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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

Local knowledge. Global power. 2

Introduction

• 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
• 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. 3

Option Pricing – Current Period Loss

• Starting Point: Assume Black Scholes delta hedging

world is best estimate model

• Risk Neutral process for stock price
• Concept of “implied volatility simp” used to describe

market condition

• Data goes out about 15 years for S&P 500

2 ) ( ) (

2 2 2 2

              rV S V S S V S q r t V Sdz Sdt q r dS s s ) , , (

imp

S t V s Price Observed 

Local knowledge. Global power. 4

S&P 500 Implied Vols at June 30, 2009 for a number of different maturities

15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0% 50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150%

Strike % Vol %

15 yrs 10 yrs 5 yrs 3 yrs 1 yrs

Local knowledge. Global power. 5

Option Pricing – Current Period Loss

• Starting Point: Black Scholes delta hedging
• Key issue is our ability to value the gain/loss in a given
• period. If S->JS then unhedged loss UHL is
• Under Black Sholes assumptions:
• Must hold capital to cover possible

– Mis estimation of the mean (parameter risk) – Unexpected large up or down movement (contagion risk)

S V S J S t V JS t V UHL       ) 1 ( ) , ( ) , ( ] exp[ t z t J     s  ) ( ] [ ... ] [

2 2 2 2 2 1

t

• UHL

VAR t S V UHL E        s

Local knowledge. Global power. 6

Option Pricing – Current Period Loss

• Choose an appropriate J and cost of capital p then

                       S V S J S t V JS t V rV S V S S V S q r t V ) 1 ( ) , ( ) , ( 2 ) (

2 2 2 2

p s

Cost of Capital Hedge Gross Loss Economic Capital Expected Loss

Local knowledge. Global power. 7

Option Pricing – Current Period Loss

• Choose a reasonable J and cost of capital p
• Equivalent to new “contagion loaded” process
• Formally a simple version of Merton’s 1973 jump

diffusion model, interpretation is new

• Reasonably compact (infinite series) closed form

solution available (See Haug’s “Option Pricing Formulas” 1997).

                       S V S J S t V JS t V rV S V S S V S q r t V ) 1 ( ) , ( ) , ( 2 ) (

2 2 2 2

p s

Sdq J Sdz Sdt J q r dS ) 1 ( )] 1 ( [        s p

Local knowledge. Global power. 8

Option Pricing – Contagion Issues

• Cost of Capital must cover frictional cost plus target

return to shareholder p = t r + b M + a

• Quantity

– is negative if option is concave rather than convex – Same as mortality/longevity issue

• For vanilla puts and calls might want to use J = .6 for

puts but J = 1.4 for calls

• Numerical examples assume we are dealing with puts

S V S J S t V JS t V UHL       ) 1 ( ) , ( ) , (

Local knowledge. Global power. 9

Large Maturity Approximation

• Over a long time (e.g. 15+ years) the jump process

can be approximated by a modified Black Scholes model

• Allows standard Black Scholes formula to be used

instead of series solution

• “Asymptotic Black Scholes Approximation”

. ) ln( )] 2 / ) ln( ) ln( 1 ( [ , ) 1 ( )] 1 ( [

2 2 2

Sdz J Sdt J J J q r dS Sdq J Sdz Sdt J q r dS p s p s p                to converges" "

Local knowledge. Global power. 10

At the Money Implied Volatility

15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0%

• 5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Maturity in Years Vol %

Single Jump Model Asymptotic Black Scholes Observed

r AA Yield Curve s 15.0% p 20.0% J 50.0% q 3.0%

Local knowledge. Global power. 11

Step 3 Parameter Risk

• Back to Black Scholes for a moment…
• Assume new information arrives that causes us to

change our best estimate volatility assumption from s2 to a new value

• Need capital to cover the loss
• New system of valuation equations

2 2 2

ˆ s s s   

V V  ˆ

   

... ˆ , ) , ( ˆ ) , ( ˆ ˆ ˆ 2 ˆ ˆ ) ( ˆ , ) , ( ) , ( ˆ 2 ) (

) 2 ( ) 2 ( 2 2 2 2 2 2 2 2

                             t V S t V S t V V r S V S S V S q r t V S t V S t V rV S V S S V S q r t V p s p s

Local knowledge. Global power. 12

Parameter Risk

• In theory, must specify volatility assumptions for entire

hierarchy of volatility assumptions

• Example: geometric hierarchy
• 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)

• Closed form solutions for some special cases e.g. a =1
• r a = 0.

,... , ,

2 2 2 2 2 2

s s s s s     ...

2 2 2 2 2 2 p p p

s s s s s       

2 1 2 1 2

s a s s     

  n n n

Local knowledge. Global power. 13

Implied Volatility - Brute Force Parameter Risk

15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0% 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 vol %

1.00 3.00 5.00 10.00 15.00

Constantly Increasing Shock Hierarchy r 5.0% s 15.0% s 15.0% p 20.0% q 3.0%

Local knowledge. Global power. 14

Parameter Risk: At the Money Implied Volatility

15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0%

• 5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Maturity In Years Vol %

Hierarchy Expected Vol Brute Force

Constantly Increasing Shock Hierarchy r 5.0% s 15.0% s 15.0% p 20.0% q 3.0%

Local knowledge. Global power. 15

Parameter Risk

• Good News! Parameter Risk is actually fairly easy to

do in practice

• Can replace shock hierarchy with a deterministic

model (mean of the hierarchy)

• Final valuation model
• Has convenient closed form solutions

2 2

s b s  

b b a p s b s                   V rV S V S S V S q r t V ] ) 1 ( 1 [ 2 ) ( ) (

2 2 2 2 2

Local knowledge. Global power. 16

Put the pieces together

• Put parameter and contagion risk together
• If we want to fit June 30, 2009 S&P 500 market data

can use

– J = 50%, p= 20%, q= 3.0% – s2 = 10%, a = 50%

• Reasonable fit for first 15 years

. ) 1 ( ) , ( ) , ( 2 ) ( ] ) 1 ( 1 [ ) (

2 2 2 2 2

                              S V S J S t V JS t V rV S V S V S V S q r t V p s b s b b a p

Local knowledge. Global power. 17

Put the pieces together

Implied Volatility - Cost of Capital Model

15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0% 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 vol %

1 3 5 10 15 r AA Yield Curve s 15.0% s 10.0% a 50.0% p 20.0% J 50.0% q 3.0%

Local knowledge. Global power. 18

Final Step: Extrapolation

• Fit not perfect but appears to capture major risk issues
• As of June 30 ,2009 we are still in financial crisis mode
• Conclusion: must respect market data for first 15 years

but can use more “reasonable” parameters after that time

• Example: Assume p goes to 10% after 15 years

Local knowledge. Global power. 19

At the Money Implied Volatility Extrapolation Assumptions

15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0%

• 5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Maturity in Years Vol %

Single Jump Model Asymptotic Black Scholes Observed Jump Model Fwd Vol

Local knowledge. Global power. 20

Asymptotic Black Scholes Implied Volatility

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Maturity in Years Vol %

Spot Volatility Fwd Volatility