where W( ) := W{i) + JQ M(s)ds defines a Brownian motion (W[t) : t [0, T}) under

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1 CHAPTERS. RISK-NEUTRAL PRICING (b) the discounted stock price S is a [local] martingale with respect to Q. The condition (a) is required because under the risk-neutral measure Q the same events A should be impossible (Q(A) = 0) which are impossible {P{A) = 0) under the original measure P. The interpretation of an arbitrage opportunity is the same as before for models in discrete time: the possibility to make profit without any risk. Thus, the definition is the same: Definition A self-financing trading strategy (F, A) -is called arbitrage opportunity if its value process V = V(I\ A) satisfies V(0) 0 P-a.s. but V(T) > 0 P-a.s. P(V{T) > 0) > 0. In discrete time models the results hold true that a model is arbitrage-free if and only if there exists an equivalent risk-neutral measure. However, the underlying argument relies heavily on the discreteness of the model. In continuous time, the no-arbitrage condition is not strict enough and one has to put some further constraints on the trading strategy, the so-called condition of no free lunch with vanishing risk. See the classical work by Delbaen and Schachermayer (1994). In this course, we avoid this rather technical part and do not introduce the meaning of no free lunch with vanishing risk but call and think of it as basically arbitrage-free. Theorem (First Fundamental Theorem of Asset Pricing) The model is basically arbitrage-free if and only if there exists an equivalent local mar tingale measure Q. In the discrete case we could construct the equivalent risk-neutral measure just by using the martingale condition. In continuous time models we derive this measure by means of Girsanov's theorem: Theorem If the volatility process (A(t) : t [0, T]) satisfies P{\(t)^0) = 1 for all t [0,T], and if the market price M obeys the Novikov condition EP expu f M2(s)ds < oo (5.2.3) then the discounted stock price S satisfies ds{t) = K(t)S(t)dW(t) for all t [0,T], where W( ) := W{i) + JQ M(s)ds defines a Brownian motion (W[t) : t [0, T}) under the measure,l], Q{A) = taexp(- f M{s)dW{s)-\ [ M2{s)ds) \ Jo Jo / (5.2.4)

2 5.3. MARTINGALE PRICING Theorem implies that the discounted share prices satisfy S(t) = 5(0) + / A(s)S(a)dW{s) for all t e [0,T]. Jo Since W is a Brownian motion under Q it follows from part (d) in Theorem that S is a local martingale under Q. Furthermore, the measure Q is equivalent to the measure P according to Remark and thus, the constructed measure Q is an equivalent local risk-neutral measure. It follows from Theorem that under the conditions in Theorem the model is (basically) arbitrage-free. However, the constructed measure Q might not be unique. Corollary Under the conditions of Theorem the share price S satisfies ds(t) = R{t)S{t) dt + A(t)S(t)dW(t) for all t e [0, T] where W( ) := W(t) -f- Jo M(s)ds for all t E [0,T]. Consequently, S can be represented by S(t) = 5(0)exp ( f (R{s) - A2(s)) ds+ I A(s)dW(«)] for all t e [0,T]. Corollary illustrates that the change from the measure P to Q changes only the mean rate of return but not the volatility of the stock price 5. It follows that for each u G [0, T) the share price can be represented by S{t) = S{u)exp ( f (R(s) - A2(s)) ds + f A{s) dw{s)\ (5.2.5) for all t e [u,t]. Example In the Black-Scholes model the market price of risk is given by M(t) = ^ ^- a forallt<e[0,t], where all parameters /x,r and a are deterministic and constant. Thus, M(t) is also deterministic and constant which yields that (5.2.3) is satisfied and the measure Q : * -> [0,1], Q(A) := EP [la exp (-^W(T) - I (^f t)\ is verified as an equivalent risk-neutral measure in the Black-Scholes model. 5.3 Martingale Pricing In the Black-Scholes model we considered so far only European call and put option. But on the market much more options are traded. Definition (a) A contingent claim is any random variable C : O > R which is ^

3 CHAPTER 5. RISK-NEUTRAL PRICING (b) A price process {Uc{t) - t [0,T]) of a contingent claim C is any adapted process. Of course, a price process {Tlc(t) : [0,T]) of a contingent claim C should be reason able, which means it does not lead to an arbitrage-opportunity. To be more precise, a price process is called arbitrage-free if the extension of the original market, which consists of the risk-free asset B and the risky asset 5, by the price process lie is arbitrage-free. In the extended market, not only the risk-free asset (S(t) : t [0, T]) and the risky-asset {S(t) : t [0,T]) but also the price process (IIc( ) : t e [0, T]) can be traded at any time t 6 [0, T]. A rigorous definition would require to consider multidimensional markets which we want to avoid. The analogue idea was exploited in the proof of Lemma The deterministic value IIc(O) is called arbitrage-free price of the contingent claim at time t = 0. Theorem 5.3,2. (Risk Neutral Pricing Formula) Assume that Q is an equivalent martingale measure and let C be a contingent claim with EQ B(T) C <oo. Then an arbitrage-free price process (Ilc(t) : t [0,T]) of the contingent claim C is given by Uc{t) = EQ \e-tfr^dsc\#t] for all te [0,T]. (5.3.6) In particular, the discounted price process Tic is a martingale under Q. Formula (5.3.6) is called risk-neutral pricing formula. The risk-neutral measure Q in Theorem is not necessarily the probability measure constructed by Giransov's The orem in Theorem In particular, it might be not unique and thus, the price process is not necessarily unique. Note, that the risk-neutral pricing formula gives an arbitragefree price process for every contingent claim C and not only for contingent claims of the form C = g{s{t)) for a function g : R+ -> R Complete markets Theorem suffers from two facts: firstly, it docs not say anything about the unique ness of the arbitrage-free price and secondly, it does not provide us with a strategy which hedges the contingent claim, i.e. which has the same value at maturity as the contingent claim. As in the discrete models these strategies are called replicating strategies. Definition (a) A contingent claim C is called attainable if there exists a self-financing strategy (F, A) such that the value process V = V(T, A) satisfies V{T) = C P-a.s. In this case (F, A) is called a replicating strategy for C. (b) A model is called complete if every contingent claim is attainable. The following theorem gives an abstract condition for the existence of a replicating strat egy.

4 5.4. COMPLETE MARKETS Theorem Assume that Q is an equivalent risk-neutral measure and let C be a contingent claim with Eq \\^j^c\\ < oo. Let {Uc{t) : t G [O.T]) be the arbitrage-free price process of C defined by Uc(t) := EQ [e" ft *< > dsc\ &t] for all t G [0, T]. // there exists an adapted process {H(t) : t G [0, T]) such that the discounted price process tic satisfies Tlc(t) = nc(0) + / H{s) ds{s) for all t G [0, T], Jo then C is attainable and the replicating strategy ((?( ), A(t)) : t [0,T]) is given by A(t) = H{t) for all t [0,T], T{t) = tlc(t) - A(t)S(t) for all t G [0,T]. By applying the martingale representation Theorem we obtain more specific condi tions for the existence of a replicating strategy. Note, that it is required that the filtration is generated by the Brownian motion which excludes stochastic volatility models as in Example Theorem (Second Fundamental Theorem of Asset Pricing) Assume that Q is an equivalent risk-neutral measure and that the volatility process (A(i) : te [0,T]) satisfies P(A(t)^0) = 1 for all t [0,T]. Let C be a contingent claim with Eq hjtfyci < and ^ (Rc{t) ' t [0, T]) be the arbitrage-free price process of C defined by Uc{t) := Eq [c~ ft Ris) dsc\ &t] for all t g [0, T]. // the filtration {^t}te[o,t] w generated by the Brownian motion W, i.e. &t = &^ for all t [0, T]} then there exists an adapted stochastic process (H(t) : t G [0,T]) such that the discounted price process tic satisfies f nc(0 = nc(0) + / H{s) ds{s) for all t G [0,T]. By applying Theorem it follows that under the conditions in Theorem the contingent claim C is attainable. Theorem states the replicating strategy in terms of the stochastic process H. However, the existence of this stochastic process H is derived in Theorem by applying the martingale representation theorem and the latter does not give an explicit construction of this stochastic process. Thus, the existence of a replicating strategy is guaranteed but still one does not know it explicitly. Concerning the uniqueness of the price process and the value process of a replicating strategy we have analogously to Theorem the following result.

5 CHAPTER 5. RISK-NEUTRAL PRICING Corollary Assume that Q is an equivalent martingale measure and let C be a contingent claim with and let (ITc(i) : t G [0,T]) denote an arbitrage-free price process ofc. If there exists a replicating strategy (F, A) for C with value process V = V(Ty A) then for all te[q,t}. In this situation, it follows that V(t) = tlc(t) = EQ [^Cl ^t] for all t e [O,Tj. (5.4.7) As in Remark if follows now that since the right hand side in (5.4.7) does not depend on a particular replicating strategy, all replicating strategies (F, A) for the same contingent claim must have the same value process K(F, A). On the other hand, since the left hand side in (5.4.7) does not depend on a particular choice of an equivalent martingale measure Q, the value process V(T, A) of a replicating strategy (F, A) is always a version of the conditional expectation Eq[-qTjxC\ &t]. It follows that the arbitrage-free price process must be unique. Consequently, if the contingent claim is attainable then its price process and the value process of every replicating strategy is unique and they coincide. As in discrete time models we have the following result: Theorem (Second fundamental theorem) Assume that the model is (basically) arbitrage-free. Then the model is complete if and only if the equivalent martingale measure is unique. Proof. See Theorem in [2]. In stochastic volatility models such as in Example the share prices 5 are in general not adapted to the filtration generated by the Brownian motion. In fact, it turns out that these models are usually not complete and that there does not exist a unique process for contingent claims in these models. Another example with the volatility A(t) = t can be found in [22, Ex. 14.2]. The analogue situation we observed already in Example in discrete time. This model can be considered as a stochastic volatility model in discrete time. The following result on conditional expectation is helpful to calculate the value process in some examples: Lemma Let 38 be a sub-a-algebra of srf. Let X be a random variable which is 3 - measurable and let Y be a random variable which is independent of 3&. If f : 1R x ]R ) IR is a function with E[\f(X, Y)\] < oo then E[f(X,Y)\@] = h(x) P-a.s., where h : IFt > H is a function defined by h(x):=e[f(x,y)}.

6 5.5. REPLICATING STRATEGIES Example In the special case of the Black-Scholes model we derived already the arbitrage-free price process for a European call option Ccau = {S(T) - K)+. In this example we derive again the arbitrage-free price process but by applying the martingale approach. Let Ccaii = 9(S(T)) with g(x) = (x - K)+ be a European call option. In the Black-Scholes model all parameters are constant and deterministic, i.e. 0(t) = fi and A( ) = a for all t e [0, T] and some constants \x G M. and a > 0 and R(t) = r for some r ^ 0. Let Q be the equivalent risk-neutral measure, see Example 5.2.6, and let (F, A) be the replicating strategy for the contingent claim C which exists according to Theorem For the value process V of (F, A) Corollary implies that V(t) = EQ [e-r^t^g{s{t))\^t] for all t e [0,T]. (5.4.8) The representation (5.2.5) yields that S(t) = S(u) exp ((r - \a2) (t - u) + a(w{t) - W(u) for all 0 ^ u ^ t. Applying this representation for t = T and u = t to (5.4.8) we obtain by Lemma that V(t) = EQ [e-'v-vg (S(i)exp ((r - ±<x2) (T - t) + a(w(t) - = h(t,s(t)) for a function h : [0, T] x R -> 1R where h(t,x) = EQ [e-r(t-^5 (xexp ((r - \a2) (T - t)+o(w{t) - = -^ [" e-r{t-vg (xexp (avf~^t y + (r - \a2)) (T - t)) e~2v2 dy. V27T./-oo V ^ * ' Using the definition g{x) = (x K)+ we can continue and evaluate the integral to obtain: h(t, x) jl ay/t^t y) - k)+ e~b* dy = xfn(d+(t - t,x)) - Ke-^T-^FN(d.(T - t,x)), where the constants d+ and d- are defined in Theorem Replicating Strategies The results on the existence of a replicating strategy rely heavily on the martingale repre sentation Theorem Since this theorem does not provide an explicit representation of the process H in Theorem one still does not know how to hedge a given contin gent claim. For the martingale approach, considered in this section, the Feynman-Kac Theorem and its Corollary enable us to derive the price function of a con tingent claim as the solution of a deterministic partial differential equation which also yields the replicating strategy.

7 CHAPTER 5. RISK-NEUTRAL PRICING It is essential in this section that the share prices (S(t) : t G [0, T]) form a Markov process. For that reason we assume in the following the classical Black-Scholes model, that is Q(t) = /*, A(t) = a and R(t) = r for all t e [0, T] and constants \x e R, r R+ and a > 0. The risk-free asset is given by whereas the risky asset 5 evolves according to db{t) - rb[t) dt for all t [0,T], 6(0) = 1, (5'5"9) ds(t) = fis{t) dt + <rs(t) dw(t) for all t e [0, T]. (5.5.10) The filtration is assumed to be generated by the Brownian motion, i.e. &t = 3?Y for all t 6 [0, T). Under these assumptions the model is arbitrage-free and complete. The measure Q constructed in Theorem is the unique equivalent risk-neutral measure. Let C be a contingent claim of the form C = g(s{t)) for a continuous function g : R+ -> R+ with EQ \wr\9{s{t))\ < oo. It follows by Theorem that there exists a replicating strategy (I\ A). Let V denote the value process of the replicating strategy (I\ A). According to Theorem 5.3.2, the arbitrage-free price process lie of C is given by Uc{t) = e-r^-^eq{c\ &t) for all t E [0,T]. Corollary implies that the value process V of the replicating strategy (F, A) and the price process n^ coincide: V(t) = Uc{t) = e-^-^eqlcl^t) for all t e [0,T]. Since (S{t) : t [0,T]) is a Markov process by Theorem and C = g(s(t)) it follows that there exists a function d : [0, T] x R+ > IR+ such that EQ[C\&t] = EQ\g(S(T))\&t] = d{t,s{t)) for all t e [0,T]. Defining a function / : [0,T] x R+ -> R+ by f{t,x) := e-r(t- )d(t,a;) yields V{t) = licit) = f(t,s(t)) for all t [0,T]. As in Chapter 3 the function / is called price function of the contingent claim C. On the other hand, Corollary guarantees that the share prices S satisfy the stochas tic differential equation ds(t) = rs(t) dt -1- (rs{t) d W(t) for all t [0, T], where (W(i) : t e [0, T]) is a Brownian motion under the measure Q. Thus, the Feynman- Kac Theorem implies that the price function / is a solution of the deterministic partial differential equation ^,x) - r/(t,«) = 0, (5 5 u) f(t,z) = g(x),

8 5.6. DIVIDEND-PAYING SHARES for all x IR+ and t [0,T]. Consequently, we have recovered the result in Theorem that the price function / in the Black-Scholes model is a solution of the partial differential equation (5.5.11). In order to obtain an explicit representation of the replicating strategy (F, A) we note that due to the self-financing condition the value process V of (r, A) satisfies dv(t) = T(t) db(t) + A(t) ds(t) = T{t)rb(t) dt + A(t) ds{t) (5.5.12) for all t [0,T]. On the other hand, since S is an Ito process of the form S{t) = S(0)+ [ rs{s)ds+ I as(s)dw{s)y Jo Jo and the price function / is in C1'2, Ito's formula implies that df(t,s(t)) = (ft(t,s(t)) + yxx(t,s(t))a2s2(t)) dt + fx(t,s(t))ds(t) (t, S(t)) - rs{t)fx(t, 5(t))) dt + /x(t, S{t)) ds(t), (5.5.13) where we used the fact in the last line that / is a solution of (5.5.11). Due to V(t) = f{t,s(t)), both equations (5.5.12) and (5.5.13) are equal and we obtain that r(t) = ^ for all te [0,T]. The derivation above verifies the price function as a solution of the partial differential equation (5.5.11). This can be used to determine the value process either explicitly or, because this is most often not possible, numerically. The application of the Feynman-Kac Theorem is restricted to contingent claims of the form g(s(t)), i.e. the pay-off depends only on the share price S(T) at expiration. But there are many other contingent claims traded on markets that are not of this form, confer the Asian options in the discrete time models. We will consider this situation in the following chapter. 5.6 Dividend-Paying Shares A dividend is a distribution of a portion of the profit of a company or of an asset to a class or all of its shareholder. Dividends are paid usually in cash or shares. We consider in this section two cases, first where the dividend is distributed continuously and secondly where it is paid in lumps Continuously Paying Dividends We assume that the dividends are paid continuously over time at a rate A(t) per unit time where {A(t) : t G [O.T]) is a nonnegative adapted stochastic process. The risk-free asset B = {B(t) : t G [0,T]) is not affected by paying dividends only the share price S whose

9 CHAPTERS. RISK-NEUTRAL PRICING value is reduced by the dividend distributed. Thus, the share price 5 = (S(t) : t [0,T]) evolves according to ds(t) = e(t)s{t) dt + A{t)S{t) dw{t) - A(t)S{t) dt for all t e [0,T], where as before (@(t) : t e [0,T]) and (A(t) : t [0,T]) are adapted stochastic processes. A trading strategy (A,F) and its value process (V(i) : t e [0,T]) are defined as before. But as the dividend payments are assumed either to purchase additional shares or to invest in the risk-free asset the notion of self-financing strategy must be modified: Definition A trading strategy ((F( ), A(t) : t [0,T]) with [ \T(s)\ ds < oo, / A(s) \A{s)\ S{s) ds < oo, / A(s) 2 ds < oo P-a.s., Jo Jo Jo is called self-financing for the dividend-paying share S if the value process V = V(A,F) satisfies dv{t) = T(t) db{t) + A(t) ds(t) + A(t)A(t)S{t) dt for all t [0, T]. Under the risk-neutral measure the discounted value process turns out to be a local martingale. This is the same situation as before for a share without paying dividends. Theorem Let Q be an equivalent risk-neutral measure and assume that the volatil ity process satisfies P(A(t) / 0) = 1 for all t G [0, T]. Then the discounted value process (V(t) : t [0, T]) of a self-financing trading strategy for the dividend-paying share S is a martingale under Q. Let Q be the equivalent risk-neutral measure constructed by Girsanov's Theorem in Theorem and let C be a contingent claim with Assume that there exists a replicating strategy (F, A) for C such that the discounted value process V of (A,F) is a martingale under the risk-neutral measure Q. Then we obtain that V(t) = EQ [V(T)\ &t] = EQ [bjtjci *t] ^ all t which we can rearrange to V{t) = EQ exp - / R{s) ds \C\ & \ Jt I for all t [0,T]. (5.6.14) We have obtained the same risk-neutral pricing formula as in Corollary of a share without paying dividends.

10 -So DIVIDEND-PAYING SHARES As in Corollary it follows that S{t) = 5(0)exp (f (R{s) - A(s) - ±A2(s)) ds + f A(s)dW(s)\ for all t G [0,T] and thus, S evolves according to = (R(t) - A(t)) S(t) dt + \{t)s(t)d\v{t) for all t e [0,T]. In contrast to the situation without paying dividends the discounted stock price S is not a (local) martingale. However, with SD(t) := exp ( f A{s) ds\ S(t) = exp(f A(a) dw(s) -if A2(s) it follows that Sd '-= (Sx>( ) : t G [0,T]) is a local martingale under Q. Example In the Black-Scholes model we consider the case of a dividend contin uously paid with constant rate a. Thus, the share price obeys ds{t) = fis{t) dt + as(t) dw{t) - as(t) dt for all t [0, T] with some constants [i G IR and a, a > 0. Let C = (S(T) K)+ be the European call option on the share 5 and let (A, T) be the replicating strategy for C with value process V. It follows as in Example that V(t) = EQ [e-^t-^(5(t) - K)+\ ] = EQ [e-r<t-*) (S{t) exp ((r - a - \a2){t - t) = h{tts(t)) for a function /i:[0,t]xr-)]r given by h(t,x) = EQ re-r<t-*> (a:exp ((r - a - \o2){t - t) +a(w{t) - W(t))) - -^)+ where d^: = j^=r(\nf + (r-a±^)(t-t)) Distributed Paying Dividends For one share it is more realistic to assume that the dividends are paid in lumps at different times 0 < t\ < < tn < T. At each time tj the dividend payment is a,js(tj-)

11 -*\ CHAPTER 5. RISK-NEUTRAL PRICING where S{tj-) denotes the share price just before the dividend distribution and aj is an &tj -measurable random variable with values in [0,1]. Thus, at time tj the share price is S{tj) = S{tj-) - ajsitj-) for j = 1,...,n. (5.6.15) Between two dividend payments the dynamic of the share price is described by ds{t) = B(t)S(t)dt + A(t)S{t)dW(t) for t e [tj,tj+1) (5.6.16) for j = 0,..., n with t0 = 0 and tn+i = T with a0 = 0 and an+i = 0. A trading strategy (F, A) with value process V = {V(t) : t [0, T]) is called self-financing if the value process V satisfies at the payment data tj-) = V(tj) forallj = l n, and between the payment data it obeys the usual self-financing condition: dv{t) = T(t)dB{t) + A{t)dS{t) for all t e [tj,tj+l) and all n = 1,...,n. Because the latter equation implies that V is piecewise continuous between the payment data and the first equation is the continuity of V at the payment data the value process V is continuous on [0, T] and satisfies dv(t) = A{t) ds{t) 4- T(t) db{t) for all t 6 [0, T]. Now we can proceed as in the situation without dividends to derive that under the riskneutral measure Q the discounted value process V is a local martingale, see Corollary If V is the value process of a replicating strategy for a contingent claim C and if the discounted value process V is a martingale under Q we obtain V(t) = EQ [V(T)\ &t] = EQ yljc\&tj for all t [0,T], which we can rearrange to V(t) = EQ exp! R(s)ds\c\ for all t [0, T]. (5.6.17) Thus, also in the case of dividends paid in lumps the risk-neutral pricing formula (5.4.7) remains the same. Example In the Black-Scholes model we consider the case of a dividend paid at the times 0 < t\ < < tn ^ T with non-random rates aj [0,1]. Thus, the share price obeys S{tj) = S(t,-) - ajsitj-) for j = 1,...,n, (5.6.18) and ds{t) = fis(t) dt -f as{t) dw(t) for t <E [t for some constants /^ee and a > 0. Let Q denote the equivalent risk-neutral measure (Girsanov measure) and W the Brownian motion under Q, see Theorem Then we have ds{t)=rs(t)dt + \S(t)dW{t) for t e [tj,tj+l), (5.6.19)

12 -n DIVIDEND-PAYING SHARES where r ^ 0 is the interest rate in our mode. We have S{t) = S(s)exp ((r - \a2){t - s) + a(w{t) - W(«))) for all tj^s^t< tj+u and in particular it follows S(tj+1-) = (1 - a^sitj-) exp ((r - \*2){tHl - td) + By using (5.6.18) and (5.6.19) we obtain for t e [0,T]: = S{0)I{t) exp ((r - ict2)i + a W(t)), where n[t) - o-j) and n[t] := max{j : tj ^ i}. Let C = (5(T) - K)+ be the European call option on the share 5 and let (F, A) be the replicating strategy for C with value process V. By defining :=^-S(t) forallt [O,T], we obtain a stochastic process (Z( ) : t e [0,T]) which we consider as the share price without dividends. The risk-neutral pricing formula implies for t [0,T]\ V(t) = EQ = I(T)EQ [e-^-0 (Z(T) - jfa By using Theorem we calculate V(t) = J(t)S(t)FN (d»d(t-t,s(t)))-e-tlt-»kfn (d «(T - t,s{t))), where - t, ar) : = j^ (In f + In J(t) + {r ± \a2) (T - t)),