BLACK SCHOLES THE MARTINGALE APPROACH

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1 BLACK SCHOLES HE MARINGALE APPROACH JOHN HICKSUN. Introduction hi paper etablihe the Black Schole formula in the martingale, rik-neutral valuation framework. he intent i two-fold. One, to erve a an introduction to expectation pricing and two, to examine thi framework in explicit mathematical detail. he reader i aumed to have fluent background in the mathematical theory of tochatic procee and calculu, but i not aumed to have background in finance. he relevant financial definition are given in ection 2. Section 2 alo etablihe the main reult (propoition 2.3) that link equivalent martingale meaure (EMM) to expectation. he key mathematical tool at work here i the martingale repreentation theorem, which guarantee the exitence of a hedging trategy under an EMM. In ection 3, we exploit thi reult by explicitly finding an EMM for geometric brownian motion (propoition 3.3). Here the key mathematical tool i Giranov theorem, which tell u how to convert to and from an EMM. Reader with financial background will recognize thi converion a ubtracting the market price of rik. For general contingent claim we of coure cannot reduce valuation to a imple formula. But the technique ued here indicate how we ought to proceed numerically. It i a two-tep proce. Firt, we back out an EMM from underlying market price. hen we compute the value of the claim a it dicounted expectation under the rik neutral meaure. here i a potential obtacle to thi procedure: an EMM might not exit. For thi we appeal to a powerful guarantee called the fundamental theorem of aet pricing, which aert the exitence of an EMM if and only if the market i arbitrage-free. 2. heory Let S be a continuouly traded aet and B be a rik free bond, often referred to a the numeraire. We aume throughout that the bond B i governed (under the phyical meaure P ) by db t = rb t dt. We ay that r i the pot rate of interet, alo called the hort rate. If we hold X amount of S and Y amount of B over a period from [, t] then the gain of our portfolio over [, t] are given by G t = X ds + Y db.

2 2 JOHN HICKSUN We call (X, Y ) a trading trategy. he value of thi trading trategy at time t i jut a linear combination V t = X t S t + Y t B t. We ay that a trategy i elf-financing if V t = V + G t for all t, and we define an arbitrage opportunity to be a elf-financing trading trategy (X, Y ) for which V =, V t, and E(V t ) > for ome t >. For any proce H we define H = B H and we ay that H ha been dicounted by the numeraire. It can be hown that if a trading trategy i elf-financing then the dicounted value proce equal the initial value plu the gain in the dicounted aet proce. We will not need thi reult but we will make ue of the convere: Propoition 2.. Suppoe that Ṽ t = Ṽ + hen the trading trategy (X, Y ) i elf-financing. X d S. Proof. Oberve that B i continuou with locally bounded variation and therefore it quadratic covariation with any other proce vanihe. Combining thi with integration by part, S t = B t St = And it follow that S db + X ds = B d S + [B, S] = B X d S + Applying integration by part to the value proce, = V t = B t Ṽ t = = B X d S + B dṽ + B X d S + herefore (X, Y ) i elf-financing. B X S db + Ṽ db = X S db + Y db = S db + X S db. B X d S + B Y B db X ds + B d S B V db Y db = G t. Suppoe there i ome meaure Q P and a Q quare-integrable martingale M uch that ds t = rs t dt + dm t. hen we ay that Q i a pot martingale meaure (for S). Likewie, if there i ome Q P uch that S i a Q martingale then we ay that Q i an equivalent martingale meaure (for S). hee definition are roughly equivalent and we will how in the next propoition that the firt implie the econd. he convere i morally true, but we don t need it and it require additionally the quare-integrability of S.

3 BLACK SCHOLES HE MARINGALE APPROACH 3 Propoition 2.2. If Q P i a pot martingale meaure then it i an equivalent martingale meaure. Proof. By claical calculu, db t Integrating by part with thi reult give u S t = B t S t = = rbt dt. B ds + S db. From our hypothei that Q i a pot martingale meaure we therefore have S t = r S d + B dm r S d = B dm. Becaue M i quare-integrable it follow that S t i a Q (quare-integrable) martingale, i.e. Q i an equivalent martingale meaure. Let C be a terminal claim on S at time. hat i C : [, ] R and C = f(s ). We ay that a elf-financing trategy replicate, or hedge, a terminal claim if V = C. If we can replicate the value of a claim at time then, to prevent arbitrage, the value of the claim at an earlier time mut equal the value of the replicating portfolio. We will now demontrate uch a hedging trategy. he following tatement form the bai for martingale pricing. Propoition 2.3. Suppoe S admit an equivalent martingale meaure Q. In the abence of arbitrage opportunitie, C t = B t E Q {B C F t }. Proof. Define M t = B t E Q {B C F t } and oberve that B M i a Q martingale. If W denote Q Brownian motion then by martingale repreentation, M t = M + H d W. By hypothei S i a martingale and therefore, for ome predictable K, we have the martingale repreentation S t = S + And by Ito lemma, uing f( S t ) = B t St = S t, S t = S + B d S = S + K d W. B K d W. Let X = H/K and Y = B (M XS) where G denote the gain of the trading trategy (X, Y ). By definition of the value proce, V t = X t S t + Y t B t = X t S t + M t X t S t B t B t = M t.

4 4 JOHN HICKSUN And in particular, at time, V = M = B E Q {B C F } = B B C E Q { F } = C. herefore (X, Y ) i a replicating portfolio for C. Oberve that V + X d S = V + X K d W = V + H d W = V + Bt M t M = Ṽt. And by propoition we ee that (X, Y ) i elf-financing. We may conclude that M t = C t, or ele there i an arbitrage opportunity. In particular, if M < C then the trategy (X, Y, M /C ) in the extended market coniting of S, B, and C ha V = M (M /C )C =, but V = M (M /C )C = C (M /C )C = C ( M /C ) >. 3. example Suppoe C i a european call with trike K and expiry on an underlying aet S. We aume that S walk like geometric brownian motion under the phyical meaure P. hat i, for contant µ and σ, and W t N (, t), ds t = S t (µdt + σdw t ). he payoff of the call i given by C = max(s K, ). By propoition 2.2, the preent value of C i given by the dicounted expectation of it payoff under an equivalent martingale meaure Q: C = e r E Q {max(s K, )}. Writing thi expreion in term of the max function make it difficult to work with; with ome imple algebra we can tranform the equation into a more ueful form: Propoition 3.. Under an equivalent martingale meaure Q, the price of a european call i given by C = e r E Q {S {S >K}} e r KQ(S > K). Proof. C = e r E Q {max(s K, )} = e r E Q {(S K) {S >K}} = e r E Q {S {S >K} K {S >K}} = e r E Q {S {S >K}} e r KE Q { {S >K}} = e r E Q {S {S >K}} e r KQ(S > K). It i clear from thi expreion that we have two difficult term to evaluate: E Q {S {S >K}} and Q(S > K). Our work will proceed a follow. Firt, we will decribe the evolution of the aet S (propoition 2). Next we will etablih the exitence of an equivalent martingale meaure Q and characterize the evolution of S under Q (propoition 3). We will then ue our reult about S to characterize the pace {S > K} (propoition 4). Finally, we will tackle the difficult term individually (propoition 5 and 6). Putting thee reult together will yield the Black-Schole formula (theorem 7).

5 BLACK SCHOLES HE MARINGALE APPROACH 5 Propoition 3.2. Let W t N (, t). If a proce S walk like geometric brownian motion then S t = S exp ( (µ σ 2 /2)t + σw t ). Proof. Firt, oberve that the given differential dynamic of S t are properly undertood a a notational horthand for the integral S t = S + µ S t dt + σ S t dw t. he former integral i newtonian and eaily handled. he latter i tochatic and require more care. By Ito lemma, f(s t ) f(s ) = Subtituting f(x) = log(x) we have log(s t /S ) = f (S t )ds t + 2 ds t S t 2 f (S t )d[s, S] t. d[s, S] t St 2. And by the integral dynamic of S t, linearity of the tochatic integral, and aociativity of the tochatic integral repectively ds t ( t t ) = d µ S t dt + σ S t dw t S t S t ( t ) ( t ) = d µ S t dt + d σ S t dw t = µ dt + σ dw t = µt + σw t. S t S t A an aide, oberve how formally ubtituting the (formal) differential dynamic for S t yield the ame reult. We can alo view aociativity a a formal cancellation of differential and integral operator. By propertie of quadratic variation, the above calculation, and Levy characterization We therefore have d[s, S] t S 2 t [ = ds t t, S t ] ds t S t = [µt, µt] + 2[µt, σw t ] + [σw t, σw t ] = σ 2 [W t, W t ] = σ 2 t. log(s t /S ) = µt + σw t σ2 2 t = And from thi the reult trivially follow. ) (µ σ2 t + σw t. 2 Propoition 3.3. Suppoe S walk like geometric Brownian motion under P. hen S admit an equivalent martingale meaure Q. Specifically, let W t = W t r µ σ t. hen W t i Q Brownian motion and ds t = rs t dt + σs t d W t.

6 6 JOHN HICKSUN Proof. Let Q be equivalent to P and define Z = dq/dp, Z t = E{Z F t }. Clearly Z t i a P martingale and by martingale repreentation, we may write Z t a an integral againt Brownian motion for ome predictable proce J t : Z t = + J dw. Auming Z i well-behaved, we can find H uch that Z H = J. herefore Z t = + Z H dw. Letting N = H W, by definition of the tochatic exponential, Z t = E(N) t. By Giranov heorem, the following expreion i a Q local martingale: [ ] σs dw d Z, σs τ dw τ. Z Our earlier work how that [ ] Z, σs τ dw τ [ = Z τ H τ dw τ, σs τ dw τ ] And by propertie of quadratic variation and Levy theorem, [ ] d Z, σs τ dw τ = Z H σs d[w, W ] = Z Z Setting H t = r µ, we ee that the following i a Q local martingale: σ σs dw H σs d = µs dt + σs dw H σs d. rs d = S t We refer to H a the market price of rik, or the Sharpe ratio. Oberve that t [ ] d[z, W ] = d Z τ H τ dw τ, dw τ = H d[w, W ]. Z Z And by Giranov theorem we proceed to define another Q local martingale W t = W t + Z d[z, W ] = W t + H t dt = W t + r µ σ t. rs d. Clearly W t i continuou, [ W t, W t ] = [W t, W t ] = t, and o by Levy characterization W t i Brownian motion under Q. Furthermore, by linearity σs t d W t + rs d = hi of coure can be expreed differentially a σs t dw t + ds t = rs t dt + σs t d W t. µs t d = S t. Brownian motion i quare-integrable and therefore Q i a pot martingale meaure. he reult follow by propoition 2.2.

7 BLACK SCHOLES HE MARINGALE APPROACH 7 Propoition 3.4. Let d = ( σ log(k/s ) (µ σ 2 /2) ). hen { } W {S > K} = > d. Proof. By Propoition 2, {S > K} = {S exp ( (µ σ 2 /2) + σw ) > K}. And olving for the random variable W give u { {S > K} = W > ( log(k/s ) (µ σ 2 /2) )}. σ he reult follow, dividing by (it i more convenient to work with a unit-variance random variable). Propoition 3.5. Let d 2 = ( σ log(s /K) + (r σ 2 /2) ). hen Q(S > K) = N (d 2 ;, ). Proof. By propoition 4 and 3 repectively we have ( ) ( W W + ( r µ ) ) Q(S > K) = Q σ > d = Q > d = Q ( ) ( ) W (r µ) W > d + σ = Q > d 2 = N (d 2 ;, ). Propoition 3.6. Let d = d 2 + σ, where d 2 i defined a in Propoition 5. hen Proof. Expanding the expectation, E Q {S {S >K}} = E Q {S {S >K}} = S e r N (d 2 + σ ;, ). S >K S dq = {S >K}S dq. By propoition 4 we can ubtitute above with {S >K}S dq = {W > σ (log(k/s ) (µ σ 2 /2) )} S exp ( (µ σ 2 ) /2) + σw dq Let U = W /. hen changing variable we have {S >K}S dq = {U >d2}s exp ((r σ 2 /2) + σ ) U dq. By the law of the unconciou tatitician (d(u Q)/dx denote the denity of U under Q) ( {S >K}S dq = {x>d2} S exp (r σ 2 /2) + σ ) d(u Q) x dx. dx R

8 8 JOHN HICKSUN And becaue U N (, ) under Q we have {S >K}S dq = S exp d 2 = S e r exp 2π Let u = x σ. hen d 2 E Q {S {S >K}} = S e r ( (r σ 2 /2) + σ x (σ ) x x2 2 σ2 2 dx. d 2 σ 2π e u2 2 du ( = S e r N ( d 2 σ ) ;, ) = S e r N (d 2 + σ ;, ) ) e x2 /2 dx 2π heorem 3.7. (Black-Schole) Let d 2 and d be defined a in Propoition 4 and 6 repectively. hen P = S N (d ;, ) e r KN (d 2 ;, ). Proof. hi reult follow trivially from Propoition, 5, and 6: P = e r E Q {S {S >K}} e r KQ(S > K) = e r S e r N (d 2 + σ ;, ) e r KN (d 2 ;, ) = S N (d 2 + σ ;, ) e r KN (d 2 ;, ).

9 BLACK SCHOLES HE MARINGALE APPROACH 9 Reference [] Protter, Philip. Stochatic integration and differential equation. Springer, 99. [2] Muiela, M, and Rutkowki, M. Martingale method in financial modelling. Vol. 36. Springer, 26. [3] [4] frey/skript-fimaii.pdf [5] zd.pdf [6] mh278/tochatic calculu.pdf