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Black-Scholes theoretical value

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Jack Jones-2

Black-Scholes theoretical value

I'd like to calculate the Black-Scholes theoretical value of an option.
I'm new to QuantLib and kind-of new to quantitative finance. I'm using
EquityOption.cpp as a base for my code.

Here is the relevant setup (questions at bottom):

//==========================

// Underlying price
shared_ptr<SimpleQuote>
    underlyingPrice(new SimpleQuote(stockPrice));

// Dividend term structure
shared_ptr<YieldTermStructure>
    dividendTS(new FlatForward(expiryDate,
                   dividend,
                   Actual360()));

// Interest rate term structure
shared_ptr<YieldTermStructure>
    interestTS(new FlatForward(expiryDate,
                   interest,
                   Actual360()));

// Historical volatility term structure
shared_ptr<BlackVolTermStructure>
    volatilityTS(new BlackConstantVol(expiryDate,
                      historicalVolitility,
                      Actual360()));

shared_ptr<StrikedTypePayoff>
    payoff(new PlainVanillaPayoff(type, strikePrice));

shared_ptr<Exercise>
    exercise(new AmericanExercise(Date::todaysDate(), expiryDate));

shared_ptr<QuantLib::GeneralizedBlackScholesProcess>
    stochasticProcess(new
   QuantLib::GeneralizedBlackScholesProcess(
        Handle<Quote>(underlyingPrice),
        Handle<YieldTermStructure>(dividendTS),
        Handle<YieldTermStructure>(interestTS),
        Handle<BlackVolTermStructure>(volatilityTS)));

shared_ptr<PricingEngine>
    engine(new BinomialVanillaEngine<CoxRossRubinstein>(150));

VanillaOption option(stochasticProcess, payoff, exercise, engine);

//==========================

I have two questions:

1. what does the "price" argument to option.impliedVolatility()
   represent?

2. now that I have this object infrastructure built up, what is the
   actual method call to get the theoretical value of the option?

Thanks for your help!

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Luigi Ballabio

Re: Black-Scholes theoretical value

On Wed, 2007-05-09 at 16:44 -0400, Jack Jones wrote:

> I'd like to calculate the Black-Scholes theoretical value of an
> option.
>
> Here is the relevant setup (questions at bottom):
>
> //==========================
>
> // Underlying price
> shared_ptr<SimpleQuote>
> underlyingPrice(new SimpleQuote(stockPrice));
>
> // Dividend term structure
> shared_ptr<YieldTermStructure>
> dividendTS(new FlatForward(expiryDate,
> dividend,
> Actual360()));

Jia.Li

Re: Black-Scholes theoretical value

> "or a couple of days later if settlement days should be considered"

Could you give an example to illustrate this phrase? What would the
discount be between today and the day "a couple of days later"? Thanks in
advance.

Jia

Luigi Ballabio
<luigi.ballabio@g
mail.com> To
Sent by: Jack Jones
quantlib-users-bo <[hidden email]>
[hidden email] cc
eforge.net [hidden email]
t
Subject
10/05/2007 08:10 Re: [Quantlib-users] Black-Scholes
theoretical value

On Wed, 2007-05-09 at 16:44 -0400, Jack Jones wrote:

This is not correct. The first argument to FlatForward should be the
evaluation date, or rather the date for which the discount equals 1.
This might be today's date, or a couple of days later if settlement days
should be considered. The same applies to the interest-rate and
volatility term structures. The rest of the setup is correct.

> I have two questions:
>
> 1. what does the "price" argument to option.impliedVolatility()
> represent?

Given an option, and once you have fixed dividend yield and risk-free
rate, you can find the price for a given volatility, or the other way
around. impliedVolatility(price) is the latter calculation; given a
target price, it returns the value of the volatility which causes the
option to have such a price.

> 2. now that I have this object infrastructure built up, what is the
> actual method call to get the theoretical value of the option?

option.NPV()

Later,
Luigi

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-- Mark Twain

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Jack Jones-2

Re: Black-Scholes theoretical value

In reply to this post by Luigi Ballabio

Thanks Luigi, that is very helpful! A follow-up questions:

On 5/10/07, Luigi Ballabio <[hidden email]> wrote:

> // Dividend term structure
> shared_ptr<YieldTermStructure>
>     dividendTS(new FlatForward(expiryDate,
>                    dividend,
>                    Actual360()));

This is not correct. The first argument to FlatForward should be the
evaluation date, or rather the date for which the discount equals 1.
This might be today's date, or a couple of days later if settlement days
should be considered. The same applies to the interest-rate and
volatility term structures. The rest of the setup is correct.

Okay, I had assumed referenceDate was relative to QuantLib::Settings::instance().evaluationDate(). I suppose the expiry is taken into account through the exercise?

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Luigi Ballabio

Re: Black-Scholes theoretical value

On Thu, 2007-05-10 at 11:05 -0400, Jack Jones wrote:
> Thanks Luigi, that is very helpful! A follow-up questions:
> [...]
> I had assumed referenceDate was relative to
> QuantLib::Settings::instance().evaluationDate(). I suppose the expiry
> is taken into account through the exercise?

Yes, it is.

Later,
Luigi

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-- Paul Graham

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Luigi Ballabio

Re: Black-Scholes theoretical value

In reply to this post by Jia.Li

On Thu, 2007-05-10 at 09:19 +0100, [hidden email] wrote:
> > "or a couple of days later if settlement days should be considered"
>
> Could you give an example to illustrate this phrase?

Yes, I wasn't very clear. And think of it, it's likely that what I wrote
didn't apply even apply to this case...

I wrote out of habit, having worked mostly in interest-rate derivatives.
For such instruments (e.g., deposits or swaps) it is usually the case
(at least in Euroland) that all instruments have a couple of days of
settlement. Since all "today's" payments will actually occur two
business days from today, it is sometimes the custom that such two days
are skipped entirely, and the discount factor is set to 1.0 at the
settlement date---two business day from today's date. In this setting,
there's no discount at all between today and the settlement date, since
those days are modeled out of existence.

But for equity options, the whole thing is probably moot and the
reference date should probably be today's date. I'll leave the answer to
someone more experienced...

Later,
Luigi

----------------------------------------

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For every action, there is an equal and opposite criticism.

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