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 .
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
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 .
![d](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab)
![S ^ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/059153ab9372a06326d2cf984574e5f3cefd24a7)
![{\ displaystyle t \ in {0, \ dotsc, T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/149981face5a7ddcdf3a2bcc317a62655d0d3c03)
![{\ displaystyle T \ in \ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba91f4b930257ad5c58dddf08d32cf028e5b894)
![{\ displaystyle S ^ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca697409cb6c118337963c06933271a6f33c031)
![{\ displaystyle t = 0, \ dotsc, T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70b601497811ba09c5a1dee02465c23e3c2f32ac)
![{\ displaystyle S_ {t} ^ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/816d2d0f0589121853483186d52a881bb2f16dfa)
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![{\ displaystyle (\ Omega, {\ mathcal {F}}, P)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2)
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 .
![\ sigma](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)
![{\ displaystyle {\ mathcal {F}} _ {0} \ subset {\ mathcal {F}} _ {1} \ subset \ ldots \ subset {\ mathcal {F}} _ {T} = {\ mathcal {F }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e62a47a0f40d0858efa6cf4554a2c633d714bbe2)
![F_ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42cf9007bfa40c26216b648363be8f67e73d38cb)
![t](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
![{\ displaystyle S_ {t} ^ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/816d2d0f0589121853483186d52a881bb2f16dfa)
![{\ mathcal F} _ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5209aeab53016ae720999b2a50e7f0bc27cb33)
![{\ displaystyle S_ {t} ^ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/816d2d0f0589121853483186d52a881bb2f16dfa)
![t](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
After all, the discounted price process is understood to mean the interest-adjusted performance of capital goods.
![{\ displaystyle X ^ {i}: = S ^ {i} / S ^ {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed746a4022e3fbc33d199fc85d43114018d0ef2)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e2636daad87d87b2c2290b8b68045a1d8217e11)
![{\ displaystyle (X_ {t}) _ {t \ in {0, \ dotsc, T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0c1f040b02b87907670c4ba6909e2c6568f38d)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
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is adapted to , ie is measurable for everyone .![{\ displaystyle ({\ mathcal {F}} _ {t}) _ {t \ in {0, \ dotsc, T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d155cd33b576c48b107269105eeaa3a27a32ba5)
![X_ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82120d04dfb3cbadc4912951dd12b5568c9cd8f3)
![{\ mathcal F} _ {t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5209aeab53016ae720999b2a50e7f0bc27cb33)
![{\ displaystyle t \ in {0, \ dotsc, T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/149981face5a7ddcdf3a2bcc317a62655d0d3c03)
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is an integrable process, ie for everyone .![{\ displaystyle X_ {t} \ in {\ mathcal {L}} ^ {1} (\ Omega, {\ mathcal {F}}, P)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9ed5d1e4f08e55a2ac18cd95b1e713c3c999ed9)
![{\ displaystyle t \ in {0, \ dotsc, T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/149981face5a7ddcdf3a2bcc317a62655d0d3c03)
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-Almost safe for everyone
Now is a probability measure on martingale if the discounted price processes for all -Martingale are.
![P ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24635cb75ac6b04bbdb6054de6df7eca458d3ef9)
![X ^ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee700b37109a25ac72a3d5587c3b8a78b149c03)
![P ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24635cb75ac6b04bbdb6054de6df7eca458d3ef9)
If is also equivalent to , then it is called equivalent martingale measure .
![P ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24635cb75ac6b04bbdb6054de6df7eca458d3ef9)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
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
![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
![2](https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f)
![{\ displaystyle 1 {,} 70}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1790d9b1814f5c4605700251a06972423db38fe)
![d = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/958b426c5ace91d4fb7f5a3becd7b21dba288d50)
![t = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c)
![{\ displaystyle t = T = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a43ddc339ec68edf0f2c31f8c293db5f36731742)
![{\ displaystyle \ Omega = \ {1,2 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b095a095061e3d68081b187f04febff99ca40c45)
![{\ displaystyle {\ mathcal {F}} _ {0} = \ {\ emptyset, \ Omega \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d65f159764341353b12efad85b3c25b36d2dcd05)
![{\ displaystyle {\ mathcal {F}} _ {1} = {\ mathcal {F}} = {\ mathcal {P}} (\ Omega)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1df06ffa68f700c5acec27400b3706a63d07803)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![{\ displaystyle X ^ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fca53d8f3bcc5469eece7dc6b23ada615470229)
![S ^ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60796c8d0c03cf575637d3202463b214d9635880)
![{\ displaystyle X_ {0} ^ {1} = 1 {,} 70}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc869e5043bc53041f3bbf27606a88fafe9957f)
![{\ displaystyle X_ {1} ^ {1} = 1 \ cdot \ chi _ {\ {1 \}} + 2 \ cdot \ chi _ {\ {2 \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92355b8c9f98d1ddfc67152769450c3e5098d630)
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
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.
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e83604e27caa196e5588ecc01503c3d51d3d80d)
![{\ displaystyle (\ Omega, {\ mathcal {F}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4af05190f6fb10d1eec88b5f66f940dc88b5ec87)
![{\ displaystyle X ^ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fca53d8f3bcc5469eece7dc6b23ada615470229)
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.
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.