4.5.2 Expected value of an option from sampled data

[ SLIDE open - scheme - code || same VIDEO as previous section: modem - LAN - DSL]

To develop your intuition, let us first define the transition probability measuring the likelyhood that an asset evolves from the present value to the terminal value : weighted by the terminal payoff of an option , this can be used to evaluate the expected return from a particular realisation of the market. Summing the weighted returns from all possible realisations with the proper Jacobian, the present value of an option could be calculated from

where the expected terminal payoff has been discounted back to the present time by multiplication of the factor . This expression can be identified with the analytical solution (4.3.2#eq.11) and shows that the price of an option can also be calculated as the present value of the expected return, using a random walk

Instead of calculating a complicated n-dimensional path-dependent integral with transition probabilities that are subject to multiple conditions

the Monte-Carlo sampling method simply uses a large number of possible realizations as an unbiased estimator for the mean price payed when the option is exercised

The realizations of the underlying asset prices are evolved using a risk-neutral random walk by setting the drift . Path dependent features such as barriers can be easily be incorporated at the end, by retaining only those prices that satisfy the conditions: the terminal payoff can for example be multiplied with a marker variable that is either equal to zero or one depending whether the condition has been fulfilled or not. The scheme that has been implemented in the

} else if(Math.abs(kappa-1.)<0.001){ //Separable log-normal if (markers){ for (k=0; k<numberOfRealisations; k++){ f[j]+= option.getValue(currentState[k][0] *x[j]/strike) *mark[k][0]; g[j]+= option.getValue(currentState[k][0] *x[j]/strike); } } else for (k=0; k<numberOfRealisations; k++) f[j]+= option.getValue(currentState[k][0] *x[j]/strike); } f[j]=Math.exp(-time*rate)*f[j]/numberOfRealisations; g[j]=Math.exp(-time*rate)*g[j]/numberOfRealisations;If the problem is separable, the random walk is first scaled according to (4.5.1#eq.2) to obtain the terminal value of the underlying with

`currentState[k][0]*x[j]/strike`

; this is then used as an argument
to accumulate the terminal payoff
from every realization
using the statement `f[j]+=option.getValue()`

and finally calculate
the discounted average of (4.5.2#eq.3) using the last two lines.
Note that two functions (`f[j],g[j]`

) have been used to compare the
price obtained with-/out barriers.
The To conclude this section with a comparison between the finite difference and Monte-Carlo methods, remember that Monte-Carlo simulations offer considerable flexibility to model path-dependent options and change the statistics of the market increments. This flexibility, however, comes at a high computing cost for reaching an acceptable precision at the percent level, this even if it is generally sufficient to calculate a single price for the option, which finite differences cannot do.

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