In this paper , a pricing method based on the Fourier-Cosine series expansion is introduced for pricing equity-indexed annuities (EIAs) under the Heston model. By means of the Fourier-Cosine series expansion, the density function of the underlying indexed is recovered from its characteristic function, and then yields an efficient way for EIAs pricing. To show the accuracy of the Fourier-Cosine expansion method, numerical experiments , we provide the numerical results of EIAs price for the classical Black-Scholes model. It is shown that the computation results obtained by the Foueier -Cosine series are as accurate as those obtained by using the Monte Carlo simulation method. The Fourier-Cosine expansion method can be used to obtain the break-even participation rate under the Heston model with or without a cap in simple ratchet EIAs.
Equity-indexed annuities (EIAs) were introduced in 1995 by Keyport Life Insurance Corporation. They are designed to provide not only certain annuity benefits, but also returns that are linked to the underlying equity index. EIAs attract investors because policyholders can share in financial market growth and their properties can be protected by the minimum guaranteed benefits provided by EIAs.
The EIA pricing problem can be traced back to 1976, which introduced modern option pricing theory and used its techniques to equity-indexed insurance pricing and minimum risk investment strategy (Brennan & Schwartz, 1976) . This theory assumed the underlying equity dynamic was the classical Black-Scholes model, a geometric Brownian motion with a constant interest rate and volatility. Under this classical Black-Scholes model, Gerber and Shiu (1994) consider derivative pricing in an incomplete market by Esscher transform. Then, three common EIA pricing, which were point-to-point, the cliquet and the lookback, were introduced by Tiong (2000) . Four path-dependence option pricing was introduced by Lee (2003) to improve the participation rate. As the classical Black-Scholes model is a simplified description of a real financial market, subsequent research began to improve the model assumption to better match real life. For example, the constant interest rate assumption is irrational in the EIAs, given the long maturity of RIA contracts, which may cause EIA pricing bias. Thus, a stochastic interest rate model was proposed by Lin and Tan (2003) , and it was shown by Monte Carlo simulation that the stochastic interest rate affects EIA premiums and break-even participation rates. Subsequently, a ratchet EIA pricing was derived under the Vasicek interest rate model (Kijima & Wong, 2008) . Ballotta (2010) studied the EIA pricing under the variance gamma model. A regime-switching model was introduced for pricing and hedging long-term equity-linked insurance products by Hardy (2001) . And Lin et al. (2009) studied the equity-indexed annuity pricing by Esscher transform under this model. In Chang (2014) , a tree method was used to price EIAs with surrender risk under the Vasicek interest rate model and CIR++ model (Wei et al., 2013) . A unified pricing framework for EIAs under a regime-switching Lévy model was presented by Kolkiewicz and Lin (2017) . In Chiu et al. (2019) , an exchange model was added to the Black-Scholes model to explain the contract and market parameter’s influence on pricing EIAs. And Hawkes jump diffusions model was studied for EIAs pricing and a numerical result was given (Sharma et al., 2021) .
The stochastic model is proposed to better match reality. In Merton (1976) , a jump-diffusion model was proposed and subsectionly, a widely jump-diffusion model, named variance gamma model, was proposed by Madan et al. (1998) . In the research on stochastic volatility, Hull and White (1987) assumed that volatility was a stochastic process and built the Hull-White model. In Stein and Stein (1991) , stochastic volatility was added to the underlying stock price. A square root model was introduced and named the Heston model (Heston, 1993) .
The Foueier-Cosine expansion method is used to price ratchet EIAs in this paper. The Foueier-Cosine expansion method was introduced by Carr and Madan (1999) and became a popular and efficient option pricing tool. Fourier-Cosine expansion method can be applied to any model with a known characteristic function for an underlying asset regardless of whether its density function is known. Previously, it was used for European options, Bermudan options (Fang & Oosterlee, 2008) and Asian options (Zhang & Oosterlee, 2013) . Recently, this method has been introduced for pricing insurance contracts in Goudenége et al. (2018) .
The contribution of this paper is two-fold. On the theoretical side, we show that the Fourier-Cosine expansion method can be applied to price EIAs under the stochastic volatility model. On the application side, we provide the formula for pricing EIAs under the Heston model and analyze the sensitivity of the price to the model and product parameters. The innovation of this paper is that we propose the stochastic volatility model in the pricing of EIAs, which allows us to consider the evolution of the volatility effect on the price of EIAs.
The rest of the paper is organized as follows. We describe the ratchet EIA and introduce the model setup in Section 2. We apply the Fourier-Cosine expansion method to the pricing EIAs in Section 3. Finally, numerical tests are presented in Section 4 to demonstrate the accuracy of the proposed method.
Assume that the payment EIA contract with maturity at T-year is based on the return of the equity index. Let S ( t ) , t ≥ 0 denote the equity index value at time t. Consider a frictionless financial market with continuous trading. Assume that taxes, transaction costs and restrictions on borrowing or short sales do not exist and that all securities are perfectly divisible.
We assume that the equity index S ( t ) underlying the EIAs contract follows the Heston model, which satisfies the following system:
( d S ( t ) = ( r − q ) S ( t ) d t + v ( t ) S ( t ) d W S ( t ) d v ( t ) = k ( v ¯ − v ( t ) ) d t + γ v ( t ) d W v ( t ) (2.1)
where r > 0 , q ≥ 0 are constants representing the riskyless interest rate and the dividend rate respectively, { v ( t ) , t ≥ 0 } is variance process of the equity value, v ¯ > 0 , γ > 0 is the mean and the variance of volatility of the equity index, k > 0 determines the speed of adjustment of the volatility towards its theoretical mean, W S ( t ) , W v ( t ) are two different Brownian motions with correlation d W S ( t ) d W v ( t ) = ρ d t under a risk-neutral measure Q.
Assume X ( t ) : = ln ( S ( t ) ) denote the log-equity index and have an initial value: X ( 0 ) = ln ( S ( 0 ) ) = x . By (2.1), X ( t ) satisfies the following system:
( d X ( t ) = ( r − q − 1 2 v ( t ) ) d t + v ( t ) d W S ( t ) d v ( t ) = k ( v ¯ − v ( t ) ) d t + γ v ( t ) d W v ( t ) (2.2)
Under Heston model (2.2), the conditional characteristic function of the log-asset price X ( T ) given that X ( t ) = x is given by Feyman-Kac formula:
ψ ( u , T ) : = E Q [ e i u X ( T ) ] = exp ( i u ( r − q ) τ + v 0 γ 2 ( 1 − e − D τ 1 − G e − D τ ) ( k − i ρ γ u − D ) ) ⋅ exp ( k v ¯ γ 2 ( τ ( k − i ρ γ u − D ) − 2 ln ( 1 − G e − D τ 1 − G ) ) ) ⋅ exp ( i u x ) , (2.3)
where τ = T − t , D = ( k − i ρ γ u ) 2 + ( u 2 + i u ) γ 2 , G = k − i ρ γ u − D k − i ρ γ u + D .
When the variance { v ( t ) , t ≥ 0 } degenerates to a constant, i.e. κ = γ = 0 , the Heston model reduces to the classical Black-Scholes model, which is given as follows:
d S ( t ) = ( r − q ) S ( t ) d t + σ S ( t ) d W S ( t ) , (2.4)
Thus, the conditional characteristic function of log-asset price X ( T ) given X ( 0 ) = x can be given by:
ψ ( u , T ) : = E Q [ e i u X ( T ) ] = exp [ i u ( x + ( r − q − 1 2 σ 2 ) T ) − 1 2 σ 2 u 2 T ] . (2.5)
In this paper, we consider the Heston model as a risky equity index model for pricing EIAs, we derive the pricing formula for evaluating EIA contracts under Heston model. To show the accuracy of proposed method, we present the price of EIAs under Black-Scholes model for comparison with the existing result, since there are only results under Black-Scholes model in former research. We also provide some numerical test for pricing EIAs under Heston models.
Consider an EIA contract with maturity at T years. A ratchet EIA is one whose equity index is reset at some frequency. We consider EIAs with an annual reset, though the model and pricing methodology can easily be extended to an arbitrary reset frequency. Denote the index return over the jth year as follows:
R j = S ( j ) S ( j − 1 ) , for j = 1 , 2 , ⋯ , T (2.6)
and denote the index return series from year 1 to year T by the following:
R 0 T : = { R j , j = 1 , 2 , ⋯ , T } .
There are two ways to compute the payment of the ratchet EIAs: the simple ratchet and the compound ratchet. The payoff of the EIAs is the greater of either the index return R j times a participation rate α or a minimum guaranteed return g. For the simple Ratchet, the payment at maturity T-year, denoted by V ( T , R 0 T ) , is defined as
V ( T , R 0 T ) = 1 + ∑ j = 1 T max [ g , α ( R j − 1 ) ] , (2.7)
For the compound Ratchet, its payment at maturity T-year is as
V ( T , R 0 T ) = ∏ j = 1 T max [ 1 + g , 1 + α ( R j − 1 ) ] . (2.8)
To reduce the cost of the EIAs, a cap design is imposing to the contract, thus the payoff of the simple Ratchet EIAs and the compound Ratchet EIAs with cap c > g take the following forms respectively:
V ( T , R 0 T ) = 1 + ∑ j = 1 T min [ max [ g , α ( R j − 1 ) ] , c ] , (2.9)
V ( T , R 0 T ) = ∏ j = 1 T min [ max [ 1 + g , 1 + α ( R j − 1 ) ] , 1 + c ] . (2.10)
Note that the simple or compound Ratchet EIAs without cap can be taken as simple or compound Ratchet EIAs with cap c = + ∞ .
Under the selected risk-neutral measure Q, the time 0 price of the EIAs with payment V ( T , R 0 T ) at maturity T-year is
V ( 0 ; α ) : = e − r T E Q [ V ( T , R 0 T ) ] . (3.1)
Assume that an initial premium of 1 unit is invested in the EIAs contract. The task of pricing the EIAs contract is to find the break-even participation rate α * such that the value of the EIAs at time 0 breaks even, that is,
V ( 0 ; α * ) = 1. (3.2)
For the simple Ratchet EIAs with and without cap whose payment as given in (2.7) and (2.9) respectively, we have
V ( 0 ; α ) = e − r T E Q [ 1 + ∑ j = 1 T max [ g , α ( R j − 1 ) ] ] = e − r T ( 1 + ∑ j = 1 T E Q [ max [ g , α ( R j − 1 ) ] ] ) (3.3)
and
V ( 0 ; α ) = e − r T E Q [ 1 + ∑ j = 1 T min [ max [ g , α ( R j − 1 ) ] , c ] ] = e − r T ( 1 + ∑ j = 1 T E Q [ min [ max [ g , α ( R j − 1 ) ] , c ] ] ) . (3.4)
For the compound Ratchet EIA with and without cap whose payment as given in (2.8) and (2.10) respectively, we have
V ( 0 ; α ) = e − r T E Q [ ∏ j = 1 T max [ 1 + g ,1 + α ( R j − 1 ) ] ] (3.5)
and
V ( 0 ; α ) = e − r T E Q [ ∏ j = 1 T min [ max [ 1 + g ,1 + α ( R j − 1 ) ] ,1 + c ] ] . (3.6)
Notice that Δ j , j = 1 , 2 , ⋯ , T is not independent under Heston model. By (3.4), R j is not independent, the computation of expectation in (3.5) and (3.6) requires the joint distribution of R j , which make it hard to use Fourier-Cosine expansion directly. Thus, we consider only the simple Ratchet EIAs with and without cap whose the time 0 price as given in (3.3) and (3.4), the case of compound Ratchet EIAs will be treated in a subsequent research.
To compute the time 0 price of the EIAs, it is essential to compute the expectation about R j in (2.7) and (2.9). In fact for any function h ( ⋅ ) about R j , its expectation can be computed as
E Q [ h ( R j ) ] = E Q [ h ( exp ( Δ j ) ) ] = ∫ 0 ∞ h ( exp ( y ) ) f Δ j ( y ) d y , (3.7)
where f Δ j ( y ) is the probability density function of Δ j . Since f Δ j ( y ) is usually not know for Black-Scholes model and Heston model, we will replace it by its Fourier-Cosine expansion
f Δ j ( y ) = ∑ k = 0 + ∞ ′ A k cos ( k π y − a b − a ) (3.8)
with
A k : = 2 b − a ∫ a b f Δ j ( y ) cos ( k π y − a b − a ) d y ≈ 2 b − a ∫ 0 ∞ f Δ j ( y ) cos ( k π y − a b − a ) d y , (3.9)
where a and b are the truncated bounds of the integration interval such that the truncated error is under control and the prime in the summation indicates that the first term in the summation is weighted by one-half. Comparing the approximation of A k in (3.9) with the characteristic function of Δ j , we have
A k ≈ 2 b − a Re [ ϕ Δ j ( k π b − a ) ⋅ exp ( − i a k π b − a ) ] , (3.10)
where ϕ Δ j ( u ) is the characteristic function of Δ j and Re { ⋅ } denotes the real part of the argument.
Subsituting f Δ j ( y ) in (3.7) by its cosine expansion, we have
E Q [ h ( R j ) ] = ∫ 0 ∞ h ( exp ( y ) ) ∑ k = 0 + ∞ ′ A k ⋅ cos ( k π y − a b − a ) d y , (3.11)
then applying the expression (3.10) of A k and interchanging the summation and integration yields
E Q [ h ( R j ) ] = ∫ 0 ∞ h ( exp ( y ) ) ∑ k = 0 + ∞ ′ 2 b − a Re [ ψ Δ j ( k π b − a ) ⋅ exp ( − i a k π b − a ) ] ⋅ cos ( k π y − a b − a ) d y ≈ ∑ k = 0 N − 1 ′ Re [ ϕ Δ j ( k π b − a ) ⋅ exp ( − i a k π b − a ) ] V k , (3.12)
where N is the truncated number for the Fourier-Cosine expansion and V k is the so-called the COS coefficient defined as
V k : = 2 b − a ∫ a b h ( exp ( y ) ) cos ( k π y − a b − a ) d y .
{ V k ; k = 1 , 2 , ⋯ } can be analytically derived as a close formula when the function h ( exp ( y ) ) is specified for simple Ratchet and with or without cap, we will derive the formulas in next section.
For the simple Ratchet EIAs with or without cap with payment as given in (2.7) and (2.9), the function h ( R j ) is specified as
h ( R j ) = min [ max [ g , α ( R j − 1 ) ] , c ] ,
thus the coefficient V k can be derived as
V k = 2 b − a ∫ a b h ( exp ( y ) ) cos ( k π y − a b − a ) d y = 2 b − a ∫ a b min [ max [ g , α ( exp ( y ) − 1 ) ] , c ] cos ( k π y − a b − a ) d y = 2 b − a [ ∫ a ln ( g α + 1 ) g cos ( k π y − a b − a ) d y + ∫ ln ( g a + 1 ) ln ( c a + 1 ) α ( e y − 1 ) cos ( k π y − a b − a ) d y + ∫ ln ( c a + 1 ) b c cos ( k π y − a b − a ) d y ] = 2 b − a [ g ϕ k ( a , ln ( g α + 1 ) ) + α χ k ( ln ( g α + 1 ) , ln ( c α + 1 ) ) − α ϕ k ( ln ( g α + 1 ) , ln ( c α + 1 ) ) + c ϕ k ( ln ( c α + 1 ) , b ) ] , (3.13)
where Equation (3.13) is obtained because the integration bound [ a , b ] is sufficiently large that a < ln ( c α + 1 ) and b > ln ( g α + 1 ) , and ∀ ( p , q ) ⊂ ( a , b ) ,
ϕ k ( p , q ) : = ∫ p q cos ( k π y − a b − a ) d y = ( [ sin ( k π q − a b − a ) − sin ( k π p − a b − a ) ] ⋅ b − a k π , k ≠ 0 q − p , k = 0 ,
χ k ( p , q ) : = ∫ p q e y cos ( k π y − a b − a ) d y = 1 1 + ( k π b − a ) 2 [ cos ( k π q − a b − a ) e q − cos ( k π p − a b − a ) e p + k π b − a sin ( k π q − a b − a ) e q − k π b − a sin ( k π p − a b − a ) e p ] . (3.14)
Based on the computation formulae for the time 0 value for the EIAs contracts given above, we can find the break-even participation rate α * by solving the equation of (3.2). The participation rate is used as a benchmark for the insurer to set the participation rate α for the EIAs contracts.
To test the accuracy and efficiency of the Fourier-Cosine expansion method, we use the formula by Fourier-Cosine expansion method in Section 3 to obtain both the time 0 price V ( 0, α ) of ratchet EIAs for some given participation rate α to compare with the existing results by Monte Carlo simulation under the Black-Scholes model given in Hardy (2004) and the break-even participation rate α * to compare them with the existing results under the Black-Scholes model given in Tiong (2000) . We also provide the break-even participation rate computed by the Fourier-Cosine expansion method Heston model for some parameter cases.
To compare the results between the Fourier-Cosine expansion method in this paper and Monte Carlo simulation in Hardy (2004) , we set the parameters of both the model and EIAs contract the same as in Hardy (2004) , which the interest rate r = 6 % , volatility σ = 0.25 , the minimum guaranteed return rate g = 0 % , the dividend rate q = 2 % , maturity time T = 7 and participation rate α and cap rate c are presented in
Absoluteerror : = AE = | FC − MC | ,
where FC is the price of EIAs in the Fourier-Cosine expansion method and MC is the price of EIAs in the Monte Carlo simulation. As it is shown in
| Price of EIA Partition Rate α | Method | Cap Rate c | |||
|---|---|---|---|---|---|
| 10% | 15% | 20% | 30% | ||
| 0.6 | FC MC AE | 83.6851 83.6853 0.0002 | 89.1147 89.1154 0.0007 | 92.8456 92.8469 0.0013 | 96.9644 96.9651 0.0007 |
| 0.8 | FC MC AE | 84.9961 84.9959 0.0002 | 91.7378 91.7387 0.0009 | 96.9181 96.9184 0.0003 | 103.7266 103.7272 0.0006 |
| 1.0 | FC MC AE | 85.8197 85.8198 0.0001 | 93.4476 93.4488 0.0012 | 99.6855 99.6871 0.0016 | 108.7400 108.7416 0.0016 |
| 1.2 | FC MC AE | 86.3831 86.3830 0.0001 | 94.6419 94.6426 0.0007 | 101.6656 101.6675 0.0019 | 112.5247 112.5220 0.0027 |
Then, the computation results of the break-even participation rate for the Black-Scholes model are presented in
For Heston model, the computation results are presented in
| q | Cap | g = 3% | |||||
|---|---|---|---|---|---|---|---|
| σ = 0.03 | σ = 0.02 | ||||||
| r = 4% | r = 5% | r = 6% | r = 4% | r = 5% | r = 6% | ||
| 2% | 10% | 0.305204 | 0.651761 | 3.239664 | 0.429140 | 0.809608 | 2.194496 |
| 12% | 0.278614 | 0.484261 | 1.195217 | 0.398136 | 0.639872 | 1.065437 | |
| 14% | 0.265895 | 0.421771 | 0.666464 | 0.383724 | 0.573124 | 0.828112 | |
| 1% | 10% | 0.293616 | 0.605957 | 2.402629 | 0.407391 | 0.740946 | 1.772472 |
| 12% | 0.269237 | 0.459479 | 0.855894 | 0.380058 | 0.598238 | 0.960387 | |
| 14% | 0.257560 | 0.403472 | 0.625091 | 0.367383 | 0.540802 | 0.766031 | |
| q | Cap | g = 3% | |||||
|---|---|---|---|---|---|---|---|
| v = 0.03 | v = 0.02 | ||||||
| r = 4% | r = 5% | r = 6% | r = 4% | r = 5% | r = 6% | ||
| 2% | 10% | 0.490737 | 0.823581 | 1.645577 | 0.551115 | 0.887698 | 1.678759 |
| 12% | 0.464632 | 0.689075 | 1.025693 | 0.524565 | 0.756356 | 1.078675 | |
| 14% | 0.453291 | 0.635331 | 0.852352 | 0.513247 | 0.703606 | 0.916962 | |
| 1% | 10% | 0.462781 | 0.752731 | 1.418702 | 0.516372 | 0.806233 | 1.394885 |
| 12% | 0.440448 | 0.641269 | 0.927894 | 0.494186 | 0.699444 | 0.971297 | |
| 14% | 0.430859 | 0.596108 | 0.786886 | 0.484881 | 0.656086 | 0.8419131 | |
In this section, we present the sensitive analysis under the Black-Scholes model and the Heston model. In Figures 1-6, the results of break-even participation rate α * to six model parameters are presented under the Black-Scholes model.In general, analysis results are intuitive. The break-even participation rate increases with the interest rate r, with dividend rate q, with maturity time T and decreases with volatility σ, with the minimum guaranteed return rate g, with cap c. In
guaranteed return rate g, the policyholder’s benefits are all from the minimum guaranteed return, which makes the break-even participation rate 0. And when r > g , as the r increases, the insurance company can obtain more benefits in the investment market, then increases the break-even participation rate. But as the volatility σ increases, the investment risk increases, and the costs of hedging will increase, then the break-even participation rate decreases. In
In Figures 7-11, we also present the sensitive analysis for five model parameters on the Heston model. Notice that the break-even participation rate decreases when the mean of volatility v ¯ increases in
Considering the long maturity of the EIAs, the volatility of the equity index might evolute as time going. This paper introduces the Heston volatility model in the pricing of EIAs and applies the Fourier-Cosine expansion method to derive an approximation formula for the price of the EIAs. The numerical test examines the accuracy of the Fourier-Cosine expansion method and analyzes the sensitivity of the break-even rate of the EIAs to the model and product parameters.
The authors declare no conflicts of interest regarding the publication of this paper.
Gu, X. C., & Wei, X. (2023). Fourier-Cosine Method for Pricing Equity-Indexed Annuity under Heston Model. Theoretical Economics Letters, 13, 650-665. https://doi.org/10.4236/tel.2023.133039