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FE545 Homework Assignment

创建日期：2024-04-17 14:10:43

所属分类：计算机computer science

标签： FE545

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FE545 Homework Assignment

Due Date: Oct. 4th (Monday). Data: An Asian option is a type of exotic option. Unlike a vanilla European option where the price of the option is depen- dent upon the price of the underlying asset at expiry, an Asian option pay-off is a function of multiple points up to and includ- ing the price at expiry. Thus it is “path-dependent” as the price relies on knowing how the underlying behaved at certain points before expiry. Asian options in particular base their price off the mean average price of these sampled points. To simplify the problem, we will consider equally distributed sample points beginning at time and ending at maturity, In this problem, we will consider Geometirc mean A of the spot prices we use the following formula: A(0, T ) = exp ( 1 N N∑ i=1 log(S(ti)) ) (1) Unlike in the vanilla European option Monte Carlo case we have learned in class, where we only needed to generate mul- tiple spot values at expiry, we now need to generate multiple spot paths, each sampled at the correct points. Thus instead of providing a double value representing spot to our option, we now need to provide a std :: vector < double > (i.e. a vector of 1 double values), each element of which represents a sample of the spot price on a particular path. We will still be modeling our asset price path via a Geometric Brownian Motion (GBM), and we will create each path by adding the correct drift and variance at each step in order to maintain the properties of GBM. Implement PayOff classes according to the following tem- plate: #ifndef __PAY_OFF__ #define __PAY_OFF__ #include

// This is needed for the std::max

// comparison function, used in the pay-off calculations
class PayOff {
public:
PayOff(); // Default (no parameter) constructor
virtual ~PayOff() {}; // Virtual destructor
// Overloaded () operator, turns the PayOff into an abstract
// function object
virtual double operator(const double& S) const = 0;
};
class PayOffCall : public PayOff {
2
private:
double K; // Strike price
public:
PayOffCall(const double& K_);
virtual ~PayOffCall() {};
// Virtual function is now over-ridden
// (not pure-virtual anymore)
virtual double operator(const double& S) const;
};
class PayOffPut : public PayOff {
private:
double K; // Strike
public:
PayOffPut(const double& K_);
virtual ~PayOffPut() {};
virtual double operator(const double& S) const;
};
#endif
For this assignment, you need to define a base pure abstract
3
class calledAsianOption and a derived class calledAsianOptionGeometric
which implements the PayOff operator according to formula (1).
#ifndef __Asian_Option__
#define __Asian_Option__
#include
#include "payoff.h"
class AsianOption {
protected:
PayOff* pay_off; // Pay-off class (in this instance call or put)
public:
AsianOption(PayOff* _pay_off);
virtual ~AsianOption() {};
// Pure virtual pay-off operator (this will determine arithmetic or geometric)
virtual double OptionPayOff(const std::vector& spot_prices) const = 0;
};
class AsianOptionGeometric : public AsianOption {
public:
AsianOptionGeometric(PayOff* _pay_off);
4
virtual ~AsianOptionGeometric() {};
// Overide the pure virtual function to produce geometric Asian Options
virtual double OptionPayOff(const std::vector& spot_prices) const;
};
#endif
Please add a function in the Random.h and Random.cpp files
which generates a Geometric Brownian Motion path according
to the following formula:
S(ti) = S(tt−1)exp[(r − 1
2
σ2∆t) + σ
√
∆t] (2)
The price path can be generated recursively from S0 from above
equation (2). The function prototype should follow the following
structure with an input argument std :: vector < double > &.
#ifndef __Option_Class__Random__
#define __Option_Class__Random__
double GetOneGaussianByBoxMuller();
void GetGBMSpotPricePath(
std::vector& spotPrices,
// Vector of spot prices to be filled in
const double& r,
5
// Risk free interest rate (constant)
const double& v,
// Volatility of underlying (constant)
const double& T
// Expiry
);
#endif /* defined(__Option_Class__Random__) */
You will also need to implement a SimpleMonteCarlo class
with a function to implement the simulation procedure:
#ifndef __Option_Class__SimpleMonteCarlo__
#define __Option_Class__SimpleMonteCarlo__
#include
#include "Vanilla1.h"
double SimpleMonteCarlo3(const AssianOption& TheOption,
double Spot,
double Vol,
double r,
unsigned long NumberOfPaths);
#endif /* defined(__Option_Class__SimpleMonteCarlo__) */
Overall, your Asian option pricer should include the following
6
files:
• Random.h
• Random.cpp
• PayOff.h
• PayOff.cpp
• AsianOption.h
• AsianOption.cpp
• SimpleMC.h
• SimpleMC.cpp
• AsianOptionMain.cpp
Please use the following parameter and generate Geometric Asian
Put and Call option prices:
T = 1
S(0) = 50
K = 50
σ = 0.30
r = 0.05
For each path you will use 250 intervals, and please run the
simulation for 1000 times to calculate expected option prices.
7
Homework Honor Policy: You are allowed to discuss the prob-
lems between yourselves, but once you begin writing up your
solution, you must do so independently, and cannot show one
another any parts of your written solutions.

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