Binomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge

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Adam McLaughlin
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1 Biomial Model Stock Price Dyamics The value of a optio at maturity depeds o the price of the uderlyig stock at maturity. The value of the optio today depeds o the expected value of the optio at maturity ad hece o the expected stock price at maturity. Objective : To specify a mathematical model for the movemets (dyamics) i the price of the uderlyig stock over time Discrete Radom Variables Probability ad Distributio Fuctios A discrete radom variable is a variable X which is to take o oe value by chace from a set of possible values The realisatio of a radom variable represets the value of X that occurs (X is o loger radom) The probabilities associated with a discrete radom variable X are give by the probability fuctio : Prob (X = x 1 ) = p (x i ) Distributio fuctio : Prob (X x 1 ) = x x 1 p(x 1 ) + p(x 2 ) + + p(x i ) Expectatios ad Momets The expected value of discrete radom variable is E[X] = x i p(x i ) If X is a radom variable ad g is a fuctio, the g(x) is also a radom variable ad i=1 E[g(X)] = g(x i )p(x i ) Mea of X is E(X) ad variace of X is Var[X] = E[(X E[X]) 2 Stochastic Processes A stochastic process is a sequece of related radom variable over time i.e there is a physical lik betwee them Let X be rv measured over time. The sequece of rv s X 0, X 1, X 2,... X T is a discrete time stochastic process The sample path of stochastic process(sp) represets the sequece of particular values of each X that occurs i.e process is o loger radom The sequece S 0,S 1, S 2 S T is a discrete time sp for the stock price over time To price optios we eed a model for the movemets i uderlyig stock over time Model movemets i stock prices over time by a stochastic process Number of differet stochastic processes may be used The Key Idea Riskless Hedge i=1 The riskless hedge is the key cocept uderlyig most of the stadard optio pricig models Icludig biomial model Based o the assumptio o o arbitrage Riskless hedge refers the fact that a combiatio of optios ad stock i appropriate proportios ca be used to replicate the payoff o a risk free bod Irrelevace of the Stock Expected Retur Same optio price whe probability upward movemet is 0.5 or 0.9. It is atural to assume that as the probability of a upward movemet as stock price icrease value of call icrease ad value of put decreases We are ot valuig the optio i absolute terms but calculatig its value i terms of the price of the uderlyig stock.

2 Probabilities of future up ad dow are already icorporated ito the price of the stock so we do ot eed to take them ito accout agai whe valuig the optio i term of stock price Implicatio of the riskless hedge approach for optio valuatio are : 1) Optio is risky but it is ot ecessary to determie the risk premium applicable to the optio i order to value it To value a risky asset we would ormally eed to kow the expected rate of retur o the asset Sice the payoff o the portfolio of stock ad optio is risk free, the retur o the portfolio is risk free rate Work out the value of the optio give curret stock price Stock is risky + optio is risky but Stock ad Optio together is riskless 2) I accordace with the o arbitrage assumptio this is a relative valuatio approach we value the optio relative to price of the stock but we do t require the stock to be fairly priced 3) If the observed market price of the optio differs from the NA price arbitrage opportuity 4) We did ot eed to kow the probability that stock price will rise or fall 5) The riskless hedge is the basis for black scholes ad biomial model Oe Period Model 1) Europea call optio o a stock with a strike K, maturity T years 2) The curret price of the stock is S ad over each period of legth years, the stock price either icreases by a factor u with probability p or decrease by a factor d with probability 1 q where u,d,q are costat 3) The stock pay o divideds over the life of the optio 4) Frictioless market 5) The omial risk free rate is costat at r% p.a where u > e rt > d 6) No arbitrage 7) Oe time period to maturity. Value of Optio at maturity If stock price rises : f u = max [us K] If stock price falls : f d = max [ds K] Value of the portfolio : Up movemet S 0 u f u Two are equal whe : Dow Movemet S 0 d f d S 0 u f u = S 0 d f d So = f u f d S 0 u S 0 d Value of the portfolio at the ed is same irrespective of whether stock price rises or falls. The value of the portfolio is therefore riskless. (ot affected by the stock price risk(stock irrelevace)) Cost of settig up the portfolio is S 0 u f Price of the value of the optio is f = pf u + (1 p)f d e rt where p = ert d u d p is the probability of a upward movemet/probability of optio price beig worth f u /f d ot probability of the stock price goig up Value of the optio does ot deped o the expected retur of the stock (depeds o the volatility expectatio) f is a relative pricig relatioship. For a give S, this is the correct f

3 is the delta of the optio ad specifies the umber of shares to be bought for each optio sold at the start of the period I practice, determiig u ad d from stock volatility u = e σ t where else d = 1/u Two Period Model 1) Europea call optio o a stock with a strike K, maturity T years 2) The curret price of the stock is S ad over each period of legth years, the stock price either icreases by a factor u with probability p or decrease by a factor d with probability 1 q where u,d,q are costat 3) The stock pay o divideds over the life of the optio 4) Frictioless market 5) The omial risk free rate is costat at r% p.a where u > e rt > d 6) No arbitrage 7) Two period of equal legth maturity Curret value of the optio f = p2 f uu + 2p(1 p)f ud + (1 p) 2 f dd e 2rT Sice u ad d are the same for each period, the tree recombies. No recombie trees explode as the umber of periods icrease The risk eutral hedge eeds to be re-balaced at the start of each period if the hedge is to be maitaied over the life of the optio. With put optio, recalculate the f u ad f d Delta ( ) of a stock optio is the ratio of the chage i the price of the stock optio to the chage i the price of the uderlyig stock. It is the umber of uits of the stock that we should old for each optio shorted i order to create riskless portfolio. This is kow as delta hedgig. Delta of a call optio is positive where as delta of put optio is egative. Purpose of valuig a optio a) The expected retur from all traded securities is the risk free iterest rate b) Future cash flows ca be valued by discoutig their expected values at the risk free iterest rate Three Period Model Risk Neutral Valuatio f = p3 f ddd + 3p 2 (1 p)f uud + 3p(1 p) 2 f ddu + (1 p) 3 f ddd e 3r t 1) Risk eutral world ivestor do ot care about risk i valuig assets. They do ot icrease the expected retur they require from a ivestmet to compesate for icreased risk. 2) The discout rate used for the expected payoff o a optio is the risk free rate. 3) I risk eutral world, the expected retur o the stock is the risk free rate 4) Risk eutral world assumes risk eutrality oly for the purpose of valuig the optio eve, though most ivestor are risk averse 5) It ca be show that value of the optio derived i risk eutral world is the same as the value of the optio i a risk averse world

4 The curret value of a optio is equal to the preset value of its expected future payoff i a risk eutral world usig the risk free rate as the discout rate f = e rt E [f T ] I the real world, it is ot easy to kow the correct discout rate to apply to the expected payoff but with risk eutral it is coveiet because we kow that i a risk eutral world, the expected retur o all assets is the risk free rate. The period model Curret value of the optio : f = j=0! j! ( j)! pf (1 p) j f,j e r t America Optio Procedure is to work back through the tree from the ed to the begiig, testig at each ode to see whether early exercise is optimal. The value of optio at the fial odes is the same as for Europea optio. At earlier otes, 1. Value is give by equatio 2. Payoff if exercise early. If payoff is more tha value of the optio, the exercise the optio Optio o other assets Cotiuous Divided Yield Optio o Stock Idices Currecies Futures p = e(r q)t d u d Stock payig a kow divided yield at rate q. The total retur from divided ad capital gais is r. So r q. Assume uderlyig idex provide a divided yield rate q. So similar assumptio as divided yield A foreig currecy regarded as a asset providig a yield at the foreig risk free rate of iterest. I this case a = e (r r f) t Cost othig to take a log or a short positio i a future cotract. It follows that i a risk eutral world, a future price should have a expected growth rate of zero. I here : p = 1 d sice future cost othig. u d Hedgig Strategies What happe if the market price of a optio differs from its theoretical price/value? The oe period biomial model is based o the 1 period riskless hedge portfolio π = S f theory So log bod = log Share + short 2 optio If we observed market price of the optio differs from the theoretical value of the optio f theory the a arbitrage opportuity exists The riskless hedge portfolio shows us how to lock i a arbitrage profit. 1) The positios of the hedge portfolio follows form the fact that you ca hedge a short optio with a equivalet log optio Sice log shares ad short bods is equivalet to log optio, it ca used to hedge a equivalet sort optio 2) Sometimes hedgig strategy calls for a reverse hedge. Here the above absolute positio are reversed but the relative positios are the same Hedge a log optio with a short stock ad log bod

5 3) The 1 period hedge states that a appropriate combiatio of optio ad stock ca be used to create a riskless portfolio This ca be exteded to that a appropriate combiatio of ay two of the securities (optios, stock ad bods) ca be used to create a portfolio equal to the third. 4) I the period model, delta chages from period to period. So riskless hedge portfolio eed to be rebalace at the start of each period 5) The π are ode depedet, the delta hedgig strategy is path depedet 6) This give a perfect hedge i theory 7) If hedge is maitaied by keepig the umber of optios costat ad buyig ad sellig bods, the miimum profit is the iitial mispricig of the optio. But if hedge is maitaied by keepig the umber of shares costat ad buyig/sellig optios ad bod a loss may result. 8) If theoretical price of optio is equal to the market price of the optio, the portfolio ca be used to hedge the optio ad is called a hedgig strategy 9) If theoretical price is ot equal to market price, the hedge portfolio ca be used to lock i a arbitrage profit ad is called a arbitrage strategy. Practical Limitatio a) Sometimes delta might be ad umber like this cause roudig error to hedge the appropriate umber b) How ofte to rebalace? I theory, cotiuously replace ad rebalace. However, i reality we have to take ito accout the trasactio cost.

Chapter 13 Binomial Trees. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull

Chapter 13 Binomial Trees. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull Chapter 13 Biomial Trees 1 A Simple Biomial Model! A stock price is curretly $20! I 3 moths it will be either $22 or $18 Stock price $20 Stock Price $22 Stock Price $18 2 A Call Optio (Figure 13.1, page