Equivalent Martingale Measures

Advanced Stochastic Processes. LECTURE 17 Equivalent martingale measures 17.1. Equivalent martingale measure We continue with the assumptions used in the previous lecture that price of a security can be modeled as an Ito process. Assume we have N price processes dXj = µj (t)dt + σj (t)dB(t), j = 1, 2, . . . , N . We assume that B is a vector of d independent Brownian motions and each σj is d­dimensional vector process as well (for simplicity we skip the · notation from here onwards). We recall that this Brownian motion is adapted to some filtration Ft on a probability space (Ω, F, P). Recall that a probability measure P∗ is equivalent to P if P(A) > 0 iff P∗ (A) > 0 for all A ∈ F, and any such measure P∗ ∗ can be obtained via some random variable ψ = ddPP . Definition 17.1. A probability measure P∗ is defined to be an equivalent martingale measure with respect to the price process X = (Xj ), 1 ≤ j ≤ N if (a) P∗ is equivalent to P. ∗ has finite variance. (b) ψ = dP dP (c) X(t) is a (componentwise) martingale with respect to P∗ . The notion of equivalent martingale measures is important since, as we shall shortly see, it implies that arbitrage is impossible and can be used in principle to compute prices of various (derivative) securities such as options. In particular, the Black­Scholes formula can be derived using the concept of equivalent martingale measure. Theorem 17.2. Consider a vector of price processes Xj , 1 ≤ j ≤ N . Suppose the price process X admits an equivalent martingale measure. Then there is no arbitrage. Proof. We fix an arbitrary self­financing strategy θ ∈ H 2 and a time horizon T . Then � T θ(T ) · X(T ) = θ(0) · X(0) + θ(t) · dX(t), 0 1 2 or � θ(0) · X(0) = θ(T ) · X(T ) − T � θ(t) · dX(t) = EP∗ [θ(T ) · X(T ) − 0 T θ(t) · dX(t)], 0 where, P∗ is an equivalent martingale measure. Note that the second equality holds trivially from the first equality. We argue that � T (17.3) EP ∗ [ θ(t) · dX(t)] = 0. 0 First, let us show how this implies the result of the theorem. This equality implies θ(0) · X(0) = EP∗ [θ(T ) · X(T )], Recall that arbitrage means θ(0) · X(0) < 0 and θ(T ) · X(T ) ≥ 0 a.s. or θ(0) · X(0) ≤ 0 and θ(T ) · X(T ) > 0 a.s. Suppose θ(T ) · X(T ) ≥ 0. Then θ(0) · X(0) ≥ 0. This rules out the first condition for an arbitrage. Now, if θ(T ) · X(T ) > 0, in other words it is non­negative a.s. and is positive on a set A with positive measure P(A) > 0, then, by equivalence, also P∗ (A) > 0, implying EP∗ [θ(T ) · X(T )] > 0 or θ(0) · X(0) > 0. We conclude that there is no arbitrage. Now we establish (17.3). Since each component Xj is a martingale w.r.t. P∗ and since it is an a.s. continuous process (by Theorem 15.1 of Lecture 15), then by Martingale Representation Theorem, Xj is an Ito process dXj = σj∗ dB ∗ for some vector process σj∗ ∈ H 2 , where B ∗ is a Brownian motion w.r.t. P∗ . Thus θj dXj = θj σj∗ dBj∗ and therefore is an Ito integral w.r.t. P∗ . We know that Ito integral �T is a martingale. Therefore E[ 0 θj (t)dXj (t)] = 0. � In the case when conditions of Girsanov theorem apply we can obtain an equivalent martingale measure explicitly. We fix a time horizon T . Suppose there exists an equivalent martingale ∗ measure given by Radon­Nykodim derivative M = ddPP . We let M (T ) = M and define M (t) = E[M (T )|Ft ] for every t ≤ T . Just as we did in Lecture 15, we can show that P∗ = M (t)P on a subfield Ft . In other words the change of measure is given by a process M (t). Problem 1. Establish that (a) M (t) is a martingale. (b) M (t) > 0 a.s. Ignoring continuity issues of M (t), we use martingale representation theorem to argue that dM = ψdB for some d­dimensional process ψ. By definition each Xj , 1 ≤ j ≤ N is a martingale under P∗ . That is M (t)Xj (t) is a martingale. Let us find the Ito representation of Xj in P∗ which is the Ito representation of M (t)Xj (t) in P: d(M Xj ) = Xj dM + M dXj + (dM dXj ) = Xj ψdB + M µj dt + M σj dB + ψσj dt. Since M Xj is a martingale under P∗ , then the non­Brownian components must be equal to zero M µj + ψσj = 0. We consider the d­dimensional process η = −ψ/M , which is allowed since M is positive a.s. We conclude that M must satisfy for every j (17.4) η(t)σj (t) = µj (t). This defines a system of N equations in d time dependent variables η. Thus we obtain the following result. LECTURE 17. APPLICATIONS TO FINANCE 3 Theorem 17.5. Given the price process X(t) = (Xj (t)) with dXj = µj dt + σj dB suppose there exists an equivalent martingale measure. Then the system of equations (17.4) must have a solution. Any such solution η is called market price of risk process for (Xj (t)). Under additional technical assumptions, the condition that (17.4) has a solution is also suf­ ficient for the existence of an equivalent martingale measure. In case η satisfies the Novikov’s condition, we can use� Girsanov’s Theorem II to conclude t ˆ = 0 η(s) + B(t) is a Brownian motion, that under the equivalent measure P∗ the process B(t) the Ito representation of Xj is given as dXj (t) = σj (t)dB̂(t) and the change of measure process �t �t is M (t) = exp(− 0 η(s)dB(s) − 12 0 η 2 (s)ds). Equivalent martingale measures can be sometimes used for price valuation of various financial instruments. For example it can be used to derive Black­Scholes formula in roughly the following way. We have N = 3. X1 (t) = S(t) is a stock process with dS = Sµdt + SσdB, Xj = β is a risk­free bond with rate r and X3 = U is a call option with strike price K and strike time T . We have established that existence of an equivalent martingale measure implies no arbitrage. Under additional technical assumptions the converse is also true. We make this assumption and suppose that there is no arbitrage and therefore and equivalent martingale measure P∗ exists which turns X(t) into a martingale. Since β is risk free then β(t) = EP∗ [β(t)]. But then martingale assumption implies β(t) = β(0) or r = 0. (In practice this is typically not the case, and the equivalent martingale measure is usually found for the discounted processes S/β). Let M = M (T ) be the random variable which does this. Since U is a martingale under P∗ then (17.6) (17.7) U (t) = EP∗ [U (T )|Ft ] = EP∗ [(S(T ) − K)+ ] E[M (T )(S(T ) − K)+ |Ft ] E[M (T )|Ft ] E[M (T )(S(T ) − K)+ |Ft ] (17.9) = M (t) Recall, that the condition of Girsanov theorem is that there exists a solution η for ησ(t) = µ(t) which satisfies Novikov’s condition. In our case σ(t) = σS(t) and µ(t) = µS(t). Therefore η is simply a constant µ/σ, it satisfies the Novikov’s condition (check this), and M (t) = exp(− σµ B(t)− µ2 t). It can be checked that the Black­Scholes formula provides a solution for the equation (17.9). 2σ 2 Equivalent martingale measure provides a method for pricing securities in cases where closed form formulae such as Black­Scholes cannot be found. One then tries to identify a measure P∗ which makes the price process a martingale via equation of the form X(t) = EP∗ [X(T )]. (17.8) = 17.2. Additional reading materials • Duffie ”Dynamic Asset Pricing Theory” [1], Chapters 5 and 6.