金融数学课件(英文版)第5讲

Fundamental theorem in continuous time Black-Scholes
L5.5
Fundamental theorem: Proof
To see that V /B is a martingale, let At := 1/Bt . Then d(At Vt ) = Vt dAt + At dVt + dAt dVt = Θt · Xt dAt + At (Θ · dXt ) + dAt Θt · dXt = Θt · (Xt dAt + At dXt + dAt dXt )
Fundamental theorem in continuous time Black-Scholes
L5.6
Fundamental theorem: Proof
No arb ⇒ ∃ equivalent martingale measure P: Intuition of proof: Same as L2.21-L2.24. Let’s simplify the problem by assuming only J possible time-T outcomes Ω = {ω1 , . . . , ωJ }. For each j = 1, . . . , J, define the jth Arrow-Debreu asset to be a contract paying: 1 unit of BT if and when ωj occurs, else 0. Let pj be the time-0 value of the jth Arrow-Debreu asset, where value is expressed in units of the asset B. So pj is how many units of B it costs today, to buy 1 unit of B in the event that ωj occurs. (But what’s “the value” of an A-D asset that does not trade? If market is complete, it’s the value of a replicating portfolio. If market is incomplete, then proof becomes more difficult.)
Black-Scholes
Fundamental theorem in continuous time
Black-Scholes
L5.3
More about martingales in Brownian settings
[Under integrability conditions. . . ]
Fundamental theorem in continuous time Black-Scholes
L5.7
Fundamental theorem: Proof
Define P({ωj }) := pj . Can check this really is an equivalent measure: pj > 0 and
N
= Θt · d(At Xt ) =
n=1
n n θt d(At Xt )
n Since each At Xt is a martingale, AV = V /B is a martingale also.
Idea: Martingales are wealth processes arising from zero-expectation games. Varying your bet size across games and across time still produces, collectively, a zero-expectation game. Can’t risklessly make something from nothing by playing zero-expectation games.
1 2 Xt Xt XN , ,··· , t Bt Bt Bt
is a martingale under P (meaning, each component is a martingale). For any self-financing trading strategy Θt with value Vt , we’ll show that V /B is a martingale, where Vt := Θt · Xt . Once we show this, we are done, because the familiar argument applies: If V0 = 0, then V0 /B0 = 0, hence E(VT /BT ) = 0. If also VT ≥ 0, then VT /BT ≥ 0, so VT /BT = 0, hence VT = 0 Conclusion: Θ is not an arbitrage.
Fundamental theorem in continuous time
Black-Scholes
L5.9
Option pricing
In L4, we did this by replication. Now let’s do it by martingale methods: Option price equals the expected discounted payoff, under a martingale measure P. Why? By the Fundamental theorem. How do we calculate P-expectations (denoted by E)? In some cases, a model is already specified under risk-neutral measure. Then simply work directly under the given measure. But what if the model is specified under physical measure? We know how S behaves with respect to physical measure P. How does it behave wrt P? W drives all the risk. So if we know what happens to W when we
L5.1
Financial Mathematics 330 Lecture 5
Roger Lee
2010 November 3
Fundamental theorem in continuous time
Black-Scholes
L5.2
Fundamental theorem in continuous time
Fundamental theorem in continuous time Black-Scholes
L5.8
Fundamental thຫໍສະໝຸດ Baiduorem: Comments
Idea: The P probability of an event is simply the price (in units of B) of a asset that pays 1 unit of B iff that event occurs. Note: In this entire proof, we never assumed that B is the bank account, and never assumed that it is riskless. It is enough to assume that B is some asset with positive price process. Later, in some applications, we will prefer to normalize using some such asset (some numeraire) that is not the bank account. By default, if we say risk-neutral or martingale measure without specifying the numeraire, it is understood to be the bank account.
j
pj = 1.
To check the MG property, let X be the price (in dollars) of any asset. Then X0 = B0
J
j=1
XT (ωj ) XT · pj = E BT (ωj ) BT
The first step simply says that the time-0 value of X equals the value of the portfolio of Arrow-Debreu assets that replicates XT (where all values are expressed in units of B). That portfolio holds XT (ωj )/BT (ωj ) units of the jth A-D asset, and the portfolio’s total value is the sum of quantity times price. The concluding step is by definition of expectation wrt P.
Fundamental theorem in continuous time
change measure, then we can find out what happens to S.
Black-Scholes
L5.10
Girsanov’s theorem
In this class we will need only one fact from the Girsanov theory of measure change. We will need only what Bj¨rk (2nd/3rd) calls the o “converse of Girsanov”. Theorem: If W is a Brownian motion under P,
W M is a martingale wrt {Ft }t≥0 if and only if there exists an
adapted process σ such that dMt = σt dWt This is a form of the martingale representation theorem. We knew that every Itˆ integral is a martingale. This says that o conversely every martingale is an Itˆ integral plus a constant. o If M is a martingale, then stochastic integrals with respect to M are martingales. That’s because dMt = σt dWt for some σt , so if dYt = ρt dMt
Fundamental theorem in continuous time
then dYt = ρt dMt = ρt σt dWt , which is driftless.
Black-Scholes
L5.4
Fundamental theorem: Proof
∃ equivalent martingale measure P ⇒ No arb: We are given that the vector of discounted asset prices 1 Xt := Bt