# HPCFinance: New Thinking in Finance. Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation

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1 HPCFinance: New Thinking in Finance Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation Dr. Mark Cathcart, Standard Life February 14, / 58

2 Outline Outline of Presentation 1 Introduction 2 Main Monte Carlo Greek Approaches 3 Main MC Greek Approaches: B-S Tests 4 Monte Carlo Greeks for More Advanced Models 5 Monte Carlo Greeks for Variable Annuities 0 / 58

3 Introduction Overview of Variable Annuity contracts Variable Annuities What are Variable Annuity (VA) Products? Types of VA GMDB,GMAB,GMIB,GMWB Nature of Liabilities of Guarantee complex multi-dimensional path-dependent Even valuing the liabilities is fairly involved - simulation. Can we construct more sophisticated estimators of the VA Greeks to aid insurer s hedging strategies? 1 / 58

4 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation There are three main approaches Bump and Revalue 2. Pathwise Approach 3. Likelihood Ratio Method 2 / 58

5 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 1. Bump and Revalue Idea is to simulate for the price of the option under some base scenarios and then again under some bumped scenarios e.g., delta: perturb the initial stock price by S(0): C(S(0) + S(0), K, σ, r, T ) C(S(0), K, σ, r, T ) S(0) i.e., a Forward Difference estimate Common Random Numbers Good practice to use same random number generator with same initial seed in base and bumped scenarios 3 / 58

6 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 2. Pathwise Approach Let α(θ) = E[Y (θ)] then we can estimate the derivative of α(θ) using: Y Y (θ + h) Y (θ) (θ) = lim h 0 h This estimator has expectation E[Y (θ)] and is an unbiased estimator of α (θ) if interchanging differentiation and taking expectations is justified, i.e., [ ] d E dθ Y (θ) = d E[Y (θ)] dθ 4 / 58

7 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 2. Pathwise Approach Let us consider a simple, illustrative example... European option under B-S model: Delta Y = e rt max(s(t ) K, 0) Payoff S(T ) = S(0)e (r 1 2 σ2 )T +σ T Z Applying the chain rule gives... dy ds(0) = dy ds(t ) ds(t ) ds(0) Dynamics 5 / 58

8 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 2. Pathwise Approach Let us consider a simple, illustrative example... European option under B-S model: Delta Y = e rt max(s(t ) K, 0) Payoff S(T ) = S(0)e (r 1 2 σ2 )T +σ T Z Applying the chain rule gives... dy ds(0) = dy ds(t ) ds(t ) ds(0) Dynamics 5 / 58

9 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 2. Pathwise Approach Let us consider a simple, illustrative example... European option under B-S model: Delta Y = e rt max(s(t ) K, 0) Payoff S(T ) = S(0)e (r 1 2 σ2 )T +σ T Z Applying the chain rule gives... dy ds(0) = e rt I{S(T ) > K} ds(t ) ds(0) Dynamics 5 / 58

10 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 2. Pathwise Approach Let us consider a simple, illustrative example... European option under B-S model: Delta Y = e rt max(s(t ) K, 0) Payoff S(T ) = S(0)e (r 1 2 σ2 )T +σ T Z Applying the chain rule gives... dy ds(0) = e rt I{S(T ) > K} ds(t ) ds(0) Dynamics 5 / 58

11 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 2. Pathwise Approach Let us consider a simple, illustrative example... European option under B-S model: Delta Y = e rt max(s(t ) K, 0) Payoff S(T ) = S(0)e (r 1 2 σ2 )T +σ T Z Applying the chain rule gives... dy ds(0) = e rt I{S(T ) > K} ds(t ) ds(0) Dynamics 5 / 58

12 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 2. Pathwise Approach Let us consider a simple, illustrative example... European option under B-S model: Delta Y = e rt max(s(t ) K, 0) Payoff S(T ) = S(0)e (r 1 2 σ2 )T +σ T Z Applying the chain rule gives... dy ds(0) = e rt I{S(T ) > K} S(T ) S(0) Dynamics 5 / 58

13 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 2. Pathwise Approach Let us consider a simple, illustrative example... European option under B-S model: Delta Thus, the Pathwise Estimator for a European call option (under the B-S model underlying dynamics) is given as: dy ds(0) = e rt I{S(T ) > K} S(T ) S(0) 5 / 58

14 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 2. Pathwise Approach What about estimating the Gamma option Greek? The Pathwise is inapplicable in this case! d ds(t ) d ds(t ) 6 / 58

15 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 2. Pathwise Approach What about estimating the Gamma option Greek? The Pathwise is inapplicable in this case! More formally, the requirement for using the PW method [ ] d E dθ Y (θ) = d E[Y (θ)] dθ is not satisfied in this instance. Indeed, [ ] dy 0 = E d ds(0) ds(0) E[Y ] 7 / 58

16 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 3. Likelihood Ratio Method Relies on differentiating prob.density rather than payoff funct., thus does not require smoothness in the payoff function Suppose we have a discounted payoff Y expressed as a function f (X), where X is a m-dimensional vector of different asset prices (or alternatively, one asset price at multiple valuation dates). Then assuming that X has a probability density g with parameter θ, and that E θ denotes an expectation taken with respect to the density g θ, taking the expected discounted payoff with respect to this density gives E θ [Y ] = E θ [f (X 1,..., X m )] = f (X)g θ (X)dX R m 8 / 58

17 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 3. Likelihood Ratio Method Now, similarly to the pathwise derivative approach, we assume the order of differentiation and integration can be interchanged. Here, however, this is not so strong an assumption, as typically densities are smooth functions, whereas payoff functions are not. This gives d dθ E θ[y ] = f (X) d R m dθ g θ(x)dx d dθ = f (X) g θ(x) R m g θ (X) g θ(x)dx d dθ = E θ [f (X) g ] θ(x) g θ (X) 9 / 58

18 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 3. Likelihood Ratio Method Then f (X) d dθ g θ(x) g θ (X) gives the Likelihood-Ratio estimator for the sensitivity with respect to the parameter θ, and this estimator is unbiased d dθ g θ(x) g θ (X) The term is often known in the statistics literature as the score function The Likelihood-Ratio method will still be applicable and robust in the case of options with discontinuous payoff functions (and estimating second-order sensitivities) 10 / 58

19 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 3. Likelihood Ratio Method Let us again consider our simple, illustrative example... European option under B-S model: Delta The log-normal density used to calculate S(T ) is given by g(x) = 1 ( ( ln x ) xσ T φ S(0) (r 1 2 σ2 )T ) σ T where φ( ) represents the standard normal density function. In this case the score function is: dg(x) ds(0) g(x) = ln ( x ) S(0) (r 1 2 σ2 )T S(0)σ 2 T 11 / 58

20 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 3. Likelihood Ratio Method Let us again consider our simple, illustrative example... European option under B-S model: Delta Evaluating this at S(T ) and multiplying by the option payoff gives the unbiased estimator of the Black-Scholes delta as: e rt max(s(t ) K, 0) ln ( S(T )/S(0) ) (r σ 2 /2)T S(0)σ 2 T Using: S(T ) = S(0)e (r 1 2 σ2 )T +σ T Z, where Z N(0,1) / 58

21 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 3. Likelihood Ratio Method Let us again consider our simple, illustrative example... European option under B-S model: Delta Evaluating this at S(T ) and multiplying by the option payoff gives the unbiased estimator of the Black-Scholes delta as: e rt max(s(t ) K, 0) ln ( S(T )/S(0) ) (r σ 2 /2)T S(0)σ 2 T Using: S(T ) = S(0)e (r 1 2 σ2 )T +σ T Z, where Z N(0,1) / 58

22 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 3. Likelihood Ratio Method Let us again consider our simple, illustrative example... European option under B-S model: Delta Evaluating this at S(T ) and multiplying by the option payoff gives the unbiased estimator of the Black-Scholes delta as: e rt Z max(s(t ) K, 0) S(0)σ T Using: S(T ) = S(0)e (r 1 2 σ2 )T +σ T Z, where Z N(0,1) / 58

23 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 3. Likelihood Ratio Method Let us again consider our simple, illustrative example... European option under B-S model: Delta Evaluating this at S(T ) and multiplying by the option payoff gives the unbiased estimator of the Black-Scholes delta as: e rt Z max(s(t ) K, 0) S(0)σ T Giving the simplified unbiased LRM estimate of Delta, for a European call option under the B-S model for the underlying 11 / 58

24 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 3. Likelihood Ratio Method And what about the case of discontinuous payoff functions... Binary (cash-or-nothing) option under B-S model Payoff = { 1 if S(T ) K 0 if S(T ) < K = I{S(T ) > K} Digital call: e rt Z I{S(T ) > K} S(0)σ T ( ) European call: e rt Z max(s(t ) K, 0) S(0)σ T 12 / 58

25 Main Monte Carlo Greek Approaches Estimating Greeks by MC simulation 3. Likelihood Ratio Method And finally what about estimating the gamma sensitivity... European option under B-S model: Gamma As the gamma is the second-order sensitivity of the payoff wrt to S 0 the likelihood ratio weight will be different to that calculated for the delta. However, it is easy to show that the LRM estimator for the gamma is given by: e rt max(s(t ) K, 0) Z 2 Zσ T 1 S(0) 2 σ 2 T 13 / 58

26 Main MC Greek Approaches: B-S Tests B-S Model Test Cases for analysis Let us look at applying the three approaches for the Delta and Gamma sensitivities of a simple European put option Case S K σ r T A % 4% 1 B % 4% 1 C % 4% 1 D % 4% 1 E % 4% 1 F % 1.5% 1 Table: Different put option set-ups considered in the tests. In all the above cases 100,000 paths and S(0) perturbation of 0.2% used. 14 / 58

27 Delta Estimates Main MC Greek Approaches: B-S Tests European put option delta estimators Case A 95% Confidence Intervals Case B Case D Case E Case F -0.6 Case C -0.7 B'nR Delta Est's LRM Delta Est's Pathwise Delta Est's Analytical Values 15 / 58

28 Delta Estimates Main MC Greek Approaches: B-S Tests European put option delta estimators Case B 95% Confidence Intervals Case E -0.4 B'nR Delta Est's LRM Delta Est's Pathwise Delta Est's Analytical Values 16 / 58

29 Gamma Estimates Main MC Greek Approaches: B-S Tests European put option gamma estimators % Confidence Intervals Case B Case C Case D Case E Case F Case A B'nR Gamma Est's Analytical Gamma Values LRM Gamm Est's 17 / 58

30 Gamma Estimates Main MC Greek Approaches: B-S Tests European put option gamma estimators Case B Case C 95% Confidence Intervals Case F B'nR Gamma Est's Analytical Gamma Values LRM Gamm Est's 18 / 58

31 Main MC Greek Approaches: B-S Tests Conclusions from B-S analysis 1. B&R and PW give very similar estimates and standard errors This is expected as the Pathwise Estimator is the small perturbation limit of the Bump and Revalue method 2. When the Pathwise estimator is available it generally gives estimators with smaller standard errors than the LRM estimator This is because Pathwise estimator is specific to a given payoff function, whereas the LRM determines a weight for a given density which can then be multiplied by any payoff function 19 / 58

32 Main MC Greek Approaches: B-S Tests Conclusions from B-S analysis 3. The LRM provides an estimate of Gamma with smaller standard error than the corresponding Bump and Revalue estimate. If the Pathwise method is inapplicable, then, essentially, the Bump&Revalue approach cannot converge to the true value Note: Although these observations have been inferred from analysis on simple European options, they generally hold for other options and models (see, e.g., Glasserman). 20 / 58

33 Main MC Greek Approaches: B-S Tests Estimating Greeks by MC simulation Mixed estimator for second-order Greeks Let us again consider our simple, illustrative example... European option under B-S model: Gamma Can form a hybrid estimator for call option Gamma by applying the PW method to the LRM estimator for Delta... ( ) d e rt Z max(s(t ) K, 0) ds(0) S(0)σ T 21 / 58

34 Main MC Greek Approaches: B-S Tests Estimating Greeks by MC simulation Mixed estimator for second-order Greeks Let us again consider our simple, illustrative example... European option under B-S model: Gamma... resulting in the LR-PW mixed estimator for Gamma: e rt Z S(0) 2 σ I{S(T ) > K}K T 22 / 58

35 Main MC Greek Approaches: B-S Tests Estimating Greeks by MC simulation Mixed estimator for second-order Greeks Let us again consider our simple, illustrative example... European option under B-S model: Gamma Can also form a PW-LR ( hybrid estimator for Gamma... e rt I{S(T ) > K} S(T ) S(0) Z S(0)σ T ) d 1 +I{S(T ) > K}S(T ) ds(0) S(0) 23 / 58

36 Main MC Greek Approaches: B-S Tests Estimating Greeks by MC simulation Mixed estimator for second-order Greeks Let us again consider our simple, illustrative example... European option under B-S model: Gamma... and after a couple of lines of maths, the PW-LR hybrid estimator for the call option Gamma is given as: ( )( ) S(T ) Z e rt I{S(T ) > K} S(0) 2 σ T 1 24 / 58

37 Gamma Estimates Main MC Greek Approaches: B-S Tests European put gamma mixed estimators % Confidence Intervals Case B Case C Case D Case E Case F Case A B'nR Gamma Est's LR-PW Mixed Gamma Est's Analytical Gamma Values LRM Gamm Est's PW-LR Mixed Gamma Est's 25 / 58

38 Gamma Estimates Main MC Greek Approaches: B-S Tests European put gamma mixed estimators Case B Case C 95% Confidence Intervals Case F B'nR Gamma Est's LR-PW Mixed Gamma Est's Analytical Gamma Values LRM Gamm Est's PW-LR Mixed Gamma Est's 26 / 58

39 Main MC Greek Approaches: B-S Tests Conclusions of mixed estimator analysis 1. Both mixed estimators provide estimates with significantly lower standard errors than the LRM estimator (which gave lower standard errors than the Bump&Revalue) 2. Mixed estimators seem to utilise the respective advantages of the PW and LRM estimators 3. Given that they are constructed from the PW and LRM approaches, the mixed estimators are also unbiased 27 / 58

40 Main MC Greek Approaches: B-S Tests Estimating Greeks by MC simulation Estimators for Greeks on path-dependent payoffs Path-dependent payoff under B-S: LRM Estimator We can see that S(0) is a parameter of the first factor, g 1 (x 1 S(0)), only. Then the score function is: log g(s(t 1 ),..., S(t m )) S(0) = log g 1(S(t 1 ) S(0)) S(0) = ζ 1(S(t 1 ) S(0)) S(0)σ Z 1 = t 1 S(0)σ t 1 where Z 1 = ζ 1 (S(t 1 ) S(0)) is the Gaussian increment which takes us from time zero to time t 1, the first monitor point. 28 / 58

41 Monte Carlo Greeks for More Advanced Models Accommodating more advanced models The problems with Black-Scholes Black-Scholes model presents many problems for use with risk-management of Variable Annuity products: Underestimates the probability of extremely low equity returns Assumes volatility and risk-free rate are constant in time Assumes stock price movements are always continuous However, B-S model tractable and widely used/assumed Thus we wish to address the problems without losing too much of the B-S structure / 58

42 Monte Carlo Greeks for More Advanced Models Accommodating more advanced models Alternative models considered 1. Heston Stochastic Volatility model (Hest): ds t = rs t dt + V t S t dw S t dv t = κ(θ V t )dt + σ V Vt dw V t where κ > 0 represents the mean reversion speed, θ > 0 denotes the mean reversion level and σ V > 0 is the volatility of the variance process. Brownian motions have correlation ρ, i.e., dwt S dwt V = ρdt 30 / 58

43 Monte Carlo Greeks for More Advanced Models Accommodating more advanced models Alternative models considered 2. Heston SV with C-I-R stochastic rates (Hest-CIR): ds t = r t S t dt + V t S t dw S t dv t = κ V (θ V V t )dt + σ V Vt dw V t dr t = κ r (θ r r t )dt + σ r rt dw r t corr(wt S, Wt V ) = ρ S,V, i.e., dwt S dwt V = ρ S,V dt corr(wt S, Wt r ) = ρ S,r, i.e., dwt S dwt r = ρ S,r dt corr(wt V, Wt r ) = ρ V,r, i.e., dwt V dwt r = ρ V,r dt 31 / 58

44 Monte Carlo Greeks for More Advanced Models Accommodating more advanced models How do the three approaches change? With the B&R and PW approaches, the estimators just require the generated equity returns (mechanics same if SV or S-IR) On the other hand, LRM uses the score function in determining estimate and for the B-S model this score function can be found With Hest or Hest-CIR the returns are no longer lognormal and thus the score function cannot be determined However, if we condition on a realisation of the variance (and interest-rate) process, the returns are (conditionally) lognormal, allowing the LRM to be applied 32 / 58

45 Monte Carlo Greeks for More Advanced Models How does the LRM change with SV? Building in equity-volatility correlation Cholesky Decomposition Brownian motions for the asset and variance processes have correlation ρ. This can be achieved by setting: dw S t = ρdw V t + 1 ρ 2 dw Ind t Get expression for asset price process (with corr. structure): ds t = rs t dt + ( V t S t ρdw V t + 1 ρ 2 dwt Ind ) 33 / 58

46 Monte Carlo Greeks for More Advanced Models How does the LRM change with SV? Itô s calculus and Geometric Brownian Motion Itô s Lemma: Given a r.v y satisfying the s.d.e dy = a(y, t)dt + b(y, t)dw t where W t is a Brownian motion and a funct. f (y, t) differentiable wrt t and twice so wrt y, then f itself satisfies: df = ( f t + a(y, t) f y b(y, t)2 2 f y 2 ) dt+b(y, t) f y dw t 34 / 58

47 Monte Carlo Greeks for More Advanced Models How does the LRM change with SV? Itô s calculus and Geometric Brownian Motion Geometric Brownian motion: dy = ds t = S t (rdt + σdw t ) Then, f (y, t) = ln y = f t = 0, f y = 1 y, 2 f y 2 Black-Scholes = 1 y 2 and d ln S t = dx t = f f dt + t y f 2 y 2 (dy 2 ) = 0 + ds t /S t + (1/2)( 1/St 2 ) ( dst 2 ) (dw ) 2 = dt = rdt + σdw t + (1/2)( 1/St 2 )St 2 σ 2 dt = (r σ 2 /2)dt + σdw t 35 / 58

48 Monte Carlo Greeks for More Advanced Models How does the LRM change with SV? Itô s calculus and Geometric Brownian Motion Geometric Brownian motion: d ln S t = (r σ 2 /2)dt + σdw t S T = S 0 exp ((r 12 ) σ2 )T + σdw T i.e., lognormal returns in the Black-Scholes model Same approach used to obtain the conditional lognormal returns from conditioning on the Heston & CIR processes 36 / 58

49 Monte Carlo Greeks for More Advanced Models How does LRM change with SV & S-IR? Heston-CIR Model: ds t S t = r t dt + V t ( C 1 dwt V + C 2 dwt r + C 3 dw Ind ) t where C 1 = ρ S,V C 2 = ρ S,r ρ S,V ρ V,r 1 ρ 2 V,r C 3 = 1 ρ 2 S,V (ρ S,r ρ S,V ρ V,r ) 2 1 ρ 2 V,r Applying Itô s Formula and collecting terms gives / 58

50 Monte Carlo Greeks for More Advanced Models How does LRM change with SV & S-IR? Heston-CIR Model: ( 1 T S T = S 0 exp(y T ) exp r t dt 1 T 0 2 C T V t dt T 0 1 T ) +C 3 V t dw Ind t T 0 where Y T = 1 2 C 2 1 +C 1 T 0 T 0 V t dt 1 2 C 2 2 Vt dw V t T 0 + C 2 T 0 V t dt Vt dw r t 38 / 58

51 Monte Carlo Greeks for More Advanced Models How does LRM change with SV & S-IR? Heston-CIR Model: ( 1 T S T = S 0 exp(y T ) exp r t dt 1 T 0 2 C T V t dt T 0 1 T ) +C 3 V t dw Ind t T Therefore, the equity returns are conditionally lognormal: S 0 S 0 e Y T σ 2 C T T 0 0 V t dt 39 / 58

52 Monte Carlo Greeks for More Advanced Models Hest-CIR Test Cases for analysis Case κ V θ V ɛ V κ IR θ IR ɛ IR ρ S-V ρ S-IR ρ V-IR A B C D E Table: Different model settings considered in tests. Heston SV parameters are denoted by a V subscript and the CIR parameters are given by an IR subscript. The correlations denoted by obvious subscripts. Asian put option: S 0 = 1, K = 1.25, T = 30. Annual monitor points. 40 / 58

53 Delta Estimates Monte Carlo Greeks for More Advanced Models Hest-CIR Asian Option Delta Case B Case A % Confidence Intervals Case C Case D Case E B'nR Delta Est's Pathwise Delta Est's LRM Delta Est's 41 / 58

54 Gamma Estimates Monte Carlo Greeks for More Advanced Models Hest-CIR Asian Option Gamma Case A Case B Case C Case D % Confidence Intervals B'nR Gamma Est's LR-PW Mixed Gamma Est's LRM Gamm Est's Case E PW-LR Mixed Gamma Est's 42 / 58

55 Monte Carlo Greeks for More Advanced Models Conclusions from analysis 1. The Likelihood Ratio Method can be extended to price Asian options under the Hest-CIR model 2. For the delta sensitivity, the PW/B&R method produced estimates with smaller standard errors than the CLRM. 3. For the Gamma sensitivity, a mixed estimator will generally be preferable to the B&R and CLRM for path-dependent payoff functions. In certain cases the CLRM estimator can become very inefficient. 43 / 58

56 Monte Carlo Greeks for Variable Annuities Variable Annuity Example Variable Annuity Assumptions We now introduce a fairly simple VA contract to demonstrate that the three MC Greek approaches can be extended to estimate sensitivities of unit-linked insurance products Assumptions in VA Example: Policyholder is male and 65 years of age at annuitisation (and static mortality rates assumed) The policyholder has saved an amount P which constitutes the initial premium of the VA contract Policyholder will take (full) withdrawals from end of year one Policyholder lapsation behaviour is static 44 / 58

57 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product Structure Fund/Account Value begins at the level of the policyholder premium and grows/falls at the same rate as the equity index with which the product is linked (at annual rebalancing dates) At each rebalancing date the policyholder withdraws some amount of Income from this Fund Value This Income amount can grow from one year to the next if the FV grows above its high watermark level. This maximum lookback level will be denoted the Guarantee Base Yearly withdrawal level given as fixed % of this GB level 45 / 58

58 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product Structure: Mathematically Fund Value t years after annuitisation F t Guarantee Base t years after annuitisation G t Income Level t years after annuitisation I t Fixed percentage of GB taken as Income w Equity Index, return from year t 1 to t R t [= S t /S t 1 ] F t = max{(f t 1 I t 1 )(1 + R t ), 0} with F 0 = P S 0 G t = max{f t, G t 1 } and I t = wg t 46 / 58

59 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product Structure: Mathematically Fund Value t years after annuitisation F t Guarantee Base t years after annuitisation G t Income Level t years after annuitisation I t Fixed percentage of GB taken as Income w Equity Index, return from year t 1 to t R t [= S t /S t 1 ] Then Value of Liability of this particular VA contract is: [ T ] L = E D t pt surv max(i t F t, 0) t=1 46 / 58

60 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product: Pathwise Estimator Now a Pathwise method for this VA product will be developed To demonstrate this consider the problem of estimating the Delta Greek [ PW = L = T ] E D t pt surv max(i t F t, 0) S 0 S 0 t=0 With static policyholder behaviour, the derivative will only act on third term above / 58

61 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product: Pathwise Estimator Now a Pathwise method for this VA product will be developed To demonstrate this consider the problem of estimating the Delta Greek [ PW = L T ] = E D t p surv t max(i t F t, 0) S 0 S 0 t=0 With static policyholder behaviour, the derivative will only act on third term above / 58

62 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product: Pathwise Estimator This derivative is given by... ( It max(i t F t, 0) = I{I t > F t } F ) t S 0 S 0 S 0... but how are It S 0 and Ft S 0 calculated? 48 / 58

63 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product: Pathwise Estimator These are given recursively by: F ( t Ft 1 = I ) t 1 (1 + R t ) with F 0 = P S 0 S 0 S 0 S 0 I t S 0 = w G t S 0 = w with I 0 S 0 = 0 = w max(f t, G t 1 ) S 0 [ F t I {Ft>Gt 1 } + I S {Ft Gt 1 } 0 G ] t 1 S 0 49 / 58

64 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product: Likelihood Ratio Method Estimator Generality of LRM means it can be applied to VA product liability Greeks without much extra effort Similar structure to Asian option est r. Uses implied shock out to first valuation date, in our case the end of year 1 Z Imp. = log(s 1/ ξ 1 S 0 ) ( r 1 σ 1 2 /2) 1 σ 1 1 [ [( LRM Z Imp. ) T ]] = E E D t p surv t max(i t F t, 0) v 1, r 1 S 0 σ 1 1 t=0 50 / 58

65 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product: Likelihood Ratio Method Estimator Generality of LRM means it can be applied to VA product liability Greeks without much extra effort Similar structure to Asian option est r. Uses implied shock out to first valuation date, in our case the end of year 1 Z Imp. = log(s 1/ ξ 1 S 0 ) ( r 1 σ 1 2 /2) 1 σ 1 1 [ [( LRM Z Imp. ) ]] = E E Liability v 1, r 1 S 0 σ / 58

66 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product: Gamma Mixed PW-LR Estimator The Pathwise and CLRM can be combined to give a PW-LR estimator for the Gamma Greek. Along each path: Γ LR-PW = LRM S 0 ([ = = Z Imp. ] T S 0 S 0 σ 1 1 t=1 [ Z Imp. ] T D t pt surv S 0 σ 1 1 t=1 [ Z Imp. ] T S0 2 σ 1 1 t=1 D t p surv t max(i t F t, 0) max(i t F t, 0) S 0 D t p surv t max(i t F t, 0) ) 51 / 58

67 Monte Carlo Greeks for Variable Annuities Variable Annuity Example VA Product: Gamma Mixed PW-LR Estimator [ Γ LR-PW Z Imp. ] T = D t p surv t max(i t F t, 0) S 0 σ 1 1 S t=1 0 [ Z Imp. ] T S0 2 σ D t pt surv max(i t F t, 0) 1 1 t=1 The two sums are already calculated in the intermediate steps to find the Pathwise Delta estimator Thus we just take the result of these sums along each path and multiply by the respective factors above A PW-LR estimator can be constructed in a similar way 52 / 58

68 Monte Carlo Greeks for Variable Annuities Hest-CIR Test Cases for analysis Case κ V θ V ɛ V κ IR θ IR ɛ IR ρ S-V ρ S-IR ρ V-IR A B C D E Table: Different model settings considered in tests. Heston SV parameters are denoted by a V subscript and the CIR parameters are given by an IR subscript. The correlations denoted by obvious subscripts. V 0 = θ V, r 0 = θ IR. VA contract has T = 30 and Ratchet Term = / 58

69 Delta Estimates Monte Carlo Greeks for Variable Annuities Variable Annuity Delta Estimators Case B Case D Case E Case A Case C 95% Confidence Intervals B'nR Delta Est's Pathwise Delta Est's LRM Delta Est's 54 / 58

70 Gamma Estimates Monte Carlo Greeks for Variable Annuities Variable Annuity Gamma Estimators 1.E-05 8.E-06 6.E-06 4.E-06 2.E-06 0.E+00-2.E-06 Case A Case B Case C Case D -4.E-06-6.E-06-8.E-06 95% Confidence Intervals B'nR Gamma Est's LR-PW Mixed Gamma Est's LRM Gamm Est's Case E PW-LR Mixed Gamma Est's 55 / 58

71 Monte Carlo Greeks for Variable Annuities Conclusions from VA analysis 1. PW est r has lower st.err. than CLRM est r for VA Delta This is expected, as the series of options in time which make up the liability has a simple, continuous payoff 2. B&R & the CLRM estimators are both relatively inefficient estimators for the VA Gamma. However, mixed estimators have much lower variance than both of these estimators. Similarly to Asian option, the CLRM estimator for Gamma does not give as low st.err s as for European options. (Uses limited part of path in shock.) However, mixed estimator again seems to provide the benefits from both approaches 56 / 58

72 Monte Carlo Greeks for Variable Annuities Conclusions/Further Research Conclusions from research to date VA Greek estimators constructed are successful, in particular the mixed second-order sensitivity estimators Future Research Variance Reduction. Control Variate, Importance Sampling, Low-Discrepancy MC Interest-rate sensitivities. wrt CIR param s More complex VA product. Two Equity Indices, Equity/Bond mix, dynamic policyholder behaviour Computational framework. GPU, Automatic Diff.? 57 / 58

73 Monte Carlo Greeks for Variable Annuities Bibliography Glasserman, P., Monte Carlo Methods in Financial Engineering, Springer-Verlag, New York. Heston, S., A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The Review of Financial Studies, Volume 6 Number 2 pp (1993). Cox, J.C., Ingersoll, J.E. and Ross, S.A., A Theory of the Term Structure of Interest Rates. Econometrica 53 pp (1985). Broadie, M. and Kaya, O., Exact Simulation of Option Greeks under Stochastic Volatility and Jump Diffusion Models, Proceedings of the 2004 Winter Simulation Conference, The Society for Computer Simulation, Hobbs, C., et. al., Calculation of Variable Annuity Market Sensitivities using a Pathwise Methodology, Life & Pensions, September / 58

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