Show that the dot product of the eigenvectors is zero and therefore orthogonal. In two to three sentences, briefly explain what one may use eigenvalue decomposition for in finance.
Financial Modeling and Engineering
Requirements: Detailed
Cholesky-Decomposition may be used to generate correlated random variables. In finance we often work with more than one underlying asset and therefore use Cholesky-Decomposition to be able to simulate multiple assets using correlated normal variates. Cholesky-Decomposition has that:
The above tells us that we can take a covariance matrix and decompose it into a lower triangular matrix and the transpose which is . Given the below covariance matrix find and its transpose . Then, take the product of and its transpose and show that your result is the covariance matrix .
. Find the eigenvalues and normalized eigenvectors (i.e. length of 1) for the following matrix:
Show that the dot product of the eigenvectors is zero and therefore orthogonal. In two to three sentences, briefly explain what one may use eigenvalue decomposition for in finance.
Use a Lagrange multiplier to optimize the following function subject to the given constraint and estimate the effect on the value of the value of the objective function from a 1-unit change in the constant of the constraint (i.e. obtain solution, substitute, and evaluate the function, take note of Lagrange multiplier value).
Then, solve the same problem with the constraint modified to be:
Explain what you find. What is the meaning of the Lagrange multiplier?
With a binomial lattice where we wish to model asset prices such that, in the limit, our Random Walk (RW) becomes Brownian Motion (BM) and our process is consistent with the Black-Scholes assumption of Geometric Brownian Motion, we basically solve a system of two equations with three unknowns (i.e. one degree-of-freedom) such that we match the mean and variance. Mathematically, we have that:
For the following values of show that we match the mean and standard deviation and therefore satisfy the system of equations (hint: plug the below values into the above equations and show that they satisfy the system of equations):
By matching the mean and the variance, and based on our returns being i.i.d, the returns converge to which are consistent with Black-Scholes.
The expected return and variance for a portfolio are as follows:
We can argue that an investor should look to maximize return while penalizing for risk via a coefficient of risk aversion. That is, maximize:
Given the above, the investor’s problem is:
Consider a three-asset model in which asset 1 is non-risky and assets 2 and 3 have correlated rates of return with correlation . Let the rate of return on the non-risky asset be denoted by , and suppose that the means for the other two assets are . The standard deviations are . Find the optimal portfolio for an investor with risk aversion 10. Show that the result is truly a maximum and compute the mean, variance, and standard deviation of the optimal portfolio.
6. Suppose that a stock’s price at time is expected to be uniformly distributed over integers in the range of 5 to 25, that is:
Further, suppose that a put option with exercise price trades on this stock with the following terminal payoff function:
Calculate the probability that the put will be exercised.
Calculate the expected value of the stock’s price contingent on it exceeding the exercise price of the put.
Calculate the conditional expected value of the put contingent on it being exercised.
Calculate the expected time value of the put.
If time what is the current value of the put?
7. Consider a one-time, two potential outcome framework where there exists Company Q stock current selling for $55 per share and a riskless $100 face value T-bill currently selling for $80. Suppose Company Q faces uncertainty, in that it will pay its owner either $30 or $75 in 1 year. Further assume that the physical probability that the stock will drop is 0.2.
List the risk-neutral probabilities for this payoff space.
Compute values for the Radon-Nikodym derivative for this change of measure.
Value call and put options on this stock, with exercise prices equal to
Does put-call parity hold for this example?
8. Suppose that to purchase a security, with potential payoffs given as follows: such that under the physical probabilities, the expected value is and the variance equals .
Find the physical probabilities in measure .
Find the risk-neutral probabilities in measure . Recall that under the risk-neutral probabilities, the expected value of the stock is equal to its current value (i.e., it is a martingale).
Calculate the Radon-Nikodym derivatives for this change of measure.
9.
Finding the Stochastic Differential Equation (SDE). Find . Hint: set , compute partial derivatives, use Ito’s Lemma with where , then set .
Finding the Stochastic Integral. Evaluate . Hint: let , compute partial derivatives, use Ito’s Lemma with where set , set the differential equal to your result from Ito and integrate from .
10. The risk-neutral process for the Constant Elasticity of Variance (CEV) model is:
Assume you have a call option () that is a function of the spot (stock price ) and time (), Ito tells us how the derivative evolves:
Using the rules of stochastic calculus (i.e. ) substitute for and and,
Show how you get the following expression:
Then, using a replicating portfolio with ,
Derive the corresponding partial differential equation (PDE) which should be:
Using a differencing scheme, replace the corresponding derivatives with appropriate approximations to write down the discretized equations for the explicit finite difference method and the implicit finite difference method (i.e., replace the partial derivatives with approximations to the derivatives).
Explain what forward, backward, and central difference are, and which is used for the time derivative for each of the explicit and implicit methods.
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