Valuation of derivative assets Lecture 8 Magnus Wiktorsson September 27, 2018 Magnus Wiktorsson L8 September 27, 2018 1 / 14
The risk neutral valuation formula Let X be contingent claim with maturity T. The risk neutral valuation formula gives [ ] B(t) Π(t, X) = E Q B(T ) X F t S, where Q is the risk neutral or the martingale measure. But we also have [ ] Λ(T ) = E P Λ(t) X F t S, where Λ is a stochastic discount factor. Magnus Wiktorsson L8 September 27, 2018 2 / 14
The Martingale measure 1 Questions about Q a) When does Q exists? b) When is Q unique? 2 How does this relate to: a) The model is free of arbitrage. b) The model is free of arbitrage and complete. 3 How can we find Q? 4 How do we relate the measures P and Q? 5 Do P impose any restriction on Q? Magnus Wiktorsson L8 September 27, 2018 3 / 14
Do P impose any restriction on Q? We want to answer questions about arbitrage possibilities in a model! The answer should be the same whether using P or Q. i) V h (0) = 0, ii) P(V h (t) 0) = 1, iii) P(V h (t) > 0) > 0, for some t > 0. This gives that if P(A) = 1 then Q(A) = 1 and vice verse. Which is the same as P(A c ) = 0 Q(A c ) = 0. So Q and P should have the same sets of mass zero. Two measures with the same zero sets are called equivalent. This is denoted as P Q. Magnus Wiktorsson L8 September 27, 2018 4 / 14
The Radon-Nikodym theorem If two probability measures P 1 and P 2 are equivalent on a σ-algebra F then there exists a random variable L on (Ω, F) such that for all A F P 2 (A) = E P 1 [I A L] = I A (ω)l(ω) dp 1 (ω), We also write this as P 1 (A) = E P 2 [I A 1 L ] = Ω Ω dp 2 dp 1 = L 1 I A (ω) L(ω) dp 2(ω). where L is called the likelihood ratio or the Radon-Nikodym derivative. Magnus Wiktorsson L8 September 27, 2018 5 / 14
Properties of L P 1 (Ω) = 1 and P 2 (Ω) = 1 this gives that 1 = P 2 (Ω) = E P 1 [I Ω L] = E P 1 [L] So E P 1 [L] = 1. Moreover since for all A F we have P 2 (A) 0 we get that P 1 (L 0) = 1. We then have that L is a non-negative random variable with P 1 -expectation one. We also get that E P 2 [1/L] = 1. Moreover we have for all random variables X on (Ω, F, P 1 ) (and (Ω, F, P 2 )) that E P 2 [X] = E P 1 [LX]. If G F then E P 2 [X G] = EP 1 [LX G] E P 1. [L G] Magnus Wiktorsson L8 September 27, 2018 6 / 14
Properties of Q We should have that Q P on FT S for each finite T where {F t S } t 0 is the filtration generated by the traded asset processes S 0, S 1,..., S N (S 0 = B is the bank account). Moreover for each traded asset S i, i = 1,..., N and each T > 0 we should have that This is equivalent to S i (t) = E Q [ B(t) B(T ) S i(t ) F S t S i (t) B(t) = EQ ], 0 t T. [ ] Si (T ) B(T ) F t S, 0 t T, which means that all discounted traded assets are Q-martingales. Magnus Wiktorsson L8 September 27, 2018 7 / 14
Equivalent measures and filtrations Let {F S t } t 0 be a filtration on (Ω, F S, P). If P Q on F S T with LR=L T then P Q on F S t, t < T since F S t F S T. What is the LR=L t on F S t? So for A F S t we should have Q(A) = E P [L t I A ] but since F S t F S T we also have Q(A) = E P [L T I A ] Tower property = E P [E P [L T I A F S t ]] Taking out what is known = E P [I A E P [L T F S t ]] So we can identify L t = E P [L T F S t ]. This also immediately gives that L t is a P-martingale w.r.t {F S t } 0 t T due to the tower property. The process {L t } 0 t T is called a likelihood ratio process. Magnus Wiktorsson L8 September 27, 2018 8 / 14
How can we find Q? It turns out that in general this is a really complicated problem. But if we assume that the asset price model is generated by SDE:s driven by Brownian motions. There is only a very specific form any likelihood ratio process can take (see next slide). Equivalent measure changes in this case only alters the drift and thus leaves the diffusion part un-altered. Magnus Wiktorsson L8 September 27, 2018 9 / 14
Girsanov transformation (Å: Thm 9.7 p. 220-221) Let F t be a filtration on (Ω, F, P) such that {W P t } t 0 is a (d-dim) Brownian motion w.r.t. F t. Let g t be a (d-dim) process adapted to F t for t [0, T ] which satisfies ( 1 T )] E [exp P g t 2 dt <, (Novikov condition). 2 0 Define the process L t by ( t L t = exp gs dws P 1 2 0 t 0 ) g s 2 ds, 0 t T. Define a new probability measure Q on F T by Q(A) = E P [1 A L T ] for A F T. Then W Q t = Wt P + t 0 g s ds is a standard (d-dim) Q-BM on [0, T ]. Magnus Wiktorsson L8 September 27, 2018 10 / 14
The new dynamics after change of measure Suppose that the market (N+1 assets) have the P-dynamics db t = r(t)b t dt, B 0 = 1, ds t = diag(s t )µ(t, S t ) dt + diag(s t )σ(t, S t ) dwt P, S 0 = s. Using the Girsanov kernel g t we get the Q-dynamics db t = r(t)b t dt, B 0 = 1, ds t = diag(s t )(µ(t, S t ) σ(t, S t )g t ) dt + diag(s t )σ(t, S t ) dw Q t, S 0 = s Magnus Wiktorsson L8 September 27, 2018 11 / 14
Conditions on the drift of (S i (t)) N i=1 One necessary condition for S i (t)/b(t) to be a Q-martingale is that the drift part vanishes if we apply Ito s-formula to S i (t)/b(t). Suppose S and B has the following Q-dynamics. db t = r(t)b t dt, B 0 = 1, ds t = diag(s i (t))µ Q (t, S i (t)) dt + diag(s i (t))σ(t, S i (t)) dw Q t, S i(0) = s i Then we get d S i(t) B(t) = S i(t) B(t) µq i (t, S t) S i(t) B(t) r(t) dt + S i(t) B(t) σ i(t, S t ) dw Q (t), = S i(t) B(t) (µq i (t, S i(t)) r(t)) dt + S i(t) B(t) σ i(t, S i (t)) dw Q (t) The drift vanishes if and only if µ Q i (t, S i(t)) = r(t)! Magnus Wiktorsson L8 September 27, 2018 12 / 14
Arbitrage and completeness If there exists at least one function g which solves the equation µ(t, S t ) σ(t, S t )g t = r(t)1 N, t [0, T ] which also satisfies the Novikov condition then the market defined in the previous slide is free of arbitrage. If this solution also is unique then the market is free of arbitrage and complete. This is the same as saying that there exist a unique martingale measure Q. Note that 1 N is a column vector consisting of N ones. Magnus Wiktorsson L8 September 27, 2018 13 / 14
The fundamental theorems of asset pricing The following is true in great generality (but not always): Theorem (First fundamental theorem) A model is free of arbitrage if and only if there exists at least one probability measure Q such that any discounted traded asset is a martingale under Q. Theorem (Second fundamental theorem) If a model is free of arbitrage then it is complete if and only if Q is unique. Magnus Wiktorsson L8 September 27, 2018 14 / 14