# Probability

Probability 150 150 Affordable Capstone Projects Written from Scratch

Home Work #1

MATH GR5360: Math Mthds-Fin Price Analysis (Spring 2018)

Distrubuted: 2/9/2018, due date: 2/23/2018.

Problem 1. Following the Basic Blackjack Strategy in the class notes (handouts) in a 6-deck blackjack, please, choose your most rational action in the following situations:

• Hit;
• Stand;
• Split;
• Double.

• Hit;
• Stand
• Split
• Double.

Please, explain how can a player have an edge in blackjack if both the player and the dealer follow nearly the same rules and outcomes (such as they can have blackjacks).

Problem 2. Using the class handouts, your notes, and/or Kelly 1956 article, please, derive and write detailed accompanying notes for optimal fractional bet size for “slightly tampered with” or “slightly unfair” coin toss betting problem. In other words, derive the Kelly-optimal fixed-fraction bet size. Try to use your own words and thoughts.

Problem 3. Following the class notes (handouts), please, spell out explicitly the following Bloomberg tickers for futures (Name/Contract Description; Expiry Month; Expiry Year):

• ESH8;
• CFG8;
• NGJ18.

Problem 4. Following the class notes, within the framework of our course, what does it mean when we say that we would like to study a financial time seriesp=p(t)? In other words, what is the basic set of steps that one should be taking to produce inference about a high-frequency financial price series? Give both brief and detailed answers. Try to use your own words and thoughts.

Problem 5. Please, explain, what is the relationship between the “energy spectrum” and the auto-correlation function? Be detailed, try using your own thoughts rather than copying the handouts.

Problem 6. Please, explain the meaning of skewness and kurtosis for any real-life financial price (or NAV) time series as compared to the Gaussian random variable. Please, be detailed.

Problem 7. Please, following the class notes, derive the formula for the probability p1(m,n) to find the large particle (or a frog) at the position a*m, at the time n*tau, for the discrete 1-dimensional Random Walk problem described in the class. Try to write your own explanations rather than copying ones from the lecture notes.

Problem 8. Using the above solution, please, derive the formulas for <m> and <m2>. Justify the intermediate steps with your own explanations.

Problem 9. Please, explain what it means that a financial time series has a:

• Short(-term) memory;
• Long(-term) memory.

Be detailed and offer original explanations and examples.

Problem 10. Please, following the class notes, derive the formula for the variance ratio VR(3) (more explicitly, VR(3)=Sigma(3*tau)/3/Sigma(tau)). By analogy, please, derive formulas for VR(4) and VR(5). Accompany your derivations with your own detailed explanations.

Problem 11. Using Matlab (or similar language) and the real-life data for the ES market (already back-adjusted 1-min frequency since inception data for the S&P 500 E-Mini and FTSE-100 futures), please, measure and plot in lin-lin (linear-linear) and log-log scales the function Sigma(tau) for tau(minutes)<=10,000 mins. The data files “ES” and “FT” are uploaded into “Lecture #4” folder on CourseWorks. Using these results and the built-into Matlab regular least squares linear regression, estimate the algebraic slope nu: Sigma(tau)=tau^nu. Provide the additional standard outputs of least squares procedure: the Slope and R^2 coefficients in regression between log10(Sigma) and log10(tau). Please,  accompany your findings with detailed notes and code that was used. You can use the handouts charts for comparisons against your results. Make possible inferences from your charts and comparisons. Write detailed explanations.

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