Finance Financial Modeling and Engineering

1. Cholesky-Decomposition may be used to generate correlated random variables. In finance we often work with more than one underlying asset and therefore use CholeskyDecomposition to be able to simulate multiple assets using correlated normal variates. Cholesky-Decomposition has that: Σ = 𝑀𝑀′ The above tells us that we can take a covariance matrix Σ and decompose it into a lower triangular matrix 𝑀 and the transpose which is 𝑀′ . Given the below covariance matrix Σ find 𝑀 and its transpose 𝑀′ . Then, take the product of 𝑀 and its transpose 𝑀′ and show that your result is the covariance matrix Σ. 25 Σ=[ −50 −50 ] 101 2. . Find the eigenvalues and normalized eigenvectors (i.e. length of 1) for the following matrix: −6 6 𝐴=[ ] 6 −3 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. 3. 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). 𝑧 = 4𝑥 2 − 2𝑥𝑦 + 6𝑦 2 𝑠. 𝑡. 𝑥 + 𝑦 = 72 Then, solve the same problem with the constraint modified to be: 𝑥 + 𝑦 = 73 Explain what you find. What is the meaning of the Lagrange multiplier? 1 4. 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: 𝐸 [𝑙𝑛 ( 𝑆𝑇 )] = 𝑞 ∙ 𝑙𝑛(𝑢) + (1 − 𝑞) ∙ 𝑙𝑛(𝑑) = 𝑟𝑇 𝑆0 𝑆𝑇 𝑢 2 𝑉𝑎𝑟 [𝑙𝑛 ( )] = 𝑞 ∙ (1 − 𝑞) ∙ [𝑙𝑛 ( )] = 𝜎 2 𝑇 𝑆0 𝑑 𝑢 √𝑞 ∙ (1 − 𝑞) ∙ 𝑙𝑛 ( ) = 𝜎√𝑇 𝑑 For the following values of 𝑞, 𝑢, 𝑎𝑛𝑑 𝑑 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): 𝑞= 1 2 𝑢 = 𝑒 (𝑟𝑇+𝜎√𝑇) 𝑑 = 𝑒 (𝑟𝑇−𝜎√𝑇) By matching the mean and the variance, and based on our returns being i.i.d, the returns converge to 𝑁(𝑟𝑇, 𝜎 2 𝑇) which are consistent with Black-Scholes. 5. The expected return and variance for a portfolio are as follows: 𝐸[𝑅𝑝 ] = 𝑤1 𝐸[𝑅1 ] + 𝑤2 𝐸[𝑅2 ] + ⋯ + 𝑤𝑛 𝐸[𝑅𝑛 ] 𝑛 𝑉𝑎𝑟(𝑅𝑝 ) = ∑ 𝑤𝑖2 𝜎𝑖2 + ∑ ∑ 𝑤𝑗 𝑤𝑘 𝐶𝑜𝑣(𝑅𝑗 , 𝑅𝑘 ) 𝑗 𝑘 𝑗≠𝑘 𝑖=1 We can argue that an investor should look to maximize return while penalizing for risk via a coefficient of risk aversion. That is, maximize: 𝜇𝑝 − 𝑎𝜎𝑝2 Given the above, the investor’s problem is: 2 𝑛 max (𝑤1 ,𝑤2 ,…,𝑤𝑛 ) 𝑛 ∑ 𝑤𝑖 𝜇𝑖 − 𝑎 ∑ 𝑤𝑖2 𝜎𝑖2 + ∑ ∑ 𝑤𝑗 𝑤𝑘 𝐶𝑜𝑣(𝑅𝑗 , 𝑅𝑘 ) 𝑖=1 𝑖=1 ( 𝑗 𝑘 𝑗≠𝑘 ) 𝑛 𝑠. 𝑡. ∑ 𝑤𝑖 𝑖=1 Consider a three-asset model in which asset 1 is non-risky and assets 2 and 3 have correlated rates of return with correlation 𝜌 = −0.5. Let the rate of return on the nonrisky asset be denoted by 𝜇1 = 𝑟 = 0.02, and suppose that the means for the other two assets are 𝜇2 = 0.06 𝑎𝑛𝑑 𝜇3 = 0.08. The standard deviations are 𝜎2 = 0.10 𝑎𝑛𝑑 𝜎3 = 0.18. 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 𝑇 is expected to be uniformly distributed over integers in the range of 5 to 25, that is: 1 𝑞(𝑥) = { 20 0 5 ≤ 𝑥 ≤ 25 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Further, suppose that a put option with exercise price 𝑋 = 16 trades on this stock with the following terminal payoff function: 𝑝𝑇 = 𝑀𝑎𝑥[𝑋 − 𝑆𝑇 , 0] a. Calculate the probability that the put will be exercised. b. Calculate the expected value of the stock’s price contingent on it exceeding the exercise price of the put. c. Calculate the conditional expected value of the put contingent on it being exercised. d. Calculate the expected time 𝑇 value of the put. e. If time 𝑇 = 2 𝑎𝑛𝑑 𝑟 = 0.01 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. a. List the risk-neutral probabilities for this payoff space. b. Compute values for the Radon-Nikodym derivative for this change of measure. c. Value call and put options on this stock, with exercise prices equal to 𝑋 = 60. 3 d. Does put-call parity hold for this example? 8. Suppose that 𝑆0 = 0.7 to purchase a security, with potential payoffs given as follows: 𝑆1 = 3, 𝑆2 = 1.5, 𝑎𝑛𝑑 𝑆3 = 0 such that under the physical probabilities, the expected value is 𝐸ℙ [𝑆] = 0.85 and the variance equals 𝐸ℙ [𝑆 − 𝐸[𝑆]]2 = 0.80. a. Find the physical probabilities in measure ℙ. b. Find the risk-neutral probabilities in measure ℚ. 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). c. Calculate the Radon-Nikodym derivatives for this change of measure. 9. 1 2 a. Finding the Stochastic Differential Equation (SDE). Find 𝑑 (𝑒 𝜎𝑊𝑡−2𝜎 𝑡 ). Hint: set 1 2 𝑓(𝑥, 𝑡) = 𝑒 𝜎𝑥−2𝜎 𝑡 , compute partial derivatives, use Ito’s Lemma with 𝑑𝑓 = 𝜕𝑓 𝜕𝑓 1 𝜕2𝑓 𝑑𝑥 + 𝜕𝑡 𝑑𝑡 + 2 𝜕𝑥 2 (𝑑𝑥)2 where (𝑑𝑥)2 = 𝑑𝑡, then set 𝑥 = 𝑊𝑡 . 𝜕𝑥 𝑡 b. Finding the Stochastic Integral. Evaluate ∫0 𝑒 −𝑠 𝑑𝑊𝑠 . Hint: let 𝑓(𝑥, 𝑡) = 𝑒 −𝑡 𝑥, 𝜕𝑓 𝜕𝑓 compute partial derivatives, use Ito’s Lemma with 𝑑𝑓 = 𝜕𝑥 𝑑𝑥 + 𝜕𝑡 𝑑𝑡 + 1 𝜕2𝑓 (𝑑𝑥)2 where (𝑑𝑥)2 = 𝑑𝑡, set 𝑥 = 𝑊𝑡 , set the differential 𝑑(𝑒 −𝑡 𝑊𝑡 ) equal to your result from Ito and integrate from 0 𝑡𝑜 𝑡. 2 𝜕𝑥 2 10. The risk-neutral process for the Constant Elasticity of Variance (CEV) model is: 𝑑𝑆 = 𝑟𝑆𝑑𝑡 + 𝜎𝑆 𝛼 𝑑𝑍 Assume you have a call option (𝑉) that is a function of the spot (stock price 𝑆) and time (𝑡), Ito tells us how the derivative evolves: 𝜕𝑉 𝜕𝑉 1 𝜕 2𝑉 (𝑑𝑆)2 𝑑𝑉 = 𝑑𝑆 + 𝑑𝑡 + 𝜕𝑆 𝜕𝑡 2 𝜕𝑆 2 Using the rules of stochastic calculus (i.e. 𝑑𝑡 ∙ 𝑑𝑡 = 0, 𝑑𝑡 ∙ 𝑑𝑍 = 0, (𝑑𝑍)2 = 𝑑𝑡) substitute for 𝑑𝑆 and (𝑑𝑆)2 and, a. Show how you get the following expression: 𝜕𝑉 𝜕𝑉 1 2 2𝛼 𝜕 2 𝑉 𝜕𝑉 𝑑𝑉 = (𝑟𝑆 + + 𝜎 𝑆 ) 𝑑𝑡 + 𝜎𝑆 𝛼 𝑑𝑍 2 𝜕𝑆 𝜕𝑡 2 𝜕𝑆 𝜕𝑆 4 Then, using a replicating portfolio with Π = 𝑉 − Δ𝑆, b. Derive the corresponding partial differential equation (PDE) which should be: 𝜕𝑉 1 2 2𝛼 𝜕 2 𝑉 𝜕𝑉 + 𝜎 𝑆 + 𝑟𝑆 − 𝑟𝑉 = 0 2 𝜕𝑡 2 𝜕𝑆 𝜕𝑆 c. 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). d. Explain what forward, backward, and central difference are, and which is used for the time derivative for each of the explicit and implicit methods. 5

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