The random network approach, developed by Andreasen and Huge (2011), is a pricing method that guarantees discrete consistency between calibration, finite difference solution, and Markov chain Monte-Carlo simulation based on the random network approach. This dissertation provides a review and application of this random network approach to pricing under the Heston model as well as the stochastic local volatility model. Consistent results are obtained for a call option under different pricing methods using parameters similar to those used in the random networks paper.
More specifically, when a Heston model is used, consistent prices are obtained for the characteristic function pricing method, the backward finite difference method, the forward finite difference method, and the Markov chain Monte-Carlo method based on the random network approach. . Similarly, consistent prices are obtained under the stochastic local volatility model for the backward finite difference method, the forward finite difference method, and the Markov chain Monte-Carlo method based on the random networks approach. First, I would like to thank my supervisor, Thomas McWalter, for his time and effort in helping me with this dissertation.
Introduction
Andreasen and Huge (2011) claim that their random grid approach achieves perfect discrete correspondence between calibration, finite difference solution, and Markov chain Monte-Carlo simulation. This is useful because the Monte-Carlo Markov chain method is developed from the FFD method. Since the Markov chain Monte-Carlo approach is derived from the FFD method, the results of the Markov chain Monte-Carlo are consistent with the results obtained by the FFD method.
Since the FFD method is consistent with the BFD method, the Monte-Carlo Markov chain method is also consistent with the BFD approach. Since the Markov chain Monte-Carlo method is derived from the FFD method, it uses a grid of discretized state values and a probability density to generate random numbers. This includes the derivation and description of the BFD method, the FFD method as well as the grid-based Markov chain Monte-Carlo approach.
Framework for Random Grids
- Share Price Process
- Option Instrument
- Framework for Finite Difference Methods
- Backward Pricing PDE for Option Price
- Forward Pricing PDE for Density
- Backwards Finite Difference Equations
- Forward Finite Difference Equations
- Framework for Monte-Carlo Methods
- Conditional Transition Probability Matrices
- Simulation Algorithm
- Characteristic Function Pricing
- Consequences of Assumptions Made
- Risk-Neutral Measure
- Forward Rates
- Use of the ADI Schemes
- Zero Correlation Between Stock Price and Volatility
Assume that the stock price is denoted by S = x and the volatility is denoted by z = y. The notation x(th)indy(th) is used to express the value assumed by the stock price and volatility at a discrete point in time. The payoff of the call option depends on the stock price at maturity, which is determined by the stochastic local volatility model described in (2.1).
Each method attempts to price the option by constructing the realization of the stock price at maturity. Note that the stochastic local volatility model has two stochastic processes - the stock price process and the volatility process. Considering (2.1) and (2.2), it is clear that the stock price process depends on the volatility process.
Therefore, based on the stock price at expiration, the option payout at expiration can be calculated for any (xi, yj) combination using (2.4). Therefore, the backward pricing PDE expresses a certain relationship between the instantaneous change of the option price and an infinitesimal change in stock price and volatility. In addition, there is zero chance that the stock price and volatility at timet0 take on a different value.
In a similar case to the backward pricing PDE, the forward pricing PDE expresses an equation in terms of the instantaneous change in the probability density over infinitesimally small changes in stock price and volatility. Therefore, instead of solving (2.8), the BFD equations corresponding to the backward pricing PDE (2.5) are described by the ADI scheme. Once the random numbers have been used to generate a stock price at expiration, the call option's payoff at expiration is calculated using (2.4).
Just as the objective of the general Monte-Carlo method was to generate the stock price at each time until expiration, the objective of the Markov-chain Monte-Carlo method based on random lattices is to generate values for the stock price (xi) and volatility (yj) at each time (th) until expiry. It is these cumulative probabilities that are used to generate stock price and volatility states. As a result, the forward strike and the forward share price are used to obtain the option prices.
Implementation
- Pricing Using the Backward Finite Difference Method
- Pricing Using the Forward Finite Difference Method
- Markov-Chain Monte-Carlo Approach
- Computing the Cumulative Transition Probability Matrices . 27
- Markov-Chain Monte-Carlo Option Price
- Determining the Model Parameters
- Efficient Algorithms
- Efficient Algorithm for Computing Matrix Inverses
- Efficient Algorithm for Solving Matrix Equations
The first step of the ADI scheme involves solving (2.11) backward in time for the half time step option price. The second step of the ADI scheme uses the solution from (2.11) to solve (2.12) backward in time with another half time step to obtain the option price at the full time stepτ−2. The second step of the ADI scheme, expressed by (2.14), uses those×1 column vectorsp(t0+1/2,·, yj) for eachnyj, obtained from the first step of the ADI scheme, to solve for the density at full-time step (p(t1,·, yj)) for each star.
Once the probability density matrix is formed at maturity, this matrix is used to calculate the call option's expected payoff at maturity. The expected payoff at time T is obtained by multiplying p(T,·,·) elementally by payoff(T,·,·) and then taking the sum of the row-wise sum as shown in Section 2.3. .4. The objective of the Markov-chain Monte-Carlo method is to simulate the paths for the stock price process (x(th)) and the volatility process (y(th)) and then use them.
The transition probability matrices are formed by taking the inverse of the relevant Ax,yj andAy operators of the finite difference approximation, as specified by (2.15) and (2.16). The purpose of the simulation process is to use the current value of xi andyj at timet0, to simulate a value for xiandyjat timet1. Since a call option depends on the value of the stock price at expiration, it is essential to continue simulating xi values until a path to time is generated.
However, it does not explain how to define some of the input parameters yourself, so that the random mesh method can be adapted to your needs. The starting point will be timet0 and the ending point will be the maturity date of the option being valued. This would ensure that most of the calculated payouts are actually measurable by the network.
This section discusses ways to improve the performance of the matrix inversion algorithm and the matrix solving algorithm used in the random mesh scheme.
Results
Heston Model
- Option Price Under Varying Strikes
- Implied Volatility as a Function of Strikes
- Option Price under a Larger Finite Difference Grid
- Effect of Sample Size on Monte Carlo Error
This is as expected due to the Monte-Carlo error that occurs in random number generation. The standard deviation gives an indication of the size of the Monte-Carlo error for the given number of samples. Note that with a standard deviation of 0.00176, the Monte-Carlo price of 0.25919 is within three standard deviations of the final difference price of 0.26125.
As the sample size increases, the price of the Markov chain in Monte-Carlo will tend towards BFD and FFD prices. Therefore, the Monte-Carlo method with the Markov chain obtained prices consistent with the BFD and FFD prices. Because Monte Carlo prices differ slightly from finite difference prices, the volatilities implied by Monte Carlo are not identical to those obtained under finite difference methods.
However, the implied volatility of 0.29586 will tend towards the finite implied volatility of the difference as the sample size tends to infinity. In addition, the Monte-Carlo prices in the Markov chain are similar to the finite difference prices and will tend towards the finite difference price as the sample size tends to infinity. Given the standard deviation of 0.04798, the Monte-Carlo price of the Markov chain of 7.53854 is also consistent with the finite difference price – although this will improve as more samples are used.
The fact that all three methods used to define the random grid approach obtain similar results to the characteristic function price indicates that the price generated by the Monte-Carlo random grid method is accurate. The Monte-Carlo estimate is calculated for a range of sample sizes to examine its convergence pattern. Since the standard deviation represents the error for prices obtained, figure 4.2 indicates that the error between the Monte-Carlo price and finite difference price decreases as the sample size increases.
In theory, it is possible to improve the accuracy of these Monte-Carlo solutions by using variance reduction techniques.
Stochastic Local Volatility Model
- Option Price Under Varying Strikes
- Implied Volatility as a Function of Strikes
In addition, the prices obtained from the Markov chain Monte Carlo are within three standard deviations of the finite difference prices. This price is quite consistent with the Monte Carlo price of 0.52878 and a standard deviation of 0.00499. As a result, it is clear that the BFD, FFD, and Monte-Carlo prices are consistent with each other, with the Monte-Carlo prices again yielding a Monte-Carlo error.
Therefore, the random networks approach under the local volatility stochastic model is shown to produce prices that are consistent with finite difference prices. Since the BFD and FFD prices are identical, it is no surprise that their respective implied volatilities are also identical. Although the implied volatility under the Markov-chain Monte Carlo method is not identical, it is relatively close and will improve as the standard deviation decreases with more samples.
To further examine how the implied volatility varies as the strike changes, the following volatility curve is obtained. Under the stochastic local volatility model, Figure 4.3 shows that the implied volatility is a function of the strike. The shape of the function is similar to that obtained from the Heston model, with the minimum implied volatility obtained at the at-the-money strike, the implied volatility of 0.26735 obtained at the 50% strike and the volatility of.
Conclusion
A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies. Characteristic Function Pricing, Stochastic Volatility and Fast Fourier Transform, Numerical Methods in Finance II Notes. Decreasing rates of inverse nonsymmetric tridiagonal and band matrices, SIAM Journal on Matrix Analysis and Applications Numerical recipes in C, Cambridge university press Cambridge.
Appendix
Decomposition of Tridiagonal Matrix to Compute Tridiagonal Inverse 42
- Using x, ˜ y ˜ and d ˜ to Compute the Transition Probability Ma-
