SERVING THE QUANTITATIVE FINANCE COMMUNITY

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Dupire equation derivation

Here I would like first to make conclusions on above discussions and add a new question to derivation of the Dupire equation itself.
I.
a)     Technically derivation looks good and the essence of the concept can be summarized as following. Observations over options data lead us to the fact that volatility coefficient of the underlying of the BS formula do not correspond to volatility coefficient of the underlying security. The volatility of the underlying that corresponds to BS formula is called implied volatility, IV.
My point is that Market prices options not fully based on no arbitrage principle. In particular, chances to make or lose money also effect on derivatives pricing. Such chances do not represented by BS pricing concept.
b)     Local volatility equation derived with the help of Dupire equation is aimed to get explicit formula for implied volatility. We cannot mix two different probabilistic representations of the BSE solution C ( t , S ; T , K ) in two coordinate space ( t , S ) and ( T , K ). First representation is BS formula for BSE solution and other is probabilistic representation of the Dupire equation solution. Thus there are three different issues: stock GBM , BS implied volatility, local volatility corresponding Dupire equation. We used stock process to get BSE in which option price is defined by the risk neutral process having the same volatility as stock process. Hence, dynamics of the call option on [ 0 , T ]  is defined by $C ( t , S ( t ) ; T , K ; \sigma_{S} )$. Now we replace risk neutral underlying process in definition $C ( t , x ; T , K ; \sigma_{iv} )$ by the implied volatility process. There is a formula that represents implied volatility by using local volatility
$\sigma^2 ( T , S ( T )) = F ( T , K ; C )$
On the other hand implied volatility is a function of time on [ 0 , T ] and Dupire representation  of the implied volatility is given only for the final moment T. Hence the attempt to present implied volatility by using local volatility does not look a complete representation.

II. Other problem is derivation of the Dupire equation derivation. For simplicity putting $r = \mu = 0$ and following Dupire’s original derivation we begin with BSE solution
$C ( t , x ; T , K ) = E max ( S ( T ; t , x ) – K , 0 )$
Differentiating twice with respect to K leads to the formula
$\frac{\partial^{2} C ( t , x ; T , K ) }{\partial K^{2} } \,=\, p ( t , x ; T , K )$
Here p is transition density of S ( T ; t , x ). Next general SDE
$dy ( t ) = b ( t , y ) dw ( t )$                                     (1)
with some diffusion coefficient b ( y , t ) and assuming that drift is zero is introduced. The forward Kolmogorov eq for density f ( t , x ; T , y ) of the process y ( t )
$\frac{\partial^{2} b^2 ( T , y )f ( t , x ; T , y ) }{\partial y^{2} } \,=\, \frac {\partial f ( t , x ; T , y )}{\partial T}$                  (2)
Next he puts y = K and is making changes
$f ( t , x ; T , K ) = p ( t , x ; T , K ) = \frac{\partial^{2} C ( t , x ; T , K ) }{\partial K^{2} }$
Here we should make a comment. We should note that p ( t , x ; T , K ) is the known density of the stock process which itself satisfies equation (2). If densities corresponding two SDEs are equal , ie f = p that implies that equations and their solutions should be identical, ie dS ( t ) = dy ( t )  and therefore
$b ( t , y ) = \sigma ( t ) y$
In such case derivation of the formula for volatility smile does not look make sense. Of course I could be wrong but I could not see my error.
Next, we can make transformations similar to Dupire but they did not lead us to $\sigma ( T , K )$ which presents volatility smile.

From my point of view the error in this derivation is based on the fact that assuming that densities p and  f are equal the fact that Kolmogorov equations for f and p should be equal too.

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: Dupire equation derivation

What does it formally mean implied volatility ?
If the constant $\sigma = \sigma_{BS}$ leads to BSE, which implies that $\sigma_{iv}$ is the true diffusion of the underlying of the $C_{BS}$ then whether the $\sigma_{iv}$ should replace the original constant $\sigma_{BS}$ in BSE? or it does not matter? , and the value of $C_BS$ does not change when we replace in BSE the constant sigma for its implied volatility counterpart.

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: Dupire equation derivation

I wish to refine my previous post question.
When we talk about BSE we mean that underlying of the BS formula is risk neutral random process, ie
$C_{BS} ( t , S ( t ) ; T , K ) = E \,exp – r ( T – t ) \,max ( S_r ( T ; t , S ( t )) , 0 )$            (1)
where
$d S_r ( t ) = \mu S ( t ) dt + \sigma S ( t ) d w ( t )$          (2)
Talking about implied volatility we mean that sigma of the underlying  (2) is replaced by implied volatility sigma, ie
$C_{BS} ( t , S ( t ) ; T , K ) = E \,exp – r ( T – t )\,max ( S_{IV} ( T ; t , S ( t )) , 0 )$
In such setting we should conclude that $C_{BS} ( t , S ; T , K )$ is the solution of the two different BSEs with different diffusion coefficients $\sigma\, S$ and  $\sigma_{IV}\, S$, ie

$\frac{\partial C_{BS} ( t , S ; T , K )}{\partial t } + ½ \sigma^2 S^2 \frac{\partial^2 C_{BS} ( t , S ; T , K )}{\partial S^ 2} + … = 0$

$\frac{\partial C_{BS} ( t , S ; T , K )}{\partial t } + ½ \sigma^2_{IV} ( t , S ) S^2 \frac{\partial^2 C_{BS} ( t , S ; T , K )}{\partial S^ 2} + … = 0$

with the same boundary condition
$C_{ BS} ( T , S ; T , K ) = max ( S – K , 0 )$
Whether or not it does true?

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: Dupire equation derivation

One, of course could start with approximation of the real stock by the model
$dS( t ) = \mu ( t , S ( t ) ) S ( t ) dt + \sigma ( t , S ( t )) S ( t ) dw ( t )$
but there is any guarantee that correspondent IV process
$dS_{IV} ( t ) = r ( t ) dt + \sigma_{IV} ( t , S_{IV} ( t )) S_{IV} ( t ) dw ( t )$
will be equal to risk neutral process specified by the eq
$dS_{rn}( t ) = r ( t ) dt + \sigma ( t , S_{rn} ( t )) S_{rn} ( t ) dw ( t )$
which appeared in correspondent BSE and again the question is the original BSE holds $\sigma ( t , S )$ and real underlying of the BSE solution should hold $\sigma _{IV} ( t , S )$ . Which sigma should be used in BSE?

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: Dupire equation derivation

list1 wrote:
One, of course could start with approximation of the real stock by the model
$dS( t ) = \mu ( t , S ( t ) ) S ( t ) dt + \sigma ( t , S ( t )) S ( t ) dw ( t )$
but there is any guarantee that correspondent IV process
$dS_{IV} ( t ) = r ( t ) dt + \sigma_{IV} ( t , S_{IV} ( t )) S_{IV} ( t ) dw ( t )$
will be equal to risk neutral process specified by the eq
$dS_{rn}( t ) = r ( t ) dt + \sigma ( t , S_{rn} ( t )) S_{rn} ( t ) dw ( t )$
which appeared in correspondent BSE and again the question is the original BSE holds $\sigma ( t , S )$ and real underlying of the BSE solution should hold $\sigma _{IV} ( t , S )$ . Which sigma should be used in BSE?

In order to answer to above question we should to verify BSE in which call price is assumed is constructed by IV. Thus we assumed that market prices of the call option are presented by BS model and its real underlying is $S_{IV} ( t )$. Fix a moment of time t and let S ( t ) = x. Then such assumptions implies that at a moment u > t the value of the call option is $C ( u , S( u ; t , x ))$ , where

$C ( t , x ) = E e^{ - r (T - t )} max ( S_{IV} ( T ; t , x ) - K , 0 )$

It is fully analogous t BS where risk neutral process for S exchanged for the risk neutral process for $S_{IV} ( t )$. Construct BS hedged portfolio. Then the change in the value at t is risk free

$d \Pi ( t ) = dC ( t , S ( t )) - C_{S}^{\prime} ( t , x ) dS ( t ) = [ C_{t}^{\prime} ( t , x ) dt + \frac {1}{2} \sigma ^{2} x^{2} C_{S S}^{\prime \prime} ( t , x ) ] dt$

and should be equal to the risk free portfolio change, ie $r \Pi ( t )$.  It leads us to expression that similar to left hand side of the BSE with stock sigma at diffusion term. Therefore we have one solution for the BSEs. One has stock diffusion coefficient and other has its IV diffusion. It can be confusing . It will not a problem if we assume that real stock on original probability space follows $S_{IV} ( t )$.
Actually it is impossible to think that implied volatility eq can be used to approximate original stock eq regardless whether it is GBM or it is more general nonlinear SDE as far as stock eq is defined only based on stock historical data without any connection to option data while IV eq is defined based on option data and BS model only.

frolloos
Posts: 1425
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: Dupire equation derivation

You know what: forget BS, forget Dupire, forget eveything, forget GBM or Ornstein-Uhlenbeck etc.

How would you price a derivative on an underlying process S?

consider the case first that S is a tradable quantity.

Any technique / idea is admissible. The only condition is that you don't want to look too stupid by giving a price that nobody wants to buy from you at, or that everybody wants to sell to you at.

outrun
Posts: 4488
Joined: April 29th, 2016, 1:40 pm

### Re: Dupire equation derivation

Texas shootout: you both scribble a price on a bit of paper, the highest bidder buys it at the lowest of the two price.

frolloos
Posts: 1425
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: Dupire equation derivation

outrun wrote:
Texas shootout: you both scribble a price on a bit of paper, the highest bidder buys it at the lowest of the two price.

42!

frolloos
Posts: 1425
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: Dupire equation derivation

I'm getting so bored with finance, vol, filtrations, Q and P, and with myself lately. Markets go up, markets go down. Calibrate here, calibrate there. Woohoo.
Last edited by frolloos on July 18th, 2017, 7:00 pm

outrun
Posts: 4488
Joined: April 29th, 2016, 1:40 pm

### Re: Dupire equation derivation

Go on...

outrun
Posts: 4488
Joined: April 29th, 2016, 1:40 pm

### Re: Dupire equation derivation

I'm sitting here..but the WiFi is good
Attachments

frolloos
Posts: 1425
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: Dupire equation derivation

That doesn't look bad at all - and I bet there's a bottle of wine or beer somewhere too! Enjoy

outrun
Posts: 4488
Joined: April 29th, 2016, 1:40 pm

### Re: Dupire equation derivation

Thanks! Even from a distance the daily struggles of list1 feel a bit like home.

Maybe you could start a project outside finance: a course, an open math problem? A price challenge?

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: Dupire equation derivation

frolloos wrote:
You know what: forget BS, forget Dupire, forget eveything, forget GBM or Ornstein-Uhlenbeck etc.

How would you price a derivative on an underlying process S?

consider the case first that S is a tradable quantity.

Any technique / idea is admissible. The only condition is that you don't want to look too stupid by giving a price that nobody wants to buy from you at, or that everybody wants to sell to you at.

Regarding question I think there is no universal pricing method as well as definition of the option premium. People on the market follow their individual plans and it is not clear whether or not say mean can be good estimate of the option premium. For example people who come to market to hedge their security are waiting for a particular price or are buying option for a particular strike. Other thinking about making money they are waiting other price of the options. We do not need that they form market supply and demand which in some way forms premium.
I think in order to present a model of premium without a lot of assumptions we should use an example that specifies next generalization. Such example is one period pricing with a finite number of stages. Ignoring outdoor market, ie looking at a stock, option, and might be risk free bond we have time t = 0 , 1 ; $S_j < S_{j+1}$ , j = 0, 1, ... N - 1. Such example can be then transform to $\Delta t = t_{i + 1} - t_j$ and continuous stage space. For me it is clear that there is no universal approach to specify option premium. In a simplest case when N = 2  we have binomial scheme and no arbitrage pricing approach to define option price. When N > 2 binomial scheme can not define option premium. Given distribution $P ( S_j < s )$ I think that more universal approach to option pricing is for example to study dynamics of the ratio of average loss to average profit, ie we assume that option premium is specified by this ratio and time.
Of course we should pay attention to all risk characteristics which would specify market risk of any choice which presents option premium whether it is no arbitrage pricing or other pricing strategy. In other words I think that dynamics of option premium do not fully or even primarily is specified by no arbitrage pricing.

list1
Topic Author
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: Dupire equation derivation

I enclosed my thoughts about implied-local volatility in a paper 'Stock, Implied, Local Volatilities and Black Scholes Pricing'. It can be found at
https://papers.ssrn.com/sol3/papers.cfm ... id=3011435
https://www.slideshare.net/list2do