Special Topic: Statistical Finance UPDATED ACTUAL
Questions and CORRECT Answers
Jensen's Inequality E((Pt - K)+) >/ (E(Pt) - K)+ since g(x) = (x - K)+ is convex in x for any convex function
g(.), E(g(x)) >/ g(E(x))
A random variable x is lognormal(mu, sigma^2) If log(x) is Normal (mu, sigma^2)
distributed ... As the variance increases the expected value will also increase due to e^x,
median always at 1
E(x) = exp(mu + sigma^)
V(x) = (exp(sigma^2) - 1) exp(2mu + sigma^2)
Brownian Motion Property Example of a Gaussian Process For any finite dimensional distribution has the
multivariate normal distrbution.
Standard Brownian Motion Stochastic Process if the following hold
1. W(0) = 0
2. W(t) is continuous as a function of t
3. W(t4) - W(t3) and W(t2) - W(t1) are independent
4. The distribution W(t2) - W(t1) depends only on |t1 - t2| distribution of change
wouldn't change for one day or 100 days apart
5. W(t) is Normal(0, t) for all t. Variance will increase with t
Brownian Motion with Drift and Scaling B(t) = mu*t + sigmaW(t)
1. W=B(0) = 0
2. B(t) is continuous as a function of t
3. B(t4) - B(t3) and B(t2) - B(t1) are independent
4. The distribution B(t2) - B(t1) depends only on |t1 - t2|
5. B(t) is Normal(mu t, sigma^2t) for all t.
Geometric Brownian Motion Implies log(Pt) is normally distributed for each time t
S(t) = S(0) * e^(B(t))
alpha = mu + sigma^2/2
1. S(t) is continuous as a function of t
2. S(t4) / S(t3) and S(t2) / S(t1) are independent
3. The distribution S(t2) / S(t1) depends only on |t1 - t2|
4. S(t) is lognormal(log(S(0)) mu t, sigma^2*t) for all t.
5. S(t) has expected value S(0) e^(alpha t)
6. log(S(t2) / S(t1)) is Normal(mu(t2 - t1), sigma^2(t2 - t1))
Kernel Density Estimator Nonparametric approach by summing a smooth kernel function centered at each
of the observed data points
Kernel function is itself a density taken to be smooth, peaked at zero, and
symmetric around zero.
Larger values of h (bandwidth) result in a density estimate that is smoother.
Smaller values of h produce and estimate that is rough or wiggly
, Bias-Variance Tradeoff V(theta-hat) + (E(theta-hat) - theta)^2
Precision + Accuracy/Bias^2
As h is larger there is lower variance but high bias, with a small h the bias will get
small but the estimator gets a large variance
Sheather Jones Approach Approach to finding the optimal h by minimizing the mean integrated square
error. With some considerations being made of the asymptotic behavior of the
criterion as n increases then minimize asymptotic mean integrated squared error
Jarque-Bera Test Formal Hypothesis test for normality. Relies on skewness (third central moment),
and kurtosis
Null: Data are sampled from the normal distribution
Alt.: Not drawn from normal
T-statistic is asymptotically chi-squared distributed with two degree of freedom.
The null would be rejected for large values of T. Suffers from low Type 1 error
prob. rate, and low power
Kurtosis Quantifies the tail and peak behaviors of a distribution
1. High Kurtosis (leptokurtic): Heavy tails, sharp peak
2. Low Kurtosis (platykurtic): Light tails, flat peak (uniform)
3. Normal Kurtosis (mesokurtic): Normal dist. has a kurtosis of 3
Shapiro-Wilk test Formal, preferred Hypothesis test for normality. Relies on order statistics and
specially-formed coefficients
Autocovariance Function (ACVF) gammax(r, s) = E(Xr, Xs) - EXrEXs
Measures the strength of linear relationship at times r and s
Stationary Properties imply that the behavior of the process (mean, variance, correlation
structure) are constant over time
Implies that the unconditional or marginal mean and variance are constant
1. V(Xt) < Infinity
2. EXt = mu for all t (mu doesn't depend on t)
3. gammax(t, t + h) = gammax( s, s +h)
Condition 3 implies ACVF is gammax(h) as the covariance would only depend on
h (lag)
Mean/Variance Reversion Implies that the mean, variance of the random variables will (over long term)
fluctuate around sound constant values of mu and sigma^2
White Noise A series {Xt} with E(Xt) = mu, V(Xt) = sigma^2 and Cov(Xs, Xt) = 0
AR(1) Model Model is autoregressive because it fits Xt like a regression model on itself using
the previous time step as a predictor. Stationary model
Xt = mu + phi(X(t-1) - mu) + epsilont
Questions and CORRECT Answers
Jensen's Inequality E((Pt - K)+) >/ (E(Pt) - K)+ since g(x) = (x - K)+ is convex in x for any convex function
g(.), E(g(x)) >/ g(E(x))
A random variable x is lognormal(mu, sigma^2) If log(x) is Normal (mu, sigma^2)
distributed ... As the variance increases the expected value will also increase due to e^x,
median always at 1
E(x) = exp(mu + sigma^)
V(x) = (exp(sigma^2) - 1) exp(2mu + sigma^2)
Brownian Motion Property Example of a Gaussian Process For any finite dimensional distribution has the
multivariate normal distrbution.
Standard Brownian Motion Stochastic Process if the following hold
1. W(0) = 0
2. W(t) is continuous as a function of t
3. W(t4) - W(t3) and W(t2) - W(t1) are independent
4. The distribution W(t2) - W(t1) depends only on |t1 - t2| distribution of change
wouldn't change for one day or 100 days apart
5. W(t) is Normal(0, t) for all t. Variance will increase with t
Brownian Motion with Drift and Scaling B(t) = mu*t + sigmaW(t)
1. W=B(0) = 0
2. B(t) is continuous as a function of t
3. B(t4) - B(t3) and B(t2) - B(t1) are independent
4. The distribution B(t2) - B(t1) depends only on |t1 - t2|
5. B(t) is Normal(mu t, sigma^2t) for all t.
Geometric Brownian Motion Implies log(Pt) is normally distributed for each time t
S(t) = S(0) * e^(B(t))
alpha = mu + sigma^2/2
1. S(t) is continuous as a function of t
2. S(t4) / S(t3) and S(t2) / S(t1) are independent
3. The distribution S(t2) / S(t1) depends only on |t1 - t2|
4. S(t) is lognormal(log(S(0)) mu t, sigma^2*t) for all t.
5. S(t) has expected value S(0) e^(alpha t)
6. log(S(t2) / S(t1)) is Normal(mu(t2 - t1), sigma^2(t2 - t1))
Kernel Density Estimator Nonparametric approach by summing a smooth kernel function centered at each
of the observed data points
Kernel function is itself a density taken to be smooth, peaked at zero, and
symmetric around zero.
Larger values of h (bandwidth) result in a density estimate that is smoother.
Smaller values of h produce and estimate that is rough or wiggly
, Bias-Variance Tradeoff V(theta-hat) + (E(theta-hat) - theta)^2
Precision + Accuracy/Bias^2
As h is larger there is lower variance but high bias, with a small h the bias will get
small but the estimator gets a large variance
Sheather Jones Approach Approach to finding the optimal h by minimizing the mean integrated square
error. With some considerations being made of the asymptotic behavior of the
criterion as n increases then minimize asymptotic mean integrated squared error
Jarque-Bera Test Formal Hypothesis test for normality. Relies on skewness (third central moment),
and kurtosis
Null: Data are sampled from the normal distribution
Alt.: Not drawn from normal
T-statistic is asymptotically chi-squared distributed with two degree of freedom.
The null would be rejected for large values of T. Suffers from low Type 1 error
prob. rate, and low power
Kurtosis Quantifies the tail and peak behaviors of a distribution
1. High Kurtosis (leptokurtic): Heavy tails, sharp peak
2. Low Kurtosis (platykurtic): Light tails, flat peak (uniform)
3. Normal Kurtosis (mesokurtic): Normal dist. has a kurtosis of 3
Shapiro-Wilk test Formal, preferred Hypothesis test for normality. Relies on order statistics and
specially-formed coefficients
Autocovariance Function (ACVF) gammax(r, s) = E(Xr, Xs) - EXrEXs
Measures the strength of linear relationship at times r and s
Stationary Properties imply that the behavior of the process (mean, variance, correlation
structure) are constant over time
Implies that the unconditional or marginal mean and variance are constant
1. V(Xt) < Infinity
2. EXt = mu for all t (mu doesn't depend on t)
3. gammax(t, t + h) = gammax( s, s +h)
Condition 3 implies ACVF is gammax(h) as the covariance would only depend on
h (lag)
Mean/Variance Reversion Implies that the mean, variance of the random variables will (over long term)
fluctuate around sound constant values of mu and sigma^2
White Noise A series {Xt} with E(Xt) = mu, V(Xt) = sigma^2 and Cov(Xs, Xt) = 0
AR(1) Model Model is autoregressive because it fits Xt like a regression model on itself using
the previous time step as a predictor. Stationary model
Xt = mu + phi(X(t-1) - mu) + epsilont
