Newbie here, please. I took an online course to learn European call option pricing using Fourier Transform, and attach my notes here to provide context and ask a question pertaining to the notes. Please refer to page 5 specifically where the Fubini rule is applied to re-arrange the inner and the outer integrals, and ultimately show that the formula for the modified call can be represented as some multiple of the characteristic function of the logarithm of the stock price.
One of the things that isn't explained on the lecture is how the inner integral changes from being k through infinity to negative infinity through s. And that's all still for the call option pricing. I would like to understand how this works for put option pricing. What would the inner integral change to? Ultimately, the goal is to represent the function for the modified call as some multiple of the characteristic function.
I have tried 3 different approaches to analytically address the inner integral, and have attached a PDF showing those 3 alternative approaches on 3 pages. The first one sets the inner integral from negative infinity through s, the second sets it from 0 through s, and the final one sets it from negative infinity to zero. However, one of those cases arrives at a form that looks like a multiple of the characteristic function.
So, I'm back to the drawing board and wondering how the integrals on Page 5 need to be modified and what limits they should pertain to, in order to price a put option instead of a call option.
Most grateful for anyone's guidance.
Kind Regards,
SK.
Serving the Quantitative Finance Community
Put Option Pricing using Fourier Transform
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- Possible Approaches.pdf
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- Option Pricing Notes - SK.pdf
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Re: Put Option Pricing using Fourier Transform
Typo above: Was meant to be
"However, none of those cases arrives at a form that looks like a multiple of the characteristic function."
"However, none of those cases arrives at a form that looks like a multiple of the characteristic function."
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- Cuchulainn
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Re: Put Option Pricing using Fourier Transform
I am trying to decipher these notes (does prof not like LATEX). and not having numbered equations doesn't help..
what I don't get is \[ \int_{a}^{s} f(s) \,ds \]
s a upper limit is a bit weird. Apologies if I misread.
yellow paper, US?
what I don't get is \[ \int_{a}^{s} f(s) \,ds \]
s a upper limit is a bit weird. Apologies if I misread.
yellow paper, US?
Re: Put Option Pricing using Fourier Transform
Hi Cuchulainn,
Thanks for your response. This isn't from a paper but from an online course offered by the Columbia University. Please see https://www.coursera.org/learn/financia ... thods/home
Thanks for your response. This isn't from a paper but from an online course offered by the Columbia University. Please see https://www.coursera.org/learn/financia ... thods/home
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- Cuchulainn
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Re: Put Option Pricing using Fourier Transform
I see. They are asking me all kinds of access stuff. Is there a pdf.Hi Cuchulainn,
Thanks for your response. This isn't from a paper but from an online course offered by the Columbia University. Please see https://www.coursera.org/learn/financia ... thods/home
Does Coursera use LATEX?
Re: Put Option Pricing using Fourier Transform
Thanks Cuchulainn -- Are you looking for a pdf of the lecture notes, please? My notes are more-or-less that.
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- Cuchulainn
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Re: Put Option Pricing using Fourier Transform
Is your problem the same as Carr and Madan?
http://faculty.baruch.cuny.edu/lwu/890/CarrMadan99.pdf
http://faculty.baruch.cuny.edu/lwu/890/CarrMadan99.pdf
Re: Put Option Pricing using Fourier Transform
Exactly.Is your problem the same as Carr and Madan?
http://faculty.baruch.cuny.edu/lwu/890/CarrMadan99.pdf
There are a couple of different approaches to option pricing by Fourier Transform. One is the Carr-Madan approach; another one is mine. The course is trying to teach you the first one.
If you get stuck, you might try mine, which is found here and also here. The two approaches are mathematically equivalent but proceed somewhat differently. Mine makes us of the generalized Fourier Transform, which is an extension of the ordinary FT to the complex plane. Option payoff functions, like puts and calls, have very simple generalized transforms w.r.t. the log-stock price: see Table 3.1 in the first link.
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- Cuchulainn
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Re: Put Option Pricing using Fourier Transform
As a follow-on, I would be curious to see the contents of that Coursera course.
Re: Put Option Pricing using Fourier Transform
Thanks Cuchulainn and Alan.
Alan -- I checked out your paper which is fairly intense
Cuchulainn -- I checked out the paper by Carr and Madan and you're right, the online course teaches exactly that! That said, that paper doesn't appear to cover put options specifically. Please see my attached one-page notes for calls. What would the formula for puts?
Alan -- I checked out your paper which is fairly intense
Cuchulainn -- I checked out the paper by Carr and Madan and you're right, the online course teaches exactly that! That said, that paper doesn't appear to cover put options specifically. Please see my attached one-page notes for calls. What would the formula for puts?
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Re: Put Option Pricing using Fourier Transform
I suppose you could do puts from 1st principles in the same way, or use put-call parity.
I'm sure Alan knows if this is feasible.
I'm sure Alan knows if this is feasible.
Re: Put Option Pricing using Fourier Transform
Indeed -- and I did use put-call parity prior to posting here. However, the surface of put premia modeled using the parity isn't anywhere close to the market surface. For calls however (priced using the formulae mentioned in the attachments above), the modeled surface very closely resembles the market surface of call premia.
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Re: Put Option Pricing using Fourier Transform
If you are using the Fourier formulas for the Black-Scholes model to extract market implied volatilities, this makes little sense to me. Just use the conventional BS formulas. Typically, this is done using just out-of-the-money quotes, as these are more liquid, so you switch from puts to calls. A good procedure is to practice on SPX options; use the VIX white paper method to first extract cost-of-carry parameters. Get some data from the CBOE DataShop.
Separately, if you really want to convince yourself that some putative Fourier version of the Black-Scholes formula is equivalent to the conventional formula, this is entirely a theoretical exercise. It neither requires -- nor is helped by -- market data. Since the conventional formulas satisfy put-call parity, so will the Fourier formulas.
My two cents.
Separately, if you really want to convince yourself that some putative Fourier version of the Black-Scholes formula is equivalent to the conventional formula, this is entirely a theoretical exercise. It neither requires -- nor is helped by -- market data. Since the conventional formulas satisfy put-call parity, so will the Fourier formulas.
My two cents.