Valuation of the Ratchet Equity Indexed Annuities with Quanto Features
2. PRODUCT SPECIFICATION AND VALUATION FRAMEWORK
2.1 Product Specification
The fundamental variable in pricing ratchet EIAs is the annual return calculated based on the linked index. Let T be the maturity of an EIA contract and S(t) be the linked index at t≤ . T Then the annual return of the linked index over the tth year would be:
T
Insurers often take averages of the index returns over sub-periods of a year when calculating the annual return to reduce the guarantee costs through dampening the return volatility. We analyze two types of geometric averaging in this paper.4
1 ,G
Rt
In the first case (which we refer as G1 hereafter), the annual return over the tth year, , is taken as the geometric average of the indexes sampled at an interval of 1/m. That is,
m
In the second case (referred as G2 hereafter), the annual return over the tth year denoted by
4 We do not consider arithmetic averaging for two reasons. Firstly, the annual return calculated using the arithmetic averaging scheme is the sum of lognormal random variables. It is well known that the options based on the sum of lognormal random variables have no closed-form pricing formulas under the B-S model (Kemma and Vorst, 1990). Secondly, the closed-form pricing formulas for options based on lognormal random variables can serve as effective control variates in pricing arithmetic-averaging-based options using the Monte Carlo algorithm (Kemma and Vorst, 1990). The pricing formulas derived later in this paper can hence be used in pricing the EIAs with arithmetic averaging.
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The next step after calculating the annual return is to calculate the return to be credited to the contract each year. The general formula is as follows:
, (4)
where Rt⋅,denotes the annual return over the tth year with or without geometric averaging, α is the participation rate in the linked index, f represents the minimum guaranteed return rate (also called the floor rate), and c stands for the cap rate. The participation rate is usually less than 100%, which is reasonable in the sense that investors sacrifice some of the upside potential for the downside protection of the minimum guarantee. When f = 0, the product provides a principal/premium guarantee. The cap rate or ceiling rate c is the maximum rate that can be credited each year. Placing a cap on the credited return is a direct way to reduce the product cost.
The annual return credited to the policy can be accumulated in two ways. For the compound version of ratchet EIAs, the total return at maturity T is calculated as:
∏
=The version without compounding but with simple adding up, which often referred as simple
5 Note that equation (1) can be deemed as the special case of setting m =1 in equations (2) and (3) that means no return averaging.
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at maturity T based on an initial premium of $1 at time 0 is.
2.2 Risk-Neutral Valuation
Since the contracts considered in this paper are quantos, we add an exchange rate model as well as a foreign risk-free rate model to the pricing framework of Black and Scholes. The linked index S(t) and exchange rate C(t) are assumed to follow geometric Brownian motions, and the interest rate r (for local currency) and rf ( for foreign currency) are assumed to be constants. More specifically,
( ) ( ) ( )
B andD
( )
t denote the domestic and foreign money market accounts, respectively.66 We do not consider stochastic interest rates in this paper for three reasons. Firstly, our aim is to provide closed-form formulas for effective contract valuation and contract analysis. Interest rates have little impact on the contract value because the payoffs of ratchet EIAs do not depend on interest rates; this is partly confirmed by the numerical results in Kijima and Wong (2007). Secondly, the models with stochastic short rates and those with constant interest rates give the same pricing formulas when the index return and short rate are driven
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The model defined in (7) is called the Black-Scholes quanto model (Baxter and Rennie, 1996). To make the model more concrete, we assume a case in which the local currency is Australian dollar and the linked index is denominated in US dollar. The model thus have three tradable assets in Australian dollar: the Australian dollar cash bondB
( )
t , theAustralian dollar worth of the US-dollar denominated bondC
( ) ( )
t D t , and the Australian dollar worth of the linked indexC( ) ( )
t S t .Based on the Girsanov theorem and the martingale representation theorem (see, for example, Bjork (2004)), there exists a unique measure Q under which both the discounted processes C
( ) ( ) ( )
t Dt B t and C( ) ( ) ( )
t S t B t are martingales. The processes S(t) and C(t) under Q can then be written as:( ) ( ) ( ) ( )
According to the risk-neutral valuation principle (see, for example, Harrison and
by independent Brownian motions. More specifically, let g(S(t): t ≤ T) be the payoff of a ratchet EIA, rt be the short rate process, and P(0,T) be the price of zero-coupon bond paying a unit amount at time T. The price of a ratchet EIA product V, under stochastic interest rates, is equal to:
[ ] [ ]
On the other hand, V under the constant interest rate assumption is:
E e −rTg s t( ( ) :t≤T)=e−rTE g s t
[
( ( ) :t≤T)]
=P(0, )T E g s t[
( ( ) :t≤T)]
when we make the common assumption (see Hull, 2006, for more detail) that both interest rate models calibrate their parameters to fit the current price P(0,T). Thirdly, pricing formulas are computationally inefficient when interest rates are stochastic. A rule of thumb in high-dimensional integral problems is to use a Monte Carlo type of algorithms when the maturity of the zero-coupon bond is longer than 3 periods. It is therefore more suitable to use numerical methods instead of pricing formulas for valuation of ratchet EIAs under stochastic interest rates.
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Kreps (1979) and Harrison and Pliska (1981)), the no-arbitrage price of the EIA contracts can be represented as:
[
*]
* E e R
V = Q −rT , (9) where EQ