Lattice Model Financial Instrument Pricing Using C++

[Michael CQF]

from the book it is using below



double dt = T/N;
double nu = r - 0.5*sig*sig;

// Up and down jumps
double dxu = std::sqrt(sig*sig*dt + (nu*dt)*(nu*dt));
double dxd = -dxu;
// Corresponding probabilities
double pu = 0.5 + 0.5*(nu*dt/dxu);
double pd = 1.0 - pu;
// Precompute constants
double disc = std::exp(-r*dt);
double dpu = disc*pu;
double dpd = disc*pd;
double edxud = std::exp(dxu - dxd);
double edxd = std::exp(dxd);

Whick look different than the common one(which we learn in CFA

Lattice model (finance) - Wikipedia

en.wikipedia.org

Wonder if they are same model, but just different presentation,
or above one is move advance?

 

[Daniel Duffy]

Looks like code from one of my books?

Which problem are you working on, exactly?

 

[Michael CQF]

Looks like code from one of my books?

Which problem are you working on, exactly?

not exact problem, but just get surprised above "u" are different than what we learned learned in CFA or even CQF

Binomial options pricing model - Wikipedia

en.wikipedia.org

[math]u=e^{\sigma {\sqrt {\Delta }}t}[/math]
[math]d=e^{-\sigma {\sqrt {\Delta }}t}={\frac {1}{u}}.[/math]
[math]p={\frac {e^{(r-q)\Delta t}-d}{u-d}}[/math]
Are this a different model?

 

[Daniel Duffy]

but just get surprised above "u" are different than what we learned learned in CFA or even CQF
LOL
what do those guys use?

u is up, d is down

 

[_quanty_]

@Daniel Duffy I really don't understand the need for a rude response. He's just asking a question.

@Michael CQF What format of u have you seen in the past?

 

[Daniel Duffy]

@Daniel Duffy I really don't understand the need for a rude response. He's just asking a question.

which part of my response was "rude"?

 

[Daniel Duffy]

not exact problem, but just get surprised above "u" are different than what we learned learned in CFA or even CQF

Binomial options pricing model - Wikipedia

en.wikipedia.org

[math]u=e^{\sigma {\sqrt {\Delta }}t}[/math]
[math]d=e^{-\sigma {\sqrt {\Delta }}t}={\frac {1}{u}}.[/math]
[math]p={\frac {e^{(r-q)\Delta t}-d}{u-d}}[/math]
Are this a different model?

There are quite a few models for u, d and p:

CRR, JR, TRG, EQP etc.

 

[Daniel Duffy]

from the book it is using below



double dt = T/N;
double nu = r - 0.5*sig*sig;

// Up and down jumps
double dxu = std::sqrt(sig*sig*dt + (nu*dt)*(nu*dt));
double dxd = -dxu;
// Corresponding probabilities
double pu = 0.5 + 0.5*(nu*dt/dxu);
double pd = 1.0 - pu;
// Precompute constants
double disc = std::exp(-r*dt);
double dpu = disc*pu;
double dpd = disc*pd;
double edxud = std::exp(dxu - dxd);
double edxd = std::exp(dxd);

Whick look different than the common one(which we learn in CFA

Lattice model (finance) - Wikipedia

en.wikipedia.org

Wonder if they are same model, but just different presentation,
or above one is move advance?

This code is from my 2018 C++ book for barrier options!

On a pedagogical advice to everyone , learn the maths of the binomial method and program it yourselves instead of copying from all kinds of (possibly dodgy) sites.
My two cents..

 

[Michael CQF]

th

This code is from my 2018 C++ book for barrier options!

On a pedagogical advice to everyone , learn the maths of the binomial method and program it yourselves instead of copying from all kinds of (possibly dodgy) sites.
My two cents..

the question is, why Barrier option need to have different formula for u, compared to the generic one shared in wiki?

 

[Michael CQF]

I think I understand now, from wiki

[math]u=e^{\sigma {\sqrt {\Delta }}t}[/math][math]d=e^{-\sigma {\sqrt {\Delta }}t}={\frac {1}{u}}[/math]Above is the original Cox, Ross, & Rubinstein (CRR) method; there are various other techniques for generating the lattice, such as "the equal probabilities" tree, see.[4][5]

So the one mentioned in code book is "other technique", wondering if we have a list of those way how lattice is generating

 

[Daniel Duffy]

I think I understand now, from wiki

[math]u=e^{\sigma {\sqrt {\Delta }}t}[/math][math]d=e^{-\sigma {\sqrt {\Delta }}t}={\frac {1}{u}}[/math]Above is the original Cox, Ross, & Rubinstein (CRR) method; there are various other techniques for generating the lattice, such as "the equal probabilities" tree, see.[4][5]

So the one mentioned in code book is "other technique", wondering if we have a list of those way how lattice is generating

Is your question "what are the other methods?"

I programmed those methods somewhere I vaguely remember.

 

[Daniel Duffy]

Do you mean?

BinomialLatticeStrategy* getStrategy(double sig, double r, double k,
                                        double S, double K, int N)
{
    cout << "\n1. CRR, 2. JR, 3. TRG, 4. EQP, 5. Modified CRR:\n6. Cayley JR Transform: 7 Cayley CRR: ";
    int choice;
    cin >> choice;

    if (choice == 1)
        return new CRRStrategy(sig,r,k);

    if (choice == 2)
        return new JRStrategy(sig,r,k);

    if (choice == 3)
        return new TRGStrategy(sig,r,k);

    if (choice == 4)
        return new EQPStrategy(sig,r,k);

    if (choice == 5)
        return new ModCRRStrategy(sig,r,k, S, K, N);

    if (choice == 6)
        return new PadeJRStrategy(sig,r,k);

    if (choice == 7)
        return new PadeCRRStrategy(sig,r,k);

}

 

[Michael CQF]

Do you mean?

BinomialLatticeStrategy* getStrategy(double sig, double r, double k,
                                        double S, double K, int N)
{
    cout << "\n1. CRR, 2. JR, 3. TRG, 4. EQP, 5. Modified CRR:\n6. Cayley JR Transform: 7 Cayley CRR: ";
    int choice;
    cin >> choice;

    if (choice == 1)
        return new CRRStrategy(sig,r,k);

    if (choice == 2)
        return new JRStrategy(sig,r,k);

    if (choice == 3)
        return new TRGStrategy(sig,r,k);

    if (choice == 4)
        return new EQPStrategy(sig,r,k);

    if (choice == 5)
        return new ModCRRStrategy(sig,r,k, S, K, N);

    if (choice == 6)
        return new PadeJRStrategy(sig,r,k);

    if (choice == 7)
        return new PadeCRRStrategy(sig,r,k);

}

Many thanks! LEt me study those model

 

[Michael CQF]

I think I understnad better, the u and pu can be different, as long as

pu+pd =1
u*pu +d*pd = r(risk free rate)

 

[Daniel Duffy]

I think I understnad better, the u and pu can be different, as long as

pu+pd =1
u*pu +d*pd = r(risk free rate)

All this stuff has been extensively documented.
I think having a look at the work of the late Mark Joshi is good

https://fbe.unimelb.edu.au/__data/assets/pdf_file/0011/2591732/MARK_JOSHI_SSRN-id2277854.pdf