is that Ito's Lemma principle?
is that Ito's Lemma principle?
I don't think so Goose,
my interperetation of Ito is that if we know the underlying behaves according to a stochastic differential,
dS = u.S.dt + @.S.dX
[where u = Drift rate, and @ = variance, most quant finance is based on this assumption]
we can use Ito to find what the u part and the @ part are for an option.
It's known as a Generalised Weiner Process, basically we are saying that the option price must follow a similar stochastic differential as the underlying, except that the drift rate and variance are a function of the underlying variable and time. Ito is a tool that lets us determine what they are, its not finance specific.
my interperetation of Ito is that if we know the underlying behaves according to a stochastic differential,
dS = u.S.dt + @.S.dX
[where u = Drift rate, and @ = variance, most quant finance is based on this assumption]
we can use Ito to find what the u part and the @ part are for an option.
It's known as a Generalised Weiner Process, basically we are saying that the option price must follow a similar stochastic differential as the underlying, except that the drift rate and variance are a function of the underlying variable and time. Ito is a tool that lets us determine what they are, its not finance specific.
Ha. No, it's a book by Prof Ricardo Rebonato and is effectively th bible when it comes to risk.
The best one is called Volatility and Correlation: the Perfect hedger and the fox.
if you're a quant i'd say it was essential reading. (to qualify i'm an options trader and by no means a quant but i am brushing up on the nuances of option pricing and hedging)
The best one is called Volatility and Correlation: the Perfect hedger and the fox.
if you're a quant i'd say it was essential reading. (to qualify i'm an options trader and by no means a quant but i am brushing up on the nuances of option pricing and hedging)
Not come across Rebonato before...
if the Stochastics are rusty, I've heard that Neftci (summit like that) is v. good. And if P.D.E's blow your skirt up, "thermodynamics" by Enrico Fermi is pretty interesting, along with "Vibrations and Waves" by A.P. French. Both are physics not finance though.
Anyway, I've highlighted where Ito appears in the derivation of the BS equation - it's very similar to a taylor series expansion, but makes a subtle change dX^2 = dt.
I will re-scan it in better quality if it's tricky
if the Stochastics are rusty, I've heard that Neftci (summit like that) is v. good. And if P.D.E's blow your skirt up, "thermodynamics" by Enrico Fermi is pretty interesting, along with "Vibrations and Waves" by A.P. French. Both are physics not finance though.
Anyway, I've highlighted where Ito appears in the derivation of the BS equation - it's very similar to a taylor series expansion, but makes a subtle change dX^2 = dt.
I will re-scan it in better quality if it's tricky
That's great Mr G-thanks for that. I decided to actually look at the minutae of the derivation of the pricing models I'm using......as I said I thought I'd left that sort of maths back in my degree but I guess it's coming in handy after all!
Cheers!
Goose,
getting up to where I finish is actually pretty simple; taking the Equation and turning it into the formula for Puts and Calls is a bit trickier, with Heaviside and Dirac Delta functions
- then you've got to model the vol. w/ GARCH, Newton-Raphson etc...However, as I come from a Physics bckground, we have a unique solution to the problem:
"for all d^3X/dX^3 and above... f*ck ém, they are too small".
Goose,
getting up to where I finish is actually pretty simple; taking the Equation and turning it into the formula for Puts and Calls is a bit trickier, with Heaviside and Dirac Delta functions
However, as I come from a Physics bckground, we have a unique solution to the problem:
"for all d^3X/dX^3 and above... f*ck ém, they are too small".
So it's fair to say I'm going to have to get my nut down!
it's a similar solution for engineers Mr G
a sort of related question? to be a market-maker or indeed to trade options proprietarily does one need to really know this level of math? do grad programs at marketmakers like mako/liquid teach you this kinda option pricing methods or focus on other more practical aspects of it?
as for the normal distribution not accounting for essentially fat tails properly, can we not use a more suitable distribution? like chi squared etc.? i remember coming across various distributions in my actuarial studies that focussed on fat tails to price excess of loss reinsurance but dont remember the details.
as for the normal distribution not accounting for essentially fat tails properly, can we not use a more suitable distribution? like chi squared etc.? i remember coming across various distributions in my actuarial studies that focussed on fat tails to price excess of loss reinsurance but dont remember the details.
Rhody Trader
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I've just started reading Mandelbrot's The (Mis)Behavior of Markets. As I understand it, he was one of the first (back in the 60s) to point out that the distribution of price changes didn't fit the standard distribution, that they have fat tails. When I get to the chapter where he explains things, I'll share.
essentially, forex market makers merely use the BSM to convert the price of options (quoted in vols) to a price that they can input in terms of value (dollars and cents or the variants). Mostly, we will tweak the vol smile instead. The general rule is to keep it as simple as possible, not every market maker is a quant but that didnt stop us from managing our option books well. then, how we estimate the fair vol smile will depend on which market we are in and the different characteristics of the markets. In pricing fairly our vol smile, we account for the skewness and kurtosis (commonly understood as the smile bias and the fat tails).
Taleb is largely an opinionated person, but, no doubt, his book "Dynamic Hedging" is still the best in the field for options. The deep math only starts to bug when we deal with exotic options. Haug is also a largely respected practioner. It is also true that BSM is not that original, option traders know how to price/trade using a similar function (i rather call the BSM a function instead of a model...), probably started from Bachelier (who first discussed the use of Stochastic Processes to price options) and Edward Thorpe (who didnt publish his works as he was using them for trading in his firm). Their differences from Black and Scholes is in how they derive the end function and their assumptions.
Having say all these, the leading practioners in option trading typically resides in fx markets. While there are options in the equities and fixed income markets, the development on pricing vol smiles still dominates in the fx world given the depth of the vanillas and exotic options markets in its world.
Taleb is largely an opinionated person, but, no doubt, his book "Dynamic Hedging" is still the best in the field for options. The deep math only starts to bug when we deal with exotic options. Haug is also a largely respected practioner. It is also true that BSM is not that original, option traders know how to price/trade using a similar function (i rather call the BSM a function instead of a model...), probably started from Bachelier (who first discussed the use of Stochastic Processes to price options) and Edward Thorpe (who didnt publish his works as he was using them for trading in his firm). Their differences from Black and Scholes is in how they derive the end function and their assumptions.
Having say all these, the leading practioners in option trading typically resides in fx markets. While there are options in the equities and fixed income markets, the development on pricing vol smiles still dominates in the fx world given the depth of the vanillas and exotic options markets in its world.
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Ft338,
Putting myself on the line, I don’t think this level of maths is necessary – it’s all computer generated (even if it’s only Excel). I would suggest knowledge of what affects the various measures – price, the greeks and implieds – and where your exposure lies are what is important. Leave the model tweaking to the quants.
I think I’ve just been excommunicated.
Can’t help re distributions. Anyone know of a better alternative?
Grant.
Putting myself on the line, I don’t think this level of maths is necessary – it’s all computer generated (even if it’s only Excel). I would suggest knowledge of what affects the various measures – price, the greeks and implieds – and where your exposure lies are what is important. Leave the model tweaking to the quants.
I think I’ve just been excommunicated.
Can’t help re distributions. Anyone know of a better alternative?
Grant.
FT,
I'm not a MM, so I can't say with any level of authority what maths is or isn't necessary. The one thing i will say, though, is that up to and including the BS equation (not the formulae', they are tricky), the math isn't anywhere near as complicated as it looks. You mention you have an actuarial history; If you can handle the math's there, you can certainly get to grips with the BS equation. It looks very flash, but it's quite straightforward if you keep all the terms in their contexts.
Co-incidentally, I have done some casual consultancy for insurance companies (offshore entities in Gib and South America on how else they can consider their risks and to what extent they should look to re-insure through lloyds etc...). Basically I explained to them how option pricing theory could be used to calculate premiums, and even simpler stuff like VaR. If an insurance salesman can get it, how hard can it be??
To answer your original question; I don't know if you need it or not - but its not that difficult anyway, it just looks it.
I'm not a MM, so I can't say with any level of authority what maths is or isn't necessary. The one thing i will say, though, is that up to and including the BS equation (not the formulae', they are tricky), the math isn't anywhere near as complicated as it looks. You mention you have an actuarial history; If you can handle the math's there, you can certainly get to grips with the BS equation. It looks very flash, but it's quite straightforward if you keep all the terms in their contexts.
Co-incidentally, I have done some casual consultancy for insurance companies (offshore entities in Gib and South America on how else they can consider their risks and to what extent they should look to re-insure through lloyds etc...). Basically I explained to them how option pricing theory could be used to calculate premiums, and even simpler stuff like VaR. If an insurance salesman can get it, how hard can it be??
To answer your original question; I don't know if you need it or not - but its not that difficult anyway, it just looks it.
TWI
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grantx has it spot on. In liquid markets the Important thing is to understand implied and historic vol relationship and also the theta, gamma and vega risk in your position and how thes changes and the relationship between these as price and implied vol moves. Once you understand this relationship I doubt you will will use BS as anything other than a function in excel or whatever other software you might be using.
If however you need to price optionality in illiquid markets or on a bespoke basis then you better get down to the number crunching, it can get very complex at this point and standard BS model is not adequate.
thanks for the info guys. its good to know that you dont have to be a quant to trade options successfully. i do understand the greeks okay and am learning more about them through the cfa but seems like i need to work on them more and learn the vol related stuff (skew, smile etc.). i guess il browse the net and get a copy of hull to help me with that.
MrGecko, i dont find BS complicated/diff to understand and actually find it quite useful to understand the impact of greeks. you're right i have come across even uglier math than that in my degree and after a while you get used to it though cant say i enjoyed it very much!
thats interesting, general insurers/reinsurers do use extreme value theory to calculate premiums and ascertain whether reinsurance is necessary. that market is becoming more quantitative and they are increasingly using actuaries though i guess that goes for most financial markets now. its the age of the math geeks!
MrGecko, i dont find BS complicated/diff to understand and actually find it quite useful to understand the impact of greeks. you're right i have come across even uglier math than that in my degree and after a while you get used to it though cant say i enjoyed it very much!
thats interesting, general insurers/reinsurers do use extreme value theory to calculate premiums and ascertain whether reinsurance is necessary. that market is becoming more quantitative and they are increasingly using actuaries though i guess that goes for most financial markets now. its the age of the math geeks!
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