Explanation Definition: Standard deviation: Suppose x is a data value in a set whose mean is x ¯ , then x − x ¯ is called x ’s deviation from mean. The formula for standard deviation of a sample n data values is s = ∑ ( x − x ¯ ) 2 n − 1 . Mean: Suppose the data set contains n data values, the mean x ¯ of the data is computing by the formula x ¯ = ∑ x n . Calculation: Standard deviation is a reliable measure of dispersion. It is a number and used to tell about the measurements for a group of data. That is the spread out from the average value or mean or expected value by using the definition mentioned above. The low standard deviation denotes that most of the numbers are close to the average. The high standard deviation denotes that the numbers are more spread out. Example: The given distribution is 8 , 6 , 7 , 6 , 4 , 5 . Compute the mean for the distribution 8 , 6 , 7 , 6 , 4 , 5 . Here, n = 8 . Compute the mean of the distribution by using the mean formula mentioned above. x ¯ = 8 + 6 + 7 + 6 + 4 + 5 8 = 36 6 = 6 Therefore, the mean of the distribution is x ¯ = 6 . Compute the squares of the deviations of the data values from the mean as shown in below table

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ISBN: 9780134434681

Author: Tom Pirnot

Publisher: PEARSON

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Problem 3. Pricing a multi-stock option the Margrabe formula The purpose of this problem is to price a swap option in a 2-stock model, similarly as what we did in the example in the lectures. We consider a two-dimensional Brownian motion given by W₁ = (W(¹), W(2)) on a probability space (Q, F,P). Two stock prices are modeled by the following equations: dX = dY₁ = X₁ (rdt+ rdt+0₁dW!) (²)), Y₁ (rdt+dW+0zdW!"), with Xo xo and Yo =yo. This corresponds to the multi-stock model studied in class, but with notation (X+, Y₁) instead of (S(1), S(2)). Given the model above, the measure P is already the risk-neutral measure (Both stocks have rate of return r). We write σ = 0₁+0%. We consider a swap option, which gives you the right, at time T, to exchange one share of X for one share of Y. That is, the option has payoff F=(Yr-XT). (a) We first assume that r = 0 (for questions (a)-(f)). Write an explicit expression for the process Xt. Reminder before proceeding to question (b): Girsanov's theorem…

Problem 1. Multi-stock model We consider a 2-stock model similar to the one studied in class. Namely, we consider = S(1) S(2) = S(¹) exp (σ1B(1) + (M1 - 0/1 ) S(²) exp (02B(2) + (H₂- M2 where (B(¹) ) +20 and (B(2) ) +≥o are two Brownian motions, with t≥0 Cov (B(¹), B(2)) = p min{t, s}. " The purpose of this problem is to prove that there indeed exists a 2-dimensional Brownian motion (W+)+20 (W(1), W(2))+20 such that = S(1) S(2) = = S(¹) exp (011W(¹) + (μ₁ - 01/1) t) 롱) S(²) exp (021W (1) + 022W(2) + (112 - 03/01/12) t). where σ11, 21, 22 are constants to be determined (as functions of σ1, σ2, p). Hint: The constants will follow the formulas developed in the lectures. (a) To show existence of (Ŵ+), first write the expression for both W. (¹) and W (2) functions of (B(1), B(²)). as (b) Using the formulas obtained in (a), show that the process (WA) is actually a 2- dimensional standard Brownian motion (i.e. show that each component is normal, with mean 0, variance t, and that their…

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The standard deviation tells about a distribution.

Solution Summary: The author explains that standard deviation is a reliable measure of dispersion.

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To explain:

The standard deviation tells about a distribution.