Simulation Methods in Finance and Insurance

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Then we move them back to the present day discounting them at the risk free rate of return. We have 5 discounted and transformed points now.

[pic 78]

[pic 79]

and so on….

We simulate the [pic 80]

We also take a 95% confidence interval. On “average” in 5 out of 100 cases the true expected value is not contained in the confidence interval.

- How does Monte Carlo integration works? What are the advantages and the drawbacks?E

- Suppose one wants to determine [pic 81]

If its density is [pic 82][pic 83]

We know that [pic 84]

Algorithm: M.C. integration

- Generate independent random numbers [pic 85][pic 86]

- Estimate [pic 87]

The sample standard deviation is: [pic 88]

Once again the strong law of large numbers ensures that the arithmetic mean converges to the correct value as the number of draws increases. [pic 89][pic 90]

- For one-dimensional integrals this method is not recommended as the convergence rate is , too slow. Whatever the dimension d is the error bound always stays the same, .[pic 91][pic 92]

The trapezoidal rule has convergence rate for , much better than the M.C. method. For higher dimensions we have convergence speed .[pic 93][pic 94][pic 95]

For the M.C. method is competitive![pic 96][pic 97]

- The curse of dimensionality in numerical integration is the fact that in the trapezoidal rule, to guarantee a prescribed level of accuracy , one needs nodes.[pic 98][pic 99]

- For higher dimensions: We have to integrate

[pic 100]

with vector of independent – random variables.[pic 101][pic 102]

Algorithm: High-dimensional integration via Monte Carlo

- Generate independent random numbers where and [pic 103][pic 104][pic 105]

, is the dimensionality and is the number of experiments[pic 106][pic 107][pic 108]

- Estimate [pic 109]

Once again the average error is , independent of d!!! M.C. method beats the curse of dimensionality! [pic 110]

Now if integration domain D is unbounded AND a suitable variable transformation is not feasible:

[pic 111]

with vector of independent random variables with density g(s) that has support exactly equal to D.[pic 112]

Algorithm: High-dimensional integration in unbounded domain

- Generate n independent random vectors with density g(s)[pic 113]

- Estimate (e.g. with multivariate normal density g(s))[pic 114]

Chapter 2

- What is the problem of generating real randomness? Which properties should an algorithm producing pseudo-random numbers fulfill?

- It is not practical since it is time consuming, expensive and difficult to implement on computer.

It is not reproducible.

- The RNG should:

Give an even distribution of random numbers

Be fast

Have long period

Pass statistical tests

Be reproducible

Be portable (algorithm deliver same RNs on every computer)

Be easily implemented

Different starting values should lead to different pseudo-random number sequences

- What is a linear congruential generator, what is the simple idea behind and what is one of its drawbacks?

- Algorithm: LCG

- Set such that[pic 115]

[pic 116]

Where

is the modulus[pic 117]

is the multiplier (or factor), [pic 118][pic 119]

is the increment, [pic 120][pic 121]

is the seed (initial point), [pic 122][pic 123]

- Numbers in are obtained via [pic 124][pic 125]

Example: [pic 126]

[pic 127]

[pic 128]

[pic 129]

, which is the seed the starting point[pic 130]

It is periodic of course. We hit the period of 5 after going to the 4th number. We are going to get , and not anything higher. The order is we actually skipped out on .[pic 131][pic 132][pic 133]

Generator often denoted by LCG [pic 134]

Note: if , the maximal possible period is . In this case we get a multiplicative generator: [pic 135][pic 136][pic 137][pic 138]

The period of a general LCG is at most .[pic 139]

- As far as choosing the coefficients are concerned:

- Often m is prime

- Mersenne primes are preferred of the form [pic 140]

The largest 32-bit signed integer is (leads to too short period)[pic 141]

- Sometimes we use . Calculations are faster but there is high correlation[pic 142]

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