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Early exercise premium method for pricing American options under the Jmodel
 Yacin Jerbi^{1}Email authorView ORCID ID profile
 Received: 22 February 2016
 Accepted: 28 November 2016
 Published: 7 December 2016
Abstract
Background
This study develops a new model called Jam for pricing American options and for determining the related early exercise boundary (EEB). This model is based on a closedform solution Jformula for pricing European options, defined in the study by Jerbi (Quantitative Finance, 15:2041–2052, 2015). The Jam pricing formula is a solution of the Black & Scholes (BS) PDE with an additional function called f as a second member and with limit conditions adapted to the American option context. The aforesaid function f represents the cash flows resulting from an early exercise of the option.
Methods
This study develops the theoretical formulas of the early exercise premium value related to three American option pricing models called Jam, BSam, and Hestonam models. These three models are based on the Jformula by Jerbi (Quantitative Finance, 15:2041–2052, 2015), BS model, and Heston (Rev Financ Stud, 6:327–343, 1993) model, respectively. This study performs a general algorithm leading to the EEB and to the American option price for the three models.
Results
After implementing the algorithms, we compare the three aforesaid models in terms of pricing and the EEB curve. In particular, we examine the equivalence between Jam and Hestonam as an extension of the equivalence studied by Jerbi (Quantitative Finance, 15:2041–2052, 2015). This equivalence is interesting since it can reduce a bidimensional model to an equivalent unidimensional model.
Conclusions
We deduce that our model Jam exactly fits the Hestonam one for certain parameters values to be optimized and that all the theoretical results conform with the empirical studies. The required CPU time to compute the solution is significantly less in the case of the Jam model compared with to the Hestonam model.
Keywords
 American option pricing
 Stochastic volatility model
 Early exercise boundary
 Early exercise premium
 Jlaw
 Jprocess
 Jformula
 Heston model
Background
The valuation of American options, while a challenge, is of interest to both academics and traders. American options are more common than their European counterparts; they allow more flexibility since they can be exercised at any time between the current time and maturity. They are presented as a compound option that includes a European option and an early exercise premium (EEP). Hence, their prices are higher than those of European options, and they are more complicated to be modeled. These prices are also significantly affected by the volatility level. The studies by Ju (1998) and Detemple and Rindisbacher (2005) regarding the American option pricing models are based on the Black and Scholes (1973) model and then cannot explain the reality of the financial markets, particularly the smile of volatility. Since the dynamics of volatility are fundamental in elaborating trading strategies for hedging and arbitrage, the pricing of the option under stochastic volatility model is important. The introduction of an additional stochastic volatility factor enormously complicates the pricing of the American options. Currently, this can be done only through numerical schemes, which involve solving integral equations as in the studies by Kim (1990), Huang et al. (1996), Sullivan (2000), Detemple and Tian (2002); performing Monte Carlo simulations as in the studies by Broadie and Glasserman (1997), Longstaff and Schwartz (2001), Rogers (2002), Haugh and Kogan (2004); or discretizing the partial differential equation as in the works of Brennan and Schwartz (1977), Clarke and Parrott (1999), and Ikonen and Toivanen (2007). The majority of the studies performed in this area are based on a numerical resolution of the problem using either parametric or nonparametric models. Currently, both academicians and practitioners need a general closedform solution as a theoretical reference for pricing American option models, considering volatility as stochastic. Such a model is useful for interpreting the results of other models. Moreover, it enables the understanding of the logic behind the option pricing process. The pricing of an American call or put has no explicit closedform solution. This is because the optimal boundary above which the American call will be exercised is unknown and is a part of the option price solution. Therefore, efforts have been focused on developing a numerical approximation scheme, allowing for a pricing of the American option that is more accurate as well as a faster one than the lattice or simulation based methods that are time consuming and computationally more demanding. These schemes are based on integral representations of the American option evaluation formula and exploit the PDE satisfied by the option price. Until now, several studies on European option pricing using stochastic volatility models that lead to a closedform solution have been conducted. These include studies by Hull and White (1987), Wiggins (1987), Stein and Stein (1991), Heston (1993), and Carr and Wu (2004). The most important study is that by Heston (1993), who considered volatility as stochastic to introduce the skewness and kurtosis effects and ensured that the model fits the reality of the financial market (“smile curve”). He considered two state variables: the underlying asset price and volatility. Jerbi (2015) developed a closed form solution Jformula based on the Jprocess considering the skewness and kurtosis effects as well as the smile effect. He showed that for the given parameter values related to the Heston volatility process, there exists a (λ*, θ*) that ensures the equivalence of the two models by minimizing the gap between the generated prices. He then matched the Heston results with a monodimensional model than with a bidimensional one. This study extends this equivalence to American options. The price of an American option is the sum of the related European option and the EEP called ε. If the European option value is a solution of a PDE, then the American option value is a solution of the same PDE with a second member equal to a function f instead of zero (PDEf). This function depends on the EEB (to be determined). It is the difference between the cash flows generated by the early exercise of the American option. In short, we can say that the European option price Ve is the solution for the homogeneous PDE and that the EEP ε is a particular solution for the PDEf. Hence, the American option price Va = Ve + ε constitutes the general solution of the PDEf. In Heston’s (1993) model, the aforesaid PDE is the Garman PDE (1976). In the case of the Jmodel, the PDE is the BS PDE, based on a dynamics of the underlying asset price under the Jprocess (please refer to Jerbi (2015)). The function f is the same for the two models. The EEP value ε is the sum of the cash flows expectancies between the current time and maturity discounted at the current time. Those cash flows are generated due to the early exercise of the option, and their calculation is based on the function f. In both cases, the solution of the PDEf is based on the optimal EEB as an optimal limit for exercising the American option. The pricing of the option relies on the determination of such a boundary. The Jprocess is an extension of the Wiener process, by considering the skewness and kurtosis effects. The use of the Jmodel instead of the Heston’s (1993), as an equivalent American option pricing model, makes the solution simpler and easier to interpret. Besides, it significantly improves the time consumed (CPU time) for a given accuracy.
This study first presents the expression of the Jformula elaborated by Jerbi (2015). We then elaborate (in detailed calculus) the expression of the EEP ε, related to Jam model. Second, we deduce the one related to the BS model by setting parameter λ equal to zero. Third, we deduce the expression of the EEP related to the Hestonam model, with a rate of dividend distribution equal to Q. Then, we detail the general methodology to determine, step by step, both the price of the American option and the limit values of the underlying asset belonging to the EEB. The determination of some points of the EEB is time consuming. In fact, these points are sufficient to determine the EEB, with good accuracy, in the form of a polynomial. This considerably reduces the time for calculating the price of an American option. This methodology can be applied to both the Jam and Hestonam models. In empirical studies, we compare the results of the two models in terms of accuracy and CPU time. This comparison helps examine the equivalence of the two models such as we can reduce the bidimensional model of Hestonam to a unidimensional model fitting the reality of the financial market, by including the skewness and kurtosis effects. This equivalence reduces the CPU time under an equivalent accuracy. After studying this equivalence, we focus on the Jam model to examine the effects of its inputs and parameters on the related option price.
Methods
Snell envelop and early exercise of option
For an American option, the EEB is defined by the curve S*(τ), representing the early exercise option limit value of the underlying asset at each time to maturity τ (please refer to Kwok (1998)). From this limit value, the early exercise of the option becomes interesting.
Let us denote the value of the American call by C, its strike price by K, and its early exercise limit value belonging to the EEB by \( {S}_c^{*}\left(\tau \right); \) this call will be exercised if we have \( S\left(\tau \right)={S}_c^{*}\left(\tau \right). \) Then, \( C\left({S}_c^{*}\left(\tau \right),\tau \right)={S}_c^{*}\left(\tau \right)K \) and \( {\left[\frac{\partial C}{\partial S}\right]}_{S={S}_c^{*}\left(\tau \right)}=1. \) If we have \( S\left(\tau \right)<{S}_c^{*}\left(\tau \right), \) the call will not be exercised.
However, if \( S\left(\tau \right)\ge {S}_c^{*}\left(\tau \right), \) the call will be exercised. We buy the underlying asset whose value is S (which generates a dividend with a rate Q) and disburse the amount K (which generates interest with a rate r). Hence, the early exercise of the call induces a cash flow QS − rK.
Let us denote the value of the American put by P and its Snell envelop by \( {S}_p^{*}\left(\tau \right), \) this put will be exercised if we have \( S\left(\tau \right)={S}_p^{*}\left(\tau \right). \) In this case, \( P\left({S}_p^{*}\left(\tau \right),\tau \right)=K{S}_p^{*}\left(\tau \right) \) and \( {\left[\frac{\partial P}{\partial S}\right]}_{S={S}_p^{*}\left(\tau \right)}=1 \).
The function f of the generated cashflow in case of the option early exercise or not
CALL  PUT  

S ≤ S*  f(S, τ) = 0  f(S, τ) = rK − QS 
S > S*  f(S, τ) = QS − rK  f(S, τ) = 0 
The related EEB is such that \( {S}_C^{*}\left(\tau, r,Q\right)*{S}_P^{*}\left(\tau, Q,r\right)={K}^2 \) (see Kwok (1998)).
American option pricing model based on the Jformula as a unidimensional approach

C1: The option value at maturity t = T is : P _{ J − am }(S _{ T }, T) = Max(K − S _{ T }; 0),

C2: \( {S}_t^{*} \) is the early exercise limit value at time t belonging to the boundary OEB, such as : \( {P}_{J am}\left({S}_t^{*},t\right)= Max\left(K{S}_t^{*};0\right), \)

C3: For S _{ t } = 0, we have P _{ J − am }(0, t) = K, and

C4:. \( {\left[\frac{\partial {P}_{J am}}{\partial S}\right]}_{S=\infty }=0 \)
Under the Jprocess, the price P _{ J − eur } is computed using the Jformula found by Jerbi (2015), and the formula of ε _{ J } is detailed, in the Appendix.
The Jformula as a solution for European option pricing under Jprocess
When the parameter λ equals zero, the Jformula is exactly the BS one.
The EEP value based on the Jformula
American option pricing model based on the Heston model as a bidimensional approach
Where : \( \left\{\begin{array}{l}{\phi}_{1,\left(\tau \omega \right)}^{(c)}={\phi}_1\left({S}_t{e}^{Q\omega },{v}_t,{S}_{\left(\tau \omega \right)}^{*},\omega, \kappa, \theta, \eta, \rho, \alpha \right)\\ {}{\phi}_{2,\left(\tau \omega \right)}^{(c)}={\phi}_2\left({S}_t{e}^{Q\omega },{v}_t,{S}_{\left(\tau \omega \right)}^{*},\omega, \kappa, \theta, \eta, \rho, \alpha \right).\end{array}\right. \)
Heston’s (1993) formula used in this study considers the dividends rate distribution Q.
Empirical studies
General methodology to determine the EEB and American option value
After determining the EEB, we can determine the American option price at the current time t in accordance with the formulas related to each of the three aforesaid models: BSam, Jam, and Hestonam, where BSam constitutes a particular case of Jam when λ equals to 0. For further work, we will consider both r and Q to equal 5%.
Results and discussion
Notations
Notation of functions = price difference between the Jmodel and the Heston model. Notation of their minimas and maximas on the moneyness space
Jmodel vs BS model  Hestonmodel vs BS model  

Function  Maxima  Minima  Function  Maxima  Minima 
Δ _{ J − eur } = P _{ J − eur } − P _{ BS − eur }  M _{ J − eur }  m _{ J − eur }  Δ _{ Heston − eur } = P _{ Heston − eur } − P _{ BS − eur }  M _{ Heston − eur }  m _{ Heston − eur } 
Δ _{ J − am } = P _{ J − am } − P _{ BS − am }  M _{ J − am }  m _{ J − am }  Δ _{ Heston − am } = P _{ Heston − am } − P _{ BS − am }  M _{ Heston − am }  m _{ Heston − am } 
ε _{ J } = P _{ J − am } − P _{ J − eur }  M _{ J }  m _{ J }  ε _{ Heston } = P _{ Heston − am } − P _{ Heston − eur }  M _{ Heston }  m _{ Heston } 
Δ _{ J − ε } = ε _{ J } − ε _{ BS }  M _{ J − ε }  m _{ J − ε }  Δ _{ Heston − ε } = ε _{ Heston } − ε _{ BS }  M _{ Heston − ε }  m _{ Heston − ε } 
These aforementioned notations are used in the following empirical study.
Comparison between the Jam model (with λ = 0) and the BSam one
American pricing model results based on the Jam model
Regarding Δ _{ J − eur }, we confirm the results found by Jerbi [1]. Regarding the value of λ, the maximum and the minimum of the difference Δ _{ J − eur } are related to θ = 0. These extreme values are close to those related to θ = 3 and θ = − 3. Regarding Δ _{ J − am }, for positive values of λ, the curve profile function of the moneyness is quite similar to the one related to Δ _{ J − eur }. For negative values of λ, the curve profile is almost the same as the corresponding Δ _{ J − eur }, with the only difference that it presents a positive maximum located in the moneyness interval [0.85; 0.9]. This maximum coincides with the maximum of Δ _{ J − ε }, as indicated in Fig. 13. In this figure, we see that the effect of θ on Δ _{ J − ε } is negligible when λ is negative. However, when λ is positive, the ε _{ J } value is sensitive to changes in the parameter θ. This is because the ε _{ J } value increases with the absolute value of this parameter, with a maximum value corresponding to the same moneyness.
Equivalence condition between the Jam model and Hestonam one
Comparison between Jam model and Hestonam one in terms of the volatility effect when the equivalence condition is satisfied
Comparison between Jam and Hestonam models in terms of the effect of the time to maturity
Finally, we can say that the comparison between the models is based on the effect of the stochastic volatility of the American option pricing. We have proved that the Jam is an extension of the model BSam, as we did with the Jmodel and BS models. Furthermore, we conclude that the Jam model gives the same results as the Hestonam one in considering equivalent parameters ensuring equivalent effects of skewness and kurtosis. The equivalence between the two models can be improved with an accurate estimation of parameters λ and θ for given values of the volatility dynamics parameters considered in the Heston model.
Comparison between the Jam and the Hestonam models in terms of CPU time
The CPU Time required at each stage for determining the American option
Time CPU (in second)  BSam  Hestonam  Jam 

CPU1: OEB determination (10 points)  1  430  5584 
CPU2: American Put value computing 21 points  0  66  918 
Improvements of the Jam model
The OEB approximate polynomial coefficients for the Jam (λ = −1.6, θ = 0.767) and Heston (v = 0.01, κ = 2,φ = 0.01 and ρ = −0.5) models, for various values of the volatility
OEB  S*(σ = 0,1)  S*(σ = 0,2)  S*(σ = 0,3)  S*(σ = 0,4)  S*(σ = 0,5)  

Jam  Hestonam  Jam  Hestonam  Jam  Hestonam  Jam  Hestonam  Jam  Hestonam  
α_{0}  100  100  100  100  100  100  100  100  100  100 
α_{1}  −88,7  −182,8  −251,4  −242,6  −357,3  −572,9  −469,3  −396,1  −577,6  −684,3 
α_{2}  454,0  3781,6  4223,9  2148,9  5524,5  13660,0  7252,0  −1198,9  8864,3  9248,4 
α_{3}  4719,5  −47754,5  −55634,7  −4162,3  −66543,0  −219163,6  −86741,4  119100,6  −104209,1  −86821,5 
α_{4}  −113190,0  358207,8  481249,7  −129020,2  526397,3  2134038,5  682209,7  −1781952,0  805895,7  532016,2 
α_{5}  957015,7  −1687800,5  −2768878,8  1470870,0  −2764903,7  −13022337,6  −3559002,7  13594552,2  −4135421,6  −2260576,2 
α_{6}  −4473462,2  5132416,9  10680702,7  −7709733,4  9728193,1  51049511,1  12413443,5  −61391738,5  14195574,1  7125696,7 
α_{7}  12559747,8  −10044517,7  −27258807,6  23187361,5  −22664547,0  −128519111,3  −28605887,8  170587764,1  −32220275,1  −16790157,4 
α_{8}  −21096551,5  12180376,5  44059429,4  −40996252,7  33535228,2  200843071,0  41770962,5  −286783820,3  46384859,5  27432124,8 
α_{9}  19564855,4  −8286347,9  −40740040,3  39687366,2  −28523653,2  −177308022,0  −34985276,1  267836101,2  −38341325,4  −26780575,1 
α_{10}  −7712384,0  2399990,3  16372793,2  −16255502,5  10614375,1  67563228,8  12792706,6  −106721471,3  13851225,0  11487040,7 
Conclusion
In this study, we have elaborated a new unidimensional model Jam for pricing American options, which is based on the Jmodel developed by Jerbi (2015). We have shown that this model is an extension of the American model BSam based on the BS model. As indicated by Jerbi (2015), we have examined the equivalence between the Jmodel and Heston one. Here, we have extended the study of this equivalence to American options. The parameters of the Jprocess ensuring this equivalence were determined as the values minimizing the squared errors between the Jprocess and CIR process used in the study by Heston (1993). We notice that these parameters can also be determined as the values minimizing the error between the Jam formula and Hestonam one. This study aimed to compare a confirmed model, the Heston’s, which is bidimensional, with an equivalent unidimensional model Jam. Assuming that we use the equivalent parameters λ* and θ*, we can say that our results regarding the Jam are totally in accordance with those of Heston’s in terms of EEB and American option pricing. The EEB and the American option price profiles generated by all the chosen models fully conform with the options theory. We have examined the similarity between the effect of λ* and ρ and the one between θ* and (κ; φ; η; v). We can say that the skewness and kurtosis effects induced by the stochastic aspect of the volatility in the bidimensional model of Heston, are equivalent to the ones generated by the extension of the Wiener process to the Jprocess. This was our conclusion, as indicated by Jerbi (2015), and we extended this conclusion to the American options. For a future work, we plan to examine the dynamic risk management related to an American option portfolio based on this model. We can also use this model to solve financial or economic problems based on American options, such as the decision optimization in an area characterized by innovation and technical progress. The Jam, as a unidimensional model, is expected to fit the reality of the financial market with a better compromise between accuracy and CPU time than the Hestonam model. The computation, based on the cumulative function F of the Jlaw, is easier than the one based on the Fourier inversion method used by Heston. A library for the function F must be constructed to ensure the optimality of the Jam model in terms of accuracy and time consumption. Moreover, the modeling of the EEB based on the polynomial approach can be carried out to significantly improve the CPU time needed to compute the American option value for a given accuracy. Finally, the results generated by the Jam model must be compared with those generated by simulations based on Malliavin calculus and using the Jprocess (see Jerbi and Kharrat (2014)).
Declarations
Acknowledgement
I have achieved all the work by myself and I have no acknowledgement to mention in the Acknowledgement section.
Competing interest
The author declares that he has no competing interests.
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Authors’ Affiliations
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