Simulation Methods in Finance and Insurance

SIMULATION METHODS IN FINANCE AND INSURANCE

THEORY QUESTIONS

Chapter 1

1. Explain the basic principle of Monte Carlo methods and its mathematical background.

• Often, the quantity of interest can be expressed as an Expected Value  of a r.v. , for example the pricing of derivative securities or even to determine probabilities of events [pic 1][pic 2]

( ).[pic 3]

We, thus, can:

1. Carry out a large number of independent experiments which all have the distribution of [pic 4]
2. Calculate the arithmetic average of the obtained results
3. Approximate the expected value  by this average value[pic 5]

• The strong law of large numbers ensures that the estimate converges to the correct value as the number of draws increases (the mean  must be finite)[pic 6]

 almost surely, where  is a sequence of n i.i.d. r.v. with .[pic 7][pic 8][pic 9]

• If, in addition to the above, X has also a finite variance  then the Central Limit Theorem provides information about the likely magnitude of the error in the estimate after a finite number of draws.[pic 10]

        [pic 11]

• So we generate n i.i.d. random numbers Xi and we introduce the Monte Carlo estimator  as:[pic 12]

 : =  [pic 13][pic 14]

        We recall that the S.L.O.L.L. ensures that    as  almost surely.[pic 15][pic 16][pic 17][pic 18]

        This estimator n is unbiased and strongly consistent.[pic 19]

α        But what can we say about the error   ?[pic 20]

        Turns out that if  is finite we look at the mean-squared error[pic 21]

                [pic 22]

thus error is bound by  for  positive real number. On the first “equal” sign we have to observe that it holds because of the unbiasedness of the Monte Carlo estimator.[pic 23][pic 24]

In other words, because of C.L.T. we have for large n:

[pic 25]

APPROXIMATELY

Thus [pic 26]

• The standard deviation of the error OR convergence rate of the method is of order

[pic 27]

It’s a measure for the (mean) accuracy of the M.C. method

Consequences:

1. Adding one decimal place of precision requires 100 times as many random numbers (M.C. runs)
2. For high accuracy very large sample sizes are needed
3. Variance reduction can improve the convergence
4. Effective method for high-dimensional integration

• Since we know that for large n   is approximately  – distributed we can construct an approximate  – confidence interval for the [pic 28][pic 29][pic 30][pic 31]

[pic 32]

        For an approximate  - confidence interval for  we have[pic 33][pic 34]

[pic 35]

where  represents the point on the x-axis for which the area under the std normal pdf is equal to [pic 36][pic 37]

The length of C.I. is  which of course tends to  as n gets larger.[pic 38][pic 39]

We see that on “average”  out of  cases the true value is not contained in the confidence interval.[pic 40][pic 41]

The length of the C.I. is  meaning that if we want to gain one digit of precision we have to use  times as many simulation steps (“slow convergence”)[pic 42][pic 43]

One problem is that  is unknown and we deal with this, introducing the sample variance which is an estimator for the variance[pic 44]

                [pic 45]

This is the definition of the sample variance. The 2 forms have different speed.

1. Discuss the Monte Carlo method using an example.

• Estimating the probability of an event

Let  be a certain event. We define the indicator function[pic 46]

                [pic 47]

The probability of  is .[pic 48][pic 49]

The M.C. estimator for  is simply the relative frequency of the occurrence of the occurrence of A in n independent experiments[pic 50]

        [pic 51]

We find an estimator for Variance. We have that [pic 52]

.[pic 53]

So [pic 54]

Approximate  confidence interval for [pic 55][pic 56]

        [pic 57]

• Option Pricing

We have that:

 are the log-returns[pic 58]

we note that  is the asset price at time ,  is the initial stock price, r is the risk free interest rate,  is the volatility and  is the time horizon[pic 59][pic 60][pic 61][pic 62][pic 63]

         is the asset price at T[pic 64]

         is the final payoff at time T of an option[pic 65]

                where  is the strike price[pic 66]

 [pic 67]

 [pic 68]

        where [pic 69]

 Take for example . We wish to simulate the price (the above expected value) of the call option. We generate  points of   (the log-returns). We then transform these points to the asset prices at . Then we take the  which are the final payoffs at T=3 of those 5 points .[pic 70][pic 71][pic 72][pic 73][pic 74][pic 75][pic 76][pic 77]