Martingale measure

The martingale measure (also risk-neutral measure ) is a term from financial mathematics . The importance of martingale measures is that, for a given market model with a probability measure , equivalent martingale measures exist precisely if there is no arbitrage option in the market model. This is exactly what the first fundamental theorem of arbitrage price theory says . ${\displaystyle P}$

Martingale measure in discrete models

Financial market model

Given a financial model consisting of assets (eg. B. shares or derivatives) , a numéraire and time points with . The performance of an asset is modeled using a stochastic process . That is, at all times corresponds to the price of the -th asset and is a non-negative random variable on a probability space . ${\displaystyle d}$ ${\displaystyle (S^{1},\dotsc ,S^{d})}$ ${\displaystyle S^{0}}$ ${\displaystyle t\in {0,\dotsc ,T}}$ ${\displaystyle T\in \mathbb {N} }$ ${\displaystyle S^{i}}$ ${\displaystyle t=0,\dotsc ,T}$ ${\displaystyle S_{t}^{i}}$ ${\displaystyle i}$ ${\displaystyle S_{t}^{i}}$ ${\displaystyle (\Omega ,{\mathcal {F}},P)}$

The information gain in the financial market under consideration can be modeled by filtering . A filtration is an ascending sequence of -algebras with . The amount describes the time to observable events. It should also apply that the prices are measurable for everyone . This is to take account of the fact that the prices are known at the time . ${\displaystyle \sigma }$ ${\displaystyle {\mathcal {F}}_{0}\subset {\mathcal {F}}_{1}\subset \ldots \subset {\mathcal {F}}_{T}={\mathcal {F}}}$ ${\displaystyle F_{t}}$ ${\displaystyle t}$ ${\displaystyle S_{t}^{i}}$ ${\displaystyle t=0,\dotsc ,T}$ ${\displaystyle {\mathcal {F}}_{t}}$ ${\displaystyle S_{t}^{i}}$ ${\displaystyle t}$

After all, the discounted price process is understood to mean the interest-adjusted performance of capital goods. ${\displaystyle X^{i}:=S^{i}/S^{0}}$

definition

Let be a filtered probability space. A stochastic process is called -Martingale if the following three properties apply: ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in {0,\dotsc ,T}},P)}$ ${\displaystyle (X_{t})_{t\in {0,\dotsc ,T}}}$ ${\displaystyle P}$

• ${\displaystyle X}$is adapted to , ie is measurable for everyone . ${\displaystyle ({\mathcal {F}}_{t})_{t\in {0,\dotsc ,T}}}$ ${\displaystyle X_{t}}$ ${\displaystyle {\mathcal {F}}_{t}}$ ${\displaystyle t\in {0,\dotsc ,T}}$
• ${\displaystyle X}$is an integrable process, ie for everyone . ${\displaystyle X_{t}\in {\mathcal {L}}^{1}(\Omega ,{\mathcal {F}},P)}$ ${\displaystyle t\in {0,\dotsc ,T}}$
• ${\displaystyle \operatorname {E} _{P}(X_{t+1}|{\mathcal {F}}_{t})=X_{t}\,P}$-Almost safe for everyone ${\displaystyle t\in {0,\dotsc ,T-1}}$

Now is a probability measure on martingale if the discounted price processes for all -Martingale are. ${\displaystyle P^{*}}$ ${\displaystyle (\Omega ,{\mathcal {F}})}$ ${\displaystyle X^{i}}$ ${\displaystyle i=1,\dotsc ,d}$ ${\displaystyle P^{*}}$

If is also equivalent to , then it is called equivalent martingale measure . ${\displaystyle P^{*}}$ ${\displaystyle P}$ ${\displaystyle P^{*}}$

example

The following game of chance is agreed: If a fair coin is tossed, the player receives euros for numbers and euros for heads . Participation in the game is set to euros. In this case, the market model consists of fixed assets and two points in time, a point in time before the throw and a point in time after the throw. The other parameters in the market model read the information in accordance with , , and is the uniform distribution on . In this case, the discounted value process of gambling corresponds to the price process and reads and . Obviously, it is not a martingale measure, as applies ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 1{,}70}$ ${\displaystyle d=1}$ ${\displaystyle t=0}$ ${\displaystyle t=T=1}$ ${\displaystyle \Omega =\{1,2\}}$ ${\displaystyle {\mathcal {F}}_{0}=\{\emptyset ,\Omega \}}$ ${\displaystyle {\mathcal {F}}_{1}={\mathcal {F}}={\mathcal {P}}(\Omega )}$ ${\displaystyle P}$ ${\displaystyle \Omega }$ ${\displaystyle X^{1}}$ ${\displaystyle S^{1}}$ ${\displaystyle X_{0}^{1}=1{,}70}$ ${\displaystyle X_{1}^{1}=1\cdot \chi _{\{1\}}+2\cdot \chi _{\{2\}}}$ ${\displaystyle P}$

${\displaystyle \operatorname {E} _{P}(X_{1}^{1}|{\mathcal {F}}_{0})=\operatorname {E} _{P}(X_{1}^{1})=1{,}50\neq 1{,}70=X_{0}^{1}}$.

In contrast, the probability measure on is an equivalent martingale measure . The discounted price process is obviously adapted and integrable (this is independent of the selected probability measure) and the following applies: ${\displaystyle P^{*}=0{,}3\cdot \delta _{1}+0{,}7\cdot \delta _{2}}$ ${\displaystyle (\Omega ,{\mathcal {F}})}$ ${\displaystyle X^{1}}$

${\displaystyle \operatorname {E} _{P^{*}}(X_{1}^{1}|{\mathcal {F}}_{0})=\operatorname {E} _{P^{*}}(X_{1}^{1})=1{,}70=X_{0}^{1}}$.

literature

• Andreas Ott: Growth-oriented valuation of derivatives. Springer-Verlag, 2007, page 18.
• Christian Mohn: Martingale measures and valuation of European options in discrete, incomplete financial markets . Dissertation, University of Oldenburg 2004.