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Move-based hedging of variable annuities: A semi-analytic approach
Insurance: Mathematics and Economics 71 (2016) 40–49
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Move-based hedging of variable annuities: A semi-analytic approach X. Sheldon Lin a,∗ , Panpan Wu b , Xiao Wang c a
Department of Statistical Sciences, University of Toronto, Toronto, Ontario M5S 3G3, Canada
b
Quantitative Engineering and Development, TD Securities, Toronto, Ontario M5K 1A2, Canada
c
Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA 52242, USA
article
info
Article history: Received February 2014 Received in revised form July 2016 Accepted 27 July 2016 Available online 16 August 2016
abstract In this paper, we propose a semi-analytic algorithm for measuring the mean and variance of the cost associated with a two-sided move-based hedging of options written on an underlying asset whose price follows a geometric Brownian motion. Numerical examples are presented to illustrate the computational accuracy and efficiency of the algorithm. We then apply the technique to a structured product-based variable annuity with buffered protection and an annual ratchet variable annuity. © 2016 Elsevier B.V. All rights reserved.
Keywords: Variable annuities Embedded guarantees Move-based hedging Laplace transform Semi analytic algorithm Maturity randomization
1. Introduction Variable annuities (VA) have been overshadowing traditional fixed annuities to become the leading form of protected investment worldwide. In the US, the total sales peaked in 2007, just before the financial crisis, at 184 billion dollars. The VA sales were 140 billion and 133 billion in 2014 and 2015, respectively. See LIMRA (2015) for more details. We also see large VA sales in other countries. For example, Towers Watson reports that UK variable annuity sales were 1.11 billion pounds and 0.87 billion pounds in 2013 and 2014, respectively. According to a report by Oliver Wyman Limited (2007), the popularity of these contracts is driven by the demographic changes taking place in most parts of the world. The over-50 population is getting larger, richer and more diverse in life styles. They not only demand access to market appreciation in order to keep abreast of the rising cost of living, but also expect protection for their assets and well-being given increased uncertainty in the volatility of asset returns. With the various investment options and multiple forms of guarantees variable annuities can offer, they are able to choose the best fit for their desired risk/return target.
The guaranteed minimum benefits in a VA contract generally fall into two categories, namely, the guaranteed minimum death benefits (GMDB) and the guaranteed minimum living benefits (GMLB). The latter can be further divided into three types, the guaranteed minimum income benefits (GMIB), the guaranteed minimum accumulation benefits (GMAB) and the guaranteed minimum withdrawal benefits (GMWB). In each case, the policyholder is promised some future payments regardless of the performance of the subaccounts. In the context of derivatives, this payoff feature is equivalent to that of a financial option and can therefore be hedged in a similar way.1 Broadly speaking, there are four approaches for VA hedging. The first is no hedging at all. For small VA blocks, running naked may be acceptable. However, for larger ones, it can be very risky due to high market volatility that has been exhibited in recent years. The second is to buy reinsurance or structured products from a third party. Such products may offer substantial protection to the insurer, but they can be expensive or even unavailable for at least two reasons: firstly, they are customized products designed particularly to meet the insurer’s needs; secondly, reinsurers may be reluctant to offer coverage for the guaranteed variable annuities given the increased market risks of these products. The third
∗
Corresponding author. E-mail addresses: [email protected] (X.S. Lin), [email protected] (P. Wu), [email protected] (X. Wang). http://dx.doi.org/10.1016/j.insmatheco.2016.07.007 0167-6687/© 2016 Elsevier B.V. All rights reserved.
1 In this paper, we are only concerned about the financial risks of the VA product. The mortality risks are assumed to be diversifiable.
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
choice is static hedging. This strategy aims to offset the embedded option in the VA contract through buying a portfolio of options from the market. Since the payoff structure of the embedded option is sometimes too exotic to be decomposed as a combination of the payoff of options available in the market, basis risk can be significant. Moreover, VA contracts usually span over a long period of time, but options longer than five years in maturity may not be available in the market for reasonable prices. The last strategy, which we will discuss in much detail here, dates back to the seminal option-pricing paper by Black and Scholes (1973), which proposed a continuous hedging strategy known as delta hedging. Since then, various refinements of delta hedging have been developed, including delta–gamma hedging, delta–vega hedging, delta–rho hedging, mean–variance hedging, local riskminimization hedging, utility based hedging, etc. See Zakamouline (2009), Cerny (2007), and references therein for the comparison of alternative hedging strategies for different contract types, market conditions and model assumptions. Though perfect in theory, continuous hedging demands rebalancing the hedging portfolio continuously in time, which is impossible in practice. Therefore, discrete hedging is employed as an approximation. Using this strategy, one constructs the same initial portfolio as continuous hedging, but adjusts it discretely in time. This difference gives rise to a non-self-financing replicating portfolio. Hence, the discrete hedging cost consists of two parts, one is the cost of constructing the initial hedging portfolio, the other is the cost associated with the subsequent rebalances. There are mainly two kinds of discrete hedging strategies, timebased and move-based. The former hedges the option at equally spaced points in time. Boyle and Emanuel (1980) is one of the first studies on the distribution of the local tracking error of time-based discrete hedging. For the global tracking error, Bertsimas et al. (2000) derived the asymptotic distribution of the tracking error at each rebalancing point, as the number of rebalancing points tends to infinity; Hayashi and Mykland (2005) generalized the result of Bertsimas et al. (2000) to continuous Itô processes and also suggested a data-driven nonparametric hedging strategy for the case of unknown underlying dynamics; Angelini and Herzel (2009) computed the mean and the variance of the error of a hedging strategy for a contingent claim when trading in discrete time, which are valid for any fixed number of trading dates (however, their methods are not applicable to the move-based hedging. See the end of Section 2 of their paper for this point); Sepp (2012) derived an approximation for the probability density function of the profit-and-loss (PnL) of the time-based delta hedging strategy for vanilla options under the diffusion model and a proposed jump–diffusion model assuming discrete trading intervals and transaction costs. Despite its analytic tractability, the time-based strategy is a plain approximation to continuous hedging with no regard to the volatility risk. When the volatility is high, the value of the subaccount fluctuates more intensively over a short period, which necessarily requires frequent rebalancing of the hedging portfolio. A wiser choice for this situation is the move-based strategy, which hedges whenever the value of the underlying asset moves outside a prescribed region. Cost estimation for the move-based discrete hedging is mathematically complex because it involves a number of dependent and right-censored hitting times as well as the values of the underlying asset at these hitting times. One approach in practice is to analyze the cost through Monte Carlo simulation. See Boyle and Hardy (1997) for example. Although straightforward by its nature, the Monte Carlo method has certain drawbacks. The pathdependency of the total hedging cost requires the generation of the whole trajectory of the asset price at each iteration, which is done by discretization. As pointed out in Glasserman (2003), this leads to bias in estimation, known as discretization error. Reducing the discretization error requires shortening the time step in
41
path generation,2 which turns out to be computationally timeconsuming (see Section 3.1 for the comments on the computational time and the convergence rate of the Monte Carlo methods). Alternatively, one may use analytic approximation. For a certain type of move-based hedging, Dupire (2005) derived the limit of the end-of-period tracking error as the bandwidth goes to zero. Henrotte (1993) found approximation formulas for expected transactions costs and the variance of the total cash flow from both time-based and move-based strategies. Toft (1996) extended the work of Henrotte (1993) by showing how these expressions can be simplified and computed efficiently. As a matter of fact, all the analytic results we mentioned above are asymptotic. Indeed, Dupire’s and Henrotte’s expressions are obtained in the limit as the bandwidth, the transactions costs and the time between rebalancing points, respectively, go to zero. These limits, however, are clearly unrealistic. Hence, there is a need to develop a method that can estimate the cost directly. In this paper, we investigate a move-based discrete hedging strategy and develop semi-analytic algorithms to compute the mean and variance of the rebalancing cost associated with the move-based strategy. Although quantifying the exact cost distribution seems infeasible, the calculation of the mean and variance is still of practical importance as they, respectively, are the proper measures of the expected return/loss and risk of the move-based hedging. We use a geometric Brownian motion model for the underlying asset and delta hedge with one hedging instrument only. One motivation for this particular choice of hedging strategy is that the execution of more advanced strategies requires hedging instruments other than the underlying subaccount. The selection of these instruments is companyand/or market-specific, due to issues such as liquidity, trading constraints and corporate policies. Hence, a universal analysis of more advanced hedging strategies is somewhat impossible. The paper is organized as follows. Section 2 gives an overview of the move-based hedging strategy and the corresponding cost. Empirical analysis is conducted on the effectiveness of the movebased hedging. Section 3 describes the semi-analytic algorithm we develop for calculating the mean and variance of the hedging cost. Section 4 applies the technique to two types of variable annuities: the structured product-based VA and the annual ratchet VA. Section 5 concludes the paper. 2. A move-based hedging strategy The main objective of this section is to provide a description of a specific move-based hedging strategy and to formulate the corresponding net hedging cost. In addition, we display some numerical examples in comparing the hedging effectiveness of the move-based and time-based strategies. 2.1. Description of move-based hedging and its cost Suppose there is a risky asset paying dividends continuously at a proportional rate η. The time-t price St is modeled as a geometric Brownian motion given by 2 St = S0 e(µ−η−σ /2)t +σ Wt ,
t ≥ 0,
(2.1)
2 The discretization error is different from the statistical error. While the latter can be reduced by increasing the number of iterations, the former has to be reduced by shortening the time step in the process generation or by high order approximation methods. If we assume the geometric Brownian motion model for the underlying asset, the formula for the first order approximation of the process is the same as those with higher orders, so the only way to reduce discretization error is through finer time steps.
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where S0 is the initial price, µ is the expected instantaneous rate of return, σ is the volatility and {Wt } is a standard Brownian motion. We also assume a money market where we can borrow and lend at a constant risk-free interest rate r > η. Consider writing one unit of a path-independent option on the risky asset with payoff Π (ST ) at maturity time T . The premium received at time 0 is P0 . To hedge the option, we construct a replicating portfolio by buying ∆0 units of the underlying asset and investing the amount of P0 − ∆0 S0 in the money market. As time passes, we replicate the payoff of the option discretely in time according to the following rule: a band [S0 eα , S0 e−α ] with a positive α is prescribed, creating a so-called no rebalancing region. Rehedging of the portfolio is not triggered until the asset price hits the boundary of this region. In particular, at the hitting time, denoted by t1 for example, the band is reset to be [St1 eα , St1 e−α ] and ∆t1 units of the underlying asset are held in the hedging portfolio with all remaining wealth put in the money market. The procedure is repeated until the maturity of the option. The parameter α serves as an indicator of the volatility risk of rehedging and can be derived from the level of the hedger’s risk aversion. More frequent rebalancing requires a smaller α . As a result, more risk-averse investors tend to specify lower value of α while more risk-tolerant investors prefer larger value of α . The consecutive hitting times can be defined recursively by tn+1 = inf{t > tn | St = Stn eα or Stn e−α },
t0 = 0,
n ≥ 0. (2.2)
We assume the hedge ratio only depends on the current asset value and the time to maturity, that is, we assume
∆t = ∆(St , T − t ),
t ≥ 0,
Time-based (weekly rebalancing)
Move-based
Mean Median Standard deviation Skewness Kurtosis Quantile 0.75 Quantile 0.9 Quantile 0.95
0.0087 0.0090 0.4307 0.1209 4.9108 0.2485 0.5120 0.7069
0.0033 0.0126 0.2531 −0.3830 5.2723 0.1486 0.2923 0.3931
Table 2.2 Comparison of hedging costs of time-based and move-based strategies for an atthe-money put option with three-year maturity. µ = 0.1, σ = 0.2, α = 0.0577. Summary statistics
Time-based (monthly rebalancing)
Move-based
Mean Median Standard deviation Skewness Kurtosis Quantile 0.75 Quantile 0.9 Quantile 0.95
0.0382 0.0329 0.8904 0.2311 4.7883 0.5359 1.0843 1.4926
0.0125 0.0516 0.5346 −0.6024 5.2632 0.3248 0.6076 0.8071
2.2. Numerical examples Now we compare the performance of the time-based and the proposed move-based strategies when hedging an at-the-money European put option by numerical analysis. The value of the asset evolves according to model (2.1). We generate sample paths of the underlying asset over a T -year period at discrete time points un = nv in the following manner: √
P (Stn , T − tn ) − ∆(Stn , T − tn )Stn
n = 0, 1, . . . , N − 1,
(2.6)
where Zn ’s are i.i.d. standard normal random variables and v = T /N is the step size. We simulate 105 sample paths. The strike price of the put option is K . The values of all common parameters are given by µ = 0.1, σ = 0.2, r = 0.02, η = 0, T = 3, S0 = K = 50. The time-0 price of the put option is P0 = Ke−rT N (−d2 (S0 , T )) − S0 N (−d1 (S0 , T )),
in the money market. Introduce the right-censored hitting times tˆn = tn ∧ T , n ≥ 0. The present value of the cumulative gains (or losses if negative) from trading the asset is ∞
Summary statistics
2 Sun+1 = Sun e(µ−η−σ /2)v+σ v Zn ,
which is true if the hedge ratio is the Black–Scholes delta. During each interval between tn and tn+1 , where tn+1 ≤ T , the move-based hedging consists of holding ∆(Stn , T − tn ) units of the underlying asset and investing the amount of
G(T ) =
Table 2.1 Comparison of hedging costs of time-based and move-based strategies for an atthe-money put option with three-year maturity. µ = 0.1, σ = 0.2, α = 0.0277.
Gn (T ),
(2.3)
n =0
where
where d 1 ( s, t ) =
ln(s/K ) + (r + σ 2 /2)t
√ σ t √ d2 (s, t ) = d1 (s, t ) − σ t ,
,
and N (·) is the standard normal distribution function. We use the Black–Scholes delta
∆t = ∆(s, T − t ) = −N (−d1 (s, T − t )).
Gn (T ) = ∆(Stˆn , T − tˆn ) e
−r tˆn+1 +η(tˆn+1 −tˆn )
Stˆn+1 − e
−r tˆn
Stˆn .
(2.4)
Note that the present value of the gains due to trading in the money market is zero. So the present value of the entire hedging portfolio is P0 + G(T ). The discrete rebalancing results in a nonself-financing replicating portfolio and consequently, a non-zero difference between the present value of the maturity payout and the hedging portfolio given by H (T ) = e−rT Π (ST ) − (P0 + G(T )).
(2.5)
This discrepancy is usually called the hedging error or the profitand-loss (PnL). In this paper, we refer to H (T ) as the (net) hedging cost of the move-based strategy.
For the time-based strategy, N is the number of rebalancing times. We study monthly and weekly rebalancing, so N = 36 and 156. For the move-based strategy, we choose N = T × 106 (v = 10−6 ) to achieve more precise simulation results. At each time point un , the asset price is examined to decide if rehedging is necessary. To conduct a fairly reasonable comparison, we determine the value of α such that the expected hedging frequency of the move-based strategy matches the number of rebalancing points of the time-based strategy. In particular, α = 0.0577 and 0.0277 correspond to N = 36 and 156 respectively in the time-based case. See Appendix C for the calculation of the expected hedging frequency of the move-based strategy. Tables 2.1 and 2.2 and Fig. 2.1 show the empirical performance of time-based and move-based strategies assuming the same
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
43
Fig. 2.1. Probability densities of hedging cost.
asset volatility and (expected) rebalancing frequency. The cost associated with the move-based hedging exhibits smaller mean, smaller standard deviation and thinner right tail. Martellini and Priaulet (2002) and Zakamouline (2009) both provided systematic comparisons of several competing hedging methods within a mean–variance framework in the presence of proportional transaction costs, which is not considered in this paper. They also found the advantage of move-based method over time-based one.
variable which is independent of {St } and has mean 1/λ. Then we have E[Gn (ϵλ )] =
∞
E[Gn (T )]λe−λT dT .
0
Hence, the Laplace transform of E[Gn (T )] is ∞
E[Gn (T )]e−λT dT =
E[Gn (ϵλ )]
λ
0
3. Measuring hedging cost: a semi-analytic algorithm In this section, we develop a semi-analytic algorithm to quantify the cost arising from the move-based discrete hedging. Specifically, the proposed method is aimed at calculating the expectation and variance (and higher-order moments) of the hedging cost. We also implement simulation studies to demonstrate the computational efficiency of the algorithm. Before proceeding with the main approach, let us first describe a distributional property of the hitting times defined by (2.2). Denote by τ the first time when the asset value {St } hits S0 eα or S0 e−α . More formally,
τ = inf{t > 0 | St = S0 eα or St = S0 e−α }. To deal with the consecutive hitting times tn , define τ (n) to be the inter-hitting time between the (n − 1)-st hit and the n-th hit, that is, τ (n) = tn − tn−1 . Hence, τ (1) = t1 = τ . By the strong Markov property and the stationary increment property of Brownian motion, τ (n) , n ≥ 1, are independent and have the same distribution as τ .
.
(3.2)
Taking advantage of the memoryless property of exponential distribution, we derive a semi-analytic formula for E[Gn (ϵλ )], which can be computed numerically. Inverting the Laplace transform in (3.2) leads to the expected value of each individual trading gain for fixed maturity time T . Throughout this paper, we employ the Matlab routine INVLAP.M to invert Laplace transforms. The routine computes the associated Bromwich integral using the quotient difference method with accelerated convergence developed in de Hoog et al. (1982). Lemma 3.1. Let f (s) be a payoff function with s being the value of the underlying asset at the time of payoff. Assume an exponential time horizon ϵλ , independent of {St } and with mean 1/λ. The hitting times tn , n ≥ 0, are defined by (2.2). Then the expected present value of a contingent claim with payoff f (Stn ) due at time tn if tn < ϵλ is given by E e−rtn f (Stn )1(tn < ϵλ )
=
n n i=0
i
−i f S0 e(2i−n)α Lir ,α Lnr ,−α ,
n ≥ 0,
(3.3)
where 1(·) is the indicator function and Lr ,±α = E e−r τ 1(Sτ = S0 e±α , τ < ϵλ ) .
3.1. The expectation of the hedging cost
Proof. See Appendix A.
From (2.3) and (2.5), the expected hedging cost is E[H (T )] = E[e−rT Π (ST )] − P0 −
∞
E[Gn (T )].
(3.1)
n =0
A direct calculation of E[Gn (T )] is a challenging task. To tackle the problem, we propose an algorithm by first determining the Laplace transform of E[Gn (T )] when the maturity time T is treated as a variable. In particular, let ϵλ denote an exponential random
Here we give a heuristic explanation of Lemma 3.1: The path of the asset price at each hitting time follows a binomial tree framework with random lengths of period τ (n+1) = tn+1 − tn , n ≥ 0, which are independent and identically distributed as τ . Thanks to the memoryless property of ϵλ , Lr ,α and Lr ,−α can be understood as the discount factors for calculating the present value provided that the asset price is the upper level and lower level, respectively. Therefore, we would expect the result in
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X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
Lemma 3.1 to resemble a binomial tree pricing formula. Using the law of iterated expectation and the independence assumption, the assertion (3.3) can be rewritten in a simpler form: E e−(r +λ)tn f (Stn ) =
n n i=0
where Lr ,±α = E e
−(r +λ)τ
f S0 e
i
(2i−n)α
−i Lir ,α Lnr ,−α ,
(3.4)
1(Sτ = S0 e±α ) . Define
τα = inf{t > 0|St = S0 eα , Su > S0 e−α for all u in (0, t )};
Table 3.1 Comparison of the expected hedging costs obtained from the semi-analytic algorithm and the Monte Carlo simulations. T
Semi-analytic
Monte Carlo (v = 10−4 )
Monte Carlo (v = 10−6 )
1 2 3 4 5
0.1395 0.1840 0.1993 0.2021 0.1992
0.1491 (0.0030) 0.1794 (0.0021) 0.1937 (0.0026) 0.2060 (0.0030) 0.2012 (0.0020)
0.1383 (0.0029) 0.1830 (0.0026) 0.1990 (0.0023) 0.2024 (0.0021) 0.1993 (0.0019)
τ−α = inf{t > 0|St = S0 e−α , Su < S0 eα for all u in (0, t )}.
By letting n = 0 in (3.11), and noting t0 = 0 and t1 = τ (1) = τ ,
Then τα is the first time when {St } hits S0 eα without hitting S0 e−α earlier, and τ−α is the first time when {St } hits S0 e−α without hitting S0 eα earlier. Note that τα = ∞ if τ = τ−α . Hence,
h(s) = E ∆(S0 , ϵλ ) e−(r −η)(τ ∧ϵλ ) Sτ ∧ϵλ − S0 |S0 = s .
e−(r +λ)τα 1(τ = τ−α ) = 0, and consequently, Lr ,α = E e−(r +λ)τ 1(τ = τα ) = E e−(r +λ)τα .
(3.5)
Lr ,−α = E e−(r +λ)τ−α .
(3.6)
Therefore, Lr ,α and Lr ,−α can be viewed as the Laplace transforms of exit times of linear Brownian motion and can be calculated explicitly using the optional sampling theorem. Their analytic forms are given, for example, by equations (13) and (14) in Lin (1998). Let θ1 and θ2 be the two roots of the quadratic equation
2
1 σ 2 θ 2 + µ − η − σ 2 θ − (r + λ) = 0.
(3.7)
2
Thus
Lr ,α Lr ,−α
(3.12)
Due to the memoryless property of ϵλ , given the event tn < ϵλ , ϵλ − tn has the same distribution as ϵλ and thus, τˆ (n+1) has the same distribution as τ ∧ ϵλ . Using the strong Markov property of Brownian motion and comparing (3.11) with (3.12), (3.11) becomes e−(r +λ)tn Etn ∆(Stn , ϵλ ) e−(r −η)(τ ∧ϵλ ) Stn +τ ∧ϵλ − Stn
= e−(r +λ)tn h(Stn ).
Likewise,
1
=
eθ1 α eθ2 α
e−θ1 α e−θ2 α
−1
1 . 1
(3.8)
Theorem 3.1. Denote by ϵλ an exponential random variable which is independent of {St } and has mean 1/λ. Individual gains Gn (T ), n ≥ 0, are defined by (2.4). Then we have E [Gn (ϵλ )] =
n n i =0
i
−i h S0 e(2i−n)α Lir ,α Lnr ,−α ,
n ≥ 0,
(3.9)
where h(s) = E [G0 (ϵλ )|S0 = s] and Lr ,±α are given by (3.5) and (3.6). Proof. By Lemma 3.1, it is sufficient to show E [Gn (ϵλ )] = E e−(r +λ)tn h(Stn ) .
(3.10)
Eq. (3.10) is trivially true when n = 0 by the definition of h(s). Given the event tn < ϵλ , define τˆ (n+1) by
τˆ (n+1) = τ (n+1) ∧ (ϵλ − tn ),
n ≥ 0,
where τ (n+1) is the inter-hitting time. Hence, tn+1 ∧ϵλ = tn +τˆ (n+1) . According to (2.4), the expression for Gn (ϵλ ) can be rewritten as
∆(Stn , ϵλ − tn ) × e−r (tn+1 ∧ϵλ )+η(tn+1 ∧ϵλ −tn ) Stn+1 ∧ϵλ − e−rtn Stn 1(tn < ϵλ ) (n+1) = e−rtn ∆(Stn , ϵλ − tn ) e−(r −η)τˆ Stn +τˆ (n+1) − Stn
The law of iterated expectation yields E [Gn (ϵλ )] = E Etn [Gn (ϵλ )] = E e−(r +λ)tn h(Stn ) ,
which completes the proof.
A closed-form formula for h(s) is difficult to derive even for plain vanilla options such as European calls and puts. Thus, we need numerical computation. In Appendix B, we discuss in detail the methodology for computing h(s). As shown in (3.2), dividing (3.9) by λ gives the Laplace transform of E [Gn (T )] for each n and fixed maturity T . The Laplace transform is then inverted numerically. There usually exist explicit expressions for E[e−rT Π (ST )] and P0 , and as a result, the expected hedging cost can be readily evaluated using (3.1). Now we implement the proposed algorithm to compute the expected cost of the move-based hedging for at-the-money European put options with different maturity times and compare the results with those given by the Monte Carlo simulations. We generate 105 sample paths according to (2.6). The values of all common parameters are given by µ = 0.2, σ = 0.2, η = 0, r = 0.02, α = 0.1, S0 = K = 50. Two step sizes v = 10−4 and 10−6 are considered. In Table 3.1, the number in the bracket represents the standard error of the Monte Carlo estimator with respect to the semi-analytic value. Note that the standard error is a measure of the statistical error, not the discretization error. The discretization error can be evaluated by the changes in the estimates when shortening the time step in the path generation. While the statistical errors shown in the brackets appear to be small, the discretization errors are large. When v = 10−4 , we still observe a relatively significant discrepancy between the simulation results and those obtained from the proposed algorithm. In order for convergence, we further refine the step size to v = 10−6 . However, this substantially increases the computational time. In fact, it takes about three and a half hours on average to compute each number in the column ‘‘Monte Carlo (v = 10−6 )’’. In contrast, the result can be obtained within seven minutes using the semi-analytic algorithm. 3.2. The variance of the hedging cost
× 1(tn < ϵλ ). Therefore, by the independence between ϵλ and {St }, the expectation of Gn (ϵλ ) conditional on tn is equal to (n+1) e−(r +λ)tn Etn ∆(Stn , ϵλ − tn ) e−(r −η)τˆ St +τˆ (n+1) − Stn tn < ϵλ . n (3.11)
This subsection is to discuss the estimation of the variance of the hedging cost. Recall that the hedging cost is H (T ) = e−rT Π (ST ) − P0 −
∞ n =0
Gn (T ),
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
where Gn (T ) is given by (2.4). A direct calculation of the variance of the above seems difficult. We have found the following approach. We decompose the hedging cost into the sum of hedging costs of individual trading periods. Define M (St , T − t ) = P (St , T − t ) − ∆(St , T − t )St to be the amount invested in the money market at time t. Then the hedging cost can be rewritten in the following alternative form: H (T ) =
∞
Hn (T ),
(3.13)
n=0
where
ˆ ˆ ˆ Hn (T ) = e−r tn+1 P (Stˆn+1 , T − tˆn+1 ) − er (tn+1 −tn ) M (Stˆn , T − tˆn )
− ∆(Stˆn , T − tˆn )eη(tˆn+1 −tˆn ) Stˆn+1
(3.14)
is the individual hedging cost for the trading period between tn and tn+1 . We may interpret each Hn (T ) as the present value of the amount of money required to be infused or withdrawn to maintain the replicating portfolio. The advantage of decomposition (3.13) for estimating the variance is that, because of the independent increment property of Brownian motion, the correlation of individual hedging costs Hn (T ) turns out to be negligible as evidenced by numerical examples. In other words, the variance of the hedging cost can be approximated by the sum of the variances of each individual cost, Var(H (T )) ≈
∞
(3.15)
The algorithm of inverting Laplace transform as in the previous section can be readily applied to calculating Var(Hn (T )), which is presented in the following theorem: Theorem 3.2. Denote by ϵλ an exponential random variable which is independent of {St } and has mean 1/λ. Individual hedging costs Hn (T ), n ≥ 0, are defined by (3.14). Then we have
Table 3.2 Comparison of the variance for different values of µ and σ .
µ/σ
Semi-analytic
Monte Carlo
Monte Carlo independence
0.05/0.2 0.10/0.2 0.15/0.2 0.05/0.3 0.10/0.3 0.15/0.3
0.9945 0.8932 0.7244 1.0092 0.9870 0.9318
0.9895 0.8912 0.7420 1.0012 0.9855 0.9326
0.9953 0.8936 0.7247 1.0100 0.9877 0.9325
the Laplace transform of the right-hand side of (3.15). The columns ‘‘Monte Carlo’’ and ‘‘Monte Carlo Independence’’ display the values of the variance obtained by Monte Carlo simulation of the left-hand side and right-hand side of (3.15), respectively. We observe that the corresponding numbers in the last two columns are very close, which indicates that the approximation is valid and accurate. We also remark that the higher-order moments of the hedging cost can be calculated in an analogous manner. 4. Applications to variable annuities In this section, we look into two types of variable annuities: structured product-based variable annuities and variable annuities with annual ratchet guarantee. Our semi-analytic algorithm is used to compute the mean and variance of the cost from hedging these two types of variable annuities with move-based strategy. 4.1. Structured product-based variable annuity
Var(Hn (T )).
n =0
E Hnk (ϵλ ) =
45
n n
i
i =0
i h(k) S0 e(2i−n)α Likr ,α Lnkr−,−α ,
k ≥ 1, n ≥ 0,
(3.16)
where h(k) (s) = E H0k (ϵλ )|S0 = s and Lkr ,±α = E e−(kr +λ)τ±α .
Dividing (3.16) by λ yields the corresponding Laplace transforms of the kth raw moments of each individual cost. The variances of individual costs can be obtained by inverting the Laplace transforms from (3.16) with k = 1, 2. The variance of the total hedging cost is then approximated by (3.15). The expressions for h(1) (s) and h(2) (s) are complicated but can be calculated numerically with reference to Appendix B. Remark 3.1. Since the total hedging cost can be rewritten as (3.13), there is obviously an alternative way of calculating the expected hedging cost according to (3.16). In short, the Laplace transform of the expectation of H (T ) is given by ∞ n 1 n (1) (2i−n)α i n−i h S0 e Lr ,α Lr ,−α . λ n=0 i=0 i
We attempt to use an at-the-money put option to justify the soundness of the proposed approximation (3.15). The results are summarized in Table 3.2. The values of all common parameters are given by r = 0.02, η = 0, α = 0.1, T = 3, S0 = K = 50. The numbers in column ‘‘Semi-analytic’’ are obtained by inverting
Recently, a new type of variable annuities with payoffs similar to those of structured products has been introduced to the market. For example, AXA Equitable made available its first batch of Structured Capital Strategies, the structured products embedded in variable annuities on October 4, 2010. This instrument allows investors to select a reference asset (the S&P 500, Russell 2000, MSCI EAFE, gold or oil), a time frame (one, three or five years) and a certain level of downside protection (10%, 20% or 30%). See AXA Equitable (2012) for a detailed description. According to Deng et al. (2013), a typical payoff of a structured product-based variable annuity (spVA) that is linked to a reference asset ST at time T is
S + S0 b T
S0 Π (ST ) = ST S0 (1 + c )
ST ≤ S0 (1 − b) S0 (1 − b) < ST ≤ S0 S0 < ST ≤ S0 (1 + c ) S0 (1 + c ) < ST
where b is the buffer level within which the payoff is buffered against losses in the reference asset and c, the cap level, defines the maximal return of the spVA. The effects of b and c are illustrated in Fig. 4.1. Since the payoff of spVA is path-independent, the semi-analytic algorithm we develop in Section 3 is readily applicable to calculate the mean and variance of the rebalancing cost of the move-based strategy for hedging such a product assuming that the dynamics of the underlying asset follows a geometric Brownian motion. Again, we use (3.15) to approximate the variance. The simulation results are provided in Tables 4.1 and 4.2. There is an interesting trend in Tables 4.1 and 4.2. As b decreases or c increases, both the mean and variance of the hedging cost increase. From Fig. 4.1, it is clear that b controls the width of the flat part to the left of the dashed line and c controls how far the payoff can go beyond S0 . Therefore, as b decreases or c increases, the flat part in the payoff diagram shrinks. In other words, the variability of the payoff increases. In response to this increased variability, both the mean and variance of the hedging cost increase.
46
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
Table 4.1 The mean and variance of the rebalancing cost of hedged spVA for different values of the buffer level b and cap level c. The values of common parameters are given by µ = 0.1, σ = 0.3, r = 0.03, η = 0.02, α = 0.1, T = 3, S0 = 50. b/c
Semi-analytic
0.1/0.2 0.1/0.3 0.1/0.4 0.2/0.2 0.2/0.3 0.2/0.4
Monte Carlo
Monte Carlo independence
Mean
Variance
Mean
Variance
Mean
Variance
−0.0010
1.0651 1.2395 1.3960 0.9688 1.2064 1.3903
−0.0008
1.0548 1.2420 1.3941 0.9683 1.2146 1.3870
−0.0008
1.0659 1.2406 1.3974 0.9695 1.2073 1.3917
0.0128 0.0209 −0.0061 0.0085 0.0157
0.0124 0.0207 −0.0057 0.0088 0.0158
0.0124 0.0207 −0.0057 0.0088 0.0158
Table 4.2 The mean and variance of the rebalancing cost of hedged spVA for different values of the buffer level b and cap level c. The values of common parameters are given by µ = 0.15, σ = 0.4, r = 0.03, η = 0.02, α = 0.1, T = 3, S0 = 50. b/c
Semi-analytic
0.1/0.2 0.1/0.3 0.1/0.4 0.2/0.2 0.2/0.3 0.2/0.4
Monte Carlo
Monte Carlo independence
Mean
Variance
Mean
Variance
Mean
Variance
−0.0001
1.0739 1.2781 1.4645 0.9630 1.1795 1.3978
−0.0001
1.0735 1.2829 1.4720 0.9667 1.1835 1.3977
−0.0001
1.0748 1.2794 1.4662 0.9639 1.1806 1.3994
0.0055 0.0163 −0.0043 0.0042 0.0088
0.0059 0.0161 −0.0039 0.0038 0.0092
0.0059 0.0161 −0.0039 0.0038 0.0092
where S0∗ = S0 , Sk∗+1 = Sk∗
(1 + g ) ∨
Sk+1
Sk
,
k = 0, 1, . . . , n − 1.
(4.2)
In light of (4.1), the hedging strategy for the annual ratchet VA can be described in the following way: in period [0, 1], we hedge S0∗ units of a put option with payoff (1 + g −
) at time 1; in period
S1 + S0
[1, 2], we hedge S1∗ (note that this strategy is feasible since S1∗ is
known at time 1) units of a put option with payoff (1 + g − S2 )+ 1 at time 2; in general, in period [k, k + 1], 0 ≤ k ≤ n − 1, we S hedge Sk∗ units of a put option with payoff (1 + g − kS+1 )+ at time k k + 1. Because of the stationary increment property, the options we hedge in each year are all identical to a vanilla put option with strike price K = 1 + g, maturity time T = 1 and initial subaccount value S0 = 1, except that the number of units hedged differs from year to year. Thus, we may use the semi-analytic algorithm to calculate the raw moments of the discounted cost over each year. In particular, let Ck , k = 0, . . . , n − 1 be the discounted hedging cost at time k for period [k, k + 1]. Then Ck ’s are independent and identically distributed. For a n-year annual ratchet VA, the total hedging cost can be expressed as S
Fig. 4.1. Payoff diagram of spVA.
4.2. Annual ratchet variable annuity In this subsection, we consider the hedging of a n-year annual ratchet variable annuity. Under the annual ratchet design, the annual return of the subaccount of the past year is reset at the end of the year and is equal to the greater of the floor rate g and the realized market return of the subaccount. Suppose the annuitant invests in the VA subaccount which has a current value of S0 . At the end of the first year, the subaccount value is guaranteed to be the maximum of S0 (1 + g ) and S1 , denoted by S1∗ . The number of units in the subaccount with the guarantee S1∗
for the second year is now S . At the end of the second year, the 1 subaccount value is guaranteed to be the maximum of S1∗ (1+g ) and S1∗
S , denoted by S2 . . .. This process of annual ratcheting continues S1 2 to the end of the n-th year, resulting in the present value of the annual ratchet guarantee to be ∗
n −1
e
−(k+1)r
Sk ( 1 + g ) − ∗
k =0 n−1
=
k=0
e−(k+1)r Sk∗
Sk∗ Sk
1+g −
+
Hratchet (n) =
n −1 k=0
∗
where Sk ’s are given by (4.2). The first two moments of the total hedging cost are respectively given by E [Hratchet (n)] =
n−1
e−kr E Sk∗ Ck =
k=0
n −1
e−kr E Sk∗ E [Ck ] ,
k =0
and 2 E Hratchet ( n) =
Sk+1
e−kr Sk∗ Ck ,
n−1
e−2kr E Sk∗2 Ck2
k=0
Sk+1 Sk
+
,
(4.1)
+2
n −1 j>i≥0
e−(i+j)r E Si∗ Ci Sj∗ Cj
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49 Table 4.3 The mean and variance of the cost for hedging annual ratchet VA with different maturity times. The values of all common parameters are given by µ = 0.1, σ = 0.3, r = 0.03, η = 0.01, α = 0.1, g = 0.05, S0 = 50. n
Semi-analytic
2 3 4 5
Monte Carlo
Mean
Variance
Mean
Variance
−0.0154 −0.0245 −0.0355 −0.0483
2.8493 5.2105 8.5558 13.2748
−0.0152 −0.0249 −0.0331 −0.0434
2.8347 5.1867 8.5334 13.2364
=
n −1
Appendix A. Proof of Lemma 3.1 Proof. We shall prove the lemma by mathematical induction on n. We use the form (3.4) and it is trivially true when n = 1. Suppose (3.4) holds for n ≥ 1, that is, suppose
E e−(r +λ)tn f (Stn ) =
k=0
n−1
+2
e−(i+j)r E Si∗ Sj E [Ci ] E Cj .
∗
The first two moments of Ck can be evaluated using the proposed semi-analytic algorithm. By the iterative expression (4.2), we have
= S0m [R(m)]i , E Si∗ Sj∗ = S02 [R(1)]j−i [R(2)]i ,
n n
−i f S0 e(2i−n)α Lir ,α Lrn,−α .
i
i=0
(A.1)
j>i≥0
and the Natural Sciences and Engineering Council of Canada (NSERC, Grant #155723-2012). The authors would like to thank Dr. Hong Xie of Manulife Financial and the referees for their comments which have helped us improve the paper greatly.
e−2kr E Sk∗2 E Ck2
47
The goal is to evaluate E e−(r +λ)tn+1 f (Stn+1 ) . Because tn+1 = tn + τ (n+1) , the conditional expectation given tn is
Etn e−(r +λ)tn+1 f (Stn+1 )
∗m
E Si
(n+1) = e−(r +λ)tn Etn e−(r +λ)τ f Stn +τ (n+1) .
j > i ≥ 0,
where
Define a function q,
m Sk+1 R(m) = E (1 + g ) ∨
q(S0 ) = E e−(r +λ)τ f (Sτ )
Sk
= (1 + g )m Pr +E
Sk+1 Sk
= (1 + g ) N m
Sk+1 Sk
= f (S0 eα )Lr ,α + f (S0 e−α )Lr ,−α , <1+g
where τ is the first passage time introduced in Section 3. Note τ (n+1) has the same distribution as τ . By the strong Markov property of Brownian motion, the conditional expectation on the right-hand side of (A.2) is equal to q(Stn ). It follows from (A.2) and the definition of q(S0 ) that
m Sk+1 1 >1+g Sk
log(1 + g ) − (µ − η − σ 2 /2)
E e−(r +λ)tn+1 f (Stn+1 ) = E e−(r +λ)tn q(Stn )
σ
+ em(µ−η−σ /2)+m σ /2 − log(1 + g ) + (µ − η + (m − 1/2)σ 2 ) ×N . σ 2
(A.2)
= E e−(r +λ)tn f (Stn eα ) Lr ,α + E e−(r +λ)tn f (Stn e−α ) Lr ,−α .
2 2
We also point out that, by multinomial expansion theorem, the high-order moments of Hratchet (n) can be calculated similarly. In Table 4.3, we present the results of the mean and variance of the hedging cost for the annual ratchet VA with different maturity times. The results suggest an almost linear growth of the mean and variance with respect to the length of the maturity.
By assumption (A.1), we have: E e−(r +λ)tn f (Stn eα ) Lr ,α =
n n
i
i=0
=
−i f S0 e(2i−n+1)α Lir+,α1 Lnr ,−α
n+1 n n+1−j f S0 e(2j−(n+1))α Ljr ,α Lr ,−α , j − 1 j=1 (A.3)
5. Concluding remarks In this paper, we investigate a move-based discrete hedging strategy for variable annuities. We identify the two-sided underlier-based hedging as the most suitable under which it produces the cost distribution with the lightest right tail for a given hedging frequency. We develop a semi-analytic algorithm for the cost analysis of the move-based hedging. The key idea behind our algorithm is the Laplace transform representation, which is analogous to the maturity randomization approach introduced for financial applications in Carr (1998). In the future, we will explore the development of similar algorithms for more general Itô processes using ideas from Wang and Pötzelberger (2007). Acknowledgments This research was supported by grants from the Committee on Knowledge Extension Research (CKER) of the Actuarial Foundation
and E e−(r +λ)tn f (Stn e−α ) Lr ,−α
=
n n
f S0 e(2j−(n+1))α Ljr ,α Lr ,−α .
j
j =0
Applying the identity
n+1−j
n j −1
+
n j
=
n +1 j
(A.4) becomes
n +1 n+1 j=0
j
f S0 e(2j−(n+1))α Ljr ,α Lr ,−α ,
which proves the lemma.
n+1−j
(A.4)
, the sum of (A.3) and
48
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
Appendix B. Calculation of h(s)
− E e−(r −η+λ)τ−α ϕ(se−α , τ−α )
This appendix is to outline the procedure of calculating h(s) properly. The calculations of h(1) (s) and h(2) (s) are similar. Recall (3.12) that h(s) is given by h(s) = E ∆(S0 , ϵλ ) e−(r −η)(τ ∧ϵλ ) Sτ ∧ϵλ − S0 |S0 = s .
Fixing S0 , we decompose h(S0 ) into three parts:
E ∆(S0 , ϵλ )e−(r −η)τ Sτ 1(τ < ϵλ )
(B.2)
− E [∆(S0 , ϵλ )S0 ] .
(B.3)
φt (Sτ , τ ) = ∆(S0 , t )e
Sτ 1(τ < t ),
t ≥ 0.
= φt (S0 eα , τα ) + φt (S0 e−α , τ−α ). It follows that the expectation (B.1) becomes E φϵλ (S0 eα , τα ) + E φϵλ (S0 eα , τ−α ) .
(B.1′ )
∞
g (t ; ak ),
k=−∞
S0
By the memoryless property of ϵλ and the strong Markov property, we have E ∆(S0 , ϵλ )e−(r −η)ϵλ Sϵλ 1(τ < ϵλ )
= E e−(r −η+λ)τ ϕ(Sτ , τ ) = E e−(r −η+λ)τα ϕ(Sτα , τα ) + E e−(r −η+λ)τ−α ϕ(Sτ−α , τ−α ) = E e−(r −η+λ)τα ϕ(S0 eα , τα )
Thus, the expectation (B.2) is equal to
, τ−α ) .
(B.2′ )
To summarize, by combining (B.1 ), (B.2 ) and (B.3), h(s) is expressed in an involved but recognizable form for numerical integration: ′
h(s) = E φϵλ (seα , τα ) + E φϵλ (seα , τ−α )
knk−1 Pr(ν ≥ n),
k ≥ 1.
(C.2)
Our approach is again to find the Laplace transform of Pr(tn < T ) with T being a variable, which is equivalent to calculating Pr(tn < ϵλ ) where ϵλ is an independent exponential random variable. Applying Lemma 3.1 and Remark 3.1 where we set r = 0 and f = 1 in (3.3), the formula for Pr(tn < ϵλ ) immediately follows: Pr(tn < ϵλ ) =
λ
′
∞ n=1
1
α
ϕ(S0 e , τα )
ϕ(S0 e
(C.1)
n n
i
i n Li0,α L0n− ,−α = (L0,α + L0,−α ) ,
(C.3)
where L0,α and L0,−α are obtained from (3.7) and (3.8) with r = 0. By Eqs. (C.1)–(C.2), the Laplace transforms with parameter λ of Pr(tn < T ) and E [ν ] respectively are
+ E e−(r −η+λ)τ−α ϕ(S0 e−α , τ−α ) .
−α
i =0
−(r −η+λ)τ−α
E νk =
−(r −η+λ)τα
We are interested in the distribution, and especially the expected value of ν . As explained before, ν is a decreasing function of the bandwidth. For each n ≥ 0, we would like to evaluate Pr(ν ≥ n). The identity below follows from the definition of ν and the monotonicity of tn : To find the moments of ν , we use the equation
−(r −η)ϵλ Sϵλ ϕ(x, t ) = E ∆(S0 , ϵλ + t )e x .
Unlike the time-based strategy which trades in the underlying asset at pre-specified time points, the move-based hedging requires rebalancing the portfolio at a sequence of random times depending on the movement of the asset value. We let random variable ν be the hedging frequency, which keeps track of how many times rehedging takes place before maturity T . More precisely, define
Pr(ν ≥ n) = Pr(tn < T ).
To evaluate the expectation (B.2), we define
−αγ − γ 2 t /2 σ2
ν = max{n ≥ 0, tn < T }.
= φt (Sτα , τα ) + φt (Sτ−α , τ−α )
−E e
Appendix C. The distribution of hedging frequency
E [φt (Sτ , τ )] λe−λt dt .
φt (Sτ , τ ) = φt (Sτα , τα )1(τ = τα ) + φt (Sτ−α , τ−α )1(τ = τ−α )
f−α (t ) = exp
0
Thus,
ϕ(S0 , 0) − E e
and
∞
φt (Sτα , τα )1(τ = τ−α ) = φt (Sτ−α , τ−α )1(τ = τα ) = 0.
g (t ; ak ),
k=−∞
2π t
Let us calculate E [φt (Sτ , τ )]. Note that τ±α = ∞ if τ = τ∓α . Since r − η > 0, we show
∞
Gaussian distribution.
Then the expectation (B.1) is equal to E φϵλ (Sτ , τ ) =
αγ − γ 2 t /2 fα (t ) = exp σ2
where γ = µ − η − σ 2 /2, ak = σα (4k + 1) and g (t ; a) = 2 √ a e−a /2t , t > 0, is the probability density function of Inverse 3
Define −(r −η)τ
We need to know the density functions of τα and τ−α , which are given by equations (24) and (25) in Lin (1998). Let fα (t ) and f−α (t ) denote these two density functions. Then
(B.1)
+ E ∆(S0 , ϵλ )e−(r −η)ϵλ Sϵλ 1(τ > ϵλ )
− E [∆(s, ϵλ )] s.
+ ϕ(s, 0) − E e−(r −η+λ)τα ϕ(seα , τα )
(L0,α + L0,−α )n ,
n ≥ 0,
and 1 L0,α + L0,−α . λ 1 − (L0,α + L0,−α ) Fig. C.1 shows how the expected hedging frequency changes with respect to the bandwidth given two different asset volatilities over a three-year time horizon. We find by trial and error the corresponding value of α at which the expected hedging frequency of the move-based strategy is equal to the number of rebalancing points of the time-based strategy.
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
Fig. C.1. The expected hedging frequency as a decreasing function of the bandwidth over a three-year period. η = 0.
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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
Move-based hedging of variable annuities: A semi-analytic approach X. Sheldon Lin a,∗ , Panpan Wu b , Xiao Wang c a
Department of Statistical Sciences, University of Toronto, Toronto, Ontario M5S 3G3, Canada
b
Quantitative Engineering and Development, TD Securities, Toronto, Ontario M5K 1A2, Canada
c
Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA 52242, USA
article
info
Article history: Received February 2014 Received in revised form July 2016 Accepted 27 July 2016 Available online 16 August 2016
abstract In this paper, we propose a semi-analytic algorithm for measuring the mean and variance of the cost associated with a two-sided move-based hedging of options written on an underlying asset whose price follows a geometric Brownian motion. Numerical examples are presented to illustrate the computational accuracy and efficiency of the algorithm. We then apply the technique to a structured product-based variable annuity with buffered protection and an annual ratchet variable annuity. © 2016 Elsevier B.V. All rights reserved.
Keywords: Variable annuities Embedded guarantees Move-based hedging Laplace transform Semi analytic algorithm Maturity randomization
1. Introduction Variable annuities (VA) have been overshadowing traditional fixed annuities to become the leading form of protected investment worldwide. In the US, the total sales peaked in 2007, just before the financial crisis, at 184 billion dollars. The VA sales were 140 billion and 133 billion in 2014 and 2015, respectively. See LIMRA (2015) for more details. We also see large VA sales in other countries. For example, Towers Watson reports that UK variable annuity sales were 1.11 billion pounds and 0.87 billion pounds in 2013 and 2014, respectively. According to a report by Oliver Wyman Limited (2007), the popularity of these contracts is driven by the demographic changes taking place in most parts of the world. The over-50 population is getting larger, richer and more diverse in life styles. They not only demand access to market appreciation in order to keep abreast of the rising cost of living, but also expect protection for their assets and well-being given increased uncertainty in the volatility of asset returns. With the various investment options and multiple forms of guarantees variable annuities can offer, they are able to choose the best fit for their desired risk/return target.
The guaranteed minimum benefits in a VA contract generally fall into two categories, namely, the guaranteed minimum death benefits (GMDB) and the guaranteed minimum living benefits (GMLB). The latter can be further divided into three types, the guaranteed minimum income benefits (GMIB), the guaranteed minimum accumulation benefits (GMAB) and the guaranteed minimum withdrawal benefits (GMWB). In each case, the policyholder is promised some future payments regardless of the performance of the subaccounts. In the context of derivatives, this payoff feature is equivalent to that of a financial option and can therefore be hedged in a similar way.1 Broadly speaking, there are four approaches for VA hedging. The first is no hedging at all. For small VA blocks, running naked may be acceptable. However, for larger ones, it can be very risky due to high market volatility that has been exhibited in recent years. The second is to buy reinsurance or structured products from a third party. Such products may offer substantial protection to the insurer, but they can be expensive or even unavailable for at least two reasons: firstly, they are customized products designed particularly to meet the insurer’s needs; secondly, reinsurers may be reluctant to offer coverage for the guaranteed variable annuities given the increased market risks of these products. The third
∗
Corresponding author. E-mail addresses: [email protected] (X.S. Lin), [email protected] (P. Wu), [email protected] (X. Wang). http://dx.doi.org/10.1016/j.insmatheco.2016.07.007 0167-6687/© 2016 Elsevier B.V. All rights reserved.
1 In this paper, we are only concerned about the financial risks of the VA product. The mortality risks are assumed to be diversifiable.
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
choice is static hedging. This strategy aims to offset the embedded option in the VA contract through buying a portfolio of options from the market. Since the payoff structure of the embedded option is sometimes too exotic to be decomposed as a combination of the payoff of options available in the market, basis risk can be significant. Moreover, VA contracts usually span over a long period of time, but options longer than five years in maturity may not be available in the market for reasonable prices. The last strategy, which we will discuss in much detail here, dates back to the seminal option-pricing paper by Black and Scholes (1973), which proposed a continuous hedging strategy known as delta hedging. Since then, various refinements of delta hedging have been developed, including delta–gamma hedging, delta–vega hedging, delta–rho hedging, mean–variance hedging, local riskminimization hedging, utility based hedging, etc. See Zakamouline (2009), Cerny (2007), and references therein for the comparison of alternative hedging strategies for different contract types, market conditions and model assumptions. Though perfect in theory, continuous hedging demands rebalancing the hedging portfolio continuously in time, which is impossible in practice. Therefore, discrete hedging is employed as an approximation. Using this strategy, one constructs the same initial portfolio as continuous hedging, but adjusts it discretely in time. This difference gives rise to a non-self-financing replicating portfolio. Hence, the discrete hedging cost consists of two parts, one is the cost of constructing the initial hedging portfolio, the other is the cost associated with the subsequent rebalances. There are mainly two kinds of discrete hedging strategies, timebased and move-based. The former hedges the option at equally spaced points in time. Boyle and Emanuel (1980) is one of the first studies on the distribution of the local tracking error of time-based discrete hedging. For the global tracking error, Bertsimas et al. (2000) derived the asymptotic distribution of the tracking error at each rebalancing point, as the number of rebalancing points tends to infinity; Hayashi and Mykland (2005) generalized the result of Bertsimas et al. (2000) to continuous Itô processes and also suggested a data-driven nonparametric hedging strategy for the case of unknown underlying dynamics; Angelini and Herzel (2009) computed the mean and the variance of the error of a hedging strategy for a contingent claim when trading in discrete time, which are valid for any fixed number of trading dates (however, their methods are not applicable to the move-based hedging. See the end of Section 2 of their paper for this point); Sepp (2012) derived an approximation for the probability density function of the profit-and-loss (PnL) of the time-based delta hedging strategy for vanilla options under the diffusion model and a proposed jump–diffusion model assuming discrete trading intervals and transaction costs. Despite its analytic tractability, the time-based strategy is a plain approximation to continuous hedging with no regard to the volatility risk. When the volatility is high, the value of the subaccount fluctuates more intensively over a short period, which necessarily requires frequent rebalancing of the hedging portfolio. A wiser choice for this situation is the move-based strategy, which hedges whenever the value of the underlying asset moves outside a prescribed region. Cost estimation for the move-based discrete hedging is mathematically complex because it involves a number of dependent and right-censored hitting times as well as the values of the underlying asset at these hitting times. One approach in practice is to analyze the cost through Monte Carlo simulation. See Boyle and Hardy (1997) for example. Although straightforward by its nature, the Monte Carlo method has certain drawbacks. The pathdependency of the total hedging cost requires the generation of the whole trajectory of the asset price at each iteration, which is done by discretization. As pointed out in Glasserman (2003), this leads to bias in estimation, known as discretization error. Reducing the discretization error requires shortening the time step in
41
path generation,2 which turns out to be computationally timeconsuming (see Section 3.1 for the comments on the computational time and the convergence rate of the Monte Carlo methods). Alternatively, one may use analytic approximation. For a certain type of move-based hedging, Dupire (2005) derived the limit of the end-of-period tracking error as the bandwidth goes to zero. Henrotte (1993) found approximation formulas for expected transactions costs and the variance of the total cash flow from both time-based and move-based strategies. Toft (1996) extended the work of Henrotte (1993) by showing how these expressions can be simplified and computed efficiently. As a matter of fact, all the analytic results we mentioned above are asymptotic. Indeed, Dupire’s and Henrotte’s expressions are obtained in the limit as the bandwidth, the transactions costs and the time between rebalancing points, respectively, go to zero. These limits, however, are clearly unrealistic. Hence, there is a need to develop a method that can estimate the cost directly. In this paper, we investigate a move-based discrete hedging strategy and develop semi-analytic algorithms to compute the mean and variance of the rebalancing cost associated with the move-based strategy. Although quantifying the exact cost distribution seems infeasible, the calculation of the mean and variance is still of practical importance as they, respectively, are the proper measures of the expected return/loss and risk of the move-based hedging. We use a geometric Brownian motion model for the underlying asset and delta hedge with one hedging instrument only. One motivation for this particular choice of hedging strategy is that the execution of more advanced strategies requires hedging instruments other than the underlying subaccount. The selection of these instruments is companyand/or market-specific, due to issues such as liquidity, trading constraints and corporate policies. Hence, a universal analysis of more advanced hedging strategies is somewhat impossible. The paper is organized as follows. Section 2 gives an overview of the move-based hedging strategy and the corresponding cost. Empirical analysis is conducted on the effectiveness of the movebased hedging. Section 3 describes the semi-analytic algorithm we develop for calculating the mean and variance of the hedging cost. Section 4 applies the technique to two types of variable annuities: the structured product-based VA and the annual ratchet VA. Section 5 concludes the paper. 2. A move-based hedging strategy The main objective of this section is to provide a description of a specific move-based hedging strategy and to formulate the corresponding net hedging cost. In addition, we display some numerical examples in comparing the hedging effectiveness of the move-based and time-based strategies. 2.1. Description of move-based hedging and its cost Suppose there is a risky asset paying dividends continuously at a proportional rate η. The time-t price St is modeled as a geometric Brownian motion given by 2 St = S0 e(µ−η−σ /2)t +σ Wt ,
t ≥ 0,
(2.1)
2 The discretization error is different from the statistical error. While the latter can be reduced by increasing the number of iterations, the former has to be reduced by shortening the time step in the process generation or by high order approximation methods. If we assume the geometric Brownian motion model for the underlying asset, the formula for the first order approximation of the process is the same as those with higher orders, so the only way to reduce discretization error is through finer time steps.
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X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
where S0 is the initial price, µ is the expected instantaneous rate of return, σ is the volatility and {Wt } is a standard Brownian motion. We also assume a money market where we can borrow and lend at a constant risk-free interest rate r > η. Consider writing one unit of a path-independent option on the risky asset with payoff Π (ST ) at maturity time T . The premium received at time 0 is P0 . To hedge the option, we construct a replicating portfolio by buying ∆0 units of the underlying asset and investing the amount of P0 − ∆0 S0 in the money market. As time passes, we replicate the payoff of the option discretely in time according to the following rule: a band [S0 eα , S0 e−α ] with a positive α is prescribed, creating a so-called no rebalancing region. Rehedging of the portfolio is not triggered until the asset price hits the boundary of this region. In particular, at the hitting time, denoted by t1 for example, the band is reset to be [St1 eα , St1 e−α ] and ∆t1 units of the underlying asset are held in the hedging portfolio with all remaining wealth put in the money market. The procedure is repeated until the maturity of the option. The parameter α serves as an indicator of the volatility risk of rehedging and can be derived from the level of the hedger’s risk aversion. More frequent rebalancing requires a smaller α . As a result, more risk-averse investors tend to specify lower value of α while more risk-tolerant investors prefer larger value of α . The consecutive hitting times can be defined recursively by tn+1 = inf{t > tn | St = Stn eα or Stn e−α },
t0 = 0,
n ≥ 0. (2.2)
We assume the hedge ratio only depends on the current asset value and the time to maturity, that is, we assume
∆t = ∆(St , T − t ),
t ≥ 0,
Time-based (weekly rebalancing)
Move-based
Mean Median Standard deviation Skewness Kurtosis Quantile 0.75 Quantile 0.9 Quantile 0.95
0.0087 0.0090 0.4307 0.1209 4.9108 0.2485 0.5120 0.7069
0.0033 0.0126 0.2531 −0.3830 5.2723 0.1486 0.2923 0.3931
Table 2.2 Comparison of hedging costs of time-based and move-based strategies for an atthe-money put option with three-year maturity. µ = 0.1, σ = 0.2, α = 0.0577. Summary statistics
Time-based (monthly rebalancing)
Move-based
Mean Median Standard deviation Skewness Kurtosis Quantile 0.75 Quantile 0.9 Quantile 0.95
0.0382 0.0329 0.8904 0.2311 4.7883 0.5359 1.0843 1.4926
0.0125 0.0516 0.5346 −0.6024 5.2632 0.3248 0.6076 0.8071
2.2. Numerical examples Now we compare the performance of the time-based and the proposed move-based strategies when hedging an at-the-money European put option by numerical analysis. The value of the asset evolves according to model (2.1). We generate sample paths of the underlying asset over a T -year period at discrete time points un = nv in the following manner: √
P (Stn , T − tn ) − ∆(Stn , T − tn )Stn
n = 0, 1, . . . , N − 1,
(2.6)
where Zn ’s are i.i.d. standard normal random variables and v = T /N is the step size. We simulate 105 sample paths. The strike price of the put option is K . The values of all common parameters are given by µ = 0.1, σ = 0.2, r = 0.02, η = 0, T = 3, S0 = K = 50. The time-0 price of the put option is P0 = Ke−rT N (−d2 (S0 , T )) − S0 N (−d1 (S0 , T )),
in the money market. Introduce the right-censored hitting times tˆn = tn ∧ T , n ≥ 0. The present value of the cumulative gains (or losses if negative) from trading the asset is ∞
Summary statistics
2 Sun+1 = Sun e(µ−η−σ /2)v+σ v Zn ,
which is true if the hedge ratio is the Black–Scholes delta. During each interval between tn and tn+1 , where tn+1 ≤ T , the move-based hedging consists of holding ∆(Stn , T − tn ) units of the underlying asset and investing the amount of
G(T ) =
Table 2.1 Comparison of hedging costs of time-based and move-based strategies for an atthe-money put option with three-year maturity. µ = 0.1, σ = 0.2, α = 0.0277.
Gn (T ),
(2.3)
n =0
where
where d 1 ( s, t ) =
ln(s/K ) + (r + σ 2 /2)t
√ σ t √ d2 (s, t ) = d1 (s, t ) − σ t ,
,
and N (·) is the standard normal distribution function. We use the Black–Scholes delta
∆t = ∆(s, T − t ) = −N (−d1 (s, T − t )).
Gn (T ) = ∆(Stˆn , T − tˆn ) e
−r tˆn+1 +η(tˆn+1 −tˆn )
Stˆn+1 − e
−r tˆn
Stˆn .
(2.4)
Note that the present value of the gains due to trading in the money market is zero. So the present value of the entire hedging portfolio is P0 + G(T ). The discrete rebalancing results in a nonself-financing replicating portfolio and consequently, a non-zero difference between the present value of the maturity payout and the hedging portfolio given by H (T ) = e−rT Π (ST ) − (P0 + G(T )).
(2.5)
This discrepancy is usually called the hedging error or the profitand-loss (PnL). In this paper, we refer to H (T ) as the (net) hedging cost of the move-based strategy.
For the time-based strategy, N is the number of rebalancing times. We study monthly and weekly rebalancing, so N = 36 and 156. For the move-based strategy, we choose N = T × 106 (v = 10−6 ) to achieve more precise simulation results. At each time point un , the asset price is examined to decide if rehedging is necessary. To conduct a fairly reasonable comparison, we determine the value of α such that the expected hedging frequency of the move-based strategy matches the number of rebalancing points of the time-based strategy. In particular, α = 0.0577 and 0.0277 correspond to N = 36 and 156 respectively in the time-based case. See Appendix C for the calculation of the expected hedging frequency of the move-based strategy. Tables 2.1 and 2.2 and Fig. 2.1 show the empirical performance of time-based and move-based strategies assuming the same
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
43
Fig. 2.1. Probability densities of hedging cost.
asset volatility and (expected) rebalancing frequency. The cost associated with the move-based hedging exhibits smaller mean, smaller standard deviation and thinner right tail. Martellini and Priaulet (2002) and Zakamouline (2009) both provided systematic comparisons of several competing hedging methods within a mean–variance framework in the presence of proportional transaction costs, which is not considered in this paper. They also found the advantage of move-based method over time-based one.
variable which is independent of {St } and has mean 1/λ. Then we have E[Gn (ϵλ )] =
∞
E[Gn (T )]λe−λT dT .
0
Hence, the Laplace transform of E[Gn (T )] is ∞
E[Gn (T )]e−λT dT =
E[Gn (ϵλ )]
λ
0
3. Measuring hedging cost: a semi-analytic algorithm In this section, we develop a semi-analytic algorithm to quantify the cost arising from the move-based discrete hedging. Specifically, the proposed method is aimed at calculating the expectation and variance (and higher-order moments) of the hedging cost. We also implement simulation studies to demonstrate the computational efficiency of the algorithm. Before proceeding with the main approach, let us first describe a distributional property of the hitting times defined by (2.2). Denote by τ the first time when the asset value {St } hits S0 eα or S0 e−α . More formally,
τ = inf{t > 0 | St = S0 eα or St = S0 e−α }. To deal with the consecutive hitting times tn , define τ (n) to be the inter-hitting time between the (n − 1)-st hit and the n-th hit, that is, τ (n) = tn − tn−1 . Hence, τ (1) = t1 = τ . By the strong Markov property and the stationary increment property of Brownian motion, τ (n) , n ≥ 1, are independent and have the same distribution as τ .
.
(3.2)
Taking advantage of the memoryless property of exponential distribution, we derive a semi-analytic formula for E[Gn (ϵλ )], which can be computed numerically. Inverting the Laplace transform in (3.2) leads to the expected value of each individual trading gain for fixed maturity time T . Throughout this paper, we employ the Matlab routine INVLAP.M to invert Laplace transforms. The routine computes the associated Bromwich integral using the quotient difference method with accelerated convergence developed in de Hoog et al. (1982). Lemma 3.1. Let f (s) be a payoff function with s being the value of the underlying asset at the time of payoff. Assume an exponential time horizon ϵλ , independent of {St } and with mean 1/λ. The hitting times tn , n ≥ 0, are defined by (2.2). Then the expected present value of a contingent claim with payoff f (Stn ) due at time tn if tn < ϵλ is given by E e−rtn f (Stn )1(tn < ϵλ )
=
n n i=0
i
−i f S0 e(2i−n)α Lir ,α Lnr ,−α ,
n ≥ 0,
(3.3)
where 1(·) is the indicator function and Lr ,±α = E e−r τ 1(Sτ = S0 e±α , τ < ϵλ ) .
3.1. The expectation of the hedging cost
Proof. See Appendix A.
From (2.3) and (2.5), the expected hedging cost is E[H (T )] = E[e−rT Π (ST )] − P0 −
∞
E[Gn (T )].
(3.1)
n =0
A direct calculation of E[Gn (T )] is a challenging task. To tackle the problem, we propose an algorithm by first determining the Laplace transform of E[Gn (T )] when the maturity time T is treated as a variable. In particular, let ϵλ denote an exponential random
Here we give a heuristic explanation of Lemma 3.1: The path of the asset price at each hitting time follows a binomial tree framework with random lengths of period τ (n+1) = tn+1 − tn , n ≥ 0, which are independent and identically distributed as τ . Thanks to the memoryless property of ϵλ , Lr ,α and Lr ,−α can be understood as the discount factors for calculating the present value provided that the asset price is the upper level and lower level, respectively. Therefore, we would expect the result in
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X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
Lemma 3.1 to resemble a binomial tree pricing formula. Using the law of iterated expectation and the independence assumption, the assertion (3.3) can be rewritten in a simpler form: E e−(r +λ)tn f (Stn ) =
n n i=0
where Lr ,±α = E e
−(r +λ)τ
f S0 e
i
(2i−n)α
−i Lir ,α Lnr ,−α ,
(3.4)
1(Sτ = S0 e±α ) . Define
τα = inf{t > 0|St = S0 eα , Su > S0 e−α for all u in (0, t )};
Table 3.1 Comparison of the expected hedging costs obtained from the semi-analytic algorithm and the Monte Carlo simulations. T
Semi-analytic
Monte Carlo (v = 10−4 )
Monte Carlo (v = 10−6 )
1 2 3 4 5
0.1395 0.1840 0.1993 0.2021 0.1992
0.1491 (0.0030) 0.1794 (0.0021) 0.1937 (0.0026) 0.2060 (0.0030) 0.2012 (0.0020)
0.1383 (0.0029) 0.1830 (0.0026) 0.1990 (0.0023) 0.2024 (0.0021) 0.1993 (0.0019)
τ−α = inf{t > 0|St = S0 e−α , Su < S0 eα for all u in (0, t )}.
By letting n = 0 in (3.11), and noting t0 = 0 and t1 = τ (1) = τ ,
Then τα is the first time when {St } hits S0 eα without hitting S0 e−α earlier, and τ−α is the first time when {St } hits S0 e−α without hitting S0 eα earlier. Note that τα = ∞ if τ = τ−α . Hence,
h(s) = E ∆(S0 , ϵλ ) e−(r −η)(τ ∧ϵλ ) Sτ ∧ϵλ − S0 |S0 = s .
e−(r +λ)τα 1(τ = τ−α ) = 0, and consequently, Lr ,α = E e−(r +λ)τ 1(τ = τα ) = E e−(r +λ)τα .
(3.5)
Lr ,−α = E e−(r +λ)τ−α .
(3.6)
Therefore, Lr ,α and Lr ,−α can be viewed as the Laplace transforms of exit times of linear Brownian motion and can be calculated explicitly using the optional sampling theorem. Their analytic forms are given, for example, by equations (13) and (14) in Lin (1998). Let θ1 and θ2 be the two roots of the quadratic equation
2
1 σ 2 θ 2 + µ − η − σ 2 θ − (r + λ) = 0.
(3.7)
2
Thus
Lr ,α Lr ,−α
(3.12)
Due to the memoryless property of ϵλ , given the event tn < ϵλ , ϵλ − tn has the same distribution as ϵλ and thus, τˆ (n+1) has the same distribution as τ ∧ ϵλ . Using the strong Markov property of Brownian motion and comparing (3.11) with (3.12), (3.11) becomes e−(r +λ)tn Etn ∆(Stn , ϵλ ) e−(r −η)(τ ∧ϵλ ) Stn +τ ∧ϵλ − Stn
= e−(r +λ)tn h(Stn ).
Likewise,
1
=
eθ1 α eθ2 α
e−θ1 α e−θ2 α
−1
1 . 1
(3.8)
Theorem 3.1. Denote by ϵλ an exponential random variable which is independent of {St } and has mean 1/λ. Individual gains Gn (T ), n ≥ 0, are defined by (2.4). Then we have E [Gn (ϵλ )] =
n n i =0
i
−i h S0 e(2i−n)α Lir ,α Lnr ,−α ,
n ≥ 0,
(3.9)
where h(s) = E [G0 (ϵλ )|S0 = s] and Lr ,±α are given by (3.5) and (3.6). Proof. By Lemma 3.1, it is sufficient to show E [Gn (ϵλ )] = E e−(r +λ)tn h(Stn ) .
(3.10)
Eq. (3.10) is trivially true when n = 0 by the definition of h(s). Given the event tn < ϵλ , define τˆ (n+1) by
τˆ (n+1) = τ (n+1) ∧ (ϵλ − tn ),
n ≥ 0,
where τ (n+1) is the inter-hitting time. Hence, tn+1 ∧ϵλ = tn +τˆ (n+1) . According to (2.4), the expression for Gn (ϵλ ) can be rewritten as
∆(Stn , ϵλ − tn ) × e−r (tn+1 ∧ϵλ )+η(tn+1 ∧ϵλ −tn ) Stn+1 ∧ϵλ − e−rtn Stn 1(tn < ϵλ ) (n+1) = e−rtn ∆(Stn , ϵλ − tn ) e−(r −η)τˆ Stn +τˆ (n+1) − Stn
The law of iterated expectation yields E [Gn (ϵλ )] = E Etn [Gn (ϵλ )] = E e−(r +λ)tn h(Stn ) ,
which completes the proof.
A closed-form formula for h(s) is difficult to derive even for plain vanilla options such as European calls and puts. Thus, we need numerical computation. In Appendix B, we discuss in detail the methodology for computing h(s). As shown in (3.2), dividing (3.9) by λ gives the Laplace transform of E [Gn (T )] for each n and fixed maturity T . The Laplace transform is then inverted numerically. There usually exist explicit expressions for E[e−rT Π (ST )] and P0 , and as a result, the expected hedging cost can be readily evaluated using (3.1). Now we implement the proposed algorithm to compute the expected cost of the move-based hedging for at-the-money European put options with different maturity times and compare the results with those given by the Monte Carlo simulations. We generate 105 sample paths according to (2.6). The values of all common parameters are given by µ = 0.2, σ = 0.2, η = 0, r = 0.02, α = 0.1, S0 = K = 50. Two step sizes v = 10−4 and 10−6 are considered. In Table 3.1, the number in the bracket represents the standard error of the Monte Carlo estimator with respect to the semi-analytic value. Note that the standard error is a measure of the statistical error, not the discretization error. The discretization error can be evaluated by the changes in the estimates when shortening the time step in the path generation. While the statistical errors shown in the brackets appear to be small, the discretization errors are large. When v = 10−4 , we still observe a relatively significant discrepancy between the simulation results and those obtained from the proposed algorithm. In order for convergence, we further refine the step size to v = 10−6 . However, this substantially increases the computational time. In fact, it takes about three and a half hours on average to compute each number in the column ‘‘Monte Carlo (v = 10−6 )’’. In contrast, the result can be obtained within seven minutes using the semi-analytic algorithm. 3.2. The variance of the hedging cost
× 1(tn < ϵλ ). Therefore, by the independence between ϵλ and {St }, the expectation of Gn (ϵλ ) conditional on tn is equal to (n+1) e−(r +λ)tn Etn ∆(Stn , ϵλ − tn ) e−(r −η)τˆ St +τˆ (n+1) − Stn tn < ϵλ . n (3.11)
This subsection is to discuss the estimation of the variance of the hedging cost. Recall that the hedging cost is H (T ) = e−rT Π (ST ) − P0 −
∞ n =0
Gn (T ),
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
where Gn (T ) is given by (2.4). A direct calculation of the variance of the above seems difficult. We have found the following approach. We decompose the hedging cost into the sum of hedging costs of individual trading periods. Define M (St , T − t ) = P (St , T − t ) − ∆(St , T − t )St to be the amount invested in the money market at time t. Then the hedging cost can be rewritten in the following alternative form: H (T ) =
∞
Hn (T ),
(3.13)
n=0
where
ˆ ˆ ˆ Hn (T ) = e−r tn+1 P (Stˆn+1 , T − tˆn+1 ) − er (tn+1 −tn ) M (Stˆn , T − tˆn )
− ∆(Stˆn , T − tˆn )eη(tˆn+1 −tˆn ) Stˆn+1
(3.14)
is the individual hedging cost for the trading period between tn and tn+1 . We may interpret each Hn (T ) as the present value of the amount of money required to be infused or withdrawn to maintain the replicating portfolio. The advantage of decomposition (3.13) for estimating the variance is that, because of the independent increment property of Brownian motion, the correlation of individual hedging costs Hn (T ) turns out to be negligible as evidenced by numerical examples. In other words, the variance of the hedging cost can be approximated by the sum of the variances of each individual cost, Var(H (T )) ≈
∞
(3.15)
The algorithm of inverting Laplace transform as in the previous section can be readily applied to calculating Var(Hn (T )), which is presented in the following theorem: Theorem 3.2. Denote by ϵλ an exponential random variable which is independent of {St } and has mean 1/λ. Individual hedging costs Hn (T ), n ≥ 0, are defined by (3.14). Then we have
Table 3.2 Comparison of the variance for different values of µ and σ .
µ/σ
Semi-analytic
Monte Carlo
Monte Carlo independence
0.05/0.2 0.10/0.2 0.15/0.2 0.05/0.3 0.10/0.3 0.15/0.3
0.9945 0.8932 0.7244 1.0092 0.9870 0.9318
0.9895 0.8912 0.7420 1.0012 0.9855 0.9326
0.9953 0.8936 0.7247 1.0100 0.9877 0.9325
the Laplace transform of the right-hand side of (3.15). The columns ‘‘Monte Carlo’’ and ‘‘Monte Carlo Independence’’ display the values of the variance obtained by Monte Carlo simulation of the left-hand side and right-hand side of (3.15), respectively. We observe that the corresponding numbers in the last two columns are very close, which indicates that the approximation is valid and accurate. We also remark that the higher-order moments of the hedging cost can be calculated in an analogous manner. 4. Applications to variable annuities In this section, we look into two types of variable annuities: structured product-based variable annuities and variable annuities with annual ratchet guarantee. Our semi-analytic algorithm is used to compute the mean and variance of the cost from hedging these two types of variable annuities with move-based strategy. 4.1. Structured product-based variable annuity
Var(Hn (T )).
n =0
E Hnk (ϵλ ) =
45
n n
i
i =0
i h(k) S0 e(2i−n)α Likr ,α Lnkr−,−α ,
k ≥ 1, n ≥ 0,
(3.16)
where h(k) (s) = E H0k (ϵλ )|S0 = s and Lkr ,±α = E e−(kr +λ)τ±α .
Dividing (3.16) by λ yields the corresponding Laplace transforms of the kth raw moments of each individual cost. The variances of individual costs can be obtained by inverting the Laplace transforms from (3.16) with k = 1, 2. The variance of the total hedging cost is then approximated by (3.15). The expressions for h(1) (s) and h(2) (s) are complicated but can be calculated numerically with reference to Appendix B. Remark 3.1. Since the total hedging cost can be rewritten as (3.13), there is obviously an alternative way of calculating the expected hedging cost according to (3.16). In short, the Laplace transform of the expectation of H (T ) is given by ∞ n 1 n (1) (2i−n)α i n−i h S0 e Lr ,α Lr ,−α . λ n=0 i=0 i
We attempt to use an at-the-money put option to justify the soundness of the proposed approximation (3.15). The results are summarized in Table 3.2. The values of all common parameters are given by r = 0.02, η = 0, α = 0.1, T = 3, S0 = K = 50. The numbers in column ‘‘Semi-analytic’’ are obtained by inverting
Recently, a new type of variable annuities with payoffs similar to those of structured products has been introduced to the market. For example, AXA Equitable made available its first batch of Structured Capital Strategies, the structured products embedded in variable annuities on October 4, 2010. This instrument allows investors to select a reference asset (the S&P 500, Russell 2000, MSCI EAFE, gold or oil), a time frame (one, three or five years) and a certain level of downside protection (10%, 20% or 30%). See AXA Equitable (2012) for a detailed description. According to Deng et al. (2013), a typical payoff of a structured product-based variable annuity (spVA) that is linked to a reference asset ST at time T is
S + S0 b T
S0 Π (ST ) = ST S0 (1 + c )
ST ≤ S0 (1 − b) S0 (1 − b) < ST ≤ S0 S0 < ST ≤ S0 (1 + c ) S0 (1 + c ) < ST
where b is the buffer level within which the payoff is buffered against losses in the reference asset and c, the cap level, defines the maximal return of the spVA. The effects of b and c are illustrated in Fig. 4.1. Since the payoff of spVA is path-independent, the semi-analytic algorithm we develop in Section 3 is readily applicable to calculate the mean and variance of the rebalancing cost of the move-based strategy for hedging such a product assuming that the dynamics of the underlying asset follows a geometric Brownian motion. Again, we use (3.15) to approximate the variance. The simulation results are provided in Tables 4.1 and 4.2. There is an interesting trend in Tables 4.1 and 4.2. As b decreases or c increases, both the mean and variance of the hedging cost increase. From Fig. 4.1, it is clear that b controls the width of the flat part to the left of the dashed line and c controls how far the payoff can go beyond S0 . Therefore, as b decreases or c increases, the flat part in the payoff diagram shrinks. In other words, the variability of the payoff increases. In response to this increased variability, both the mean and variance of the hedging cost increase.
46
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
Table 4.1 The mean and variance of the rebalancing cost of hedged spVA for different values of the buffer level b and cap level c. The values of common parameters are given by µ = 0.1, σ = 0.3, r = 0.03, η = 0.02, α = 0.1, T = 3, S0 = 50. b/c
Semi-analytic
0.1/0.2 0.1/0.3 0.1/0.4 0.2/0.2 0.2/0.3 0.2/0.4
Monte Carlo
Monte Carlo independence
Mean
Variance
Mean
Variance
Mean
Variance
−0.0010
1.0651 1.2395 1.3960 0.9688 1.2064 1.3903
−0.0008
1.0548 1.2420 1.3941 0.9683 1.2146 1.3870
−0.0008
1.0659 1.2406 1.3974 0.9695 1.2073 1.3917
0.0128 0.0209 −0.0061 0.0085 0.0157
0.0124 0.0207 −0.0057 0.0088 0.0158
0.0124 0.0207 −0.0057 0.0088 0.0158
Table 4.2 The mean and variance of the rebalancing cost of hedged spVA for different values of the buffer level b and cap level c. The values of common parameters are given by µ = 0.15, σ = 0.4, r = 0.03, η = 0.02, α = 0.1, T = 3, S0 = 50. b/c
Semi-analytic
0.1/0.2 0.1/0.3 0.1/0.4 0.2/0.2 0.2/0.3 0.2/0.4
Monte Carlo
Monte Carlo independence
Mean
Variance
Mean
Variance
Mean
Variance
−0.0001
1.0739 1.2781 1.4645 0.9630 1.1795 1.3978
−0.0001
1.0735 1.2829 1.4720 0.9667 1.1835 1.3977
−0.0001
1.0748 1.2794 1.4662 0.9639 1.1806 1.3994
0.0055 0.0163 −0.0043 0.0042 0.0088
0.0059 0.0161 −0.0039 0.0038 0.0092
0.0059 0.0161 −0.0039 0.0038 0.0092
where S0∗ = S0 , Sk∗+1 = Sk∗
(1 + g ) ∨
Sk+1
Sk
,
k = 0, 1, . . . , n − 1.
(4.2)
In light of (4.1), the hedging strategy for the annual ratchet VA can be described in the following way: in period [0, 1], we hedge S0∗ units of a put option with payoff (1 + g −
) at time 1; in period
S1 + S0
[1, 2], we hedge S1∗ (note that this strategy is feasible since S1∗ is
known at time 1) units of a put option with payoff (1 + g − S2 )+ 1 at time 2; in general, in period [k, k + 1], 0 ≤ k ≤ n − 1, we S hedge Sk∗ units of a put option with payoff (1 + g − kS+1 )+ at time k k + 1. Because of the stationary increment property, the options we hedge in each year are all identical to a vanilla put option with strike price K = 1 + g, maturity time T = 1 and initial subaccount value S0 = 1, except that the number of units hedged differs from year to year. Thus, we may use the semi-analytic algorithm to calculate the raw moments of the discounted cost over each year. In particular, let Ck , k = 0, . . . , n − 1 be the discounted hedging cost at time k for period [k, k + 1]. Then Ck ’s are independent and identically distributed. For a n-year annual ratchet VA, the total hedging cost can be expressed as S
Fig. 4.1. Payoff diagram of spVA.
4.2. Annual ratchet variable annuity In this subsection, we consider the hedging of a n-year annual ratchet variable annuity. Under the annual ratchet design, the annual return of the subaccount of the past year is reset at the end of the year and is equal to the greater of the floor rate g and the realized market return of the subaccount. Suppose the annuitant invests in the VA subaccount which has a current value of S0 . At the end of the first year, the subaccount value is guaranteed to be the maximum of S0 (1 + g ) and S1 , denoted by S1∗ . The number of units in the subaccount with the guarantee S1∗
for the second year is now S . At the end of the second year, the 1 subaccount value is guaranteed to be the maximum of S1∗ (1+g ) and S1∗
S , denoted by S2 . . .. This process of annual ratcheting continues S1 2 to the end of the n-th year, resulting in the present value of the annual ratchet guarantee to be ∗
n −1
e
−(k+1)r
Sk ( 1 + g ) − ∗
k =0 n−1
=
k=0
e−(k+1)r Sk∗
Sk∗ Sk
1+g −
+
Hratchet (n) =
n −1 k=0
∗
where Sk ’s are given by (4.2). The first two moments of the total hedging cost are respectively given by E [Hratchet (n)] =
n−1
e−kr E Sk∗ Ck =
k=0
n −1
e−kr E Sk∗ E [Ck ] ,
k =0
and 2 E Hratchet ( n) =
Sk+1
e−kr Sk∗ Ck ,
n−1
e−2kr E Sk∗2 Ck2
k=0
Sk+1 Sk
+
,
(4.1)
+2
n −1 j>i≥0
e−(i+j)r E Si∗ Ci Sj∗ Cj
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49 Table 4.3 The mean and variance of the cost for hedging annual ratchet VA with different maturity times. The values of all common parameters are given by µ = 0.1, σ = 0.3, r = 0.03, η = 0.01, α = 0.1, g = 0.05, S0 = 50. n
Semi-analytic
2 3 4 5
Monte Carlo
Mean
Variance
Mean
Variance
−0.0154 −0.0245 −0.0355 −0.0483
2.8493 5.2105 8.5558 13.2748
−0.0152 −0.0249 −0.0331 −0.0434
2.8347 5.1867 8.5334 13.2364
=
n −1
Appendix A. Proof of Lemma 3.1 Proof. We shall prove the lemma by mathematical induction on n. We use the form (3.4) and it is trivially true when n = 1. Suppose (3.4) holds for n ≥ 1, that is, suppose
E e−(r +λ)tn f (Stn ) =
k=0
n−1
+2
e−(i+j)r E Si∗ Sj E [Ci ] E Cj .
∗
The first two moments of Ck can be evaluated using the proposed semi-analytic algorithm. By the iterative expression (4.2), we have
= S0m [R(m)]i , E Si∗ Sj∗ = S02 [R(1)]j−i [R(2)]i ,
n n
−i f S0 e(2i−n)α Lir ,α Lrn,−α .
i
i=0
(A.1)
j>i≥0
and the Natural Sciences and Engineering Council of Canada (NSERC, Grant #155723-2012). The authors would like to thank Dr. Hong Xie of Manulife Financial and the referees for their comments which have helped us improve the paper greatly.
e−2kr E Sk∗2 E Ck2
47
The goal is to evaluate E e−(r +λ)tn+1 f (Stn+1 ) . Because tn+1 = tn + τ (n+1) , the conditional expectation given tn is
Etn e−(r +λ)tn+1 f (Stn+1 )
∗m
E Si
(n+1) = e−(r +λ)tn Etn e−(r +λ)τ f Stn +τ (n+1) .
j > i ≥ 0,
where
Define a function q,
m Sk+1 R(m) = E (1 + g ) ∨
q(S0 ) = E e−(r +λ)τ f (Sτ )
Sk
= (1 + g )m Pr +E
Sk+1 Sk
= (1 + g ) N m
Sk+1 Sk
= f (S0 eα )Lr ,α + f (S0 e−α )Lr ,−α , <1+g
where τ is the first passage time introduced in Section 3. Note τ (n+1) has the same distribution as τ . By the strong Markov property of Brownian motion, the conditional expectation on the right-hand side of (A.2) is equal to q(Stn ). It follows from (A.2) and the definition of q(S0 ) that
m Sk+1 1 >1+g Sk
log(1 + g ) − (µ − η − σ 2 /2)
E e−(r +λ)tn+1 f (Stn+1 ) = E e−(r +λ)tn q(Stn )
σ
+ em(µ−η−σ /2)+m σ /2 − log(1 + g ) + (µ − η + (m − 1/2)σ 2 ) ×N . σ 2
(A.2)
= E e−(r +λ)tn f (Stn eα ) Lr ,α + E e−(r +λ)tn f (Stn e−α ) Lr ,−α .
2 2
We also point out that, by multinomial expansion theorem, the high-order moments of Hratchet (n) can be calculated similarly. In Table 4.3, we present the results of the mean and variance of the hedging cost for the annual ratchet VA with different maturity times. The results suggest an almost linear growth of the mean and variance with respect to the length of the maturity.
By assumption (A.1), we have: E e−(r +λ)tn f (Stn eα ) Lr ,α =
n n
i
i=0
=
−i f S0 e(2i−n+1)α Lir+,α1 Lnr ,−α
n+1 n n+1−j f S0 e(2j−(n+1))α Ljr ,α Lr ,−α , j − 1 j=1 (A.3)
5. Concluding remarks In this paper, we investigate a move-based discrete hedging strategy for variable annuities. We identify the two-sided underlier-based hedging as the most suitable under which it produces the cost distribution with the lightest right tail for a given hedging frequency. We develop a semi-analytic algorithm for the cost analysis of the move-based hedging. The key idea behind our algorithm is the Laplace transform representation, which is analogous to the maturity randomization approach introduced for financial applications in Carr (1998). In the future, we will explore the development of similar algorithms for more general Itô processes using ideas from Wang and Pötzelberger (2007). Acknowledgments This research was supported by grants from the Committee on Knowledge Extension Research (CKER) of the Actuarial Foundation
and E e−(r +λ)tn f (Stn e−α ) Lr ,−α
=
n n
f S0 e(2j−(n+1))α Ljr ,α Lr ,−α .
j
j =0
Applying the identity
n+1−j
n j −1
+
n j
=
n +1 j
(A.4) becomes
n +1 n+1 j=0
j
f S0 e(2j−(n+1))α Ljr ,α Lr ,−α ,
which proves the lemma.
n+1−j
(A.4)
, the sum of (A.3) and
48
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
Appendix B. Calculation of h(s)
− E e−(r −η+λ)τ−α ϕ(se−α , τ−α )
This appendix is to outline the procedure of calculating h(s) properly. The calculations of h(1) (s) and h(2) (s) are similar. Recall (3.12) that h(s) is given by h(s) = E ∆(S0 , ϵλ ) e−(r −η)(τ ∧ϵλ ) Sτ ∧ϵλ − S0 |S0 = s .
Fixing S0 , we decompose h(S0 ) into three parts:
E ∆(S0 , ϵλ )e−(r −η)τ Sτ 1(τ < ϵλ )
(B.2)
− E [∆(S0 , ϵλ )S0 ] .
(B.3)
φt (Sτ , τ ) = ∆(S0 , t )e
Sτ 1(τ < t ),
t ≥ 0.
= φt (S0 eα , τα ) + φt (S0 e−α , τ−α ). It follows that the expectation (B.1) becomes E φϵλ (S0 eα , τα ) + E φϵλ (S0 eα , τ−α ) .
(B.1′ )
∞
g (t ; ak ),
k=−∞
S0
By the memoryless property of ϵλ and the strong Markov property, we have E ∆(S0 , ϵλ )e−(r −η)ϵλ Sϵλ 1(τ < ϵλ )
= E e−(r −η+λ)τ ϕ(Sτ , τ ) = E e−(r −η+λ)τα ϕ(Sτα , τα ) + E e−(r −η+λ)τ−α ϕ(Sτ−α , τ−α ) = E e−(r −η+λ)τα ϕ(S0 eα , τα )
Thus, the expectation (B.2) is equal to
, τ−α ) .
(B.2′ )
To summarize, by combining (B.1 ), (B.2 ) and (B.3), h(s) is expressed in an involved but recognizable form for numerical integration: ′
h(s) = E φϵλ (seα , τα ) + E φϵλ (seα , τ−α )
knk−1 Pr(ν ≥ n),
k ≥ 1.
(C.2)
Our approach is again to find the Laplace transform of Pr(tn < T ) with T being a variable, which is equivalent to calculating Pr(tn < ϵλ ) where ϵλ is an independent exponential random variable. Applying Lemma 3.1 and Remark 3.1 where we set r = 0 and f = 1 in (3.3), the formula for Pr(tn < ϵλ ) immediately follows: Pr(tn < ϵλ ) =
λ
′
∞ n=1
1
α
ϕ(S0 e , τα )
ϕ(S0 e
(C.1)
n n
i
i n Li0,α L0n− ,−α = (L0,α + L0,−α ) ,
(C.3)
where L0,α and L0,−α are obtained from (3.7) and (3.8) with r = 0. By Eqs. (C.1)–(C.2), the Laplace transforms with parameter λ of Pr(tn < T ) and E [ν ] respectively are
+ E e−(r −η+λ)τ−α ϕ(S0 e−α , τ−α ) .
−α
i =0
−(r −η+λ)τ−α
E νk =
−(r −η+λ)τα
We are interested in the distribution, and especially the expected value of ν . As explained before, ν is a decreasing function of the bandwidth. For each n ≥ 0, we would like to evaluate Pr(ν ≥ n). The identity below follows from the definition of ν and the monotonicity of tn : To find the moments of ν , we use the equation
−(r −η)ϵλ Sϵλ ϕ(x, t ) = E ∆(S0 , ϵλ + t )e x .
Unlike the time-based strategy which trades in the underlying asset at pre-specified time points, the move-based hedging requires rebalancing the portfolio at a sequence of random times depending on the movement of the asset value. We let random variable ν be the hedging frequency, which keeps track of how many times rehedging takes place before maturity T . More precisely, define
Pr(ν ≥ n) = Pr(tn < T ).
To evaluate the expectation (B.2), we define
−αγ − γ 2 t /2 σ2
ν = max{n ≥ 0, tn < T }.
= φt (Sτα , τα ) + φt (Sτ−α , τ−α )
−E e
Appendix C. The distribution of hedging frequency
E [φt (Sτ , τ )] λe−λt dt .
φt (Sτ , τ ) = φt (Sτα , τα )1(τ = τα ) + φt (Sτ−α , τ−α )1(τ = τ−α )
f−α (t ) = exp
0
Thus,
ϕ(S0 , 0) − E e
and
∞
φt (Sτα , τα )1(τ = τ−α ) = φt (Sτ−α , τ−α )1(τ = τα ) = 0.
g (t ; ak ),
k=−∞
2π t
Let us calculate E [φt (Sτ , τ )]. Note that τ±α = ∞ if τ = τ∓α . Since r − η > 0, we show
∞
Gaussian distribution.
Then the expectation (B.1) is equal to E φϵλ (Sτ , τ ) =
αγ − γ 2 t /2 fα (t ) = exp σ2
where γ = µ − η − σ 2 /2, ak = σα (4k + 1) and g (t ; a) = 2 √ a e−a /2t , t > 0, is the probability density function of Inverse 3
Define −(r −η)τ
We need to know the density functions of τα and τ−α , which are given by equations (24) and (25) in Lin (1998). Let fα (t ) and f−α (t ) denote these two density functions. Then
(B.1)
+ E ∆(S0 , ϵλ )e−(r −η)ϵλ Sϵλ 1(τ > ϵλ )
− E [∆(s, ϵλ )] s.
+ ϕ(s, 0) − E e−(r −η+λ)τα ϕ(seα , τα )
(L0,α + L0,−α )n ,
n ≥ 0,
and 1 L0,α + L0,−α . λ 1 − (L0,α + L0,−α ) Fig. C.1 shows how the expected hedging frequency changes with respect to the bandwidth given two different asset volatilities over a three-year time horizon. We find by trial and error the corresponding value of α at which the expected hedging frequency of the move-based strategy is equal to the number of rebalancing points of the time-based strategy.
X.S. Lin et al. / Insurance: Mathematics and Economics 71 (2016) 40–49
Fig. C.1. The expected hedging frequency as a decreasing function of the bandwidth over a three-year period. η = 0.
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