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FE621 geometric Brow- nian motion model
创建日期：2024-07-20 16:41:22
所属分类：金融Finance
标签： FE621
geometric Brow- nian motion model
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Final Exam FE621

Please note: for the problems using data we are not interested in all the minor steps performed when analyzing the data. However, you should

Name:

  • Please note: for the problems using data we are not interested in all the minor steps performed when analyzing the data. However, you should explain your broad reasoning and hand in a final report.
  • Do not include any failed attempts at modeling just to give volume to the final report.
  • You need to argument with graphs or selected numerical results all the conclusions you draw. Put important plots and tables within the report and relegate any non essential ones to an appendix at the end of the document.
  • Please add supporting arguments (e.g., output) in the appendix. The whole report should be constructed as a regular journal article.
  • The final write-up (excluding the appendix) should not be more than 10-15 pages.
  • Communication with other students either physical or virtual is strictly forbidden.

For  instructor’s  use only

Problem Points Score
A 100
B 100
C 100
Bonus 20
Total 300

Problem A: Asian Option Pricing using Monte Carlo Control Variate.

The payoff of an arithmetic Asian call option is:

Its value may be computed using straight Monte Carlo simulations. However, in order to obtain a small standard error, the number of simulations must be very high. To solve this computationally extensive problem, we will use the payoff of a geometric Asian call option as the control variate:

The idea is to use the known analytic price of the geometric Asian and the distance between MC simulations to obtain an approximate for the analytical formula for the arithmetic Asian price.

In this problem we consider r = 3%, σ = 0.3, S0 = 100, and assume the goal

is to price an arithmetic Asian call option with strike K = 100 and maturity

T = 5.

We also assume the asset follows the standard log-normal/geometric Brow- nian motion model:

S(∆t) = S(0)e((µ− σ2 )∆t+(σ√∆t)s)

(a)The price of a geometric Asian option in the lack-Scholes model is given by:

Pg = e−rT .S0eρT N (d1) − KN (d2)Σ

where:

such  that  σˆ  is  adjusted  sigma  and  N  is  the  total  number  of  trading  days

Use the above formula to price this geometric Asian call option.

(b)Implement a Monte Carlo scheme to price an arithmetic Asian call option (P sim). Use M = 1, 000, 000 simulations. Record the answer, a confidence intervaland the time it takes to obtain the

(c)Implement a Monte Carlo scheme to price a geometric Asian Call option (Psim).

(d)Using M = 10, 000 simulations and the same exact random variablescreate:

numbers Xi which are M replications for the arithmetic Asian Option price

numbers Yi which are M replication for the geometric Asian Option price

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