Help Session 2. David Sovich. Washington University in St. Louis

Transcription

1 Help Session 2 David Sovich Washington University in St. Louis

2 TODAY S AGENDA Today we will cover the Change of Numeraire toolkit We will go over the Fundamental Theorem of Asset Pricing as well

3 EXISTENCE OF A UNIQUE RISK-NEUTRAL MEASURE The Fundamental Theoreom of Asset Pricing states that if 1. There is no arbitrage 2. Markets are complete then there exists a unique risk-neutral measure Q that prices all assets No arbitrage implies a set of positive state prices, and completeness ensures these are unique

4 SO WHAT DOES THE FTAP MEAN? The FTAP states that for an asset X with dividend process {δ s }, we have that [ T ] X t = Et Q e s t r x dx δ s ds + e T t r x dx X T t - e.g. prices are expected discounted future cash flows Example: The time t price of a riskless zero-coupon bond with payoff P(T,T) is given by ] P(t,T) = E Q t [e T t r s ds P(T,T) = E Q t [e ] T t r s ds so if rates are constant, then P(t,T) = e r(t t).

5 CHOICE OF CONVENIENT NUMERAIRE Our first attempt at pricing is usually done under the risk-neutral measure However, for some securities Q may not be the most natural measure to take the expectation under To handle this issue, we employ change Change of Numeraire techniques

6 NUMERAIRES A numeraire is any non-dividend paying asset with positive price The risk-neutral measure is actually the measure associated with numeraire B(t) = e t 0 r s ds

7 NUMERAIRE INVARIANCE PRINCIPLE Numeraire Invariance Principal: If there exists a numeraire N such that any asset divided by N is a martingale under the measure P N associated with N, then for any other numeraire U there exists a probability measure P U equivalent to P N, and any asset divided by U is also a martingale under P U Note that the FTAP implies that setting N(t) = B(t) satisfies the requirements for the theorem (re-write expectation as m.g.) The results of this theorem (then... part) are informally stated via two FACTS in the textbook

8 CHANGE OF NUMERAIRE - FACT ONE Fact One: The price of any asset divided by a numeraire is a martingale under the numeraire s measure Practical Use: e rt S t is an asset (S) divided by a numeraire (B). So, under Q and diffusion assumptions we have de rt S t = re rt dt + e rt ds t + 0 = e rt ( rs t + µ Q (t,s) ) dt + (...)dw Q t = µ Q (t,s) = rs t

9 CHANGE OF NUMERAIRE - FACT ONE Example: What is the drift of ds t /S t = µdt + σdw t under the S measure? The quantity e rt /S t is a martingale under S so d ert = r ert 1 S t S t St 2 ds t + 2 2St 3 dst 2 which implies and thus e rt [ ] r µs (t,s) + σ 2 dt = 0 S t S t S T = S t e (r+ 1 2 σ 2 )(T t)+σw S T t

10 CHANGE OF NUMERAIRE - FACT TWO Fact Two: The time t risk-neutral price is invariant by Change of Numeraire - e.g. for a non-dividend paying asset V [ ] [ ] V t = E Q Bt t V T = E Y Yt t V T B T Y T Again, Fact One and Fact Two are just a restatement of the Numeraire Invariance Principle The FTAP gives us the martingale property (existence of a martingale numeraire) required for the theorem Then Fact One and Fact Two state the result of the theorem

11 CHANGE OF NUMERAIRE - FACT TWO The T-forward measure is the measure that uses the time T maturity riskless bond as numeraire Fact Two tells us that (for a non-dividend paying asset V) ] [ ] V t = E Q t [e T t r s P(t,T) ds V T == E T t P(T,T) V T = P(t,T)E T t [V T ] The Forward rate F(t,S,T) is a martingale under T forward measure F(t,S,T)P(t,T) P(t, T) = τ 1 [P(t,S) P(t,T)] = F(t,S,T) P(t, T) }{{} tradableassetdividedbynumeraire

12 CHANGE OF NUMERAIRE - FACT THREE For most complicated derivatives, closed-form solutions under any choice of numeraire do not exist Fact Three tells us how to find the drift of the process under the choice of arbitrary numeraire This allows us to conduct pricing via Monte-Carlo simulation (diffusion remains the same)

13 CHANGE OF NUMERAIRE - FACT THREE Proposition: Assume two numeraires, S and U, evolve under Q U according to ds t = (...)dt + σ S t CdW Q t du t = (...)dt + σ U t CdW Q t where C is the correlation matrix CC = ρ Then the drift of any asset X under Q U is ( σ u U t (X t ) = u S S t (X t ) σ t (X t )ρ t σ U ) t S t U t

14 CHANGE OF NUMERAIRE - FACT THREE If we know the drift of X under S, then we know its drift under U However, if we knew the λ term to connect the measures in Girsanov s Theorem, then we could just set µ U = µ S λ Example: Under the risk-neutral measure Q, the asset S t grows at S t r. Hence, under any numeraire S t, S t has drift of ( 0 u S t (S t ) = S t r σ t (S t )ρ σ S ) t B t S t ( = S t r S t σ 1 σs ) t ( = S t r + σ 2 ) S t