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[問數] Probability theory題目
1. Let `X~N(mu, psi^2)` and `K` be a constant. Calculate
(a) `E(I{X>K})` where `I` is the indicator function.
(b) `E("max"{K-X,0})`.
Hint Use `(K-X) = (K-mu) - (X-mu)`.
(c) `E(e^{tx})`.
Leave answers where appropriate in terms of the normal cumulative distribution function
`Phi(·)`.
2. A random variable `Y` is said to be lognormally distributed with parameters `mu` and `sigma` if
`log Y ~ N(mu, sigma^2)`. We sometimes write `Y ∼"lognormal"(mu, sigma^2)`.
(a) Using your answer to Question 1(c), calculate `E(Y)` and `Var(Y)`.
(b) Suppose `mu = log f - 1/2 sigma^2` for a constant `f>0`.
Calculate `E(Y)` and show that `Var(Y) = f^2 sigma^2` (`1+{sigma^2}/2 + {sigma^4}/6 + `higher order terms).
3. A sequence of random variables `X_0, X_1, ..., X_n, ...` is defined by `X_0 = 1` and `X_n = X_{n-1}xi_{n-1}`, where `xi_i`, `i=0,1,...`, are independent and identically distributed with
\[
\xi_i =
\cases{
1+u & \text{with probability } p\cr
1+d & \text{with probability } 1 – p,
}
\]
where `u>d`.
(a) Calculate `E(X_n|X_{n-1})`.
(b) Let `Y_n = {X_n}/(1+r)^n` for a constant `r>0`.
Find in terms of `p`, `u`, and `d` the value of `r` such that `E(Y_n|Y_{n-1}) = Y_{n-1}`.
Hence find in terms of `u` and `d` the range of such possible values for `r`.
(c) Suppose `E(Y_n|Y_{n-1}) = Y_{n-1}` for all `n`.
Use any result concerning conditional expectation to prove that `E(Y_n|Y_m) = Y_m` for all `n>m>=0`.
4. Show that the first three non-zero terms of the Taylor series for the normal cumulative distribution function `Phi(x)` around zero are
`Phi(x) = 1/2 + x/sqrt{2pi} - {x^3}/{6sqrt{2pi}}`.
Thanks
(a) `E(I{X>K})` where `I` is the indicator function.
(b) `E("max"{K-X,0})`.
Hint Use `(K-X) = (K-mu) - (X-mu)`.
(c) `E(e^{tx})`.
Leave answers where appropriate in terms of the normal cumulative distribution function
`Phi(·)`.
2. A random variable `Y` is said to be lognormally distributed with parameters `mu` and `sigma` if
`log Y ~ N(mu, sigma^2)`. We sometimes write `Y ∼"lognormal"(mu, sigma^2)`.
(a) Using your answer to Question 1(c), calculate `E(Y)` and `Var(Y)`.
(b) Suppose `mu = log f - 1/2 sigma^2` for a constant `f>0`.
Calculate `E(Y)` and show that `Var(Y) = f^2 sigma^2` (`1+{sigma^2}/2 + {sigma^4}/6 + `higher order terms).
3. A sequence of random variables `X_0, X_1, ..., X_n, ...` is defined by `X_0 = 1` and `X_n = X_{n-1}xi_{n-1}`, where `xi_i`, `i=0,1,...`, are independent and identically distributed with
\[
\xi_i =
\cases{
1+u & \text{with probability } p\cr
1+d & \text{with probability } 1 – p,
}
\]
where `u>d`.
(a) Calculate `E(X_n|X_{n-1})`.
(b) Let `Y_n = {X_n}/(1+r)^n` for a constant `r>0`.
Find in terms of `p`, `u`, and `d` the value of `r` such that `E(Y_n|Y_{n-1}) = Y_{n-1}`.
Hence find in terms of `u` and `d` the range of such possible values for `r`.
(c) Suppose `E(Y_n|Y_{n-1}) = Y_{n-1}` for all `n`.
Use any result concerning conditional expectation to prove that `E(Y_n|Y_m) = Y_m` for all `n>m>=0`.
4. Show that the first three non-zero terms of the Taylor series for the normal cumulative distribution function `Phi(x)` around zero are
`Phi(x) = 1/2 + x/sqrt{2pi} - {x^3}/{6sqrt{2pi}}`.
Thanks
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我一看就知這些題目出自 Blyth 那本 Introduction to Quantitative Finance.....
1(a) digital option, (b) put option, (c) mgf
2. lognormal
3. Martingale....
4. saddlepoint approximation
畀你發現到![[sosad]](/face/hkg/sosad.gif)
![[banghead]](/face/hkg/banghead.gif)
究竟呢啲數係咪prob theory入門就會學到？我最近為咗呢幾題(因為想鑽研下QF)，專登走去睇Ross嗰本Introduction to Probability Models，睇咗Expectation嗰個章節都唔識做第一題
相關major的intro就差唔多係呢d
不過呢d係香港的程度係略為難的程度
qf聞說係香港唔係做research其實優勢不大
btw果位巴打連出自邊都知
我純粹係為興趣先想計下QF啲數，睇下可唔可以應用喺market度
好多financial math 書 都要 prob theory / measure theory 底
先睇 david williams, probability with mart
再睇 shreve stochastic calculus for finance i & ii 就perfect 池 la
上面D Statistics題目有2nd year底應該可以KO九成
Undergrad d quant finance多數係旁門左道, skip晒d measure theory 同stochastic analysis
典型例子就係麥當當本derivatives markets, brownian motion個到寫得好含糊
sor9ly ,有 D 199
我意思係你打好個底先諗 Quant fin
如果唔係，就睇麥當當本書，開頭ok，後面數學一多，求其就過左
麥當當本書即係邊本？
http://www.math.hcmus.edu.vn/~tdhien/Financial Modelling/derivative market.pdf
我建議你再補下D prob同 stat底先
唔記得講，我得yr1程度，即係無深入讀過set theory, measure theory, algebra, prob theory, stochastic processes嗰啲
我前幾日已經開始刨緊probi書，而家都係識少少mgf同埋joint pdf
我前幾日已經開始刨緊probi書，而家都係識少少mgf同埋joint pdf
我一看就知這些題目出自 Blyth 那本 Introduction to Quantitative Finance.....
1(a) digital option, (b) put option, (c) mgf
2. lognormal
3. Martingale....
4. saddlepoint approximation
畀你發現到
究竟呢啲數係咪prob theory入門就會學到？我最近為咗呢幾題(因為想鑽研下QF)，專登走去睇Ross嗰本Introduction to Probability Models，睇咗Expectation嗰個章節都唔識做第一題
相關major的intro就差唔多係呢d
不過呢d係香港的程度係略為難的程度
qf聞說係香港唔係做research其實優勢不大
btw果位巴打連出自邊都知
我純粹係為興趣先想計下QF啲數，睇下可唔可以應用喺market度
好多financial math 書 都要 prob theory / measure theory 底
先睇 david williams, probability with mart
再睇 shreve stochastic calculus for finance i & ii 就perfect 池 la
上面D Statistics題目有2nd year底應該可以KO九成
Undergrad d quant finance多數係旁門左道, skip晒d measure theory 同stochastic analysis
典型例子就係麥當當本derivatives markets, brownian motion個到寫得好含糊
sor9ly ,有 D 199
我意思係你打好個底先諗 Quant fin
如果唔係，就睇麥當當本書，開頭ok，後面數學一多，求其就過左
麥當當本書即係邊本？
http://www.math.hcmus.edu.vn/~tdhien/Financial Modelling/derivative market.pdf
我建議你再補下D prob同 stat底先
有咩書啱補底？
我睇緊ross本intro to probi models，唔知佢點解無啦啦將joint pdf同唔知乜柒Jacobian matrix扯埋一齊來講
重有就係睇緊all of statistics同hogg本 probi and stat inference，呢兩本好似易入口啲
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
佢應該學緊 change of variable，其實只要識計determinant，同轉d coordiate
然后照番個form sub d數入去就得。jacobian你當係個matrix個名就得
undergrad 都係take 多D pure好，pure好，自學prob同stat其實好容易
唔記得講，我得yr1程度，即係無深入讀過set theory, measure theory, algebra, prob theory, stochastic processes嗰啲
我前幾日已經開始刨緊probi書，而家都係識少少mgf同埋joint pdf
讀下discrete structure先啦
![[flowerface]](/face/hkg/flowerface.gif)
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
佢應該學緊 change of variable，其實只要識計determinant，同轉d coordiate
然后照番個form sub d數入去就得。jacobian你當係個matrix個名就得
undergrad 都係take 多D pure好，pure好，自學prob同stat其實好容易
between， stat撚想學quant fin，數底又唔夠，有d 似精算路線
咩係INDICATOR FUNCTION?
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
我見到有啲partial derivatives，但係唔知用來做乜鳩
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
佢應該學緊 change of variable，其實只要識計determinant，同轉d coordiate
然后照番個form sub d數入去就得。jacobian你當係個matrix個名就得
undergrad 都係take 多D pure好，pure好，自學prob同stat其實好容易
咁食得屎啦，我year1剩係take咗個無咁難嘅maths，無take難啲個maths course，諗住屈下人哋機
其實pure野要學calculus同inequalities同埋啲乜？
咩係INDICATOR FUNCTION?
我睇書講Indicator function係個function of an outcome，如果事件A有`omega`呢個outcome，咁`I_A(omega) = 1`，無呢個outcome咁`I_A(omega) = 0`
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
我見到有啲partial derivatives，但係唔知用來做乜鳩
做multivariate transformation 會用到 Determinant of partial derivatives
http://wwwf.imperial.ac.uk/~ayoung/m2s1/Multivariatetransformations.PDF
lecture note p.34 有MULTIVARIATE TRANSFORMATION 用到 JACOBIAN同 PARTIAL DERIVATIVES
http://wwwf.imperial.ac.uk/~ayoung/m2s1/M2S12010.PDF
所以呢 PURE都係讀多D好
,目標係Quant finance 既話 pde都好重要，希望我無拖得太遠 咁如果我想學topology
algebraic general
有咩書岩睇
algebraic general
有咩書岩睇
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
我見到有啲partial derivatives，但係唔知用來做乜鳩
做multivariate transformation 會用到 Determinant of partial derivatives
http://wwwf.imperial.ac.uk/~ayoung/m2s1/Multivariatetransformations.PDF
lecture note p.34 有MULTIVARIATE TRANSFORMATION 用到 JACOBIAN同 PARTIAL DERIVATIVES
http://wwwf.imperial.ac.uk/~ayoung/m2s1/M2S12010.PDF
所以呢 PURE都係讀多D好
有咩pure書？
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
佢應該學緊 change of variable，其實只要識計determinant，同轉d coordiate
然后照番個form sub d數入去就得。jacobian你當係個matrix個名就得
undergrad 都係take 多D pure好，pure好，自學prob同stat其實好容易
咁食得屎啦，我year1剩係take咗個無咁難嘅maths，無take難啲個maths course，諗住屈下人哋機
其實pure野要學calculus同inequalities同埋啲乜？
你4年制Y1就乖乖地讀返人地ASSIGN比你嘅COURSE先啦
PURE深果D COURSE全部都係3500+LV
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
我見到有啲partial derivatives，但係唔知用來做乜鳩
做multivariate transformation 會用到 Determinant of partial derivatives
http://wwwf.imperial.ac.uk/~ayoung/m2s1/Multivariatetransformations.PDF
lecture note p.34 有MULTIVARIATE TRANSFORMATION 用到 JACOBIAN同 PARTIAL DERIVATIVES
http://wwwf.imperial.ac.uk/~ayoung/m2s1/M2S12010.PDF
所以呢 PURE都係讀多D好
有咩pure書？
Yr 1, analysis, linear algebra, Multivariable Calculus, Differential equation, geometry, number theory, probability
Yr 2, 繼續analysis ,metric space, complex analysis, vector calculus, algebra 1&2, pde, stat inference, stochastic process
Yr 3, Functional analysis, measure theory, probability theory, mathematical finance, actuarial maths
之後再自學左d stochastic analysis
但係我final year 揀科避開晒d 高階 algebra
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
佢應該學緊 change of variable，其實只要識計determinant，同轉d coordiate
然后照番個form sub d數入去就得。jacobian你當係個matrix個名就得
undergrad 都係take 多D pure好，pure好，自學prob同stat其實好容易
咁食得屎啦，我year1剩係take咗個無咁難嘅maths，無take難啲個maths course，諗住屈下人哋機
其實pure野要學calculus同inequalities同埋啲乜？
你4年制Y1就乖乖地讀返人地ASSIGN比你嘅COURSE先啦
PURE深果D COURSE全部都係3500+LV
我其實都係想打pure底啫，無諗住讀到咁深，老老竇竇Algebra呢啲濕鳩野我都唔諗住讀
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
我見到有啲partial derivatives，但係唔知用來做乜鳩
做multivariate transformation 會用到 Determinant of partial derivatives
http://wwwf.imperial.ac.uk/~ayoung/m2s1/Multivariatetransformations.PDF
lecture note p.34 有MULTIVARIATE TRANSFORMATION 用到 JACOBIAN同 PARTIAL DERIVATIVES
http://wwwf.imperial.ac.uk/~ayoung/m2s1/M2S12010.PDF
所以呢 PURE都係讀多D好
有咩pure書？
Yr 1, analysis, linear algebra, Multivariable Calculus, Differential equation, geometry, number theory, probability
Yr 2, 繼續analysis ,metric space, complex analysis, vector calculus, algebra 1&2, pde, stat inference, stochastic process
Yr 3, Functional analysis, measure theory, probability theory, mathematical finance, actuarial maths
之後再自學左d stochastic analysis
但係我final year 揀科避開晒d 高階 algebra
咁我睇analysis, linear algebra, Multivariable Calculus, Differential equation算
Differential equation係咪會學晒ODE同PDE?
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
佢應該學緊 change of variable，其實只要識計determinant，同轉d coordiate
然后照番個form sub d數入去就得。jacobian你當係個matrix個名就得
undergrad 都係take 多D pure好，pure好，自學prob同stat其實好容易
咁食得屎啦，我year1剩係take咗個無咁難嘅maths，無take難啲個maths course，諗住屈下人哋機
其實pure野要學calculus同inequalities同埋啲乜？
你4年制Y1就乖乖地讀返人地ASSIGN比你嘅COURSE先啦
PURE深果D COURSE全部都係3500+LV
我其實都係想打pure底啫，無諗住讀到咁深，老老竇竇Algebra呢啲濕鳩野我都唔諗住讀
唔讀algebra讀少好多basic野 而且algebra好好玩
我一看就知這些題目出自 Blyth 那本 Introduction to Quantitative Finance.....
1(a) digital option, (b) put option, (c) mgf
2. lognormal
3. Martingale....
4. saddlepoint approximation
畀你發現到
究竟呢啲數係咪prob theory入門就會學到？我最近為咗呢幾題(因為想鑽研下QF)，專登走去睇Ross嗰本Introduction to Probability Models，睇咗Expectation嗰個章節都唔識做第一題
相關major的intro就差唔多係呢d
不過呢d係香港的程度係略為難的程度
qf聞說係香港唔係做research其實優勢不大
btw果位巴打連出自邊都知
我純粹係為興趣先想計下QF啲數，睇下可唔可以應用喺market度
好多financial math 書 都要 prob theory / measure theory 底
先睇 david williams, probability with mart
再睇 shreve stochastic calculus for finance i & ii 就perfect 池 la
上面D Statistics題目有2nd year底應該可以KO九成
Undergrad d quant finance多數係旁門左道, skip晒d measure theory 同stochastic analysis
典型例子就係麥當當本derivatives markets, brownian motion個到寫得好含糊
再搭單睇埋
Steele, Stochastic Calculus and Financial Applications
Oksendal, Stochastic differential equations
PURE撚表示完全唔撚明題目
數佬唔係都有學少少prob theory咩？
我STAT剩係讀到2000-LV 仲要F撚左
我讀其他野根本唔需要STAT
屌，我記得你好似講過肥咗probi
你有冇學過Jacobian matrix?
Jacobian唔係STAT學 係Multivariable Calculus學嘅
佢應該學緊 change of variable，其實只要識計determinant，同轉d coordiate
然后照番個form sub d數入去就得。jacobian你當係個matrix個名就得
undergrad 都係take 多D pure好，pure好，自學prob同stat其實好容易
咁食得屎啦，我year1剩係take咗個無咁難嘅maths，無take難啲個maths course，諗住屈下人哋機
其實pure野要學calculus同inequalities同埋啲乜？
你4年制Y1就乖乖地讀返人地ASSIGN比你嘅COURSE先啦
PURE深果D COURSE全部都係3500+LV
我其實都係想打pure底啫，無諗住讀到咁深，老老竇竇Algebra呢啲濕鳩野我都唔諗住讀
唔讀algebra讀少好多basic野 而且algebra好好玩
L jer bra
我一看就知這些題目出自 Blyth 那本 Introduction to Quantitative Finance.....
1(a) digital option, (b) put option, (c) mgf
2. lognormal
3. Martingale....
4. saddlepoint approximation
畀你發現到
究竟呢啲數係咪prob theory入門就會學到？我最近為咗呢幾題(因為想鑽研下QF)，專登走去睇Ross嗰本Introduction to Probability Models，睇咗Expectation嗰個章節都唔識做第一題
相關major的intro就差唔多係呢d
不過呢d係香港的程度係略為難的程度
qf聞說係香港唔係做research其實優勢不大
btw果位巴打連出自邊都知
我純粹係為興趣先想計下QF啲數，睇下可唔可以應用喺market度
好多financial math 書 都要 prob theory / measure theory 底
先睇 david williams, probability with mart
再睇 shreve stochastic calculus for finance i & ii 就perfect 池 la
上面D Statistics題目有2nd year底應該可以KO九成
Undergrad d quant finance多數係旁門左道, skip晒d measure theory 同stochastic analysis
典型例子就係麥當當本derivatives markets, brownian motion個到寫得好含糊
再搭單睇埋
Steele, Stochastic Calculus and Financial Applications
Oksendal, Stochastic differential equations
thx
果然finance都係離唔開stochastic calculus
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