1 Introduction

Options, as widely used financial contracts, can play an important role in reducing the risk of investors and practitioners in the financial markets. A new recently developed type of option entitled as high dimensional or multi-asset (basket) options, are used in some real-world applications. The option, which could be considered as an exotic option, encompasses a group of securities, commodities, or currencies whose owner/s could exercise it with a predefined price. The price of the option and the weights of tradable assets are officially determined at the time of issuing that option. Obviously, the weighted values of assets or portfolio in the basket options influences the option payoff.

The main advantage of having a basket option would be that it is cheaper than a portfolio including single vanilla options. Furthermore, the risk management of a portfolio having a basket option would be more comfortable than of a portfolio including single vanilla options because of their different early exercise times. Finally, the transaction cost of trading a basket would be remarkably lowered as a buyer/seller only pays a single transaction cost for trading the respective basket option while he/she should pay multiple transaction costs for trading vanilla options.

There are two main types of basket options in the real-world applications in terms of the expiry time of option: American and European basket options. Pricing and valuation of American option, even the single asset option, is a hard problem in the field of quantitative finance (Mitchell et al. 2014).

There are also some research studies in which lower and upper bounds are suggested for the prices of European basket options. D’aspremont and El Ghaoui (2006) found the bound using linear programming. They used an efficient linear programming relaxation based on an integral transform interpretation of the call price function. Caldana et al. (2016) proposed a general closed-form bound which is applicable for all types of European options without having any new assumption on the characteristic function. Cho et al. (2016) also found the bounds using an efficient solution procedure based on the separation problem.

To the best of our knowledge, there is no simple closed-form solution for pricing/valuation of American options. Because of practical importance of this type of options in real-world problems, there are few attempts to develop approximation and numerical methods for pricing/valuation of this type of option (Moon et al. 2008; Cortazar et al. 2008; Bandi and Bertsimas 2014; Breitner 2000; Samimi et al. 2017).

Generally speaking, the developed methods for pricing/valuation of American options would be classified in three main categories: partial-differential-equations-, lattice- and simulation-based methods.

In the first category, the dynamic programming is initially used for finding the optimal stopping point and the partial equations based on Black–Scholes model are derived for the pricing of American option. One of the limitations of these methods is the high computational efforts to reach the reasonable accuracy (Huang et al. 1996). There are a lot of research papers in this category in the literature; see for example: Benth et al. (2003), Wang et al. (2015) and Dehghan and Bastani (2017).

Lattice-based method can be also categorized in three branches: binomial trees (Breen 1991), trinomial trees (Yuen and Yang 2010), and finite difference approaches (Cont and Voltchkova 2005). These approaches have been widely used for single-asset options. However, when the dimension of problems increases, usually up to four, the extensions of binomial and trinomial methods would be computationally expensive. In other words, even though the lattice-based methods are usually easy especially where the early exercise feature exists, there is a remarkable computational burden where the dimension of problems grows up (Cox et al. 1979; Barraquand and Martineau 1995).

In contrary to most lattice-based methods, Monte Carlo simulation methods would be more appropriate to cope with pricing/valuation in high dimensional American options leading to more suitable outcome. However, some type of simulation-based methods such as forward versions have a main drawback in dealing with the issue of early exercise features for these options. To overcome this problematic issue, few research works have been accomplished (Stentoft 2011; Kohler and Krzyżak 2012; Lian et al. 2015; Antonelli et al. 2013).

Dynamic programming is a popular and well-known technique which has been widely used in the literature to tackle with the early exercise features of American options (Cornuejols and Tütüncü 2007). However, DP-based methods mostly suffer from the curse of dimensionality issue in real-world applications (Powell 2011). Approximate Dynamic Programming (ADP), as an efficient simulation-based method used in many real problems, could remarkably alleviate the curse of dimensionality (Powell 2007). To even alleviate the issue of dimensionality, Barraquand (1995) used a stratified state aggregation technique to generate a single-dimensional problem. In spite of the fact that the proposed method is extremely fast in calculating an approximation for the price, its accuracy mainly depends on the way of finding the optimal exercise policy, which is usually obtained through the single payoff function. Broadie and Glasserman (2004) demonstrated that the stratification algorithm will not converge to the correct value.

Longstaff and Schwartz (2001) presented a least squares regression with basis functions to approximate the value function. Ben-Ameur et al. (2007). proposed a DP-based method for pricing options embedded in bonds. A main advantage of their method is that it could be applied to the problems considering interest rates. Gagliardini and Ronchetti (2013) developed a semi-parametric estimator of American option prices in discrete time. They suggested a semiparametric estimator for Stochastic Discount Factor (SDF) parameter and then used dynamic programming for pricing American option.

Generally, using DP in pricing/valuation is mostly limited to small- or mid-size problems. Therefore, it could be a challenging issue to apply this method in large-scale problems, such as pricing American basket option included with different assets. In this situation, different versions of approximate DP, each of which might be called a simulation-based DP, could be suitable methods for these types of problems (Powell 2011; Gosavi 2014).

In spite of different advantages of using options in hedging, in many cases practitioners in a financial market really believe that the mispricing of derivatives could be the main reason of the collapse in some financial institutions. Mispricing or incorrect valuation of derivatives might be because of not considering risk measurements in the process of option pricing (Elliott and Siu 2011).

Considering risk in pricing American option would be very attractive topic, which has been rarely investigated in the literature. The motivation of this study is to develop new simulation-based models to consider the worst-case (pessimistic/risk-averse) and best-case (optimistic/risk-taking) scenarios for pricing the American basket options with respect to the level of risk aversion for each investor. Moreover, there are some degree of auto-correlation and, more importantly, heteroscedasticity in most financial time series. GARCH-type models have been extensively used, specifically for modeling auto-correlation and heteroscedasticity of financial data (Hajizadeh et al. 2015; Tian and Hamori 2015; Gouriéroux 2012; Davari-Ardakani et al. 2016; Maciel et al. 2016). To generate scenario for a specific basket of assets, we use a method based on GARCH and Copula models to consider both auto-correlation of each time series and the correlations between assets.

The remainder of this study organized into the following sections: Sect. 2 presents the proposed models for pricing/valuation of the American basket option; third Section expresses the data description and scenario generation performed using Copula and GARCH models; the two last sections are dedicated to the computational results and conclusions.

2 Pricing American Basket Option Using Approximate Dynamic Programming (ADP)

2.1 Basket Options

A basket option is an option on a basket of assets including securities, commodities or currencies. A vanilla option on a linear combination of securities is a frequently traded basket option. To formulate a basket option payoff, suppose:

  • s i t is the price of stock \( i\left( {i = 1, \ldots , m} \right) \) at the time step \( t\left( {t\, = \,1, \ldots , T} \right) \) during the maturity of the option,

  • st = ∑  m i=1 wis i t is the average price of assets in the basket at time t,

  • X = exercise price,

  • wi= weight of asset i in the basket.

The payoff of the basket put option can be calculated as follows:

$$ \hbox{max} (X - s_{t} ,0). $$
(1)

2.2 The Formulation of the American Basket Option Pricing with Approximate Dynamic Programming

In American option problem, decisions should be taken in a sequential manner. Moreover, earlier decisions can affect the performance and feasibility of later decisions. In this situation, for finding the optimal strategies, the current and future decisions should be considered concurrently.

Stochastic Dynamic programming (SDP) could be used for modeling and solving those problems that follow Markov Decision Processes (MDPs) and sequential decision making under uncertainty.

SDP, a powerful solution approach, could be also applied for both pricing and early exercise of American option. The optimal policy in SDP is derived in terms of the value function of stochastic control problems (Cornuejols and Tütüncü 2007). Using the Bellman’s optimality, as a main equation in SDP, it can be proved that all the remaining decisions are optimal for given state variables.

The respective value functions in SDP should be updated in a backward movement. In this situation, the value function should be calculated for all states st, over time horizon T.

Before getting into the detail of mathematical formulation of the problem, we should initially explain the main components of the SDP. There are four main components in SDP including state (S), stage (K), decision variables (A), and the performance function called cost-to-go function (g).

The situation in a Markov Decision Processes (MDPs) is specified with state variables which are changing over time. The next state is only dependent to the current state which means that the Markov chain is a process without memory. Assuming that S is a finite state space and s ϵ S, there exist some admissible actions (decisions) for each state, As. The cost-to-go function (the action-value function, ga(s)) for all pairwise of states, s ϵ S, and their respective admissible actions, a ϵ As, could be calculated with sufficient amount of observations. Finally, after converging the action-value functions or observing enough instances for all respective pairwises, an operational policy would be derived.

In our problem in this study, without loss of generality, it is assumed that each day is considered as stage variable, therefore, the total number of stage variables would be equal to the number of days in the option life of the basket \( \left( {K = 0, 1,. \ldots , L} \right). \) It is worth mentioning that the time interval could be changed to less or higher than 1 day. The state variable in each stage can be defined as current price of basket, s.

Other parameters used to find the cost-to-go function are as follows:

  • r: is a risk free interest rate,

  • It(st): immediate payoff of put option at time step t, which can be described by:

    $$ I_{t} (s_{t} ) = \hbox{max} (X - s_{t} ,0). $$
    (2)

This formula could be rewritten for call option as follows:

$$ I_{t} (s_{t} ) = \hbox{max} (s_{t} - X,0). $$
(3)

In pricing American option using SDP, both the immediate payoff and the optimal exercise strategy should be simultaneously taken into account. In SDP, this could be accomplished through computing value functions in each state. The value functions should be therefore calculated using two different terms including the immediate payoff (It(st)) and the discounted expected value of future payoff in period t. The expected value in period t is written as:

$$ E_{t} \left[ {e^{ - r.\Delta t} V_{t + 1} (s_{t + 1} )\left| {s_{t} } \right.} \right]. $$
(4)

Thus, the value function used in pricing American option is defined as:

$$ V_{t} (s_{t} ) = \hbox{max} \left( {I_{t} (s_{t} ),E_{t} \left[ {e^{ - r.\Delta t} V_{t + 1} (s_{t + 1} )\left| {s_{t} } \right.} \right]} \right). $$
(5)

As it is obvious, the value function is obtained in a recursive way. However, in contrary to classical stochastic DP, the value function is not additive. Therefore, when we experience the point that the intrinsic value of option is more than the continuation value, the best decision would be to exercise the option. Another challenging issue originates from the fact that the expected value of the future option value should be estimated through finding the optimal strategy in the future. We know that the only control variable in each state is to or not to exercise the option (Cornuejols and Tütüncü 2007).

Although SDP can end up with the global solution even for highly nonlinear and nonconvex optimization problems such as pricing American option, it suffers from the issue of double curses: dimensionality and modelling. That is, if the number of underlying assets in the case of American option increases, traditional dynamic programming might become intractable and computationally expensive. To overcome this challenging issue, many efforts have been made and many approximation methodologies based on DP have been proposed.

Approximate Dynamic Programming (ADP) is a powerful substitute method for large-scale problems, which can cope with the issue of curse of dimensionality in SDP.

One of techniques in ADP would be the use of both simulation and regression-based models to estimate the respective value functions for the corresponding state (Bellman and Dreyfus 1959). This simulation-based method has been widely used in high dimensional problems such as the cases in financial engineering (Longstaff and Schwartz 2001; Brandt et al. 2005; Tsitsiklis and Roy 2001).

One of these techniques is to introduce suitable basis functions and to approximate the respective value functions using them in an iterative way. These basis functions would be linearly combined to each other to approximate Q-value functions (Haugh and Kogan 2007). As a result, the coefficients of these basis function should be suitably trained through a proper learning process. In other words, as it is demonstrated in Eq. 6, the discounted expected value of future payoff E(), could be obtained through a linear regression with proper basis functions, φi(s), in which all the respective parameters, κi, should be appropriately estimated. This is expressed as follows:

$$ E\left[ {e^{ - r\Delta t} V_{t + 1} (s_{t + 1} )\left| {s_{t} } \right.} \right] \approx \sum\limits_{i = 1}^{o} {\kappa_{i} \varphi_{i} (s),} $$
(6)

where o is the number of the basis functions.

It should be mentioned that the basis functions by themselves could be nonlinear, however, they are linearly combined to each other using proper coefficients, κi.

To find the price using ADP, we should initially generate sufficient scenario paths \( \left( {n = 1, \ldots , N} \right). \) The method of generating scenarios is explained in detail later in the paper. In the next step, for each scenario path n the exercise time of option should be obtained using Eq. 5. Given the time exercise, \( PV_{Exp}^{n} , \) could be calculated through the following equation:

$$ PV_{Exp}^{n} = e^{{ - r{ \cdot }t}} \left( {X - s_{t}^{n} } \right). $$
(7)

Given the prices for all individual paths through Eq. 7, the final price of the American basket option (PADP) is then calculated as follows:

$$ P_{ADP} = \mathop E\limits_{n} \left( {PV_{Exp}^{n} } \right). $$
(8)

3 Risk-Based Basket Option Pricing

Some financial analysts and practitioners have emphasized that derivatives could be the main cause of the collapse of some financial companies. The main reason for such a reality is to misprice the derivatives due to not properly considering their risks (Elliott and Siu 2011). Up to now, most of existing option pricing models solely consider the expected value for pricing (e.g., binomial and, trinomial, and Black–Scholes methods). However, in real-world option pricing problems, decision makers should also consider suitable risk measure. For this reason, considering risk in the option pricing would be a vital and important issue, especially for an American option in which a financial practitioner should decide whether or not to exercise the American option in every time interval within its maturity. As we mentioned in the previous section, one of the efficient method that could be used for pricing a high dimensional option, is ADP. The respective value function in ADP, in contrary to the classical stochastic DP, which is usually defined as the discounted expected value of future payoff, is not additively obtained in American option pricing problems. In other words, the value functions of ADP used for pricing an American option is the maximum of immediate payoff and the discounted future payoff. In fact, to find the value function, a comparison between both the real value (i.e., immediate payoff) and an expected value should be accomplished. This comparison, which is also used for finding the time of exercising an American option during its maturity, would not be sufficiently fair, and it seems that another criterion considering risk should be taken into account.

In the following, we propose a new approach to incorporate risk into the pricing of American options in which two pessimistic (worst-case) and optimistic (best-case) prices are obtained

One of conservative ways to cope with the aforementioned challenge is to use a proper simulation approach to generate a large number of scenario paths (i.e., possible prices of assets in basket). In the next step, the discounted intrinsic value of option payoff (\( PVI_{t}^{n} (S_{{_{t} }}^{n} ) \)) at time step t for scenario n is calculated by:

$$ PVI_{t}^{n} \left( {S_{{_{t} }}^{n} } \right) = e^{ - rt} \hbox{max} \left( {X - S_{t}^{n} ,0} \right),\quad \forall \quad t = 1, \ldots ,T\,\,\,\& \,\,\,n = 1, \ldots ,N. $$
(9)

For the worst-case situation, the minimum amount of all discounted intrinsic payoff for each scenario is calculated as follows:

$$ PV_{\hbox{min} }^{n} = \mathop {\hbox{min} }\limits_{t} \left( {PVI_{t}^{n} (s_{t}^{n} )} \right)\quad \forall \,\,\,n = 1, \ldots ,N. $$
(10)

This procedure is repeated for all scenarios and finally, the expected value of all minimum discounted intrinsic values of the option payoff in each scenario would be considered as the pessimistic (worst-case) price for the basket option:

$$ P_{{worst} \text{-} {case}} = \mathop E\limits_{n} \left( {PV_{\hbox{min} }^{n} } \right). $$
(11)

It should be noted that if a person is sufficiently risk averse, it is more beneficial to use this approach for pricing the option.

For the optimistic (best-case) situation, the similar justification for “min” operator in Eq. 10 could be repeated for “max” operator, in which the best (maximum) likely discounted intrinsic values of the option payoff is obtained:

$$ PV_{\hbox{max} }^{n} = \mathop {\hbox{max} }\limits_{t} \left( {PVI_{t}^{n} (s_{t}^{n} )} \right)\quad \forall \,\,\,\,n = 1, \ldots ,N. $$
(12)

In this case, similar to the risk-averse approach, the procedure is repeated for all scenarios and finally, the expected value of all maximum discounted intrinsic values of option payoff in each scenario would be considered as an optimistic (best-case) price of the basket option:

$$P_{{best} \text{-} {case}} = \mathop E\limits_{n} \left( {PV_{\hbox{max} }^{n} } \right). $$
(13)

If a buyer of an option is sufficiently risk taking, it is more beneficial to use this approach for option pricing.

Indeed, using these changes, the optimistic (best-case) and pessimistic (worst-case) prices of the option could be obtained. Given these approaches, a decision-maker can more suitably find out the domain of its payoff for each option and could make a more proper decision (i.e., the time of exercising option).

The interesting point is that the price obtained through the best-case approach in Eq. 13 might be less than the one calculated by the expected value approach in Eq. 8 in some cases. Of course, this situation seems to occur for a remarkably big risk-free rate (e.g., r   > = 50%), which rarely happens in daily market. However, where the risk-free rate is remarkably low (e.g., close to zero), it can be demonstrated that the worst- and best-case prices could be also two proper bounds for the real price of American option. The following Lemma proves this claim:

Lemma

for r = 0, the below equation is always true:

$$ P_{{worst} \text{-} {case}} \le P_{ADP} \le P_{{best} \text{-} {case}} $$
(14)

Proof

In each scenario path where e−r.t = 1, the price based on expected value \( \left( {PV_{Exp}^{n} } \right) \) should be obtained through Eq. 7 while the prices for the worst-case \( \left( {PV_{\hbox{min} }^{n} } \right) \) and the best-case \( \left( {PV_{\hbox{max} }^{n} } \right) \) situations are obtained through Eqs. 10 and 12, respectively.

As it was previously mentioned, the price of American option in each path is obtained using one of immediate payoffs which are discounted using e−r.t. However, as the risk-free rate is zero, the discounted value of the immediate payoff remains unchanged for all three mentioned methods. Furthermore, the minimum and maximum payoff in each path are chosen in the worst-case and best-case approaches, respectively. Therefore, the payoff for each scenario path n obtained through the expected value approach \( \left( {PV_{Exp}^{n} } \right) \) is between the corresponding payoffs achieved through two other methods \( (PV_{min}^{n} \,\,\& \,\,\,PV_{\hbox{max} }^{n} ) \). This is quite obvious that the expected values of these three sequences, which are the final prices of American option for these three computational methods (i.e., PVworst-case, PADP, and Pbest-case), should necessarily comply with the inequality in Eq. 14.

4 Scenario Generation Method for Assets Prices in Basket

One of the important problems in the simulation-based optimization methods is how to generate scenarios. As the scenarios could be generated by Monte Carlo simulation methods, choice of true probability distributions for the random variable inputs is necessary. Choosing a probability distribution for each input variables is often straightforward, however, modeling the dependencies between variables would be a challenging task.

In this study, for modeling and simulation of dependencies between variables, we use a Monte Carlo simulation techniques using a t-student copula and Extreme Value Theory (McNeil and Frey 2000).

Most financial time series exhibit some degree of autocorrelation and, more specifically, heteroscedasticity. We initially extract the residuals from each stock return series with an asymmetric GARCH model. In order to use Generalized Pareto Distribution (GDP) for modeling the tails of a distribution, the observations should be approximately independent and identically distributed.

To produce independent and identically distributed observations, we fit a first order autoregressive model for the returns and an asymmetric GARCH model (GJR model) (Glosten et al. 1993) to the conditional variance of each asset in the basket:

$$ r_{t} = \beta_{0} + \beta_{1} r_{t - 1} + \varepsilon_{t} . $$
(15)

The GJR(p, q) model can be written as:

$$ \sigma_{t}^{2} = \omega + \sum\limits_{i = 1}^{p} {\beta_{i} \sigma_{t - i}^{2} + \sum\limits_{i = 1}^{q} {\alpha_{i} \varepsilon_{t - i}^{2} + \sum\limits_{i = 1}^{q} {\gamma_{i} S_{t - i}^{ - } \varepsilon_{t - i}^{2} } } } , $$
(16)

where, S − t– i  = 1 if ɛt–i < 0, and S − t– i  = 0 if ɛt–i ≥ 0. The model is well defined if the following conditions are satisfied:

$$ \sum\limits_{i = 1}^{p} {\alpha_{i} + \sum\limits_{i = 1}^{q} {\beta_{i} + \frac{1}{2}\sum\limits_{i = 1}^{q} {\gamma_{i} < 1} } } ,\quad \omega \,{\ge}\, 0,p \,{\ge}\, 0, ,q \,{\ge}\, 0, \alpha_{i} \ge 0,\beta_{i} \,{\ge}\, 0\,\,\,{\text{ and}}\,\,\,\,\alpha_{i} + \gamma_{i} \,{\ge}\, 0. $$

4.1 Estimation of the Semi-Parametric Cumulative Distribution Function for Each Stock

After obtaining independent and identically distributed residuals, we estimate the empirical Cumulative Distribution Function (CDF) of each stock with Gaussian kernel. For better estimating the distribution tails of each input variable, Extreme Value Theory has been applied to those residuals that fall in each tail. Then, we use maximum likelihood method to fit a parametric Generalized Pareto Distribution for those extreme residuals, which are beyond a predefined threshold (Nystrom and Skoglund 2002).

4.2 t Copula Calibration

There are many research studies in the area of copula modeling and simulation, see for example Bouyé et al. (2000), Daul et al. (2003), Lee and Kim (2015), Li and Yang (2013). For t copula formulation, suppose \( Z \sim N(0,\sum ) \) and \( R \sim \sqrt v /\sqrt S \) be independent that S = χ 2 v that has a Chi Square distribution with ν degrees of freedom. Moreover, the Rd-value random vector Y written by

$$ Y = RZ, $$
(17)

where it has a t distribution with ν degrees of freedom. Moreover, for ν > 2, Cov(Y) can be written as \( \frac{\nu }{\nu - 2}\sum \). According to the Sklar’s Theorem (1959), the copula of Y can be written as

$$ C_{v,p}^{t} (u) = t_{v,p}^{d} (t_{v}^{ - 1} (u_{1} ), \ldots ,t_{v}^{ - 1} (u_{d} )). $$
(18)

That \( \rho_{ij} = \sum_{ij} /\sqrt {\sum_{ii} \sum_{jj} } \) for i, j ϵ {1, …, d}, t d v, p represents the distribution function of \( \sqrt v Z > \sqrt S \), where \( S \sim \chi_{v}^{2} \),\( Z \sim N_{d} (0,\rho ) \) are independent and tv represents the marginal distribution function of \( t_{v,p}^{d} \) (Daul et al. 2003).

We can find the parameters of t copula using maximum likelihood approach. After finding the parameters of t copula, by simulating the corresponding dependent standardized residuals, we can generate jointly dependent assets returns. After that, through the extrapolating into the Generalized Pareto tails and interpolating into the smoothed interior, using the inversion of the semi-parametric marginal cumulative distribution function of each asset, uniform variables could be transformed to the standardized residuals.

5 Numerical Results

In this section, we present the results of applying proposed models on the real and simulated data. At the first step, we generate 50,000 scenarios with copula-based GARCH method. For implementation of the models, we work with real data. Historical data of five assets which including Apple, IBM, Microsoft, Yahoo and Amazon from 2-January-2008 up to 31-December-2014 have been gathered. Figures 1 and 2 illustrate the relative price movements and daily logarithmic returns of each asset.

Fig. 1
figure 1

Price movement of assets

Full size image
Fig. 2
figure 2

Daily logarithmic returns of each asset

Full size image

Table 1 presents the basic statistical characteristic of the five time series. The results of ARCH test (Engle 1982) and Ljung–Box Q-test (Ljung and Box 1978) demonstrate the degree of persistence in variance, and implies that GARCH-type models might be suitably matched to these data.

Table 1 Data description and descriptive statistics of each asset in basket
Full size table

As it is demonstrated in Fig. 3, the empirical CDF of the upper tail is closely matched with the CDF obtained through GDP.

Fig. 3
figure 3

Comparison between fixed Generalized Pareto CDF and empirical CDF

Full size image

In the following experiments, we perform all these proposed methods for finding the prices of one unique American put basket option including five equally-weighted assets.

Table 2 demonstrates the information of five assets and their respective put option prices with exercise price = 140 and interest rate = 0.05, and time to maturity = 2 months (44 days).

Table 2 The information of five assets and their respective put option prices with exercise price = 140 and interest rate = 0.05, and time to maturity = 2 months
Full size table

The Table 3 illustrates the price of basket options using Monte Carlo simulation and compare these prices with those ones obtained through the weighted summation of option prices for all five individual assets with the same exercise prices, maturity time, and interest rate. Moreover, to verify that the Monte Carlo outputs are sufficiently accurate, the prices of this basket (i.e., an equally weighted portfolio using assets in Table 2) in different exercise prices are calculated using Black–Scholes method. In fact, the basket is assumed to be a new asset whose price is obtained through Black–Scholes method.

Table 3 The advantage of making basket option
Full size table

As it is illustrated in Table 3, the results of simulation-based and Black Sholes are quite close to each other, which verifies the accuracy of prices obtained through simulation-based method. There are also significant differences between the prices of basket option and the weighted option prices obtained through the BS and simulation-based methods. This could be justified based on the main concepts in Modern Portfolio Theory (MPT) in which making a portfolio would remarkably decrease the risk in terms of different measurements such as variance. It means that combining multiple risky assets with quite volatile behaviors and negative or zero correlations between underlying assets in the portfolio would end up with an asset (i.e., a portfolio) with low volatile behavior. This would be a known fact that the volatility is of important parameter along with the time to maturity and the interest rate for pricing or valuation of European or American put option (Cvitanić and Zapatero 2004; Hull 2006) Therefore, the price of European or American put option with no dividend obtained through Black–Scholes or simulation-based method increases where the volatility goes up.

Moreover, the expected payoff of the basket option would be usually less than the weighted summation of payoffs of individual assets especially where their current stock prices are quite different (it ranges between 46.45 and 315.35 in our samples in Table 2). This would be another justification for the gap between the prices of the basket options compared to the weighted prices

It should be noted that the gap between the prices of basket option compared the weighted summation of individual options prices in the stock market would be due to more flexibility in exercising all options in different times. It means that all respective options on individual assets in the basket could be independently exercised or traded. Therefore, the total weighted price for these options could be reasonably higher than the price of the basket option.

Moreover, buying a basket option would be better in terms of transaction cost as there is only single transaction payment for each trading.

To investigate the performance of the proposed method, the new results obtained through the developed methods in this study are also compared with those ones obtained for European basket option through Monte Carlo simulation and Nengjiu Ju model. The Nengjiu’s model is an approximation model for the basket option pricing via Taylor expansions. As the mathematics of the model is quite complex, one may refer to Ju (2002) for more detail. This is known a fact that European option prices are always a lower bound for American option prices (Hull 2006). Moreover, to investigate the real-world conditions and robustness of our models, all these models are performed with different exercise prices, maturity dates, and interest rates for the basket option.

Table 4 demonstrates the results of the proposed models with different exercise prices. The parameters in these experiments are, time to maturity (T) is 44 days, interest rate (r) is 0.05, and the current basket price (S0) is 137.5 $. As it can be seen, by increasing the value of exercise prices the value of put option increases. Furthermore, as it is expected, there is a significant difference between the prices obtained through the pessimistic and optimistic approaches.

Table 4 The results of the proposed models with different exercise prices
Full size table

Tables 5 also illustrates the results of the proposed models with different times to maturity for the option. In this case, the parameters are set as follows: time to maturity (T) is 44 days, exercise price (X) is 150 $, and the current basket price (S0) is 137.5 $. As it can be seen, the price calculated using ADP grows up while the times to maturity increase (e.g., price of American option is 13.763 where T = 22 and it is 13.987 where T = 132). This is totally in contrary to Monte Carlo approach in which price goes down where times to maturity increase (e.g., price of European option is 12.022 where T = 22 and it is 9.747 where T = 132). This indicates that ADP could more suitably consider the flexibility of American option and it therefore affects the price of option.

Table 5 The results of the proposed models with different times to maturity of option
Full size table

As a general rule, this can be claimed that as the time of maturity increases, the price of risk-averse approach (pessimistic) will decrease and the price of risk-taking approach (optimistic) will increase.

Table 6 also presents the results of the proposed models with different interest rates of option (r). In this case, the parameters of options are set as, time to maturity (T) is 44 days, exercise price (X) is 150 $, and the current basket price (S0) is 137.5 $. As it is illustrated in this table, the put option prices are negatively affected where the interest rates increase.

Table 6 The results of the proposed models with different interest rates
Full size table

Actually, it could be concluded that our proposed models could properly convey a message to the investors and practitioners in the financial markets to take proper decisions (i.e., the time of exercise and the price) according to the level of their risk aversion. Indeed, the investor/s could find prices for the American basket options according to worst-case and best-case scenarios, which are called pessimistic and optimistic prices in this study.

Furthermore, backtesting using historical data is a method, which has been widely used to test a trading strategy or to validate a new developed model. The objective of backtesting in this paper is to explore what would happen if investors bought the option with the obtained prices through three mentioned strategies: risk-averse, risk-neutral, and risk-taking approaches. Using 5000 out-sample sequences of data the respective backtests are performed for two cases. In the first one exercise price = 160 $, Interest rate = 0.05, time to maturity = 44 days, risk-Averse option price = 12.146, ADP-based option price = 23.734, risk-taking option price = 29.701. In the second one exercise price = 165 $, Interest rate = 0.05, time to maturity = 44 days, risk-averse option price = 16.408, ADP-based option price = 28.728, risk-taking option price = 34.457.

The backtesting for each sequence of generated data is performed through the following steps.

  1. 1.

    The investor should find the price of option (P) based on one of three above-mentioned methods.

  2. 2.

    Assuming that the option with the price (P) is available in the market, the time of exercise for each scenario (tn) is extracted in each sequence and the respective immediate payoff is calculated (X − s n t ).

  3. 3.

    The net payoff could be obtained through subtracting the net present value of the respective immediate payoff (NPV(payoff)) and the price of option (P) obtained in the first step for each sequence (e−rt(X − st) − P).

    1. 4.

      Repeating all three above steps for all sequences, the distribution for net payoff could be derived using suitable statistical software.

As it is demonstrated in Figs. 4 and 5, if an investor chooses the risk-averse approach, the distribution function of the net payoff has a negative skewness that means the investor earns profit in most of the scenarios. It should be mentioned that this price occurs less often in real-world. However, if the investor chooses the ADP-approach, the distribution function of net payoff has close to the normal distribution. Of course, there is no mathematical proof for this conclusion and we cannot therefore mathematically prove this issue. Finally, if the investor chooses the risk-taking approach, the distribution function of the net payoff has a positive skewness, that means the investor loses in most of the scenarios. The risk-taking investor really hopes to gain net payoffs whose probabilities are small; however, their values would be quite large.

Fig. 4
figure 4

Backtesting results (exercise price = 160 $, Interest rate = 0.05 and time to maturity = 44 days)

Full size image
Fig. 5
figure 5

Backtesting results (exercise price = 165 $, Interest rate = 0.05 and time to maturity = 44 days)

Full size image

6 Conclusion

High dimensional or multi-asset options as a widely used options contracts called basket options in real-world applications, encompass a group of securities, commodities, or currencies. The pricing/valuation for such an option with multiple assets would be a challenging and problematic issue that traditional methods such as binomial tree cannot be coped with. To overcome this issue there have been many attempts using simulation-based techniques such as Approximate Dynamic Programming (ADP). This technique utilizes simulation to alleviate the curse of dimensionality. However, almost all these techniques do the pricing without risk consideration. To the best of the authors’ knowledge, American option pricing with risk consideration has been quite rarely investigated in the literature.

In this paper, for pricing/valuation the American high dimensional options we proposed a simulation-based approach with risk consideration. Therefore, we should use a suitable technique to generate sufficient scenarios. The main challenging issue in scenario generation is to handle the nonlinear dependencies between the returns of the respective assets in the basket option. Therefore, after finding the proper distributions for all assets in the basket, we used a Monte Carlo simulation technique using a t-student copula-GARCH method and Extreme Value Theory to tackle with nonlinear dependencies between variables.

In this study, as a pioneering computational research work for pricing the American basket option, we found two risk-based prices called pessimistic and optimistic prices for the American basket options according to worst-case and best-case scenarios. This might make the decision-making process more comfortable for financial practitioners as they have multiple prices for an American option based on different values of risk-aversion factors.

To demonstrate the computational efficiency and the accuracy of the proposed methodology, we compared the prices with those ones achieved using the Monte Carlo method and the Nengjiu’s model for the European basket options. All the experimental results have been repeated with different maturity dates and exercise prices.

Other risk measures such as VaR or CVaR could be considered in pricing of American options. These are left for future studies.