Black-Scholes model under Arithmetic Brownian Motion

Black-Scholes model under Arithmetic Brownian Motion Marek Kolman Uniersity of Economics Prague December 22 23 Abstract Usually in the Black-Scholes world it is assumed that a stock follows a Geometric Brownian motion. he aim of our research is to present Black-Scholes model in a world where the stock is attributed an Arithmetic Brownian motion. Although Arithmetic Brownian motion is simpler due to lack of the geometric terms as it is shown the option model is eentually analytically less tractable than under Geometric Brownian motion. JEL Classification: C C2 G3 Keywords: Black-Scholes Arithmetic Brownian Motion Risk-neutral aluation his research has been supported by the Czech Science Foundation Grant P42/2/G97 Dynamical Models in Economics

Introduction Black and Scholes in their seminal paper Black and Scholes 973 deried an option pricing model of which one of main assumptions was that underlying stock follows a Geometric Brownian motion GBM. In this article we are presenting an alternatie model that gies one a notion about how would the Black-Scholes look like if Arithmetic Brownian motion ABM was used. We are fully aware that in the real-world there is no reason to assume that stock follow ABM but from academic perspectie it is an interesting subject to study. We structure this article as follows. Firstly we equip the reader with simplified main building blocks that determine most of obtained results of the whole article. hese are standard theorems and do not differ from those in classical textbooks. Secondly we present a classical model of Black-Scholes we derie a closed-form formula for call option price. Among many others we apply martingale approach since that is where one can see the beauty of playing with martingales. Of course we could omit this part related to well-known results but we deem it is a good benchmark for the reader who is interested in the differences in deriation of Black-Scholes under GBM and ABM. Finally in the third part we derie a call option pricing model on a stock under ABM. We also mention special features of the model and demonstrate closed-form formula for basket options within the ABM framework. 2 Building blocks of option price deriation 2. Risk-neutral aluation formula Although there exist many ways how to derie stock option prices for the purpose of this study we will use a martingale approach. Martingale approach is based on risk-neutral aluation formula [ V = E X V X where option payoff at maturity V = ηk S is some deterministic function and V is option alue as of time t =. X is a so-called numeraire asset used for relatie prices Vt X t and also induces a measure X. ypically we use a money market account as a numeraire B t = exp t X r s ds = e rt where we assume r is constant. Such numeraire implies a spot martingale measure Q which will be used through this study. In the world of ariety interest-rate deriaties it is often useful to use so-called - forward measure Q see Jamshidian 989 for the original paper i.e. a measure where a bond B is a numeraire. In our case of constant interest rates both approaches coincide. Since we use the money market account B as the numeraire the risk-neutral aluation formula that we use through this study is [ V = E Q V. 2. B B Bear also in mind that by definition B = and due to constant risk-free rate B = e r. he formula 2. can thus be rewritten as V = e r E Q [V. 2.2 Since we derie a call option price we set in terms of stock option pricing this is an acceptable compromise V = max[s K. 2.3 2

2.2 Stock motion S under measure Q Normally the stock motion S under empirical measure P has the dynamics ds t = µs t tdt + S t tdw P t. 2.4 We will specialize further this Ito s process so that it models either GBM or ABM but now it is prefereable to work with the generic form 2.4. he question implied by 2. 2 or by 2.2 respectiely is what is the Q-dynamics of S? here are seeral useful definitions of spot martingale measure Q in Björk 29 and of these we will suffice with the definition that under Q the price process St B t must be a martingale which means a driftless process. Set U t = Bt and consider a differential of the process Z t = ZS t U t = S t U t. Ito s product rule see Joshi 23 Eq. 5.34 yields dz t = S t du t + U t ds t + ds t du t = rs t e rt dt + e rt ds t + [ }{{} = rs t e rt dt + e rt µs t tdt + S t tdwt P = e rt µs t tdt rs t dt + S t tdw P t = e rt [µs t t rs t dt + S t tdw P t. In order to kill the drift of Z t one must therefore ensure that µs t t rs t =. his is done by a measure change from P to Q ia a Girsano theorem. Since the change of measure for ABM differs from that for GBM we specialize the form of Girsano transformation in the sections related to ABM and GBM models. o this end let dq L t = dp F t denote the Radon-Nikodym deriatie. hen we hae the following Radon-Nikodym process dl t = L t ϕtdw P t where ϕt is a Girsano transformation kernel. Girsano kernel gies us a new Q-Wiener process gien by dw P t = ϕtdt + dw Q t. 2.5 We will use 2.5 quite extensiely because it allows to migrate from P to Q. 2.3 Change of numeraire Consider two geometric processes under measure Q S is a numeraire where Q is a martingale measure: hen the Radon-Nikodym deriatie takes the form d = µ dt + dw t ds t = µ S t dt + S t dw t. 2.6 dl t = L t dw t and ia a Girsano transformation with kernel defines a new martingale measure Q with S under Q as a numeraire. 2 we know that V is by 2.3 a deterministic function of S 3

Proof: Consider general Ito s processes under measure Q dst = µ St tdt + St tdwt dst = µ St tdt + St tdwt. 2.7 We know that the Radon-Nikodym deriatie that takes us from Q to Q is dl t = L t ϕtdw t which is obiously a Q martingale. he change of numeraire is induced when the Radon-Nikodym deriatie takes the form L t = S St S St. 2.8 o get its differential we must use Ito s lemma. We firstly define an auxiliary process Z t = S t and so 3 dl t = S S dz t. 2.9 dz t now has to be obtained ia Ito s quotient rule see e.g. Fries 27 Appendix C and it yields that dz t Z t = ds t S t ds t ds t S t d [ ds 2 + t which is after a substitution for Z t dz t = S t ds t S t ds t ds t S t d [ ds 2 + t St. By substituting this into 2.9 for dz t we get dl t = S St S St ds t S t ds t ds t S t d [ ds 2 + t St which can be after substitution from 2.8 written as dst dl t = L t St ds t St ds t St d [ ds 2 + t St. Now the problem requires an analysis. We know L t must be a Q martingale see for example Musiela and Rutkowski 2 for more details and so we need to separate and collect coefficients of dt in the last expression to drop the drift term. Doing this results into dl t = L t S t tdw t S t = L t S t t S t tdw t t dw t. he processes in 2.7 howeer are further specified in 2.6 by St t = St and plugging these terms into the last equation gies dl t = L t dwt which proes the gien result. and S t t = S t 3 note that S S is nothing but a scaling constant 4

3 Black-Scholes classics: Geometric Brownian Motion Define ds t = µs t dt + S t dw P t 3. to be Geometric Brownian Motion of a stock S under the empirical measure P. Our objectie is to price a call option V we will henceforth assume t = on a gien non-diidend stock S. he risk-neutral aluation formula B as numeraire gien in 2. says that we need to obtain dynamics of S under Q. Following the Section 2.2 with µs t t = µs t and S t t = S t from 3. we hae the differential of Z as follows dz t = e rt [µs t rs t dt + S t dw P t = e rt [µ rs t dt + S t dw P t. 3.2 It is clear that if µ = r then dz t has zero drift and so Z would be a martingale under the measure Q that makes that drift zero. Again following Section 2.2 we set the Girsano kernel to be ϕt = µ r and thus by Girsano change-of-measure we hae dwt P = µ r dt + dw Q t. 3.3 Placing dwt P from 3.3 into 3.2 gies [ e rt [µ rs t dt + S t dwt P = e rt [µ rs t dt + S t µ r dt + dw Q t = e rt S t dw Q t and so indeed Z is a Q-martingale. he change of measure also affects the dynamics of the stock in the SDE 3. so that ds t = µs t dt + S t µ r dt + dw Q t and thus finally after the µ-terms cancel out ds t = rs t dt + S t dw Q t 3.4 which is a Q-dynamics of S crucial to sole the expectation in 2.. In order to ealuate 2. we could use its form 2.2 so we would be challenged to sole V = e r max[s K f S S ds. 3.5 where f S is a probability density of a random ariable S. Distribution properties of S are well-known we omit its deriation here and we know that 4 ln S N ln S + r 2 2 4 we write Nµ for Normal distribution with mean µ and standard deiation 5

so Now set so that S = S exp r 2 2 + W Q. 3.6 Y = Y N r 2 2 + W Q r 2 2 which means we can rewrite S as Now we plug 3.7 into 3.5 and it turns out that V = e r S = S e Y. 3.7 max[s e Y K f N Y dy. he max max[ can easily be eliminated when we consider when identify the alue of Y when the expression starts to take a positie alue. his is alid for Y > ln K. Now we can rewrite the expression S once again eliminating the max[ and with changed lower bound of integration to hae V = e r ln K S S e Y Kf N Y dy. A tedious algebra would yield the well-known result. We omit this and use a smarter approach of changing a numeraire. he trick relies on rewriting 2. using indicator functions so that V B = E Q his equation can be further separated into two terms [ V = E Q S {S K} B [ S K {S K} B B. KE Q [ {S K} B he expression aboe can be easily represented in terms of J and J 2 such that. V B = J B K J 2 B 3.8 where J B = E Q J 2 B = E Q [ S {S K} B [ {S K} B 3.9 3. are actually digital options both expressed in terms of Q-expectation. J 2 is easy to sole and we will do this later. Let us now focus on option J. he product S {S K} is not triial so its expectation can 6

not be directly computed. A trick with change of numeraire can simplify it. Let us now choose the stock S under Q as the numeraire asset. Now consider our two geometric processes db t = rb t dt ds t = rs t dt + S t dw Q t. Section 2.3 says that the Radon-Nikodym deriatie to switch from Q induced by numeraire B to S induced by numeraire S under Q is dl t = L t dw Q t where we identify Girsano kernel ϕt =. hen it follows that dw Q t = dt + dw S t. Substituting the aboe defined dw Q t into 3.4 gies us SDE for S under measure with a numeraire S: ds t = r + 2 S t dt + S t dw S t. Next by replacing B by S and B by S we hae changed the numeraire from B to S in 3.9 we hae [ J = E S S {S K} = E S [ S {S K} S so by the change of numeraire we hae effectiely eliminated S from the product S {S K}. Since E S [ {S K} = S[S K 3. we can sole this in the following manner. Let us determine the distribution of S under S by Ito s lemma for dzs t t = ln S t. It gies dzs t t = S t ds t 2 [ds t 2 S 2 t = r + 2 dt + dwt S 2 2 dt = r + 2 2 dt + dwt S. his means S = S exp r + 2 2 + W S. Going back to 3. let us substitute X X N for W S and we hae E S [ [ {S K} = S S exp r + 2 2 + X K ln K r + S 2 = S X 2. Since normal distribution is symmetric it holds that Pr[X x Pr[X < x and so E S [ [ {S K} = S X ln S K + r + 2 2 ln S K + r + 2 = N 2. 7

Eentually J = S N Secondly in similar fashion we tackle 3. which can be rewritten as We are about to sole ln S K + r + 2 2. 3.2 J 2 = B e r E Q [ {S K} = e r E Q [ {S K}. E Q [ {S K} = Q[S K where S under Q has already been defined in 3.6. Analogously to the expectation problem aboe now just setting W S = X X N we sole Finally E Q [ {S K} [ = Q S exp r 2 2 + X K ln K r S 2 = Q X 2 = N J 2 = e r N ln S K + r 2 2. ln S K + r 2 2. 3.3 We can now assemble the Black-Scholes formula. Since from 3.8 if follows that V = J KJ 2 we just substitute for J and J 2 from 3.2 and 3.3 respectiely to hae V = S Nd Ke r Nd 2 where as usual d = ln S K + r + 2 2 d 2 = d. 4 Black-Scholes under Arithmetic Brownian motion In the preious section we hae deried a classical result - call option price when the underlying asset stock follows GBM. Objectie of the current section is to derie a call option price under adjusted settings namely we relax assumption of GBM and adopt assumption of ABM followed by the underlying stock S. Let us start with SDE for ABM under P ds t = µdt + dw P t. 4. Moing on to the risk-neutral aluation formula we must make discounted process S a martingale under Q. In order to fully apply rules justified in Section 2.2 we must identify our SDE 4. with the generic Ito one in 2.4. It is clear that in the case of ABM µs t t = µ S t t = 8

so the differential of Z t = ZBt S t takes the form dz t = e rt µ rs t dt + dw P t It is obious that in order to eliminate the drift the Girsano kernel that makes Z t a Q-martingale will not be the same as in the case of GBM. Set the Girsano kernel to ϕt = µ rs t which ia Girsano theorem implies dw P t =. µ rs t dt + dw Q t. 4.2 Again recall the differential dz t with dwt P as defined in 4.2 to hae [ dz t = e µ rt rs t dt + µ rs t dt + dw Q t = e rt dw Q t from which it is clear that Z t = St B t is a Q-martingale. Combining 4. and 4.2 gies us a Q-dynamics of S ds t = rs t dt + dw Q t. 4.3 We are howeer also interested in the Q-distribution of S associated with the SDE 4.3. he problem is that 4.3 is not geometric so the application of Ito s lemma to d ln S t as it is usually done for GBM will not yield a result. Instead of using logarithm 5 set Z t = ZS t t = S t e rt and compute the differential dz t. By irtue of Ito s lemma 6 we hae dz t = Z t t dt + Z t ds t + 2 Z t S t 2 S t 2 [ds t 2 4.4 he partials are easily computed as Z t t = rs t e rt Z t S t = e rt 2 Z t S t 2 =. Substituting for the partials in 4.4 it turns out that dz t = rs t e rt dt + e rt rs t dt + dw Q t = e rt dw Q t. Integrating the aboe between and we hae Z Z = S e r S = e rt dw Q t = e rt dw Q t 5 we use basically the same trick that is usually used in Vasicek model when soling for distribution of r t. See Nawalkha et al. 27 Example.3 6 we can indeed do this because the S under Q-ABM 4.3 complies with the definition of Ito s process 2.4 9

and so S = S e r + e r e rt dw Q t. 4.5 In the last expression we stuck because we can not integrate a deterministic function of t with respect to W. We can oercome this by using Björk 29 Lemma 4.5 by inestigating properties of stochastic integral. he lemma says that if there is some deterministic function ft such that then M is normally distributed with M = E[M = Var[M = ftdw t f 2 tdt. Using this rule for our integral in 4.5 gies us e S = S e r + e r 2r Y Y N 2r from which is easy to read off that e S = N S e r e r 2r. 4.6 2r For our later calculations it will be easier to work with e := e r 2r 2r so that 4.6 is simply just S N S e r. Haing the Q-distribution of S we can return to the fundamental pricing problem inoling Q-expectation of S. When using our indicator function trick and when we express B and B explicitly in terms of r we hae V = e r E Q [max[s K = e r E Q [ {S K} max[s K = e r E Q [ {S K}S Ke r E Q [ {S K}. 4.7 his reminds the J and J 2 problem soled in the chapter of GBM. Here we will not make any further change-of-measure since it is not possible. he reason is that change of measure can relate to positie processes only which rules out ABM. Let us firstly sole the expectation of the second indicator which is E Q [ {S K} = Q[S K = Q [ S e r + X K = Q [X K S e r [ = Q X S e r K S e r K = N. 4.8

In the aboe we used a dummy ariable X N such that X = e r Y. Now we hae to tackle the trickier first expectation of indicator E Q [ {S K}S. In the case of GBM we soled this problem by a change-of-numeraire technique which eliminated S from the Q-expectation E Q [. his technique can not be applied for ABM model because the numeraire price process is required to positie see Schönbucher 23 Section 4.2 or Epps 27 Definition 3. In plain words we can interpret it as expected alue of S in the interal [K. Expressing this using an integral yields E Q [ {S K}S = S 2π exp S S e r 2 2 2 ds. 4.9 We make a substitution h = S S e r which implies a new lower bound of the last integral K and also note that K S e r dh = ds. By irtue of the substitution the integral 4.9 changes to E Q [ {S K}S = = K S e r S 2π exp 2 h2 dh [ h + S e r exp 2π 2 h2 dh K S e r = S e r exp 2π 2 h2 dh + h exp 2π 2 h2 dh. K S e r K S e r 4. he first integral is actually a probability Pr [H K S e r [ = Pr H < S e r K S e r K = N 4. because from the first integral in 4. it is clear that we were integrating oer standard normal density. he second integral is just a normal density at the lower bound. hus f N K S e r Using these two last results 4. and 4.2 we hae S e r K = f N f N x = exp x2. 4.2 2π 2 E Q [ {S K}S = S e r S e r K K S e r N + f N. 4.3

Putting together 4.7 with 4.8 and 4.3 yields the final result where V = S N d + e r [f N d KN d d = S e r K. It can be proed that the PDE has the form V V + rs t S + 2 2 2 V = rv. 4.4 S2 his is same as in the case of GBM model except for the coefficient second-order partial deriatie which is 2 2 rather than 2 2 S 2 in the case of GBM. he deriation of the PDE 4.4 is showin in Appendix. 4. Special features of ABM model Contrary GBM model the proposed option alue under ABM suggests that there is no solution for the case when r =. his is because r is in the numerator of. Since zero interest-rate is not an uncommon obseration in the market we need to seek some solution. Let d depend explicitly on r so that we for a moment use the notation d = dr. aking limit we obtain and so the solution for the option alue is simply Ω = d = lim r dr = S K V = Nd S K + f N d. Finally we show how simple analytical solution exists for basket options under ABM compared to the case of GBM. he GBM basket option is a demanding pricing problem mostly for the reason that sum of log-normal distributions is not log-normal anymore. his implies that closed-form solution does not exists and so for GBM one has to use some approximation. For instance Beisser 999 uses conditional expectation approach Gentle 993 approximates the arithmetic aerage by a geometric aerage or Ju 993 that uses expansion of ratio of characteristic functions. he good property of ABM is that we can sum distributions of indiidual assets in the basket. Haing a basket on n assets S = S S 2... S n for which corresponding weights in the basket are w = w w 2... w n we hae the following distribution of the basket portfolio Sw. he coariance matrix is e 2r 2 e 2r 2r ρ 2 e 2r e 2r 2 2r ρ 2 e 2r e 2r n 2r ρ 2 e 2r 2 e 2r 2r. ρ n e 2r e n 2r 2r hus the basket standard deiation is e 2r 2 2 e 2r 2r ρ 2n e 2r e 2 2r n 2r.... ρ n2 e 2r e n 2r 2 2r e 2r n 2 e 2r 2r = wωw. Now one can alue the basket option using the closed-form formula from the preious section with S = Sw and defined aboe in terms of the coariance matrix Ω.. 2

5 Conclusion In this short article we hae departed from the usual assumption of GBM for a stock S in Black-Scholes world. Instead we inestigated a pseudo B-S model in which a non-diidend paying stock follows ABM. Indeed it has been shown that mathematical properties of ABM are less desirable than that of GBM and so the closed-form formula for call option when the stock is following ABM is slightly more difficult to derie. In particular for the ABM case we were unable to inoke the second change of measure as we did in the case of GBM but we had to analytically tackle an integral representing expectation. We hae also inestigated special features of the ABM model and shown how easily a basket option can be priced analytically. 3

A Deriation of the PDE under ABM Consider a contingent claim V that is function of both stock price S and time t so that we may write V := V S t. Using Ito s lemma we may express differential of V as dv = r V S + V t + 2 2 2 V S 2 dt + V S dw Q t Form a hedging portfolio Π such that we short one option and buy V S Compute the differential of the portfolio alue After substitution for dv we hae dπ = V S ds = V V rdt + S S dw Q t V = t + 2 2 2 V S 2 Π = V S S V. dπ = V ds dv. S V S r + V t + 2 2 2 V S 2 dt V V S dt. rdt V S dw Q t S dw Q t delta shares S: V t + 2 2 2 V S 2 Since there is no source of randomness W in dπ anymore the Π portfolio must earn risk-free rate. 7 hus Substituting for dπ and Π we obtain which is after rearrangement of the terms V t + 2 2 2 V S 2 dπ = rπdt. V dt = r S S V dt dt V t + rs V S + 2 2 2 V S 2 = rv. S 7 here we assume the risk-free rate is earned geometrically. his does not coincide with our assumption of ABM for the stock 4

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