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## Northeastern University Binomial & Quadratic Programming Discussion

### Question Description

I need two comments, each part is around 100 words.

Part 1:

As noted in the discussion post prompt, quadratic programming is an optimization model whose objective function has a quadratic form. The model includes n decision variables and m linear constraints. A quadratic programming model can be utilized in the real world in various fields, such as finance, agriculture, economics, production operations, marketing, and public policy. Furthermore, an example of a quadratic programming model in finance for portfolio optimization from Numerical.net will be utilized in the following explanation. If a financial analyst has assets with a known historical average return, correlation, and variance for each asset, quadratic programming can be utilized to analyze the combination of assets that best fits the objectives with the end goal of minimizing risk while achieving a minimal return. Diving into the model details, the unknowns include the amount invested for each asset, xi. Furthermore, all the values together form the vector x. The assumption is that xi ? 0. The variance of the value of each portfolio is the measure of risk, meaning that as the variance increases, the probability of a large loss also increases. R is denoted by the variance-covariance matrix of the assets, while the variance of the asset value is xTRx, which the analyst wants to minimize. If the budget is \$10,000 and the minimum expected return on our investment is 10%. The summary of the model is:

Minimize xTRx
Subject to ? xi ? 10000
? ri xi ? 1000 xi ? 0

As stated by Numerical.net, A quadratic program in standard form looks like this:

Minimize c0 + cTx + ½xTHx
Subject to AEx = bE
AIx ? bI
l ? x ? u

where c0 is a scalar, H, AE, and AI are matrices, and all other values are vectors. The subscript E states quality constraints, while the subscript I denotes inequality constraints. However, there are no equality constraints in this specific problem. To conclude, a practical quadratic programming model includes four assets with returns 5%, -20%, 15% and 30%, respectively. The budget is \$10,000, and the minimum return is 10%. The produce is done using a statistical platform such as Excel. Numerical.net notes that the results are as followed: the vector x will contain the optimal amounts for each asset. It gives us: \$3,453, \$0, \$1,069, \$2,223. To conceptualize the results, the second asset showcased a negative return on average and should not be included in the portfolio. But, the asset with the lowest positive return should have the largest share.

Modern Portfolio Theory (MPT) and Option Pricing Volatility analysis both relate to quadratic programming. MPT allows for the construction of a portfolio that minimizes risk for a given level of return, which can be also done in quadratic programming, as seen in the example I provided above. Furthermore, MPT is built upon optimization. Quadratic programming is utilized to solve optimization problems while either minimizing or maximizing a quadratic function of several variables subject to linear constraints. The MPT constraints include equality constraints, inequality constraints, and integer constraints. The decision variables are the assets, and the objective function is either a loss function or its negative. An Option Pricing Volatility Analysis relates to a quadratic programming model in that options can be used to hedge assets and portfolios in order to control the risk due to movement in the share price. A binomial option pricing model shows the price of the underlying asset fluctuating positively or negatively in the period. Given the possible prices of the underlying asset and the strike price of an option, calculating the payoff of the option under the scenarios showcases discounted payoffs and finds the value of that option. Furthermore, this goes back to the idea of optimization in quadratic programming. The decision variables include either the stock price, exercise price, volatility, interest rate, and the time to expiration to value the option. The objective function is the calculated probability that the option will be exercised, and the constraints for each model include the market prices.

Part 2:

Question 1:

The company sets a price (Z) for a new variety of grape juice. The price is affected by factors such as the state’s weather x1, the state’s minimum wage x2, and the cost of transporting grape juice nationwide x3.

The relationship could be stated as Z = Q (x1, x2, x3), where Q is the function that defines Z in terms of the xi.

As an example, say

Z = (5/2) x12 – 2 x1 x2 – x1 x3 + 2 x22 + 3 x2x3 + (5/2) x32 + 2 x1 – 35 x2 – 47 x3 + 5

And in this case,

Q =
5 -2 -1
-2 4 3
-1 3 5

c T = (2 -35 -47) and ? = 5. This example assumes no constraints, so set the gradient(G) to 0.

G = Q x + c = 0

Hence, Q x = -c

After calculation, all the eigenvalues of the Q matrix are greater than 0; in other words, the Q matrix is positive definite, so there is a global minimum. If the Q matrix is not positive definite, there may not be a global minimum, and only a local minimum can be found.

Next?[A](x) = (b), where A is Q and b is -c.

The solution is x T = (3 5 7)

In other words, Z takes a minimum value when
x1 = 3, x2 = 5 and x3 = 7

Question 2

The basic idea of the Markowitz Portfolio Theory is to assume that all investors are risk-averse, use the expectation (mean) of return (rate) to measure the return on investment, and use the variance of return (rate) to measure the risk of investment return. Markowitz’s mean-variance model uses probability theory and optimization techniques to model investment behavior under uncertain conditions (Chen, 2020). The decision-making method is to minimize risk under the conditions of a given expected return, or to maximize the expected return given the risk.

If the investor chooses n types of investable risk assets for portfolio investment.

R ?? r? ?r? ? ?rn ??is the expected return vector of risk assets.

V ???ij ?n × nis the covariance matrix of the return on risky assets.

x ?? x? ?x? ? ?xn ??is the vector of risk asset investment ratio.

l ??? ?? ? ????is a vector with all 1 components.

The expected return of the portfolio x

The risk of portfolio x

Then the decision variable in the quadratic programming model is the proportion of asset investment

The objective function: min x?Vx

The constraint: s.t. R?x ? ?, l?x ? ?

? is the expected return level given by the investor.

The key idea of Option Pricing and Volatility models is to hedge options by buying and selling basic assets in the correct way, thereby eliminating risks. In their model known as Black-Scholes, the value of an option depends on the future volatility of a stock rather than on its expected return (Mezofi & Szabo, 2019). With options investing your initial loss is limited to the amount spent on purchasing the option. You are then required to buy or sell the underlying stock if the option is exercised in the money. The decision variable is whether the option will create an overall profit or loss. The objective function is to maximize profit spending the least amount of money. One constraint could be whether they can short the stock or not.

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