1 The Black-Scholes model

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1 The Blac-Scholes model. The model setup I the simplest versio of the Blac-Scholes model the are two assets: a ris-less asset ba accout or bod)withpriceprocessbt) at timet, adarisyasset stoc) withpriceprocess St) at time t The price dyamics of the two price processes uder the objective probability measure P are { dbt) = rbt)dt ) B0) = where r is a costat ad { dst) = µst)dt+σst)dwt) S0) = s 0 2) where µ ad σ are costats ad W is what is ow as Wieer process. Defiig the local rate of retur of the ris-less asset B as dbt) Bt)dt = r, we see that it is determiistic, whereas the rate of retur o the stoc give by dst) St)dt = µ+σdwt), dt is stochastic, sice there is the white oise dwt)/dt term. We see that µ represets the local mea rate of retur of the stoc. The costat σ is ow as the volatility of the stoc ad determies how much ifluece the radom term dw has i 2). Both Equatios ) ad 2) are explicitly solvable ad the solutios are Bt) = e rt, St) = s 0 exp {µ 2 } )t+σw σ2 t. The solutio for the ris-less asset offers o problems, but what ca we say about the solutio of the stoc price equatio? Well, the distributio of the expoet is ormal with ] E [µ 2 σ2) t+σw t = µ 2 σ2) t, ] V [µ 2 σ2) t+σw t = σ 2 t. The distributio of the stoc price uder the objective measure P is therefore log-ormal with the parameters give above..2 The Blac-Scholes formula Just as for the biomial model the model will be free of arbitrage if there exists a martigale measure, ad complete if the martigale measure is uique. The defiitio of a martigale measure is basically the same as before

2 Defiitio Cosider the maret above ad fix the asset B as the umerarire asset. A probability measure Q is said to be a ris eutral) martigale measure if. Q P. 2. The ormalized stoc price process is a martigale uder Q. Z t = S t B t, It turs out that there is a uique martigale measure Q for the Blac-Scholes maret, which meas that the maret is free of arbitrage ad complete. Uder Q the local mea rate of retur of the stoc has to be equal to the local rate of retur of the ris-less asset. This meas that uder Q the dyamics of the stoc price process will loo as follows dst) = rst)dt+σst)dŵt) S0) = s 0 3) with the solutio St) = s 0 exp {r 2 ) } σ2 t+σŵt. where Ŵ is a Q-Wieer process. Now, suppose that we wat to compute the price of a Europea call optio with strie price K ad expiratio date T. Just as before it turs out that to esure that there is o arbitrage o the maret we should mae sure that all ormalized price processes are Q-martigales. This meas that we get the followig pricig formula Propositio Let X = φs T ) be a simple) cotiget T-claim. The price of X at time t, Π t X), is give by Π t X) = e rt t) E Q [φs T ) F t ]. 4) Proof: We wat Π t X) B t to be a Q-martigale. This meas that we should have Π t X) B t [ = E Q ΠT X) F t Usig that B t = e rt ad that Π T X) = X for o arbitrage reasos, we obtai the pricig formula i the propositio. B T ]. For a Europea call optio we have X = max{s T K,0} ad the price at time zero is therefore Π 0 X) = e rt E Q [max{s T K,0}]. 2

3 where ST) = s 0 e Z ad Z is ormally distributed with with expecatio m = r 2 σ2) T ad stadard deviatio s = σ T. Let ϕ deote the desity fuctio of the distributio of Z. The we have E Q [max{s T K,0}] = E Q [max{s 0 e Z K,0}] = 0 Qs 0 e Z K)+ = s 0 lk/s 0 ) e z 2πs 2 e z m)2 /2s 2) dz KQ lk/s 0 ) K Z > l s 0 e z K)ϕz)dz )). By completig the square i the expoet of the itegrad we see that the first term ca be writte as s 0 e z /2s 2) 2πs 2 e z m)2 dz lk/s 0 ) = s 0 e s2 /2+m )) 2 /2s 2) lk/s 0 ) 2πs 2 e z m+s2 dz K = s 0 e s2 /2+m QY > l ), s0) where Y id ormally distributed with expectatio m+s 2 ad stadard deviatio s. Now use the followig relatios.. For ay stochastic variable X: QX > x) = QX x). 2. If X is ormally distributed with expectatio a ad stadard deviatio b we have ) x a QX x) = N, b where N deotes the cumulative distributio fuctio of the stadard ormal distributio. 3. Nx) = N x). You will the arrive at the Blac-Scholes formula. Propositio 2 Blac-Scholes formula) The price of a Europea call optio with strie price K ad expiry date T at time t is give by ct,s t ), where Here ct,s) = sn[d t,s)] e rt t) KN[d 2 t,s)]. { d t,s) = σ l s ) } T t K + r + 2 σ2) T t), d 2 t,s) = d σ T t. ad N deotes the cumulative distributio fuctio of the stadard ormal distributio. Remar Note that you ca clearly see from the above propositio that arbitrage pricig is pricig is i terms of the uderlyig asset. Give that we ow the stoc price today we ca compute the price of the optio, otherwise we are at a loss. s 0 3

4 .3 The Blac-Scholes formula as the limit of the biomial model formula The objective of this sectio is to show that if a biomial model with carefully chose parameters is used to compute the price of a Europea call optio, the price will coverge to Blac-Scholes formula as we let the umber of time steps i the biomial tree go to ifiity. Cosider pricig a Europea call optio with strie price K ad exercise time T o a stoc with a yearly volatility of σ. Divide each year ito periods. This gives a biomial model with T periods. I this tree, which we label the th tree, we choose the followig parameters { } u = exp σ, { } d = exp σ { r +R = exp. } = u, Here r deotes the ris-free iterest rate per aum with cotiuous compoudig ad R deotes the oe period iterest rate i the biomial model. From Propositio 4 i lecture 4 we obtai that the price at time t = 0 of the optio computed i the th model, c, is give by c = +R ) T T T where S 0 deotes the stoc price at time t = 0 ad We will ow proceed i a umber of steps. ) q q ) T max{s 0 u dt q = +R ) d u d. K,0},. To start off with we use that max{s K,0} = S K) I{S > K} where I{A} deotes the idicator fuctio of the set A, i.e. {, if ω A, I{A}ω) = 0, otherwise. to show that c ca be writte i the followig way ) T T c = S 0 q ) q )T I{S 0 u dt > K} where First c = K +R ) T T T ) T T +R ) T ) T T +R ) T ) q q ) T I{S 0 u d T q = u +R q. > K}, q ) q ) T S 0 u dt I{S 0 u dt > K} q q ) T KI{S 0 u d T 4 > K},

5 or c = S 0 T T K +R ) T ) q u T +R ) T ) q )d +R ) T I{S 0 u d T > K} q q ) T I{S 0 u dt Sice q = u q = +R u q +R +R ad the defiitio of q tells us that we have that ad thus where 2. Now use that to see that c = S 0 T u q = +R d +d q, q = +R +R d +d q ) +R = d q ) +R T K +R ) T where S T) = S 0 u Y dt Y Q. Usig that ) q ) q ) T I{S 0 u d T > K} T T ) q q ) T I{S 0 u dt q = u +R q. E P [I{A}] = PA) c = S 0 Q ls T) > lk) Ke rt QlS T) > lk), > K}. > K}, ad Y BiT,q ) uder Q ad Y BiT,q ) uder E P [I{A}] = PA), ad the defiitio of R we readily see that c = S 0 Q S T) > K) Ke rt QS T) > K), where S T) = S 0 u Y dt Y ad Y BiT,q ) uder Q, ad Y BiT,q ) uder Q. Sice the logarithm is a strictly icreasig fuctio we have that QS T) > K) = QlS T) > lk), ad thus c = S 0 Q ls T) > lk) Ke rt QlS T) > lk). 5

6 3. It should come as o surprise that the Cetral Limit Theorem will come i to play evetually. I order to use it we will eed to compute the followig quatities M Q = E Q [ls T)], V Q = Var Q [ls T)], M Q = E Q [ls T)], V Q = Var Q [ls T)]. What we really eed to ow is what these quatities ted to as teds to ifiity. Straight forward calculatios give M Q = E Q [ls T)] = E Q [ls 0 +Tld +Ylu ld )] = ls 0 +Tq lu + q )ld ), M Q = E Q [ls T)] = ls 0 +Tq lu + q )ld ), V Q = Var Q [ls T)] = Var Q [ls 0 +Tld +Ylu ld )] = Tq q )lu ld ) 2, V Q = Var Q [ls T)] = Tq q )lu ld ) 2. Now rewrite the expressio for M Q usig the defiitios of u, d, R, ad q. σ M Q e r/ e σ/ ls 0 = T e σ/ e σ e σ/ e r/ ) σ/ e σ/ e σ/ = T 2e r/ e σ/ e σ/ ) σ. e σ/ e σ/ The MacLauri expasio of the secod order of e x reads e x = +x+x 2 /2+Ox 3 ) so e r/ = +r/+o/ 2 ), e ±σ/ = ±σ/ +σ 2 /2)+O/ 3/2 ). Isertig this ito the expressio for M Q yields M Q ls 0 = T ) 2r/ σ 2 /+O/ 3/2 ) σ 2σ/ +O/ 3/2, ) 2r σ 2 +O/ ) ) = Tσ 2σ +O/) ) T r σ2 as. 2 Similar calculatios for V Q, M Q, ad V Q give ) M Q ls 0 T r + σ2, 2 V Q σ 2 T, V Q σ 2 T. 6

7 4. Aother way to thi of ls T) is as a sum of T idepedet variables with possible outcomes lu,ld ) ad associated probabilities q ad q or q ad q ) use that lab = la+lb to see this!). This meas that we have a sum of idepedet radom variables for which the first ad secod momet coverge. This puts you i a positio to use a versio of the Cetral Limit Theorem to determie what the distributio of ls T) teds to as teds to ifiity. Use the result to compute lim c = lim {S 0Q ls T) > lk) Ke rt QlS T) > lk)}. What a versio of) the Cetral Limit Theorem tells us is that the limit of the sum is ormally distributed, i.e. ls T) Q/Q NlS 0 +r ±σ 2 /2)T,σ 2 T). Thus, we should substitute X NlS 0 +r ±σ 2 /2)T,σ 2 T)) for ls T) whe computig the limits of the probabilities o the right had side. We therefore obtai lim c = lim {S 0Q ls T) > lk) Ke rt QlS T) > lk)} = S 0 Q X > lk) Ke rt QX > lk) X = S 0 Q ls0 r +σ 2 /2)T σ > lk ls ) 0 r +σ 2 /2)T T σ T X Ke rt ls0 r σ 2 /2)T Q σ > lk ls ) 0 r σ 2 /2)T T σ. T After some fial rewritig multiply both sides iside Q ad Q by, thus reversig the iequality, ad use that N0, )-distributed variables are symmetric ad cotiuous, ad that lx/y) = lx ly) we obtai lim c = S 0 Nd ) Ke rt Nd 2 ). Here N deotes the distributio fuctio of a stadard ormal distributio, ad { ) d = σ S0 l + r+ ) } T K 2 σ2 T, d 2 = d σ T..4 Parity relatios To actually replicate a claim usually requires cotiuous rebalacig of the portfolio. Problems the arise due to tarasactio costs amog other thigs). The questio is ow which cotracts ca be replicated usig costat buy-ad-hold) portfolios? To ivestigate this we will use that a simple T-claim i this model is a radom variable of the form X = φs T ), where the give fuctio φ is ow as the cotract fuctio. You caot mae do with the versio you ow from Saolihetsteori sice that versio requires a scaled sum of idetically distributed radom variables. Not to worry though, the Lideberg-Feller-versio of the Cetral Limit Theorem will apply to the situatio at had. 7

8 Fix a maturity T. We cosider usig the follwog assets i our portfolio: Asset T Bod Stoc Call optio Cotract fuctio φ B x) = φ S x) = x φ c,k x) = max{x K,0} Price at time t e rt t) S t c K t,s t ) The call optios used should be Europea ad we will allow usig several call optios with differet strie prices K hece the sub idex K i the pricig fuctio). Fix a simple T-claim X = φs T ). If the cotract fuctio φ satisfies φx) = αφ B x)+βφ S x)+ γ i φ c,k x), i= for some real umbers α, β, γ i, i =,...,, the the price of X ca be obtaied as Π t X) = απ t φ B )+βπ t φ S )+ γ i Π t φ c,k ), i= ad X is replicated by the costat portfolio α,β,γ,...,γ ). To prove it use theris eutral valuatio formula 4) ad that coditioal expectatio is liear. Now cosider a Europea put optio with strie price K. This has a cotract fuctio φ p,k give by φ p,k x) = max{k x,0}. If you draw a picture you will fid that which gives us the followig. φ p,k x) = Kφ B φ S +φ c,k. Propositio 3 Put-call parity) Cosider a Europea call optio ad a Europea put optio, both with strie price K ad expiratio date T. Deotig the pricig fuctios ct, s) ad pt, s), respectively, we have the followig relatio pt,s t ) = Ke rt t) +ct,s t ) S t. This meas that we ca replicate a put optio by buyig K zero coupo bods with maturity T, ad a call optio with the same strie ad maturity as the put optio, ad sellig the uderlyig stoc. I fact oe ca show the followig. Propositio 4 All cotiuous cotract fuctios with compact support ca be replicated with arbitrary precisio i sup-orm) usig bods, call optios ad the uderlyig stoc. Remar 2 The problem with the above propositio is that i geeral the portfolio will cotai a huge amout of assets. If you have a piecewise liear cotract fuctio you will do fie though! 8

9 .5 Volatility I order to use the Blac-Scholes formula for pricig you eed the values of t, T, r, σ, ad S t. Obtaiig t, T or rather T t), ad S t does ot preset a problem, ad proxies for r are available. That leaves us with the volatility σ. There are two basic approaches, amely to use historic volatility or implied volatility..5. Historic volatility Usig historic volatility meas that we estimate σ usig historical stoc price data. Suppose that we will observe the stoc price at equidistat poits i time t 0,t,...,t, where t i t i = t. Now let ) Sti ) Z i = l, i =,...,. St i ) The Z,...,Z are idepedet, ormally distributed radom variables with E[Z i ] = µ ) 2 σ2 t, ad VZ i ) = σ 2 t, uder P. Note that we ca ot mae observatios uder Q! From stadard statistical theory we ow that S Z = Z i Z) 2, where Z = Z i i= i= is a estimate of the stadard deviatio of Z i, i.e. of σ t. Therefore σ = S Z t, is a estimate of σ. A estimate of the stadard deviatio of the estimate σ ca be show to be Dσ ) σ 2. A argumet agaist usig historical volatility is that for pricig purposes we wat to ow what will happe i the future, ad this may ot be reflected i historical data..5.2 Implied volatility We wat to price i such a way that the prices we compute are cosistet with the prices already observable i the maret. This meas that we should really use the maret expectatio of the volatility for the time to maturity of iterest. We ca achieve this by gettig maret data for a bechmar optio writte o the same stoc as the optio we wat to value. For this optio the price p should be ow, ad therefore we ca bac out σ from the relatio p = ct,t,r,σ,k,s t ) where ct,t,r,σ,k,s t ) is give by the Blac-Scholes formula. The value of σ obtaied i this way is called the implied volatility 9

10 Implied volatilities ca be used to test the Blac-Scholes model. If we compute the implied volatility for Europea call optios writte o the same stoc with the same maturity, but with differet strie prices, ad plot the implied volatility as a fuctio of the strie price it should come out as a horizotal lie the Blac-Scholes model assumes costat volatility!). Empirically it is ofte observed that optios far out of the moey or deep ito the moey are traded at higher volatilities, tha optios at the moey. The graph of the implied volatilities the loos lie a smile, ad for this reaso the implied volatility curve is termed the volatility smile. 0

0.1 Valuation Formula:

0.1 Valuation Formula: 0. Valuatio Formula: 0.. Case of Geeral Trees: q = er S S S 3 S q = er S S 4 S 5 S 4 q 3 = er S 3 S 6 S 7 S 6 Therefore, f (3) = e r [q 3 f (7) + ( q 3 ) f (6)] f () = e r [q f (5) + ( q ) f (4)] = f ()