Valuation of guaranteed annuity options in affine term structure models. presented by. Yue Kuen KWOK. Department of Mathematics

Transcription

1 Valuation of guaranteed annuity options in affine term structure models presented by Yue Kuen KWOK Department of Mathematics Hong Kong University of Science & Technology This is a joint work with Chi Chiu CHU.

2 Product nature of guaranteed annuity option (GAO) Provides the policyholder the right to either receive at retirement anassuredaccumulatedfundor a life annuity at a fixed rate. A single premium equity-linked policy whose maturity date T coincides with the retirement age R of the policyholder. The premium is invested in equity whose value is S t. Write a R (t) as the market value at time t of a life annuity of one dollar per annum starting at age R. Let g denote the guaranteed conversion rate (say, g = 9). Provided that the policyholder survives to age R, he receives either S T or S T g a R(T ) at policy maturity date T (same as R). Terminal value of GAO = S T g (a R(T ) g) +, where x + denotes max(x, 0).

3 Notations np R = probability that a person aged R survives n years D T +n (t) = market value of unit par default free zero-coupon bond at time t with maturity date T + n ω = maximum age in the mortality table By constructing a portfolio of default free bonds that match exactly with the expected cash flows of the annuity, we have a R (t) = ω R 1 n=0 np R D T +n (t) The annuity stream is like a coupon-payment bond with coupon payment n p R at T + n.

4 Assumptions actuarial and financial deterministic mortality rates may extend to hazard rate model of arrival of death rational behavior of policyholders no distinction between insurer s and insured s viewpoint about expectations on mortality, interest rate and equity. two-factor affine class short rate model to model interest rate uncertainties one factor interest rate model would imply full correlation of all future interest rates equity value follows the lognormal distribution may extend to stochastic volatility or regime switching model so as to model better the volatility risk.

5 Decomposition of the payoff structure of GAO max(a R (T ) g, 0) resembles an option on a coupon bond with strike g and coupon payment of amount n p R at time T + n, n = 0, 1,. The factor S T behaves like the exchange rate factor in a quanto g option. In a quanto option, the underlying asset is a foreign stock but the payoff is the domestic currency. Why embedded GAO may become an uncontrollable liability? 1. Current interest rates stay at lower level so that annuity value becomes higher. 2. Accumulated equity value may increase substantially with a strong return of the stock market. 3. Improvement of mortality compared to the anticipated mortality assumption at contract inception date.

6 Three major sources of risk 1. Equity risk Payoff is essentially in units of stock rather than in cash. Long-term stock price dynamics is difficult to be correctly specified. GAO value is highly sensitive to the correlation between the stock price and interest rate movement. 2. Mortality risk Hedged by selling more life insurance policies. When people live longer, the losses on the option to annitize is offset by profits on the life policies sold (difficulties: cashflows may not match exactly). 3. Interest rate risk Long-dated receiver swaptions are traded instruments to hedge interest rate exposure.

7 Hedging interest rate exposure with swaptions Bt entering a receiver swaption, the institution protects itself against the risk of falling interest rates. For example, if the market swap rate is 5% and the strike rate is 7%, the swaption holder will choose to exericse. Difficulties 1. The nominal amount to be purchased will depend on the assumed growth of the policyholder s fund. 2. The term of the annuity and the fixed term of the swaption may not match exactly. The policyholder may choose his retirement date within a range of ages. The company may have liabilities beyond 15 years, when swaptions are not available.

8 Multi-factor affine term structure model r t = short rate; the risk neutral processes of r t and the l-component vector of risk factors x(t) are governed by where a(t) = r t = a(t) T x(t) +b(t) dx(t) = µ(x,t) dt + σ(x,t) dz(t) a 1 (t) a 2 (t). a l (t) b(t) is a scalar function µ 1 (x,t) drift rate vector µ(x,t)= µ 2 (x,t). µ l (x,t) volatility matrix σ(x,t)= is a deterministic vector function, σ 11(x,t). σ 1m (x,t). σ l1 (x,t) σ lm (x,t) The m components in the random vector Z(t) are independent Wiener processes under risk neutral measure Q..

9 Discount bond price D T (t) = exp( A T (T t) T x(t) B T (t, T t)) where A T (t) andb T (t) are governed by a system of Ricatti differential equations. The volatility vector of the bond price σ D (x,t; T )= Equity value dynamics: σ D,1 (x,t; T ) σ D,2 (x,t; T ). σ D,m (x,t; T ) = σ(x,t)t A T (T t). ds t S t =(r q) dt + σ S (t) T dz.

10 Risk neutral valuation of GAO value Assuming zero pre-retirement benefit [ V (S, x,t)= T t p t E Q e T t r u du S T g (a R(T ) g) + where T t p t is the probability of survival of the policyholder over the next T t years. ] The above expectation calculations can be simplified by the method of change of numeraire. Define [ F n (S, x,t) = E Q e ] T t r u du ST D T +n (T ) = time-t value of the security that pays S T D T +n (T ) at time T

11 Next, suppose we use F n (S, x,t),n = 0, 1, 2,,ω R as the numeraire, then = E Q ω R 1 n=0 e T t r u du S T g ω R 1 n=0 np R D T +n (T )1 {ar (T )>g} np R g F n(s, x,t)p QF n [a R(T ) >g]. Gaussian type model σ(x,t) is a function of t only, then σ D (t; T )= σ(t) T A T (T t). Under the Gaussian assumption, the bond prices are lognormally distributed. However, the future annuity payments can be seen as a portfolio of discount bonds and the density of the sum of lognormal distributions has no closed form representation.

12 Stochastic differential equations The SDE of F n (S, x,t) under the risk neutral measure Q is given by df n F n = r t dt + [σ S (t)+σ D (t; T + n) σ D (t; T )] T dz = r t dt + { σ S (t)+σ(t) T [ A T (T t) A T +n (T t) ]} T dz. The solution to F n (S, x,t) is readily found to be F n (S, x,t) = D T +n(t) D T (t) S te exp ( T t q(t t) [σ S (u) σ D (u; T )] T [σ D (u; T + n) σ D (u; T )] du. The SDE of x under Q Fn can be deduced to be dx = {µ(x,t)+σ(t)σ S (t) + σ(t)σ(t) T [ A T (T t) A T +n (T t) ]} dt + σ(t) dz. QF n )

13 Three analytic approximation methods 1. Method of minimum variance duration Use a single zero-coupon bond as a proxy for the original couponbearing bond. The aproximation error is minimized by choosing the maturity of the zero-coupon bond equal to the stochastic duration. 2. Edgeworth approximation Compute the Edgeworth expansion of the probability distribution of the value of the annuity stream. 3. Affine approximation The exercise probability of the annuity option is approximated through an approximation of the exercise region. This is achieved by the linearization of the exercise region, whose boundary is approximated by a hyperplane.

14 Method of minimum variance duration Related to the method of stochastic duration. Recall that the stochastic duration of a coupon bond in a multi-factor diffusion model is the time to maturity of the zero-coupon bond with the same relative volatility as that of the coupon bearing bond. Consider a numeraire that corresponds to the security F τ (S, x,t) that pays S T D T +τ (T )att. Here, τ is the time to maturity of the underlying bond at time T. Choose τ such that the error in the approximate solution is minimized in some sense.

15 Let Q Fτ denote the pricing measure when F τ (S, x,t)isusedasthe numeraire. Under Q Fτ,thetime-t value of the GAO is given by V (S, x,t)= T t p t F τ (S, x,t)e QF τ ( ar (T ) gd T +τ (T ) 1 D T +τ (T ) ) +. Nice analytic tractability can be achieved if we set some constant K. a R (T ) gd T +τ (T ) be ConditioningonmeanK is judiciously chosen to be the mean a of R (T ) gd T +τ (T ) under Q F τ. Approximate solution to V (S, x,t) is takentobe V a (S, x,t)= T t p t F τ (S, x,t)e QF τ ( K 1 D T +τ (T ) ) +, where K = E QF τ [ a R (T ) gd T +τ (T ) ].

16 1. K can be readily found to be K = 1 g = 1 g = ω R 1 n=0 q(t t) S t e F τ (S, x,t) 1 gf τ (S, x,t) np R E QF τ ω R 1 n=0 ω R 1 n=0 2. The expectation is found to be E QF τ ( K 1 D T +τ (T ) [ D T +τ (T ) > 1 K ) + [ DT +n (T ) D T +τ (T ) ] np R E QS [D T +n (T )] np R F n (S, x,t). ] S q(t t) te = KP QF τ F τ (S, x,t) P Q S By combining the results together, we obtain V a (S, x,t) = T t p t ω R 1 n=0 np R F n (S, x,t) P QF g τ S t e q(t t) P QS [ D T +τ (T ) > 1 K [ D T +τ (T ) > 1 ]. K [ D T +τ (T ) > 1 ] K ]}.

17 The mean of ln D T +τ (T ) under Q Fτ and Q S are obtained as follows: where c(τ) = E QF τ [ln D T +τ(t )] = c(τ)+ v2 (τ) 2 E QS [ln D T +τ (T )] = c(τ) v2 (τ) 2 T t [ ] DT +ln +τ (t) D T (t) [ ] DT +ln +τ (t), D T (t) [σ S (u) σ D (u; T )] T [σ D (u; T + τ) σ D (u; T )] du v 2 (τ) = var[lnd T +τ (T )] = T t σ D (u; T + τ) σ D (u; T ) 2 du.

18 Also, we may express F n (S, x,t)andk in the following forms: F n (S, x,t)= D T +n(t)s t D T (t) K = e c(τ) gd T +τ (t) Here, the quantity ω R 1 n=0 q(t t)+c(n) e np R D T +n (t)e c(n) = e c(τ) gd T +τ (t) a R(t). a R (t) = ω R 1 n=0 np R D T +n (t)e c(n) can be interpreted as the equity-risk-adjusted annuity.

19 The two probability values are found to be P QF τ [ D T +τ (T ) > 1 ] K ln 1 = N K ln DT +τ (t) D T (t) v(τ) c(τ) v2 (τ) 2 = N(d) and [ P QS D T +τ (T ) > 1 ] K = N(d v(τ)), where ln a R(t) gd d = T (t) + v2 (τ) 2. v(τ) Finally, the analytic expression for V a (S, x,t) is found to be [ V a (S, x,t)= T t p t Se q(t t) ar (t) N(d) N(d v(τ)) gd T (t) ].

20 2. Determination of τ using minimization of variance duration The error in the approximation of V (S, x,t)byv a (S, x,t) is quantified by E QFτ [ Y ], where ( ) + ( ) + ar (T ) Y = gd T +τ (T ) 1 1 K. D T +τ (T ) D T +τ (T ) The pricing error is minimized by choosing τ so as to minimize the variance of da R (t) a R (t) dd T +τ(t), that is, the optimal value of τ is given by D T +τ (t) τ = argmin ( dar (t) varqfτ τ 0 a R (t) dd ) T +τ(t). D T +τ (t) Proof ( ) ( ) ar (T ) Let m =min gd T +τ (T ),K ar (T ) and M =max gd T +τ (T ),K. The following three events are mutually exclusive and exhaustive: { } { } { } 1 E 1 = D T +τ (T ) 1 M, E 2 = m < D T +τ (T ) < 1 M and E 3 = D T +τ (T ) m,

21 ] (i) E QF [ Y 1 τ E1 =0sinceY becomes zero when E 1 occurs. ] { [ (ii) E QF [ Y 1 τ E2 E QF Y 2]} 1/2 PQF [E τ τ 2] and P [E QF τ 2] has a a smaller value when R (T ) stays closer to its mean K. This gd T +τ (T ) ( ) ar (T ) occurs when var QF is minimized. τ gd T +τ (T ) (iii) E QF τ [ Y 1 E3 ] = E QF τ = [ a R (T ) gd T +τ (T ) K 1 E 3 ( ) 2 ar (T ) gd T +τ (T ) K P [E QF τ 3] ( ) 1/2 ar (T ) P QF gd T +τ (T ) [E τ 3]}. E Q F τ { var QF τ ] 1/2

22 ( ar (T ) ) The pricing error is minimized by minimizing var QF τ gd T +τ (T ) over the choice of τ. We minimize the relative change of value in a R (T ) D T +τ (T ), which can be measured by the variance of da R(t) a R (t) dd T +τ (t) D T +τ (t). Under Q, the dynamics of a R (t) andd T +τ (t) are given by da R (t) a R (t) = r t dt + σ a (t; T ) T dz dd T +τ (t) D T +τ (t) = r t dt + σ D (t; T + τ) T dz, where the volatility vector of annuity σ a and σ D (t; T +n) are related by σ a (t; T )= ω R 1 n=0 np R D T +n (t) σ D (t; T + n). a R (t)

23 For an one-factor interest rate model, the solution to τ is given by σ a (t; T )=σ D (t; T + τ ), which is just the stochastic duration of the annuity. For the general multi-factor case, the minimization of ( dar (t) var QF τ a R (t) dd ) T +τ (t) D T +τ (t) leads to the following non-linear algebraic equation for τ: [σ a (t; T ) σ D (t; T + τ)] T σ D(t; T + τ) τ =0.

24 Two-factor Gaussian model To compute V a (S, x,t) using the two-factor Gaussian interest rate model (G2++). For the G2++ model, the interest rate r t is given by r t = x 1,t + x 2,t + b(t) where the dynamics of the risk factors are governed by dx 1 = κ 1 x 1 dt + σ 1 dz 1 dx 2 = κ 2 x 2 dt + σ 2 (ρdz ρ 2 dz 2 ). Here, b(t) is a function which is determined by fitting the current interest rate term structure and ρ is the correlation coefficient between the risk factors.

25 and A T (T t) = B T (t, T t) = ln D T (0) D t (0) σ2 1 σ2 2 κ 2 1 e κ 1 (T t) κ 1 1 e κ 2 (T t) κ 2, ( ) 1 e 2κ 1 t 1 e 2κ 1(T t) κ 1 2κ 1 ( ) 1 e 2κ 2 t 1 e 2κ 2(T t) 2κ 1 2κ 2 2κ 2 ( 1 ρσ 1 σ ) 1 e (κ 1+κ 2 )t 1 e (κ 1+κ 2 )(T t) κ 1 κ 2 κ 1 + κ 2 κ 1 + κ 2 ( σ ρσ )( ) 1σ 2 1 e κ 1 t 1 e κ 1(T t) κ 1 κ 2 κ 1 κ 1 ( )( ) ρσ1 σ σ2 2 1 e κ 2 t 1 e κ 2(T t). κ 1 κ 2 κ 2 κ 2

26 For the G2++ model, we obtain T c(τ) = [σ S σ D (u; T )] T [σ D (u; T + τ) σ D (u; T )] du ( t = σ2 1 1 e κ 1 )[ τ 1 e 2κ 1 ] ( (T t) + σ2 2 1 e κ 2 )[ τ 1 e 2κ 2 ] (T t) κ 1 κ 1 2κ 1 κ 2 κ 2 2κ ( 2 2 e κ 1 )[ τ e κ 2τ 1 e (κ 1 +κ 2 ] )(T t) + ρσ 1 σ 2 κ 1 κ 2 κ 1 + κ ( 2 σ 2 + ρσ )( )[ 1σ 2 1 e κτ 1 e κ 1 ] (T t) + σ 1 σ S ρ S1 κ 1 κ 2 κ 1 κ [ 1 σ ρσ ]( 1σ 2 1 e + σ 2 σ S (ρ S1 ρ + ρ S2 1 ρ 2 κ 2 )[ τ 1 e κ 2 ] (T t) ), κ 2 κ 1 κ 2 κ 2 and T v 2 (τ) = σ D (u, T + τ) σ D (u, T ) 2 du ( t 1 e κ 1 ) τ 2 [ 1 e 2κ 1 ] ( (T t) 1 e κ 1 )( τ 1 e κ 2 ) τ +2ρσ 1 σ 2 = σ1 2 κ 1 2κ 1 κ [ 1 1 e (κ 1 +κ 2 ] ( )(T t) 1 e + σ 2 κ 2 ) τ 2 [ 1 e 2κ 2 ] (T t) 2. κ 1 + κ 2 κ 2 2κ 2 κ 2

27 Edgeworth expansion 1. Approximation of P QFn [a R (T ) >g] in terms of cumulants Let π (n) (a) denote the density function of a R (T ) under Q Fn. denote the characteristic function of a R (T ) under Q Fn,where Π (n) (λ) =E QFn [e iλa R(T ) ]= e iλa π (n) (a) da. Let Π (n) (λ) The cumulants are defined as the coefficients of a Taylor series expansion of the logarithm of the characteristic function, where ln Π (n) (iλ) j (λ) = c j. j! By taking the Fourier inversion of Π (n) (λ) and keeping cumulants only up to the third order, we obtain π (n) (a) = 1 2π = 1 2π 1 2π j=1 e iλa Π (n) (λ) dλ ( exp iλa + iλc (n) ( exp i(a c (n) 1 1 c(n) 2 c(n) 2 )λ 2 λ2 ) 2 λ2 i c(n) 3 6 λ3 + o(λ 3 ) dλ )( ) 1 ic(n) 3 6 λ3 dλ.

28 Cumulants There exists an one-to-one relationship between moments and cumulants. Let m (n) j and c (n) j denote the j th moment and j th cumulants of a R (T ) under Q Fn. c (n) 1 = m (n) 1,c(n) 2 = m (n) 2 (m (n) 1 )2, c (n) 3 = m (n) 3 3m (n) 1 m(n) +2+2(m (n) 1 )3, etc. The first two cumulants are simply the mean and variance. Procedures 1. Use cumulants to approximate π (n) (a). From which, we approximate P QF n [a R(T ) >g]. 2. Cumulant are computed through the computation of moments of a R (T ) under Q Fn.

29 After some tedious integration procedure, we obtain π (n) (a) 1 c (n) 2 Also we deduce that where c(n) 3 (a c(n) 1 ) 2(c (n) + 2 )5/2 P [a(t ) >g] = QF π (n) (a) da n g c (n) 3 (a c(n) 6(c (n) 2 )7/2 1 )3 n a c (n) 1 c (n) 2 N(z 1 )+ c(n) 3 6(c (n) 2 )3/2(z2 1 1)n(z 1), z 1 = c(n) 1 g. c (n) 2,

30 2. Determination of the moments of a R (T ) under the measure Q Fn We would like to find the j th moment of a R (T ) under the measure Q Fn as defined by Note that a(t ) j = so that = m (n) j = ω R 1 n=0 ω R n 1,n 2,,n j =0 m (n) j = E QF n [a R(T ) j ]. np R D T +n (T ) ω R 1 n 1,n 2,,n j =0 E QF n j ( n1 p R n 2 p R nj p R )[D T +n1 (T )D T +n2 (T ) D T +nj (T )] exp ( n1 p R n 2 p R nj p R ) j k=1 [ AT +nk (T ) T x(t )+B T +nk (T ) ] The moments are seen to have the exponential affine representation..

31 We assume that the drift term µ(x,t) takes the linear form µ(x,t)=µ 0 (t)+µ 1 (t)x, where µ 0 (t) isal-component vector and µ 1 (t) isal l matrix. E QF n exp j k=1 = exp ( G T (t) T x(t) G 0 T (t)) A T +nk (T ) T x(t ) B T +nk (T ) where G T (t) and G 0 T (t) have dependence on n 1,n 2,,n j and n, and they satisfy the following systems of Ricatti equations.

32 (i) (ii) dg T (t) dt G T (T )= dg 0 T dt + µ 1 (t) T G T (t) =0 j k=1 A T +nk (T ); + G T (t) T { µ 0 (t)+σ(t)σ S (t)+σ(t)σ(t) T } [A T (t) A T +n (t)] = 1 2 G T (t) T σ(t)σ(t) T G T (t) G 0 T (T )= j k=1 B T +nk (T ).

33 Let Φ T (t) be the solution to the following system of differential equations dφ T (t) = µ 1 (t) T Φ T (t) dt Φ T (T )=I where Φ T (t) isal l matrix and I is the l l identity matrix. It can be shown that Φ T (t) = exp ( T t µ 1 (u) T du Now, the closed form solution to G T (t) andg 0 T (t) can be expressed in terms of Φ T (t) as follows G T (t) = Φ T (t)g T (T )= j k=1 exp ( T t ). µ 1 (u) T du G 0 T T (t) = G0 T (T )+ G T (u) {µ T 0 (u)+σ(u)σ S (u) t [ + σ(u)σ(u) T A T (u) A T +n (u) G T (u) 2 ) A T +nk (T ) ]} du.

34 Two-factor Gaussian model The volatility matrix σ(t) is given by ( σ1 0 0 σ(t) = σ 2 ρ σ 2 1 ρ 2 0 so that ( ) σ(t)σ(t) T σ 2 = 1 ρσ 1 σ 2 ρσ 1 σ 2 σ2 2. The solution to Φ T (t) is found to be ( e κ 1 (T t) 0 Φ T (t) = 0 e κ 2(T t) ) )

35 Finally, the solution to G T (t) andg 0 T (t) are given by ( GT,1 (T )e G κ ) ( ) 1(T t) GT,1 (T ) T (t) = G T,2 (T )e κ 2(T t) where G T (T )=, G T,2 (T ) and G 0 T (t) = G0 T (T )+σ 1σ S ρ S1 G T,1 (T ) 1 e κ 1(T t) + σ 2 σ S (ρ S1 ρ + ρ S2 1 ρ 2 )G T,2 (T ) 1 e κ 2(T t) [ σ1 2 1 e κ 1 n G T,1(T ) + G ] T,1(T ) 1 e 2κ 1(T t) κ 1 2 2κ 1 [ σ2 2 1 e κ 2 n G T,2(T ) + G ] T,2(T ) 1 e 2κ 2(T t) κ 2 2 2κ 2 ( ) ( 1 e κ 2 n 1 e κ 1 n ρσ 1 σ 2 [G T,1 (T ) + G T,2 (T ) + G T,1 (T )G T,2 (T ) κ 2 κ 1 ] 1 e (κ 1+κ 2 )(T t) κ 1 + κ 2. κ 2 κ 1 )

36 Affine approximation approach Linearization of the exercise boundary by fitting a hyperplane β T x = 1 that approximates the exercise boundary a R (x,t)=g. The probability of exercising P [a QF n R(x,t) >g] is then approximated by either P QF n [βt x > 1] or P QF n [βt x < 1] (whose choice depends on the location of the exercise region). For the Gaussian type models, β T x(t )=β 1 x 1 (T )+ +β l x l (T ) is normally distributed whose mean and variance are given by β T µx and β T σxβ, whereµx and σx are the conditional mean vector and covariance matrix of x(t ) given x(t) under Q Fn.

37 Affine approximation to the exercise boundary in the relevant region for a 5-year at-the-money call on a 30-year 10% coupon bond.

38 Fitting algorithm Consider a two-factor interest rate model with two risk factors, the fitting algorithm involves the following steps. 1. Choose a level of significance α (say, 1%), then find the two values x 2,α/2 and x 2,1 α/2 such that P QF n [x 2,α/2 <x 2(T ) <x 2,1 α/2 ]=1 α. 2. Once x 2,α/2 and x 2,1 α/2 are known, solve for x 1,α/2 and x 1,1 α/2 so that the two points (x 1,α/2,x 2,α/2 ) and (x 1,1 α/2,x 2,1 α/2 ) fall on the exercise boundary: a(x,t)=g.

39 3. Fit a hyperplane (a line in the case of a two-factor interest rate model) β 1 x 1 + β 2 x 2 =1 to the two points determined in Step 2 by solving for the parameters β 1 and β 2 through β = ( ) β1 β 2 ( ) 1 ( ) x1,α/2 x = 2,α/2 1. x 1,1 α/2 x 2,1 α/2 1 Choose the appropriate region {β T x > 1} or {β T x < 1} so as to approximate the exercise region {a R (x,t) >g}.

40 Running time comparison Accuracy comparison

41 3 2 Simulation Edgeworth Affine Minimum variance duration 1 precentage error a/[gd (t)] T Fig. 1. Comparison of the pricing error (in percentage) of various analytic approximation methods. Good numerical accuracy is a revealed when > 1 (the annuity option is currently in-themoney). gd T (t)

42 60 T t=10 T t=15 T t= GAO value g Fig. 2. Plot of GAO value against guaranteed conversion rate g at varying values of time to expiry T t. The rate of decrease of GAO value is higher at a lower value of g. This is related to the time value of money since smaller T t means shorter time horizon over which the annuity paymemts are discounted. This effect counteracts the usual theta effect of option value (longer-lived option has a higher value).

43 4.5 ρ S1 =0.5, ρ S2 =0.5 ρ S1 = 0.5, ρ S2 =0.5 ρ S1 =0.5, ρ S2 = GAO value ρ Fig. 3. Price sensitivity of the GAO value to the correlation coefficients in the pricing model. It is quite disquieting to observe that the GAO value is highly sensitive to the correlation coefficients.

44 Summary 1. Method of minimum variance duration Starts with a judicious analytic approximation so that closed form formula can be obtained. Pricing error is minimized by choosing period τ of a reference bond such that the variance of the value of the annuity payment normalized by the price of the (T + t)-maturity bond is minimized. 2. Edgeworth expansion Seeks the Edgeworth approximation of the probability distribution of the annuity value at option s maturity. 3. Affine approximation The exercise probability of the annuity option is approximated through the approximation of the exercise boundary, where the concave exercise boundary is approximated by a hyperplane.

45 Conclusion In terms of numerical accuracy and computational efficiency, the method of minimum variance duration seems to have the best performance. When the annuity option is in-the-money or slightly out-of-themoney, the pricing errors are within a few percentage points. GAO value has strong dependence on the conversion rate and correlation of risk factors Future improvements on the model 1. More accurate model to characterize the equity return process. 2. Incorporate other market factors (company expenses, tax effects, pre-retirement death benefits, etc.) 3. Better understanding of the exercise behaviors of the policyholders.