We obtain a Hull and White type option price decomposition for a general local volatility model. We apply the obtained formula to CEV model. As an application we give an approximated closed formula for the call option price under a CEV model and an approximated short term implied volatility surface. These approximated formulas are used to estimate model parameters. Numerical comparison is performed for our new method with exact and approximated formulas existing in the literature.
In [
The model presented here assumes the volatility is a deterministic function of the underlying stock price, and therefore, there is only one source of randomness in the model. These models are sometimes called local volatility models in the industry and GARCHtype volatility models in financial econometrics. Recall that these models are different from the socalled stochastic volatility models, like Heston model, where the volatility process is driven by an additional source of randomness, not perfectly correlated with the stock price innovations.
As an application, for the particular case of CEV model, we obtain an approximation of the atthemoney (ATM) implied volatility curve as a function of time and an approximation of the implied volatility smile as a function of the logmoneyness, close to the expiry date. We use these approximations to calibrate the CEV model parameters.
Let
The following notation will be used in all the paper:
We define the BlackScholes function as a function of
where
Note that the price of a plain vanilla European call under the classical BlackScholes theory is
We will denote frequently by
We use in all the paper the notation
In our setting, the call option price is given by
Recall that from the FeynmanKac formula, the operator
satisfies
We define the operators
Then, for any
For any
Here we obtain a general abstract decomposition formula for a certain family of functionals of
Assume we have a functional of the form
Then we have the following lemma.
For all
Applying the Itô formula to process
Now, applying FeynmanKac formula for
On the other hand, using Itô calculus rules, it is easy to see that
Finally, substituting this expression in (
For the BlackScholes function previous lemma reduces to the following corollary.
For all
Applying Lemma
For clarity, in the following we will refer to terms of the previous decomposition as
In [
In this section we obtain an approximation formula to plain vanilla call price by approximating terms (I)–(IV). The main idea is to use again Lemma
For all
We apply Lemma
(I)
(II)
(III)
(IV)
The constant elasticity of variance (CEV) model is a diffusion process that solves the stochastic differential equation
There exists a closed form formula for call options; see [
Applying Corollary
For all
We will write
The exact formula can be difficult to use in practice, so we will use the following approximation.
For all
The proof is a direct consequence of applying Lemma
In this section, we compare our numerically approximated price of a CEV call option with the following different pricing methods:
The exact formula, see [
The Singular Perturbation Technique, see [
The results for a call option with parameters
Call option
Parameters  Exact formula  Approximation  HW  


Price  Price  Error  Price  Error 
0.25  0.2882882  0.2882884 

0.2882019 

1  1.0103060  1.0103070 

1.0100377 

2.5  2.4709883  2.4709894 

2.4708310 

5  4.8771276  4.8771278 

4.8771099 

The results in the case that
Call option
Parameters  Exact formula  Approximation  HW  


Price  Price  Error  Price  Error 
0.25  0.5356736  0.5356765 

0.5354323 

1  1.3886303  1.3886529 

1.3868801 

2.5  2.8506826  2.8507669 

2.8450032 

5  5.1658348  5.1660433 

5.1543092 

The results in the case that
Call option
Parameters  Exact formula  Approximation  HW  


Price  Price  Error  Price  Error 
0.25  1.3887209  1.3887438 

1.3883284 

1  3.0389972  3.0391797 

3.0359001 

2.5  5.2954739  5.2961870 

5.2835621 

5  8.2781049  8.2800813 

8.2459195 

Finally, the results in the case that
Call option
Parameters  Exact formula  Approximation  HW  


Price  Price  Error  Price  Error 
0.25  2.6404164  2.6404455 

2.6401025 

1  5.5191736  5.5194053 

5.5166821 

2.5  9.1446125  9.1455159 

9.1349142 

5  13.5553379  13.5578351 

13.5286009 

Note that the new approximation is more accurate than the approximation obtained in [
In Figure
Error surface between exact formula and our approximation for
We calculate also the speed time of execution (in seconds) of every method running the function timeit of Matlab 1.000 times. The computer used is an Intel Core i7 CPU Q740 @1.73 GHz 1.73 GHz with 4 GB of RAM with a Windows 10 (×64). The results are presented in Table
Call option
Measure  Exact formula  Approximation  HW 

Average 



Standard deviation 



Max 



Min 



We observe that singular perturbation method is the fastest method to calculate the price of CEV call option. The method developed in this work is a little more expensive in computation time. But to compute the exact price is much more expensive than any of the other two methods. Note that, in our method, we also are able to calculate at the same time the price and the Gamma of the lognormal price.
In the above section we have computed a bound for the error between the exact and the approximated pricing formulas for the CEV model. Now, we are going to derive an approximated implied volatility surface of second order in the logmoneyness. This approximated implied volatility surface can help us to understand better the volatility dynamics. Moreover we obtain an approximation of the ATM implied volatility dynamics.
In this section, for simplicity and without losing generality, we assume
Using the results from the previous section, we are going to derive an approximation to the implied volatility as in [
Let
On the other hand we can consider the Taylor expansion of
Then, equating this expression to
Note that
Note that the pricing formula has an error of
We calculate now the short time behavior of the approximated implied volatility
For
Note that
Note that (
Note that, in stochastic volatility models, the implied volatility depends homogeneously on the pair
The behavior of the approximated implied volatility when the option is ATM is easy to obtain:
In this section, we compare numerically our approximated implied volatilities with implied volatility computed from call option prices calculated with the exact formula and with the ones obtained using the following formula obtained in [
In Figure
Comparative of implied volatility approximations for
Comparing the ATM volatility structure, we have the following graphics.
In Figure
Comparative of ATM implied volatility approximations for
Now, we put the implied volatility approximation found in (
Call option
Parameters  Exact formula  BS with implied volatility ( 
HW  


Price  Price  Error  Price  Error 
0.25  0.2882882  0.2882882 

0.28820185 

1  1.0103060  1.0103057 

1.010037675 

2.5  2.4709883  2.4709880 

2.470830954 

5  4.8771276  4.8771275 

4.877109923 

The results in the case that
Call option
Parameters  Exact formula  BS with implied volatility ( 
HW  


Price  Price  Error  Price  Error 
0.25  0.5356736  0.5356732 



1  1.3886303  1.3886267 



2.5  2.8506826  2.8506672 



5  5.1658348  5.1657911 



The results in the case that
Call option
Parameters  Exact formula  BS with implied volatility ( 
HW  


Price  Price  Error  Price  Error 
0.25  1.3887209  1.3887176 



1  3.0389972  3.0389707 



2.5  5.2954739  5.2953686 



5  8.2781049  8.2778040 



And the results in the case that
Call option
Parameters  Exact formula  BS with implied volatility ( 
HW  


Price  Price  Error  Price  Error 
0.25  2.6404164 




1  5.5191736 




2.5  9.1446125 




5  13.5553379 




Our approximation is better than Hagan and Woodward one.
We compare also execution times (see Table
Call option
Measure  HW  BS with implied volatility ( 

Average 


Standard deviation 


Max 


Min 


We can observe that both formulas are similar in computation time, with the new approximation formula being a bit faster.
Following the ideas of [
Using a set of options with the same maturity and the parameters
Using the same procedure, for
Using the same procedure, for
Using the same procedure, for
Using a set of ATM options with the same maturity and parameters
Using the same procedure, for
Using the same procedure, for
Using the same procedure, for
We have seen that to do a quadratic regression is enough to recover a good approximation of the parameters.
In this paper, we notice that ideas developed in [
In the following appendices we obtain the error terms of the decomposition in Theorem
The term (I) can be decomposed by
The term (II) can be decomposed by
The term (III) can be decomposed by
The term (IV) can be decomposed by
The authors declare that there are no conflicts of interest regarding the publication of this article.
Josep Vives was partially supported by Grant MEC MTM 2013 40782 P.