Pricing Variable Annuity Contracts with High-Water Mark Feature

In this paper, we focus on valuation of VA contract value where guaranteed minimum payoff is calculated from set of fund values over the entire term of contract. High-water mark values are calculated at predetermined observation dates. Usually, frequencies of these reset dates are fixed and they can be monthly, quarterly or yearly. Because of this feature, some of maximal fund values can be missed and it reduces value of the contract. This effect needs to be taken into account to price correctly VA contract.

We consider the following lookback contract, which pay-off at expiration time T can be presented in the following form

where N is a notional1 ; V (0) = 1;

Hw(t) = M ax(S(t)/S(0), Hw(t -∆t));

is a High Watermark; S(t) is price of underlying at time t; ∆t is time interval between observation dates; γ is a rate of guaranteed minimum payoff.

In continuous limit ∆t → 0 equation ( 1) can be presented in the following form

From eq.( 3) we can see that the probability distribution of the VA contract value is distribution of extreme (maximal) values of underlying

So, we can use distribution of extreme (maximal) values to calculate present value of this VA contract.

Consider the following dynamics for underlying S(t)

where

is a risk-neutral drift; r and y are short term interest rate and dividend yield; v is volatility; W (t) is a standard Brownian motion

Notice, that to model distribution of underlying S(t) ( 4), all we need to do, is just to redefine dividend yield as

Probability distribution function of returns

in Black-Scholes world (5) satisfies the following Fokker-Plank equation

with initial boundary condition

This solution is

To find probability distribution function of maximal values x(t) for time interval 0 < t < T it is convenient to calculate probability of event that x(t) < h for all t in the time interval [0, T ]. This problem is equivalent to problem with absorbing wall at x = h and therefore we need to find solution which satisfies eq.( 10) for all 0 < t < T and x < h and to impose the following boundary conditions:

Solution of eq.( 13) with boundary conditions (13) can be found from the following anzatz:

where A and x 0 are constants and x 0 > h. Eqs. (13,14) lead us to the following equation

and we obtain that

Then, probability of the event, that return x does not cross the border x = h by time t = T can be calculated

where

are error functions.

4 High Watermark Liability Value

In the limit ∆t → 0 present value of VA contract (3) can be calculated from maximal value distribution (19) as

where µ = ryγ ; X = S H ≥ S(0) is a strike; S H is maximal observed price of underlying from issue date to current time (t = 0). Call(S, X, v, r, y, t) is a Black-Scholes formula for European Call Option price. Here, to calculate present value of that contract we take into account discount factor e -rT . High Watermark contract value depends on size of time interval between observation dates (∆t > 0). Value of contract with ∆t > 0 is lower than in the limit ∆t → 0 just because selected dates of resets (observation dates) can miss the actual extremal values of underlying stochastic process.

To take into account finite size of time steps, let us consider Brownian motion (5) on time interval T with N equal time intervals between observation dates: 0, t 1 , t 2 , . . . , t n , . . . , T ;

Then, the average maximal value near the point t i is

It means that in the limit of small time steps ∆t → 0, distributions of extreme values can be presented in the following form

where

Then, the integral in (20) can be transformed to the following form

where ǫ = v 2 ∆t π ; X = Xe ǫ . And finally, from (20) we obtain the following formula for VA contract value

where X = S H e γt h ; t h is a time from the observed underlying value to current time which gives the highest guaranteed payoff at time t = 0.

At issue date, the initial contract value is

In this paper we consider VA contract which guaranteed minimum payoff is calculated from a set of maximal fund values over the entire term of contract. In the limit of small time intervals between observation dates (∆t → 0) we derived explicit pricing formula for this type of VA contracts. Finite time interval corrections are calculated and compared with Monte-Carlo results. Good agreement between analytical formula and numerical calculations of VA values is demonstrated.

Below we assume that N = 1 and S(0) = 1