Applying the Cost of Capital Approach to Extrapolating an Implied - PowerPoint PPT Presentation

Local knowledge. Global power. Applying the Cost of Capital Approach to Extrapolating an Implied Volatility Surface August 1, 2009 B John Manistre VP Risk Research

Introduction o AEGON Context: European based life insurer that needs to develop market consistent financial statements o 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 o o Can we apply the cost of capital concepts developed for insurance liabilities to fill the “holes”? o Key ideas 1. Assume Law of Large Numbers Applies where appropriate 2. Start with simple Best Estimate (Black Scholes) 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. 2

Option Pricing – Current Period Loss o Starting Point: Assume Black Scholes delta hedging world is best estimate model o Risk Neutral process for stock price    s dS ( r q ) Sdt Sdz   s  2 2 2 V V S V      ( r q ) S rV 0    2 t S 2 S o Concept of “implied volatility s imp ” used to describe market condition  s imp Observed Price V ( t , S , ) o Data goes out about 15 years for S&P 500 Local knowledge. Global power. 3

S&P 500 Implied Vols at June 30, 2009 for a number of different maturities 45.0% 40.0% 35.0% Vol % 30.0% 25.0% 20.0% 15.0% 50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150% Strike % 15 yrs 10 yrs 5 yrs 3 yrs 1 yrs Local knowledge. Global power. 4

Option Pricing – Current Period Loss o Starting Point: Black Scholes delta hedging o Key issue is our ability to value the gain/loss in a given period. If S->JS then unhedged loss UHL is  V     UHL V ( t , JS ) V ( t , S ) ( J 1 ) S  S     s  o Under Black Sholes assumptions: J exp[ t z t ]  2 V  s   2 1 E [ UHL ] t ...  2 2 S   2 VAR [ UHL ] o ( t ) o Must hold capital to cover possible – Mis estimation of the mean (parameter risk) – Unexpected large up or down movement (contagion risk) Local knowledge. Global power. 5

Option Pricing – Current Period Loss o Choose an appropriate J and cost of capital p then   s     2 2 2 V V S V V       p    ( r q ) S rV V ( t , JS ) V ( t , S ) ( J 1 ) S         2 t S 2 S S Gross Loss Hedge Expected Cost of Loss Capital Economic Capital Local knowledge. Global power. 6

Option Pricing – Current Period Loss o Choose a reasonable J and cost of capital p   s     2 2 2 V V S V V       p    ( r q ) S rV V ( t , JS ) V ( t , S ) ( J 1 ) S         2 t S 2 S S o Equivalent to new “contagion loaded” process    p   s   dS [ r q ( J 1 )] Sdt Sdz ( J 1 ) Sdq o Formally a simple version of Merton’s 1973 jump diffusion model, interpretation is new o Reasonably compact (infinite series) closed form solution available (See Haug’s “Option Pricing Formulas” 1997). Local knowledge. Global power. 7

Option Pricing – Contagion Issues o Cost of Capital must cover frictional cost plus target return to shareholder p = t r + b M + a  V     o Quantity UHL V ( t , JS ) V ( t , S ) ( J 1 ) S  S – is negative if option is concave rather than convex – Same 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

Large Maturity Approximation o Over a long time (e.g. 15+ years) the jump process can be approximated by a modified Black Scholes model    p   s   dS [ r q ( J 1 )] Sdt Sdz ( J 1 ) Sdq , " converges" to    p     s  p 2 2 2 dS [ r q ( J 1 ln( J ) ln( J ) / 2 )] Sdt ln( J ) Sdz . o Allows standard Black Scholes formula to be used instead of series solution o “Asymptotic Black Scholes Approximation” Local knowledge. Global power. 9

At the Money Implied Volatility 45.0% AA Yield Curve r s 15.0% 40.0% p 20.0% 50.0% J q 3.0% 35.0% Vol % 30.0% 25.0% 20.0% 15.0% - 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Maturity in Years Single Jump Model Asymptotic Black Scholes Observed Local knowledge. Global power. 10

Step 3 Parameter Risk Back to Black Scholes for a moment… o o Assume new information arrives that causes us to change our best estimate volatility assumption from s 2 to a new value s  s   s ˆ 2 2 2 ˆ V  o Need capital to cover the loss V o New system of valuation equations     s  2 2 2 V V S V ˆ       p  ( r q ) S rV V ( t , S ) V ( t , S ) ,    2 t S 2 S   ˆ ˆ ˆ   s  ˆ 2 2 2 V V S V ˆ ˆ ˆ       p  ( 2 ) ( r q ) S r V V ( t , S ) V ( t , S ) ,    2 t S 2 S ˆ  ( 2 ) V  ...  t Local knowledge. Global power. 11

Parameter Risk o In theory, must specify volatility assumptions for entire hierarchy of volatility assumptions s s   s s   s 2 2 2 2 2 , , ,... 2  s   s  a   s 2 2 n 1 2 o Example: geometric hierarchy  n n 1 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) p p p s  s   s  s   s  2 2 2 2 2 ... 2 o Closed form solutions for some special cases e.g. a =1 or a = 0. Local knowledge. Global power. 12

Implied Volatility - Brute Force Parameter Risk 45.0% 40.0% Constantly Increasing Shock Hierarchy 35.0% 5.0% r s 15.0% vol % s 15.0% 30.0% p 20.0% 3.0% q 25.0% 20.0% 15.0% 50.0% 60.0% 70.0% 80.0% 90.0% 100.0% 110.0% 120.0% 130.0% 140.0% 150.0% Strike as % of current price 1.00 3.00 5.00 10.00 15.00 Local knowledge. Global power. 13

Parameter Risk: At the Money Implied Volatility 45.0% 40.0% Constantly Increasing Shock Hierarchy 35.0% 5.0% r s 15.0% Vol % s 15.0% 30.0% p 20.0% 3.0% q 25.0% 20.0% 15.0% - 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Maturity In Years Hierarchy Expected Vol Brute Force Local knowledge. Global power. 14

Parameter Risk o Good News! Parameter Risk is actually fairly easy to do in practice o Can replace shock hierarchy with a deterministic s  b  s 2 2 model (mean of the hierarchy) o Final valuation model   s  b  s   2 2 2 2 V V ( ) S V V       p   a b ( r q ) S rV [ 1 ( 1 ) ]     b 2 t S 2 S o Has convenient closed form solutions Local knowledge. Global power. 15

Put the pieces together o Put parameter and contagion risk together    s  b  s  2 2 2 2 V V V ( ) S V    p   a b   ( r q ) S [ 1 ( 1 ) ] rV    b  2 t S 2 S    V   p    V ( t , JS ) V ( t , S ) ( J 1 ) S .      S o If we want to fit June 30, 2009 S&P 500 market data can use – J = 50%, p = 20%, q= 3.0% – s 2 = 10%, a = 50% o Reasonable fit for first 15 years Local knowledge. Global power. 16

Put the pieces together Implied Volatility - Cost of Capital Model 45.0% AA Yield Curve r s 15.0% s 10.0% 40.0% a 50.0% p 20.0% J 50.0% 3.0% q 35.0% vol % 30.0% 25.0% 20.0% 15.0% 50.0% 60.0% 70.0% 80.0% 90.0% 100.0% 110.0% 120.0% 130.0% 140.0% 150.0% Strike as % of current price 1 3 5 10 15 Local knowledge. Global power. 17

Final Step: Extrapolation o Fit not perfect but appears to capture major risk issues o As of June 30 ,2009 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 p goes to 10% after 15 years Local knowledge. Global power. 18

At the Money Implied Volatility Extrapolation Assumptions 45.0% 40.0% 35.0% Vol % 30.0% 25.0% 20.0% 15.0% - 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Maturity in Years Single Jump Model Asymptotic Black Scholes Observed Jump Model Fwd Vol Local knowledge. Global power. 19

Asymptotic Black Scholes Implied Volatility 40.0% 35.0% 30.0% 25.0% Vol % 20.0% 15.0% 10.0% 5.0% 0.0% 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 Maturity in Years Spot Volatility Fwd Volatility Local knowledge. Global power. 20