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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

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SLIDE 1

Robustness, Model Uncertainty and Pricing

Antoon Pelsser1

1Maastricht 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

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SLIDE 2

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

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SLIDE 3

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:

1

Robust agent prices unhedgeable risks using a “worst case” drift

2

Novel Interpretation of “Good Deal Bound” pricing

3

Drift depends on type of liability: leads to non-linear pricing

  • A. Pelsser (Maastricht U)

Robust Pricing 3-4 Feb 2011 – Paris 3 / 18

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SLIDE 4

Introduction

Outline of This Talk

1

Robustness & Model Uncertainty

2

Applications

3

Summary & Conclusion

  • A. Pelsser (Maastricht U)

Robust Pricing 3-4 Feb 2011 – Paris 4 / 18

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SLIDE 5

Robustness & Model Uncertainty

Tree Setup

Suppose we have a stock price S with return process x = ln S: dx = m dt + σ dWx, Discretisation in binomial tree: x(t + ∆t) = x(t) + +σ √ ∆t with prob. 1

2(1 + m σ

√ ∆t) −σ √ ∆t with prob. 1

2(1 − m σ

√ ∆t). Model uncertainty as m ∈ [mL, mH]. 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

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SLIDE 6

Robustness & Model Uncertainty

Tree Setup (2)

Introduce additional non-traded process y: dy = a dt + b dWy, with dWx dWy = ρ dt. “Quadrinomial” discretisation:

State: y + b √ ∆t y − b √ ∆t x + σ √ ∆t p++ = (1+ρ)+( m

σ + a b )

√ ∆t 4

  • p+− =

(1−ρ)+( m

σ − a b )

√ ∆t 4

  • տ ր
  • ւ ց

x − σ √ ∆t p−+ = (1−ρ)−( m

σ − a b )

√ ∆t 4

  • p−− =

(1+ρ)−( m

σ + a b )

√ ∆t 4

  • A. Pelsser (Maastricht U)

Robust Pricing 3-4 Feb 2011 – Paris 6 / 18

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SLIDE 7

Robustness & Model Uncertainty

Model Uncertainty

Model uncertainty in both m and a. Additional notation: µ := m a

  • ,

Σ := σ2 ρσb ρσb b2

  • .

Describe uncertainty set as ellipsoid: K := {µ0 + ε | ε′Σ−1ε ≤ k2}. 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

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SLIDE 8

Robustness & Model Uncertainty

Ellipsoid Uncertainty Set

  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5 rift of Insurance Process

  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5

  • 4%

0% 4% 8% 12% 16% Drift of Insurance Process Return on Financial Market ConfInt mu0 r a*

  • A. Pelsser (Maastricht U)

Robust Pricing 3-4 Feb 2011 – Paris 8 / 18

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SLIDE 9

Robustness & Model Uncertainty

Robust Optimisation Problem

Consider derivative f with payoff f

  • t + ∆t, x(), y()
  • at time t + ∆t.

Consider hedged position: f (t + ∆t) + θ

  • ex(t+∆t)−x(t) − er∆t

Robust rational agent solves the following optimisation problem for a time-step ∆t: max

θ

min

µ∈K e−r∆t

f1 +

  • f ′

xµ + θ(e′ 1µ − r + 1 2σ2) + 1 2 tr(fxxΣ)

  • ∆t
  • ,

where fx denotes gradient (fx, fy)′ and e1 denotes the vector (1, 0)′. Reformulate & simplify problem max

θ

min

ε

θq + ε′(fx + θe1) s.t. ε′Σ−1ε ≤ k2. with q = (e′

1µ0 − r + 1 2σ2) is excess return

  • A. Pelsser (Maastricht U)

Robust Pricing 3-4 Feb 2011 – Paris 9 / 18

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SLIDE 10

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 (fx + θe1): ε∗ := −

  • k
  • (fx + θe1)′Σ(fx + θe1)
  • Σ(fx + θe1).

If we use this value for ε∗ we obtain the reduced optimisation problem for the agent: max

θ

θq − k

  • (fx + θe1)′Σ(fx + θe1).

Maximise expected excess return θq minus k times st.dev. of hedged portfolio.

  • A. Pelsser (Maastricht U)

Robust Pricing 3-4 Feb 2011 – Paris 10 / 18

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SLIDE 11

Robustness & Model Uncertainty

Optimal Response for Agent

Solution to reduced optimisation problem for agent: θ∗ := −

  • fx + bρ

σ fy

  • +

q/σ

  • k2 − (q/σ)2

b

  • 1 − ρ2

σ |fy|.

Note, switch of notation: back to scalar expressions fx and fy!

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 fy

  • A. Pelsser (Maastricht U)

Robust Pricing 3-4 Feb 2011 – Paris 11 / 18

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SLIDE 12

Robustness & Model Uncertainty

Agent’s Valuation of Contract

If we substitute optimal ε∗ and θ∗ into original expectation, we obtain “semi-linear” pde ft + fx(r − 1

2σ2) + fya∗ + 1 2σ2fxx + ρσbfxy + 1 2b2fyy − rf = 0,

where the drift term a∗ for the insurance process is given by a∗ =

  • a0 − q ρb

σ

  • + b
  • 1 − ρ2 ·

      

  • −
  • k2 − (q/σ)2
  • for fy > 0,
  • +
  • k2 − (q/σ)2
  • for fy < 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

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SLIDE 13

Robustness & Model Uncertainty

Agent’s Valuation of Contract – Graphical

  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5 rift of Insurance Process

  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5

  • 4%

0% 4% 8% 12% 16% Drift of Insurance Process Return on Financial Market ConfInt mu0 r a*

  • A. Pelsser (Maastricht U)

Robust Pricing 3-4 Feb 2011 – Paris 13 / 18

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SLIDE 14

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

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SLIDE 15

Applications

Trend Uncertainty Equity

800% 1000% 1200% 1400% 1600% 1800%

Trend Uncertainty Equity

0% 200% 400% 600% 800% 1000% 1200% 1400% 1600% 1800% 1950 1970 1990 2010 2030 2050

Trend Uncertainty Equity

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

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SLIDE 16

Applications

Trend Uncertainty Life Expectancy

85 90 95

Trend Uncert LifeExpect (Male)

70 75 80 85 90 95 1950 1970 1990 2010 2030 2050

Trend Uncert LifeExpect (Male)

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

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SLIDE 17

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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SLIDE 18

Summary & Conclusion

Summary & Conclusion

Use model uncertainty and ambiguity aversion as “umbrella” Results:

1

Price contracts in incomplete markets in a “market-consistent” way

2

Robust agent prices unhedgeable risks using a “worst case” drift

3

Actuarial notion of “prudence”

4

Novel Interpretation of “Good Deal Bound” pricing

Netspar Panel Paper: work in progress

  • A. Pelsser (Maastricht U)

Robust Pricing 3-4 Feb 2011 – Paris 18 / 18