Applying the Cost of Capital Approach to Extrapolating an Implied Volatility Surface

Transcription

1 Local knowledge. Global power. Applying the Cost of Capital Approach to Extrapolating an Implied olatility urface August 1, 009 B John Manistre P Risk Research

2 Introduction o o o o o AEGON Context: European based life insurer that needs to develop market consistent financial statements Basic idea: use observed market prices for hedgeable risk use cost of capital to price non-hedgeable risk Practical Problem: Holes in observed market data Can we apply the cost of capital concepts developed for insurance liabilities to fill the holes? Key ideas 1. Assume Law of Large Numbers Applies where appropriate. tart with simple Best Estimate (Black choles 3. Consider risk of current period loss (Contagion Event 4. Consider potential future losses (Parameter Risk 5. Revise Best Estimate assumptions if appropriate Local knowledge. Global power.

3 Option Pricing Current Period Loss o tarting Point: Assume Black choles delta hedging world is best estimate model o Risk Neutral process for stock price d = ( r q dt σdz t ( r q σ r = 0 o Concept of implied volatility σ imp used to describe market condition Observed Price = ( t,, σ o Data goes out about 15 years for &P 500 Local knowledge. Global power. 3 imp

4 45.0% &P 500 Implied ols at June 30, 009 for a number of different maturities 40.0% 35.0% ol % 30.0% 5.0% 0.0% 15.0% 50% 60% 70% 80% 90% 100% 110% 10% 130% 140% 150% trike % 15 yrs 10 yrs 5 yrs 3 yrs 1 yrs Local knowledge. Global power. 4

5 Option Pricing Current Period Loss o tarting Point: Black choles delta hedging o Key issue is our ability to value the gain/loss in a given period. If ->J then unhedged loss UHL is UHL = ( t, J ( t, ( J 1 o Under Black holes assumptions: 1 E[ UHL] = σ Δt... AR[ UHL] = o( Δt o Must hold capital to cover possible = exp[ µ Δt σz Δt] Mis estimation of the mean (parameter risk Unexpected large up or down movement (contagion risk J Local knowledge. Global power. 5

6 Option Pricing Current Period Loss o Choose an appropriate J and cost of capital π then t σ ( r q r = π ( t, J ( t, ( J 1 Expected Loss Cost of Capital Gross Loss Hedge Economic Capital Local knowledge. Global power. 6

7 Option Pricing Current Period Loss o Choose a reasonable J and cost of capital π t σ ( r q r = π ( t, J ( t, ( J 1 o Equivalent to new contagion loaded process d = [ r q π ( J 1] dt σdz ( J 1 dq o Formally a simple version of Merton s 1973 jump diffusion model, interpretation is new o Reasonably compact (infinite series closed form solution available (ee Haug s Option Pricing Formulas Local knowledge. Global power. 7

8 Option Pricing Contagion Issues o Cost of Capital must cover frictional cost plus target return to shareholder π = τ r β M α o Quantity UHL = ( t, J ( t, ( J 1 is negative if option is concave rather than convex ame as mortality/longevity issue o For vanilla puts and calls might want to use J =.6 for puts but J = 1.4 for calls o Numerical examples assume we are dealing with puts Local knowledge. Global power. 8

9 Large Maturity Approximation o Over a long time (e.g. 15 years the jump process can be approximated by a modified Black choles model d = [ r q π ( J 1] dt σdz ( J 1 dq, "converges" to d = [ r q π ( J 1 ln( J ln( J / ] dt σ π ln( J dz. o Allows standard Black choles formula to be used instead of series solution o Asymptotic Black choles Approximation Local knowledge. Global power. 9

10 At the Money Implied olatility 45.0% 40.0% 35.0% r AA Yield Curve σ 15.0% π 0.0% J 50.0% q 3.0% ol % 30.0% 5.0% 0.0% 15.0% Maturity in Years ingle Jump Model Asymptotic Black choles Observed Local knowledge. Global power. 10

11 Local knowledge. Global power. 11 tep 3 Parameter Risk o Back to Black choles for a moment o Assume new information arrives that causes us to change our best estimate volatility assumption from σ to a new value o Need capital to cover the loss o New system of valuation equations ˆ σ σ σ Δ = ˆ [ ] [ ]... ˆ,, ˆ(, ( ˆ ˆ ˆ ˆ ˆ ( ˆ,, (, ˆ( ( ( ( = = t t t r q r t t t r q r t π σ π σ

12 Parameter Risk o In theory, must specify volatility assumptions for entire hierarchy of volatility assumptions σ, σ Δσ, σ Δσ o Example: geometric hierarchy,... Δσ n Δσ n1 = α n1 Δσ o Formal solution is a stochastic volatility model where volatility jumps from one level to the next with transition intensity equal to cost of capital (Brute Force σ π σ Δσ π σ Δσ... o Closed form solutions for some special cases e.g. α =1 or α = 0. π Local knowledge. Global power. 1

13 Implied olatility - Brute Force Parameter Risk 45.0% 40.0% vol % 35.0% 30.0% 5.0% Constantly Increasing hock Hierarchy r 5.0% σ 15.0% Δσ 15.0% π 0.0% q 3.0% 0.0% 15.0% 50.0% 60.0% 70.0% 80.0% 90.0% 100.0% 110.0% 10.0% 130.0% 140.0% 150.0% trike as % of current price Local knowledge. Global power. 13

14 Parameter Risk: At the Money Implied olatility 45.0% ol % 40.0% 35.0% 30.0% 5.0% Constantly Increasing hock Hierarchy r 5.0% σ 15.0% Δσ 15.0% π 0.0% q 3.0% 0.0% 15.0% Maturity In Years Hierarchy Expected ol Brute Force Local knowledge. Global power. 14

15 Parameter Risk o Good News! Parameter Risk is actually fairly easy to do in practice o Can replace shock hierarchy with a deterministic model (mean of the hierarchy σ βδσ o Final valuation model t ( σ βδσ ( r q r = π [1 (1 α β ] β o Has convenient closed form solutions Local knowledge. Global power. 15

16 Local knowledge. Global power. 16 Put the pieces together o Put parameter and contagion risk together o If we want to fit June 30, 009 &P 500 market data can use J = 50%, π= 0%, q= 3.0% Δσ = 10%, α = 50% o Reasonable fit for first 15 years. 1 (, (, ( ( ] (1 [1 ( = Δ J t J t r q r t π σ β σ β β α π

17 Put the pieces together Implied olatility - Cost of Capital Model 45.0% 40.0% 35.0% r AA Yield Curve σ 15.0% Δσ 10.0% α 50.0% π 0.0% J 50.0% q 3.0% vol % 30.0% 5.0% 0.0% 15.0% 50.0% 60.0% 70.0% 80.0% 90.0% 100.0% 110.0% 10.0% 130.0% 140.0% 150.0% trike as % of current price Local knowledge. Global power. 17

18 Final tep: Extrapolation o Fit not perfect but appears to capture major risk issues o As of June 30,009 we are still in financial crisis mode o Conclusion: must respect market data for first 15 years but can use more reasonable parameters after that time o Example: Assume π goes to 10% after 15 years Local knowledge. Global power. 18

19 45.0% At the Money Implied olatility Extrapolation Assumptions 40.0% 35.0% ol % 30.0% 5.0% 0.0% 15.0% Maturity in Years ingle Jump Model Asymptotic Black choles Observed Jump Model Fwd ol Local knowledge. Global power. 19

20 Asymptotic Black choles Implied olatility 40.0% 35.0% 30.0% 5.0% ol % 0.0% 15.0% 10.0% 5.0% 0.0% Maturity in Years pot olatility Fwd olatility Local knowledge. Global power. 0