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4.5.2 Expected value of an option from sampled data $ \spadesuit$

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To develop your intuition, let us first define the transition probability $ p[S,t;S^\prime,T]$ measuring the likelyhood that an asset evolves from the present value to the terminal value $ (S,t) \rightarrow (S^\prime,T)$ : weighted by the terminal payoff of an option $ \Lambda(S^\prime)$ , 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

$\displaystyle V(S,t) = \exp[-r(T-t)]\int_0^\infty p[S,t;S^\prime,T] \Lambda(S^\prime) \frac{dS^\prime}{{S^\prime}^\kappa}$ (4.5.2#eq.1)

where the expected terminal payoff has been discounted back to the present time $ t$ by multiplication of the factor $ \exp[-r(T-t)]$ . 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 in a risk-neutral world where the drift is replaced by the spot rate minus the dividend yield $ \mu=r-D_0$ . (Note the analogy with the delta hedging, where the risk has been eliminated to obtain a Black-Scholes equation that is also independent of the drift $ \mu$ .)
Instead of calculating a complicated n-dimensional path-dependent integral with transition probabilities $ p[S_i,t_i;S_j,t_j\vert\mathcal{C}_j]$ that are subject to multiple conditions $ \mathcal{C}_j$

\begin{displaymath}\begin{split}V(S,t) = \exp[-r(T-t)] &\int_0^\infty \frac{dS_1...
...{n-1},t_{n-1};S_n,T\vert\mathcal{C}_n] \Lambda(S_n) \end{split}\end{displaymath} (4.5.2#eq.2)

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

$\displaystyle V(S,t) = \exp[-r(T-t)]\frac{1}{N}\sum_{k=1}^N \Lambda(S_k)$ (4.5.2#eq.3)

The realizations of the underlying asset prices $ \{S_1,S_2,\dots S_N\}$ are evolved using a risk-neutral random walk by setting the drift $ \mu=r-D_0$ . 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 VMARKET class reads
    } 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);
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 $ S_j$ with currentState[k][0]*x[j]/strike; this is then used as an argument to accumulate the terminal payoff $ \Lambda(S_k)$ 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 VMARKET applet below illustrates the result in the case of a simple vanilla put option.

VMARKET applet:  press Start/Stop to simulate the price of a super-share option assuming first a separable log-normal random walk.

Virtual market experiments: Monte-Carlo expectation
  1. Adjust the number of Walkers to achieve a precision of only about 10%.
  2. Change the parameter LogNkappa=1.002 to keep a nearly log-normal distribution of the increments and yet force the applet to use a new random walk for every value of the underlying asset. The ``noise'' between adjacent prices can then be used as a measure or the precision of the calculation. How far were you from the previous 10% target?
  3. Vary the parameter LogNkappa to study how different distributions of the market increments affect the price of an option.

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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