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**Instructions**

Answer each of the below questions.Please write neatly or feel free type your answers but if you do, please use Equation Editor for any formulas. The exam is obviously open book, open notes, and I have referenced certain texts and pages in the questions to give you hints for which texts may be useful in answering a question. A grade of “A” is well-written with ** formulaic expressions** where applicable and explanations of their meaning. A grade of “A” is not copying word-for-word from a text but rather demonstrating an understanding and translating into your own words. The exam and paper each make up 50% of your final grade. Upon completing the exam please send it back to me by

**Questions on Key Concepts from Stochastic Calculus**

- (20 points). Explain each of the following concepts. You need not write more than 2 – 3 sentences for each. Use
to explain where feasible. For example, a Martingale is a conditional expectation and you can write it down. You can write the*math*for an Ito process, Ito’s lemma, etc. and then explain.Hint: You should likely look at the text entitled Risk Neutral Pricing and Financial Mathematics, A Primer. I posted it under the Syllabus link on Blackboard. The password is UNCC_2015.Or, you may look at the reading Risk-Neutral Probabilities Explained, Gisiger.*mathematical expressions*

- State prices or Pure Securities

- Risk-neutral probabilities

- Martingale

- Markov Process

- Equivalent Probability measure

- Radon-Nikodym derivative

- Girsanov’s Theorem

- Ito Process

- Ito’s Lemma
- Complete Market

- Incomplete Market

**Questions related to Black-Scholes**

- (10 points). The Black-Scholes closed-form solution for European style call and put options are given by:

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** **

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** **

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Compute (see course notes pages 62 – 67, in particular, page 64 which has important relationship). What are the signs for each of and do the signs correspond to your intuition?

Finally, what does mean in the Black-Scholes model (see course notes page 60)? What is a binary or digital option? How is related to the value of a binary or digital option?

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**Question on Replicating Portfolios and Deriving a PDE for the Finite Difference Method**

- (15 points). 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:

Then, using a replicating portfolio with (see class notes, page 76), 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 (see the article Introduction to the Numerical Solution of Partial Differential Equations in Finance, Monk (2007), this was one of the readings for the final). Be sure to explain what forward, backward, and central difference are, and which is used for the time derivative for each of the explicit and implicit methods.

Finally, how would you determine the values for in using the CEV model to value derivatives? Write down the expression you may use to include your decision variables. (Hint: see Elements of Financial Risk Management, Christoffersen, pages 233, 237 – recall that we discussed Calibration in class where Calibration is an optimization problem).

**Questions on the Binomial Lattice**

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- (15 points). Suppose we have the following risk-neutral probability and up and down movements for a binomial lattice:

Show that the discounted asset value is a martingale. Hint, you can substitute and use basic algebra to show that under the risk-neutral measure the expectation (i.e. that is, under the risk-neutral measure , the value one-period prior is equal to the discounted value one-period hence). Hint (see Extra Notes 1), set , substitute for and algebraically manipulate.

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 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 specification, show that we match the mean and standard deviation and therefore 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 is consistent with Black-Scholes.

**Questions on Monte Carlo Simulation**

- (15 points).Under the physical measure we have the following Geometric Brownian Motion consistent with Black-Scholes:

In the risk-neutral measure, we have:

What is the theorem that is used to change the measure and induce a shift in the drift of the above process (provide the name and brief explanation)?

Using the above risk-neutral process, write down the Euler discretized equation one would use for simulation:

Assuming we have a derivative that is a function of the spot underlying and time with the following function . Use Ito’s lemma and solve for the solution to the stochastic differential equation (hint: see course notes page 28):

Take your solution for , and plug it into the payoffs for a call option and put option and write down (no need to solve) the integrals you would solve to derive the closed-form solution for the Black-Scholes model (hint: see course notes page 48):

For Monte Carlo Simulation, we often use variance reduction techniques to reduce the standard error of the estimate of the value we are simulating. Describe what an antithetic variate and include a brief mathematical explanation about how it serves to reduce the standard error of the estimate (see Extra Notes 2 page 3):

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

**Questions on GARCH**

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- (15 points). (Hint: see Chapter 4 of Elements of Financial Risk Management, Christoffersen). GARCH models are used to forecast volatility, risk modeling, and even in the pricing of financial derivatives. Explain the drawbacks to computing volatility (or variance) using the standard historical measure which has as a multiplier outside its summation operator:

What is the Exponentially Weighted Moving Average (EWMA) estimator for variance (or standard deviation) and what makes it different from the standard historical measure? What is the main drawback to using (EWMA) for purposes of forecasting? Does the EWMA estimator take account of the “leverage effect” or does it treat large positive and negative returns the same?

Write down the equation for the standard GARCH(1,1) model. What is the long run unconditional variance? What are the drawbacks to the standard GARCH(1,1) model? If variance (or volatility) reacts differently to large positive returns and large negative returns, what GARCH models may be used to take account of this phenomenon?

How are GARCH models generally estimated? Briefly explain the estimation procedure to include any mathematical expression that may be useful.

**Questions on Portfolio Optimization**

- (10 points). (Hint: see Asset Management lecture notes).For the case where one wishes to minimize the variance subject to the sum of the weights equaling one and the target return on a portfolio is .12, write down the Markowitz optimization problem in matrix notation:

For the same case, write down the Lagrangian:

How does one solve the problem using standard calculus and Lagrange? What do the Lagrange multipliers tell us about the objective function and constraints?

With the introduction of a risk-free rate, how does one solve for the tangency where the ray emanating from the *y*-axis touches the efficient frontier (see Asset Management lecture notes). Explain the mathematical approach taken to find where the slope of the line is at a maximum and tangent to the efficient frontier.

What are some of the common alternative objective functions as opposed to minimizing variance?

Briefly explain why one may wish to consider portfolio resampling as described by Michaud. Is the resampled efficient frontier above, equal to, or below its corresponding Markowitz efficient frontier? Why, explain?

With Markowitz we generally start by estimating a vector of returns and covariance matrix with the weights the decision variables in the optimization. Black-Litterman proceeds differently. What does one solve for in the Black-Litterman model? What is meant by reverse or inverse optimization and why may it be beneficial relative to the standard Markowitz approach?

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