate from i) by tis risk-free rate Forexample, supposeweexaminetevalueoftestock, S, in terms of te risk-neutral valuation meto. Define r as te continuously-compoune interest rate (te rate of return on a risk-free asset). Ten we ave S = e r t Ê [S t+ t ] (2) = e r t [pus +(1 p)s] were Ê [ ] is te expectations operate uner te conition tat te expecte rate of return on all assets equals te risk-free interest rate, wic is not necessarily te assets true expecte rates of return. Re-arranging (2) we obtain e r t = pu +(1 p) (3) 1

2 wic implies p = er t u (4) Tis is te same formula for p as was erive earlier (wit te risk-free return now in terms of a continuously-compoune interest rate). Hence, risk-neutral valuation is consistent wit tis simple example. To use te CRR moel to value actual options, te parameters u an must be calibrate to fit te variance of te unerlying stock. Wen estimating a stock s volatility, it is often assume tat stock prices are lognormally istribute. Tis implies tat te continuously-compoune rate of return on te stock over a perio of lengt t, given by ln (S t+ t ) ln (S t ), is normally istributewitaconstantper-periovarianceof tσ 2. Tis constant variance assumption is also consistent wit te Black-Scoles option pricing moel. Tus, te sample stanar eviation of a time-series of istorical log stock price canges provies us wit an estimate of σ. Base on tis value of σ, approximatevaluesofu an tat result in te same variance for a binomial stock price istribution are 1 u = e σ t (5) = 1 u = e σ t Hence, conition (5) provies a simple way of calibrating u an to te stock s volatility, σ. Now consier te pat of te stock price. Because we assume u = 1, te binomial process for te stock price as te simplifie form: 1 Tat te values of u an in(5)resultinavarianceofstockreturnsgivenbyσ 2 t for sufficiently small t can be verifie by noting tat, in te binomial moel, te variance of te en-of-perio stock price is E St+ t 2 E [St+ t ] 2 = qu 2 S 2 +(1 q) 2 S 2 [qus +(1 q) S] 2 = S 2 qu 2 +(1 q) 2 [qu +(1 q) ] i 2ª = S 2 e ³e α t σ t + e σ t 1 e 2α t,wereq = e α t an α is te (continuously-compoune) expecte³ rate of return on te stock per i unit time. Tis implies tat te variance of te return on te stock is e α t e σ t + e σ t 1 e 2α t. Expaning tis expression in a series using e x = 1 + x x x an ten ignoring all terms of orer ( t) 2 an iger, it equals tσ 2. 2

3 u 4 S u 3 S u 2 S u 2 S us us S S S (6) S S 2 S 2 S 3 S 4 S Given te stock price, S, an its volatility, σ, te above tree or lattice can be calculate for any number of perios using u = e σ t an = e σ t. Nowwecannumericallyvalueanoptionontisstockbystartingattelastperioan working back towar te first perio. Recall tat a put option tat is not exercise early will ave a final perio (ate T )value P T =max[0,x S T ] (7) TevalueofteputatateT t is ten te risk-neutral expecte value iscounte by e r t. P T t = e r t Ê [P T ] (8) i = e r t ppt u +(1 p) PT However, wit an American put option, we nee to ceck weter tis value excees te value 3

4 of te put if it were exercise early, wic is ii P T t =max X S T t,e r t ppt u +(1 p) PT (9) Let us illustrate tis binomial valuation tecnique wit te following example: A stock as a current price of S =$80.50 an a volatility σ =0.33. If t = 1 9 year, ten u = e.33 9 = e.11 = an = 1 u = Tuste3-periotreefortestockpriceis Date : S = Next, consier valuing an American put option on tis stock tat matures in τ = 1 3 (4 monts) an as an exercise price of X =$75. Assumetatterisk-freerateisr =9. Tis implies p = er t u = e =.5181 Wecannowstartatate3anbeginfilling in te tree for te put option. 4

5 Date : P uuu P P u P P uu P u P P uu P u P Using P 3 =max[0,x S 3 ], we ave Date : P P u P P uu P u P

6 Next, using ii P 2 =max X S 2,e r t pp3 u +(1 p) P3 Date : P u P 1.37 P Note tat at P te option is exercise early since ii P =max X S 2,e r t pp3 u +(1 p) P3 =max[ , 9.65]=$10.40 Next, using ii P 1 =max X S 1,e r t pp2 u +(1 p) P2 6

7 Date : P Note tat te option is not exercise early at P since ii P =max X S 1,e r t pp2 u +(1 p) P2 =max[ , 5.66]=$5.66 Finally, we calculate te value of te put at ate 0 using ii P 0 =max X S 0,e r t pp1 u +(1 p) P1 = max[ 5.5, 3.03] = $3.03 an te final tree for te put is 7

8 Date : One can generalize te above proceure to allow for te stock (or portfolio of stocks suc as a stock inex) to continuously pay iviens tat ave a per unit time yiel equal to δ, tat is, for t sufficiently small, te owner of te stock receives a ivien of δs t. For tis case of a ivien-yieling asset, we simply reefine p = e(r δ) t u (10) Tis is because wen te asset pays a ivien yiel of δ, its expecte risk-neutral appreciation is e (r δ) t rater tan e r t. For te case in wic a stock is assume to pay a known ivien yiel, δ, atasingle point in time, tenifatei t is prior to te stock going ex-ivien, te noes of te stock price tree equal u j i j S j =0, 1,..., i. (11a) If te ate i t is after te stock goes ex-ivien, te noes of te stock price tree equal 8

9 u j i j S (1 δ) j =0, 1,...,i. (11b) Te value of an option is calculate as before. We work backwars an again ceck for te optimality of early exercise. 9