Financial mathematics lecture notes
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Each point is worth one cent, but she is only able to redeem them for cash on multiples of 5,000. Introduce a random variable Y taking 1 values +1 Heads and? You withdraw that and put it in another account for n years, after which your capital has grown to 1 + i n 1 + i m C. This shows that money has a time value: the value of money depends on the time. Modelling many Random Variables together. The bond is to be delivered in two years immediately after the dividend has been paid. The present term structure is given by the following rates of interest on a yearly basis 6 months 5. The rationale for this is that forward and futures prices for any goodâ€”also consumption goodsâ€” exhibit a Martingale property on an arbitrage free market, whereas this is not true in general for spot prices other than for pure investment assets.

The idea behind compound interest is that in the second year, you should get interest on the interest you earned in the first year. These are the units that are used most often. Covariance Properties of Variance and Covariance 1. Deposit this on the money market account, and increase 1 2 r +r +r 1 2 3 the futures position to e contracts. Z 2 , by the previous result? As the name implies, simple interest is easy to understand, and that is the main reason why we talk about it here. E X 2 It is always nonnegative 5. In financial mathematics, all payments must have a date attached to them.

In other words, the interest you earn in the first year is combined with the principal, and in the second year you earn interest on the combined sum. Assume that there are no other dividends or other convenience T yield during the time up to T. Cov X, Y denotes the matrix with elements Cov Xi, Yj , Cov? Einstein in 1905 explained Brownian motion as a result of molecular bombardment by the molecules of the? Nevertheless, simple interest is sometimes used, especially in short-term investments. C for any n, then for any stopping time? This avoids some mathematical techni- calities that seem irrelevant to the reality we are modelling. But it 0 0 0 would give the trader the dividend of the share for free, for this dividend goes to the holder of the share, not the holder of the forward.

With the same setting as in example 2, assume that there are dividend yield payments at several points in time t. Conditional Expectation as the best predictor Expectation as best predictor Let X denote a random variable. Find the accumulation of Â£5000 after seven years in this account. Brown described the motion of a pollen particle suspended in? In the second year, he would get Â£98. Exercise: Give a proof that Optional stopping applies in the martingale above.

The time when something happens for the? Core reading for the 2009 examinations. The interest rate of the Euro is 4. The next day, day 3, the balance r +r +r 1 2 3 has grown to e , where r is the short interest rate from day 2 3 to day 3; it is random as seen from day 0 and day 1, and its outcome r +Â·Â·Â·+r 1 T occurs day 2, and so on. Say you gamble from 8pm to 11 pm, and? How much do you need to 1 invest to get a capital C after one time unit? Consider again the equivalent payments in Figure 1. Compare the following three loans: a loan charging an annual effective rate of 9%, a loan charging 8 43 % compounded quarterly, and a loan charging 8 12 % payable in advance and convertible monthly.

The proof is left as exercise. Provided the interest rate is not too big, the discount factor is close to one. A careful comparison shows that for periods less than a year simple interest pays out more, while compound interest pays out more if the period is longer than a year. A share is valued at present at 80 dollars. It may seem more logical to quote the rate as 8. On the other hand, calculating the expectation directly E X? Random Walk and Discrete time Martingales 7.

What is the forward rate from 6 to 12 months? This implies x 1 that its graph lies over its tangent. There is an alternative but equivalent definition of the symbol i p , which leads naturally to the valuation of annuities described in the next chapter. Financial Mathematics Notes Financial Mathematics topic-wise notes. A payment of i at the end of the year is equivalent to a payment of d at the start of the year. Successive sums give simulated path. Note that B 2 and B 1 are not independent, therefore this probability cannot be calculated as a product P B 1? A capital of Â£1000, invested for half a year at 9%, grows to Â£1045 under simple interest and to Â£1044.

E X Y are uncorrelated. The rate of interest i is the interest paid at the end of a time unit divided by the capital at the beginning of the time unit. Assume we want to buy a foreign currency in t years time at an exchange rate, the forward rate, agreed upon today. We can in a meaningful way assign probabilities to outcomes of experiments that can be repeated under simi- lar circumstances, or where there are strong symmetries between possible outcomes. N 0, tv and Yk? Instead, savings accounts in banks pay compound interest, which will be introduced in the next section. N is a stopping time, which is bounded by N. Flat is good for distance vision.

For instance, a farmer may lend his tractor to a neighbour, and get 10% of the grain harvested in return. Compound interest has the desirable property that this does not make a difference. However, that is not necessary. Wiener in 1931, hence it is also called the Wiener process. If you leave it for only half a year, then you get 12 Â·0. In the analysis below we assume we use exact time.