代写Barrier option、Matlab程序代写、代做Matlab设计

日期：2019-04-09 11:27

PROJECT. COMPUTING BARRIER OPTION PRICES UNDER LOCAL

VOLATILITY MODEL WITH FINITE DIFFERENCE AND MONTE CARLO

SIMULATION

Assume the stock price S paying a continuous dividend yield q follows the process below dSt = (r q)Stdt + σ(St, t)StdWt, (1)

where r is the risk free interest rate and W is a Wiener process under the risk-neutral

probability measure. The local volatility surface has been calibrated with the form below:

We know that the price of an option, V (S, t), on the above stock satisfies the PDE below:

In this project you are asked to price some very popular weakly path dependent option,

such as a Barrier option using both Finite Difference and Monte Carlo Methods.

Barrier options (also called knock-in or knock-out options) are standard calls or puts except

that they disappear (knock-out) or come into existence (knock-in) if the underlying asset

price is found to have crossed a predetermined (barrier) level, B, anytime before the maturity

of the option. They have a fairly standard naming convention which describes whether

the barrier is below or above the current asset price (“down” or “up”), whether the option

disappears or appears when the barrier is crossed (“out” or “in”) and whether they have a

standard call or put pay-off. For instance, an up-and-out call option with zero rebate will

have a payoff

hup-and-out call(ST ) = (max(ST K, 0), if max0≤t≤T St < B0, otherwise

The aim of this project is to use the following pricing parameters, spot price S0 = ￡100,

strike price K = ￡100, a Barrier level B = ￡130, risk-free rate r = 3%, dividend yield

q = 5%, time to maturity T = 0.5 year, and α in the local volatility function α = 0.35,

solve the Black-Scholes PDE numerically to price an up-and-out call option by means of

(1) Explicit finite differences

(2) Implicit finite differences

Date: Current Version February 8, 2018.

1

2 PROJECT

(3) Crank-Nicolson finite differences

Since there is no analytic solution to the PDE, you need to compare the results from finite

difference method with the prices calculated from Monte Carlo simulation. Please finish

the tasks below:

a (35% of project mark) Implement Monte Carlo Simulation using Euler Scheme to discretize

equation (1) with 500 time steps and 1 million paths to compute the up-and-out

call option prices and 95%? confidence intervals of those prices. Please use the antithetic

method to reduce the variance of the results.

b (15% of project mark) Discuss proper domain and boundary conditions you should use

in the finite difference schemes based on the properties of the up-and-out call option.

c (40% of project mark) For each of three finite difference schemes, namely each scheme

in (1) - (3), compare the prices with different grid sizes with those from the Monte Carlo

methods and report the convergence properties of each scheme.

d (10% of project mark) In the case of the explicit finite difference scheme demonstrate

that the scheme is conditionally stable and find the condition under which the scheme

would be stable.

The code should be accompanied by detailed documentation, split into:

code developer documentation: presenting the structure of the code, available functions,

main variables and any other information to help to understand the code,

including directions on how to extend the code to handle more general features; the

test runs of your code and timings should be reported in a separate section of the

developer documentation;

end-user’s instructions: how to use the .m program, how to input data and how the

results are presented, and a brief description of the methods implemented. Also the

discussions of parts (b), (c) and (d) should be included here.

Each of the marks in the above categories (a) (d) will be split as follows: 60%

for coding style, clarity and accuracy of computation, and 40% for documentation

(including comments within the code).

This project will contribute 40% towards the final mark for the module.

Submit your work by uploading it in Moodle by 23:55 on Sunday, 15 April 2018. Submit

the code as a single compressed .zip file, including all Matlab (.m) files all residing in a single

parent directory, whose name should contain your name and student code. The .zip file

should preserve the subdirectory structure of the parent directory (that is, the subdirectory

structure must be automatically recreated when unzipping the file).

14th April 2019