Girsanov s theorem and the risk-neutral measure

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1 Chapter 7 Girsanov s theorem an the risk-neutral measure Please see Oksenal, 4th e., pp Theorem.52 Girsanov, One-imensional Let B t T, be a Brownian motion on a probability space F P. Let F t T, be the accompanying filtration, an let t T, be a process aapte to this filtration. For t T, efine t eb = u u + B t =exp ; u Bu ; 2 t 2 u u an efine a new probability measure by fip = T IP 8 2F: Uner f IP, the process e B t T, is a Brownian motion. Caveat: This theorem requires a technical conition on the size of. If everything is OK. We make the following remarks: is a matingale. In fact, IE exp 2 T 2 u u < =; B B B ; 2 2 t = ; B: 89

2 9 fip is a probability measure. Since =, we have IE = for every t. In particular fip = T IP = IET = so f IP is a probability measure. fie in terms of IE. Let f IE enote expectation uner f IP.IfX is a ranom variable, then fie = IE [T X] : To see this, consier first the case X =, where 2F. We have fiex = IP f = T IP = T IP = IE [T X] : Now use Williams stanar machine. fip an IP. The intuition behin the formula fip = is that we want to have T IP fip! =T!IP! 8 2F but since IP! =an f IP! =, this oesn t really tell us anything useful about f IP. Thus, we consier subsets of, rather than iniviual elements of. Distribution of e BT. If is constant, then T = exp ebt =T + BT : n o ;BT ; 2 2 T Uner IP, BT is normal with mean an variance T,soBT e is normal with mean T an variance T : IP BT e 2 ~ b= p exp ; ~ b ; T 2 ~ b: Removal of Drift from e BT. The change of measure from IP to f IP removes the rift from e BT. To see this, we compute fie BT e =IE [T T + BT ] h n o = IE exp ;BT ; 2 2 T i T + BT = p T + b expf;b ; 2 2 T g exp ; = p b + T 2 T + b exp ; b ; y = T + b = p =: y exp ; ; y2 2 ; b2 y Substitute y = T + b b

3 CHPTER 7. Girsanov s theorem an the risk-neutral measure 9 Because We can also see that f IE e BT =by arguing irectly from the ensity formula n o IP eb 2 ~ b = p exp ; ~ b ; T 2 T = expf;bt ; 2 2 T g ~ b: = expf; e BT ; T ; 2 2 T g = expf; e BT T g we have n o n o n o fip ebt 2 ~ b = IP ebt 2 ~ b exp ; ~ b T = p exp ; ~ b ; T 2 ; ~ b T = p exp ; ~ b 2 ~ b: ~ b: Uner f IP, e BT is normal with mean zero an variance T. Uner IP, e BT is normal with mean T an variance T. Means change, variances on t. When we use the Girsanov Theorem to change the probability measure, means change but variances o not. Martingales may be estroye or create. Volatilities, quaratic variations an cross variations are unaffecte. Check: e B e B = t + B 2 = B:B = t: 7. Conitional expectations uner f IP Lemma.53 Let t T.IfX is F-measurable, then fiex = IE[X:]: Proof: fiex = IE[X:T ] = IE [ IE[X:T jf] ] = IE [X IE[T jf] ] = IE[X:] because t T, is a martingale uner IP.

4 92 Lemma.54 Baye s Rule If X is F-measurable an s t T, then fie[xjfs] = IE[XjFs]:. s Proof: It is clear that IE[XjFs] is Fs-measurable. We check the partial averaging s property. For 2Fs, we have s IE[XjFs] f IP = IE f s IE[XjFs] = IE [ IE[XjFs]] Lemma.53 = IE [IE[ XjFs]] Taking in what is known = IE[ X] = IE[ f X] Lemma.53 again = X f IP: lthough we have prove Lemmas.53 an.54, we have not prove Girsanov s Theorem. We will not prove it completely, but here is the beginning of the proof. Lemma.55 Using the notation of Girsanov s Theorem, we have the martingale property fie[ e BjFs] = e Bs s t T: Proof: We first check that e B is a martingale uner IP. Recall Therefore, e B = t + B =; B: e B= e B + e B + e B Next we use Bayes Rule. For s t T, fie[ e BjFs] = = ; e B B + t + B; t =; e B + B: s IE[ e BjFs] = ebss s = e Bs:

5 CHPTER 7. Girsanov s theorem an the risk-neutral measure 93 Definition 7. Equivalent measures Two measures on the same probability space which have the same measure-zero sets are sai to be equivalent. The probability measures IP an IP f of the Girsanov Theorem are equivalent. Recall that IP f is efine by fip = T IP 2F: If IP =, then R T IP =: Because T > for every!, we can invert the efinition of IP f to obtain IP = T f IP 2F: If f IP =, then R IP =: T 7.2 Risk-neutral measure s usual we are given the Brownian motion: B t T, with filtration F t T, efine on a probability space F P. We can then efine the following. Stock price: S = S t + S B: The processes an are aapte to the filtration. The stock price moel is completely general, subject only to the conition that the paths of the process are continuous. Interest rate: r t T. The process r is aapte. Wealth of an agent, starting with X = x. We can write the wealth process ifferential in several ways: X = S + r[x ; S] t Capital gains from Stock Interest earnings = rx t +[S ; rs t] = rx t + ; r S t +S B Risk premium 2 3 = rx t + S ; r 6 t + B Market price of risk=

6 94 Discounte processes: R e ; t ru u S R e ; t ru u X Notation: = e ; R t ru u [;rs t + S] R = e ; t ru u [;rx t + X] = e ; R t ru u S : R t =e ru u =r t R t = e; ru u = ; r t: The iscounte formulas are S X Changing the measure. Define = [;rs t + S] = [ ; rs t + S B] = S[ t + B] S = = S [ t + B]: eb = t u u + B: Then S X = S e B = S e B: Uner IP f, S X an are martingales. Definition 7.2 Risk-neutral measure risk-neutral measure sometimes calle a martingale measure is any probability measure, equivalent to the market measure IP, which makes all iscounte asset prices martingales.

7 CHPTER 7. Girsanov s theorem an the risk-neutral measure 95 For the market moel consiere here, fip = T IP where = exp ; t u Bu ; 2 2F t 2 u u is the unique risk-neutral measure. Note that because = ;r we must assume that 6=. Risk-neutral valuation. Consier a contingent claim paying an FT -measurable ranom variable V at time T. Example 7. V =ST ; K + V =K ; ST + V = T T V = max S tt Su u ; K European call European put! + Look back sian call If there is a heging portfolio, i.e., a process t T, whose corresponing wealth process satisfies XT =V, then X = IE f V : T This is because X is a martingale uner f IP,so X = X = f IE XT = IE f V : T T