Financial Monte Carlo Simulation with Matlab
DKWC
2020-05-24
1 Q1
1.1 Answer Q1-a
- This part is adapting the concept of Widynski (2020) for the construction of middle square Weyl sequence generator.
One of the disadvantage of Middle Square Weyl Sequence Pseudorandom Generator is that irrational initial seed number should be used to span over the number of iteration space. Otherwise, the simulated x will fast and quick drop to zero in Middle Square Weyl Sequence Pseudorandom Generator.
The seed number(13737373) is the irrational number that can generate non-zero Weyl Sequence over the iteration space.(see Figure 1)
If seed number(84848484) is non-irrational number, it can generate Weyl Sequence of which drop to zero quickly over the iteration space.(see Figure 2)
1.2 Answer Q1-b
- Middle square Weyl sequence simulation of a standard Gumbel distribution with cumulative distribution function \(F(x)=e^{-e^{-x}}\). The inverse of \(F(x)=e^{-e^{-x}}\) is \(F^{-1}(x) = -log(-log(x))\). The following simulation is to use the direct simulation of inversed-method. (see Figure 3)
1.2.1 Matlab Code for Q1-a and Q1-b
2 Q2
2.1 Answer Q2-a and Q2-b
2.1.1 Disussion and Summary
Assume that X and Y are bivariate Guassian Random Variables. The simulation samples N = is 10,000. Their uniform distribution denoted as \(U_X\) and \(U_Y\)
By using the acceptance -rejection method for the bivariate independent joint distribution, it found that the acceptance lower triangle region(red in color)is \(50.04\%\) (see Figure 4)
Whereas the rejection upper triangle region(green in color) is \(49.96\%\)
The 3-dimensional simulated plot(see Figure 4) and numerical plot(see Figure 5) are shown below. Both the simulated plot combines acceptance region output(red in color) and rejection rejection output(green in color).
Compared the simulation plot with the numerical plot(see Figure 5) over the Square Space Region, the two plots are very close and similar provided that the sample size N of simulation are sufficient large.
3 Q3
3.1 Answer
- Tasks: 1) Compute Asian Put-Option Price; and 2) Investigate how change in the number of monitoring dates changes the price of this option
- Averaging three antithetic simulated Stock Price Projectories (S1, S2, S3) over Time:
- Matlab code(Monte Carlo Simulation Asian Put Option)
3.1.1 Disussion and Summary
This exercise is to simulate Asian Put Option with both non-antithetic and antithetic sampling method.
First is to generate three correlated GBM stock prices trajectories. All GBM simulatuion are using the Euler Scheme for both Antithetic Sampling and Non-Antithetic Sampling separetely.
Second is to averge three simulated Stock Price Projectories (S1, S2, S3) for non-antithetic sampling method and antithetic sampling method separately:
Third is to compute the Asian Put Option and 95% confident interval based on the simulated Stock Price Projectories for non-antithetic sampling method and antithetic sampling method separately.
The replication sampling numbers are NReplication = [100, 1000, 10000]
The number of observations here means the sampling points of stock prices participated for evaluating GBM Asian put option pricing before expiration time (0.5 years).
It found that the effect of variance reduction is more significant with antithetic sampling method when both the numbers of observations (say, Nobservation = 60) and the number of replication simulation(say N Replication = 100) are relatively low.
However, when both the numbers of observations (say, Nobservation = 360) and the number of replication simulation(say NReplication = 10000) are sufficient large, the effect of variance reduction for both antithetic and non-antithetic sampling method is not very significant different. It is because the the computational result will converge to the same and true value for both methods only if the number of replication is sufficient large.
4 Q4
4.1 Answer
- Tasks:
- Estimate the price of the compound call-on-call option with strike price K = 36 and maturing in 3 months, written over that call option.
- By considering several different values of L below and above 400, investigate how L influences the price of the compound option.
4.1.1 Black Scholes Model: European call on a call compound option
- Details of closed-form solution of Compound Option can refer to the paper of Clewlow, Llanos, and Strickland (1994)
4.1.2 Disussion and Summary
Question 4 is to compute the current payoff of European call on a call compound option wiith Black Scholes Model.
In this exercise, I introduce a variance reduction method with control variate of \(\rho\) range = (0.33 , 0.96) to compute the payoff of the European call on a call compound option at \(t = 0\). The control variate of \(\rho\) is a baised technique for variance reduction. Therefore, any confidence interval estimate is meaningless for the baised control variate techinque.
With the given following parameters, the payoff of the European call on a call compound option at \(t = 0\) is estimated to be \(V_{ECOC}(S_0, 0) = 2.444\) at the fixed \(\rho = 0.8944\).
The effect of first strike price \(X1\) on the payoff value of \(V_{ECOC}(S_0, 0)\) starts to monotonically decrease at \(X1 = 10\) to the minmium value at \(X1 = 280\) and then montonically increase the payoff value of \(V_{ECOC}(S_0, 0)\) to \(X1 = 600\) for any fixed \(\rho\).
By changing the \(\rho\) from 0.33 to 0.96, it can observe that the payoff value of \(V_{ECOC}(S_0, 0)\) is monotonically deceasing over \(\rho\) for each fixed strike price \(X1\). Hence, the higher the correlation coefficient, lower the payoff value of \(V_{ECOC}(S_0, 0)\) is.