Robustness, Model Uncertainty and Pricing Antoon Pelsser 1 1 Maastricht University & Netspar Email: a.pelsser@maastrichtuniversity.nl 3-4 Feb 2011 Conference Longevity and Pension Funds – Paris A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 1 / 18
Introduction Motivation Pricing contracts in incomplete markets Examples: Pricing very long-dated cash flows T ∼ 30 − 100 years Pricing long-dated equity options T > 5 years Pricing pension & insurance liabilities Actuarial premium principles typically “ignore” financial markets Actuarial pricing is “static”: price at t = 0 only Financial pricing considers “dynamic” pricing problem: How does price evolve over time until time T ? Financial pricing typically “ignores” unhedgeable risks A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 2 / 18
Introduction Main Ideas Pricing contracts in incomplete markets in a “market-consistent” way Use model uncertainty and ambiguity aversion as “umbrella” Agent does not know the “true” drift rate of stochastic processes Agent does know confidence interval for drift Agent is worried about model mis-specification Agent can trade in financial markets Agent is “robust”; i.e. tries to maximise worst-case expected outcome Results: Robust agent prices unhedgeable risks using a “worst case” drift 1 Novel Interpretation of “Good Deal Bound” pricing 2 Drift depends on type of liability: leads to non-linear pricing 3 A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 3 / 18
Introduction Outline of This Talk Robustness & Model Uncertainty 1 Applications 2 Summary & Conclusion 3 A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 4 / 18
Robustness & Model Uncertainty Tree Setup Suppose we have a stock price S with return process x = ln S : dx = m dt + σ dW x , Discretisation in binomial tree: � + σ √ √ with prob. 1 2 (1 + m ∆ t ∆ t ) √ √ σ x ( t + ∆ t ) = x ( t ) + with prob. 1 2 (1 − m − σ ∆ t ∆ t ) . σ Model uncertainty as m ∈ [ m L , m H ]. Change mean ⇐ ⇒ change probability ⇐ ⇒ stoch. discount factor √ “Local Volatility” of stoch. discount factor: m /σ ∆ t Conf.Intv. on mean ⇐ ⇒ bounds on discount factor volatility Bounds on discount factor ⇐ ⇒ likelihood ratio test A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 5 / 18
Robustness & Model Uncertainty Tree Setup (2) Introduce additional non-traded process y : dy = a dt + b dW y , with dW x dW y = ρ dt . “Quadrinomial” discretisation: √ √ State : y + b ∆ t y − b ∆ t � (1+ ρ )+( m � (1 − ρ )+( m √ √ √ σ + a � σ − a � b ) ∆ t b ) ∆ t x + σ ∆ t p ++ = p + − = 4 4 տ ր • ւ ց √ � (1 − ρ ) − ( m � (1+ ρ ) − ( m √ √ σ − a � σ + a � b ) ∆ t b ) ∆ t x − σ ∆ t p − + = p −− = 4 4 A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 6 / 18
Robustness & Model Uncertainty Model Uncertainty Model uncertainty in both m and a . Additional notation: � σ 2 � m � � ρσ b µ := , Σ := . b 2 ρσ b a Describe uncertainty set as ellipsoid: K := { µ 0 + ε | ε ′ Σ − 1 ε ≤ k 2 } . Motivated by shape of confidence interval of estimator ˆ µ Motivated by set of “indistinguishable” models under LR test A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 7 / 18
Robustness & Model Uncertainty Ellipsoid Uncertainty Set 0.5 0.5 0.4 0.4 Drift of Insurance Process rift of Insurance Process 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.5 -4% 0% 4% 8% 12% 16% Return on Financial Market ConfInt mu0 r a* A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 8 / 18
Robustness & Model Uncertainty Robust Optimisation Problem � � Consider derivative f with payoff f t + ∆ t , x () , y () at time t + ∆ t . e x ( t +∆ t ) − x ( t ) − e r ∆ t � � Consider hedged position: f ( t + ∆ t ) + θ Robust rational agent solves the following optimisation problem for a time-step ∆ t : µ ∈K e − r ∆ t � 2 σ 2 ) + 1 f ′ x µ + θ ( e ′ 1 µ − r + 1 � � � max min f 1 + 2 tr( f xx Σ) ∆ t , θ where f x denotes gradient ( f x , f y ) ′ and e 1 denotes the vector (1 , 0) ′ . Reformulate & simplify problem θ q + ε ′ ( f x + θ e 1 ) max min ε θ ε ′ Σ − 1 ε ≤ k 2 . s.t. with q = ( e ′ 1 µ 0 − r + 1 2 σ 2 ) is excess return A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 9 / 18
Robustness & Model Uncertainty Optimal Response for Mother Nature Two-player game: agent vs. “mother nature” Worst-case choice for Mother Nature given any θ is “opposite direction” of vector ( f x + θ e 1 ): � � k ε ∗ := − Σ( f x + θ e 1 ) . � ( f x + θ e 1 ) ′ Σ( f x + θ e 1 ) If we use this value for ε ∗ we obtain the reduced optimisation problem for the agent: � max θ q − k ( f x + θ e 1 ) ′ Σ( f x + θ e 1 ) . θ Maximise expected excess return θ q minus k times st.dev. of hedged portfolio. A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 10 / 18
Robustness & Model Uncertainty Optimal Response for Agent Solution to reduced optimisation problem for agent: � 1 − ρ 2 � f x + b ρ � q /σ b θ ∗ := − + | f y | . σ f y k 2 − ( q /σ ) 2 � σ Note, switch of notation: back to scalar expressions f x and f y ! Nice economic interpretation: Left term is “best possible” hedge Perfect hedge for “pure financial” risks Leads to market-consistent pricing Right term is “speculative” position, which is product of: Function of Sharpe ratio q /σ Residual unhedgeable risk Absolute value of f y A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 11 / 18
Robustness & Model Uncertainty Agent’s Valuation of Contract If we substitute optimal ε ∗ and θ ∗ into original expectation, we obtain “semi-linear” pde 2 σ 2 ) + f y a ∗ + 1 f t + f x ( r − 1 2 σ 2 f xx + ρσ bf xy + 1 2 b 2 f yy − rf = 0 , where the drift term a ∗ for the insurance process is given by � � � k 2 − ( q /σ ) 2 − for f y > 0 , � a 0 − q ρ b � a ∗ = � 1 − ρ 2 · + b σ � � � k 2 − ( q /σ ) 2 + for f y < 0 . Again, nice economic interpretation for a ∗ . Same result as Good Deal Bound pricing. A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 12 / 18
Robustness & Model Uncertainty Agent’s Valuation of Contract – Graphical 0.5 0.5 0.4 0.4 Drift of Insurance Process rift of Insurance Process 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.5 -4% 0% 4% 8% 12% 16% Return on Financial Market ConfInt mu0 r a* A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 13 / 18
Robustness & Model Uncertainty Different Interpretations for k Equivalence between Good Deal Bound pricing & Model Uncertainty Parameter k is: Width of confidence interval for trend Volatility of pricing kernel / stoch.disc.factor Sharpe-ratio of risks Cost-of-Capital times # of st.dev’s for unhedgeable risk Calculation of k : Sharpe-ratio for equity: k ≈ (8% − 4%) / 16% = 0 . 25 √ Conf.intv.: k ≈ 2 / 25 = 0 . 4 Cost-of-Cap: k ≈ 0 . 06 ∗ 2 . 5 = 0 . 18 A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 14 / 18
Applications Trend Uncertainty Equity Trend Uncertainty Equity Trend Uncertainty Equity 1800% 1800% 1600% 1600% 1400% 1400% 1200% 1200% 1000% 1000% 800% 800% 600% 400% 200% 0% 1950 1970 1990 2010 2030 2050 S&P equity index from 1950 until 2009 Conf.intv. for mean: [3 . 7% , 17 . 2%] per annum. A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 15 / 18
Applications Trend Uncertainty Life Expectancy Trend Uncert LifeExpect (Male) Trend Uncert LifeExpect (Male) 95 95 90 90 85 85 80 75 70 1950 1970 1990 2010 2030 2050 Life Expectancy (at birth) for Dutch Males 1950 until 2006 Conf.intv. for trend: [0 . 9 , 2 . 8] months per annum. A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 16 / 18
Applications ALM with Model Uncertainty Toy-model: stylised pension-fund with fixed investment mix Only Equity risk & Longevity risk; fixed interest rate of 4% % ‐ beleg.mix 0% 5% 10% 15% 20% 25% 185% Aandelen 0.0% ‐ 5.8% ‐ 6.4% ‐ 6.6% ‐ 6.7% ‐ 6.7% A0 Levensverw. 1.49 0.78 0.46 0.32 0.25 0.20 5.00 E(surpl) 9.292 10.572 10.694 10.645 10.544 10.422 130% Aandelen 0.0% ‐ 5.2% ‐ 6.2% ‐ 6.5% ‐ 6.6% ‐ 6.6% A0 Levensverw. 1.49 0.96 0.61 0.44 0.35 0.28 3.50 E(surpl) 1.863 3.002 3.231 3.263 3.231 3.170 102% Aandelen 0.0% ‐ 4.7% ‐ 5.9% ‐ 6.3% ‐ 6.5% ‐ 6.6% A0 Levensverw. 1.49 1.08 0.72 0.54 0.43 0.35 2.75 E(surpl) ‐ 1.852 ‐ 0.825 ‐ 0.541 ‐ 0.460 ‐ 0.453 ‐ 0.480 74% Aandelen 0.0% ‐ 3.9% ‐ 5.4% ‐ 6.0% ‐ 6.3% ‐ 6.4% A0 Levensverw. 1.49 1.21 0.88 0.68 0.55 0.46 2.00 E(surpl) ‐ 5.567 ‐ 4.700 ‐ 4.366 ‐ 4.231 ‐ 4.178 ‐ 4.165 Worst case model depends on agent’s investment mix Worst case model depends on initial coverage ratio Robustness ⇒ reduce “model optimism” A. Pelsser (Maastricht U) Robust Pricing 3-4 Feb 2011 – Paris 17 / 18
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