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Part A – Expected payoffs
Part A will be marked by Stephen Taylor.
You must not use Monte Carlo methods to obtain your results.
1. A die has 6 faces, numbered from 1 to 6 inclusive. A die is fair if the numbers thrown in a sequence of rolls are independently and identically distributed, with all integers from 1 to 6 having the same chance of being thrown. Suppose you throw a fair die three times and obtain the total X. Your financial payoff is then:
£ X if the total X is an even number, £ − X if X is an odd number.
For example, you are paid £10 if you throw 4, 1, 5 but you have to pay £11 if you throw 2, 6, 3.
What is your expected payoff?
- A fair die has 6 faces, numbered from 1 to 6 inclusive. You throw the die once and get some number Y. You then throw the die another Y times and get a total Z. For example, you might throw a 3 and then have 3 more throws, obtaining 1, 4 and 6 soY = 3 and Z =11. Your financial payoff is £ Z. What is your expected payoff?
- An integer N is a prime number if it has exactly 2 factors, namely 1 and N. For example 2, 3 and 5 are prime numbers, while 1, 4 and 6 are not prime numbers. The integers N and N + 2 are prime-pair numbers if both numbers are prime. For example, 3, 5, 11 and 13 are prime-pair numbers, while 23 and 37 are not prime- pair numbers.Game 1 is played between yourself and an opponent as follows: (1) a random integer X is selected by a computer, (2) all integers between 1 and M inclusive are
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equally likely to be selected, (3) you are paid £X if X is a prime-pair number, (4)
you have to pay £10 if X is not a prime-pair number.
Should you play Game 1 if M equals 100 and your objective is to maximize your
expected payoff?
Game 2 is the same as Game 1, except you are allowed to choose any positive value
of M less than 1000. What is your best strategy?
Game 3 is the same as Game 2 except your opponent now only pays you the
minimum of £X and £100 if X is a prime-pair number. What is your best strategy?
You must provide reasons for your answers.
Required for Part A:
All text must be typed in font size 12. The line spacing must be either “1.5 lines” or
“At least 18pt”.
You must include your VBA code as part of your report. All code should be formatted using the Courier New font.
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Part B – Volatility Analysis
Part B will be marked by Sandra Nolte.
You must use Monte Carlo methods to obtain your results.
Let us assume that the continuous price process, , is a noisy signal for an underlying continuous process ∗
= ∗ + ,
where is the noise term, and t denotes the time measured in days. The noise process satisfies the following conditions:
a) ∗ ⊥ , for all and (exogeneity), b) ⊥ for all ≠ (independence), c) [ ] = 0, [ 2] = 2 for all .
While this assumption is unrealistic from an empirical point of view, it is convenient for our analysis. Moreover let us assume that ∗ follows a Brownian motion given by the following stochastic equation:
∗ = + .
where μ is the expected rate of return, σ is the constant volatility of the stock price ∗,
and is a Wiener process. The discretised version of this process can be written as Δ ∗ = Δ + √Δ ,
where Δ ∗ is the change in the stock price, ∗, in a small interval of time Δt. For example, Δt represents a one second interval in a trading day. is a random drawing from a standardised normal distribution.
The Integrated Variation (IV) of ∗ is given by
=� 2. −1
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The intra-daily returns for the t-th day are given by
+− +� −1�
, = ,j = 1,2,…, ,
Questions:
+� −1�
where M denotes the number of equally spaced time intervals in one day. For example, assume that you would like to construct one minute returns, and that trading occurs during 6.5 hours during the day, then you would observe = 390 one minute returns during that day.
The realised volatility (RV) for the t-th day with a sampling frequency M is defined as
the sum of the squared intra-daily returns over the day:
=� 2.
=1
,
Assume that the market is open 252 days per year, and that trading goes on for 6.5 hours each day, between 09:30 and 16:00, and assume that we only consider one trading day in this Monte Carlo experiment.
Assume that the initial price 0∗ = 50, = 7 per annum, and = 10 per annum, The noise process is iid normally distributed N(0, 2). Consider two different scenarios, where 2 can either take the value 3 (medium noise) or 6 (high noise).
Using 5000 Monte Carlo simulations, compare the accuracy of the one and five minutes RV estimators with respect to IV for both scenarios, by using the mean squared error statistic. What results do you obtain? Can you explain why?
In order to perform the Monte Carlo simulation, first simulate the true one second price process, ∗, and then compute one and five minutes returns.
Required for Part B:
All text must be typed in font size 12. The line spacing must be either “1.5 lines” or “At least 18pt”.
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You must include your VBA code as part of your report. All code should be formatted using the Courier New font.