Risk 1 - Hans Buehler

Equity Derivatives Teach-In 

Modeling Risk 

Dr. Hans Buehler, Head of EDG QR Asia 

Singapore, August 1 st , 2009 

hans.x.buehler@jpmorgan.com 

0

English_General 

[ P R E S E N T A T I O N T I T L E ] 

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1

Contents 

• Introduction, Overview, Product Lifecycle 

• Products 

• Modeling Risk 

— Basics 

— Local Volatility 

— Stochastic Volatility 

— Crash Risk 

• Numerical Methods 

2

Contents 

Challenges in Equity Modelling 

• Deterministic 

— Implied Volatility 

— Interest rates 

— Dividend marks and handling 

(proportional vs. discrete) 

— FX Volatilities 

— Equity Correlation 

— Equity/FX Correlation 

— Repo, borrow, basis, etc 

• First Order 

— Local Volatility 

— CDS/Credit risk 

• Second Order 

— Stochastic Volatility / Jumps 

— Stochastic Dividends 

— Stochastic Interest rates 

— Equity/IR correlation 

• Third Order 

— Stochastic Correlation 

— Stochastic Credit Risk 

— Equity/Credit ―correlation‖ 

• Fourth Order 

— Transaction Costs, Liqudity 

— Forward curve models 

3


Basics 

4

Modeling Risk – Basics 

Setting 

• Given is a Stochastic Differential Equation 

Brownian 

Motion 

dS( 

t) 

S( 

t) 

= ( 

t) 

dt s ( t, 

S( 

t)) 

dW ( t) 

Drift 

Volatility 

Forward 

— In the risk-neutral world, the drift is implied by E[S(t)] = F(t) as 

dF( 

t) 

( t) 

dt = r( 

t) 

 

( t) 

F( 

t) 

— Black & Scholes [3]: s= s BS is a constant number. 

5


Black & Scholes 

1400 

S&P500 versus Geometric Brownian Motion 

1300 

1200 

1100 

1000 

900 

800 

One of these paths 

is the actual S&P 

500, the other two 

are simulated using 

Black & Scholes. 

28/06/2003 06/10/2003 14/01/2004 23/04/2004 01/08/2004 09/11/2004 17/02/2005 28/05/2005 05/09/2005 14/12/2005 

6



• For this particular choice, the SDE has the analytical solution 

S 

BS 

1 2 

( t) 

= F( 

t)exp 

s 

BSW 

( t) 

s 

BS 

t 

2 

which allows computing vanilla option prices and, indeed, many more 

option prices analytically. 

The famous Black&Scholes formula for the value of an European call is: 

FV( T,Stoday) 

: = 

DF( T) 

= DF( T) 

E 

 

 

 

S 

BS 

( T) 

K 

F(T)N ( d 

 

 

 

 

) KN( 

d 

 

) 

d 

 

= 

ln( F( 

T) / K) 

 

s 

BS 

T 

1 

2 

s 2 

BS 

T 

7



• The value of each payoff under Black&Scholes is a function of the current 

spot level S today and time-to-maturity T: 

FV( T,Stoday) : = DF( T) 

E F( 

S ( T) 

 

t, S) : = FV( t, 

S) 

S 

— The seminal paper of Black&Scholes has shown that if the market is 

behaving like the model predicted, then perfect replication is possible: 

 

BS 

 

Payoff at 

maturity T 

 

S( 

T) 

K = FV(0, S ) 

Fair Value of 

the option 

today (time 0) 

T 

today 

( t, 

S( 

t)) 

dS( 

t) 

0 

Delta-Hedging 

(*) This simplified 

formula holds if 

•Interest rates are zero 

•The stock pays mo 

dividends 

•The stock earns/pays 

no borrow 

8



• What does this formula mean? 

S( 

T) 

K = FV(0, S ) ( t, 

S( 

t)) 

dS( 

t) 

— It tells us how to execute the hedge: 

FV( 

t 

 

today 

1 

dt, 

S( 

t1)) 

= FV(0, Stoday) 

(0, 

Stoday) 

S( 

t1) 

T 

0 

 

S 

today 

 

Buy units of 

shares today 

and sell them 

tomorrow. 

Tomorrow’s 

option value 

— For two days: 

Today’s option 

value 

9



• What does this formula mean? 

— For two days: 

FV( t 

2 

= FV( t 

, S( 

t 

1 

, S( 

t 

= FV(0, S 

 

S( 

T) 

K = FV(0, S ) ( t, 

S( 

t)) 

dS( 

t) 

2 

)) 

1 

today 

)) ( 

t 

1 

, S( 

t 

) (0, 

S 

1 

)) 

today 

) 

 

S( 

t 

2 

today 

) S( 

t 

) 

T 

0 

 

S 

( t ) S ( 

t , S( 

t )) S 

( t ) S( 

t ) 

1 

1 

today 

1 

1 

2 

1 

10



• The formula is understood as 

Change of 

option value 

Change of 

Delta position 

value 

FV( t 

dt, 

S( 

t 

dt)) 

FV( t, 

S( 

t)) 

dFV( 

t) 

= 

= 

( 

t, 

S( 

t)) 

 

S( 

t 

( 

t) 

dS( 

t) 

dt) 

S( 

t) 

 

• Practitioners often look at ―cash delta‖ rather than delta. 

It reflects the cash value of the delta position: 

— This gives 

 

$ 

( t) : = 

dFV( 

t, 

S( 

t)) 

= 

( 

t) 

S( 

t) 

 

$ 

dS( 

t) 

( t) 

S( 

t) 

Return on the 

stock 

investment 

11

Fair Value 

-200% 

-180% 

-160% 

-140% 

-120% 

-100% 

-80% 

-60% 

-40% 

-20% 

0% 

20% 

40% 

60% 

80% 

100% 

120% 

140% 

160% 

180% 

200% 


4 

Hedging a European Call 

(3d to maturity, 45% Volatility) 

Price T 

3 

Price(S0) + Delta T 

Price T+1 

2 

1 

Break-Even 

Point at 

-100% 

Break-Even 

Point at 

+100% 

0 

Stock price move in standard deviations 

(relative to one-day volatility) 

12

Daily Return / Initial Fair Value 


40% 

Naked European Call (matures 99% OTM) 

30% 

20% 

10% 

0% 

-10% 

-20% 

Using actual 

-30% 

HSI market 

04-Jul-08 04-Sep-08 04-Nov-08 04-Jan-09 04-Mar-09 data 04-May-09 

13



40% 

Delta-Hedging an European Call 

30% 

20% 

Unhedged 

Hedged 

10% 

0% 

-10% 

-20% 

-30% 

04-Jul-08 04-Sep-08 04-Nov-08 04-Jan-09 04-Mar-09 04-May-09 

14



40% 

Delta-Hedging an European Call 

500% 

30% 

20% 

LogSqRet 

Unhedged 

450% 

400% 

350% 

10% 

Hedged 

300% 

250% 

0% 

200% 

-10% 

-20% 

High daily 

volatility leads 

to deterioration 

of hedging 

performance. 

150% 

100% 

50% 

-30% 

04-Jul-08 04-Sep-08 04-Nov-08 04-Jan-09 04-Mar-09 04-May-09 

0% 

15

Probability 


70% 

Effect of Delta Hedging on the Return Distribution for ATM Calls 

60% 

50% 

40% 

30% 

Unhedged 2w 

Unhedged 1m 

Unhedged 2m 

Hedged 2w 

Hedged 1m 

Hedged 2m 

20% 

10% 

Marked 

reduction 

in tail risk 

Marked 

reduction 

in tail risk 

0% 

-50% -43% -37% -30% -23% -17% -10% -3% 3% 10% 17% 23% 30% 37% 43% 50% 

__ 

Period Feb 06 – April 07 

16



• What about the error if we only hedge daily (not instantaneous) ? 

— Black&Scholes PDE: 

dFV( 

t) 

 

$ 

( t) 

dS( 

t) 

S( 

t) 

= 

1 

2 

 

$ 

2 

dS( 

t) 

 

 

S( 

t) 

 

dt 

“Cash Gamma” 

$ := S 2 

Theta 

• Key message 

— Theta ―compensates‖ for Gamma (theta is negative – the option value 

decays in time) 

— The hedge works perfectly if Theta and the Gamma term cancel. 

17


Black & Scholes – Break Even Vol 

• In a Black&Scholes world, the hedge works perfectly, 

• Hence, Theta must cancel if the asset squared return is just as anticipated 

by the Black&Scholes model – this means: 

dFV( 

t) 

 

( t) 

dS( 

t) 

S( 

t) 

• Define the ―Break Even Volatility‖ 

$ 

s 

2 

break even 

= 

= 

1 

 

2 

1 

 

2 

$ 

$ 

2 

dS( 

t) 

 

 

S( 

t) 

 

 

s s 

2 

realized 

 

2 

dt 

: = 

$ 

dt 

2 

BS 

dt 

18


Black & Scholes – Break Even Vol 

• Break Even Vol 

— Note that for practical computation, needs to be computed carefully; 

the so-called ―optionality theta‖ is the raw theta 

 

optionalit y 

= 

simpleshift 

{dividends 

funding 

client cashflows} 

• In other words, 

s 

2 

break even 

 

optionalit y 

: = 

$ 

2 

dt 

19


Black & Scholes – Summary Break Even Vol 

• The most basic equation of Modeling Risk 

dFV( 

t) 

 

$ 

( t) 

dS( 

t) 

S( 

t) 

= 

1 

2 

 

$ 

 

s 

2 

realized 

s 

2 

model 

dt 

— Holds for all one-factor stock price models (*) 

dS( 

t) 

S( 

t) 

= ( 

t) 

dt s ( t, 

S( 

t)) 

dW ( t) 

— A similar formula will be shown for stochastic volatility 

We say ―we pay Theta‖ for the Gamma of the option. 

__ 

(*) assuming no daily recalibration. 

20


Local Volatility 

21

Modeling Risk – Local Volatility 

Black & Scholes – Implied Volatility 

• Recall the Black&Scholes formula for a European call: 

FV( T,Stoday) 

: = 

DF( T) 

= DF( T) 

E 

 

 

 

S 

BS 

( T) 

K 

F(T)N ( d 

 

 

 

 

) KN( 

d 

 

) 

d 

 

= 

ln( F( 

T) / K) 

 

s 

BS 

T 

1 

2 

s 2 

BS 

T 

22

Implied Vol 

8346 

9737 

11128 

12519 

13910 

15301 

16692 

18083 

19474 



• Using the BS volatility s BS as a free parameter, we can invert the BS 

formula to back out the so-called ―BS implied volatility‖ surface of the 

Market 

market. 

70% 

60% 

50% 

40% 

30% 

1Y 

2Y 

3Y 

3M 

Strike 

Example: HSI Mar 25, 2009 @ 13,910 

23



s Τ 

Κ 

= s 

ATM 

Τ 

Κ 

2 

K 1 

K 

skew* ln 

convexity *ln 

 

F( 

T) 

2 

F( 

T) 

 

approximately 

s ( T,105%) 

s ( T,95%) 

skew = 

10% 

for 95/105 strikes relative to forward (what we used before). 

Formula does not hold for far OTM strikes. 

24


Recall: Pricing with Skew 

0.7 

Skew Impact on ATM Digital Puts (95-105 skew @ -5%) 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 

Digital 

Put Spread 0.01% 



Put Spread 1% 

1w 1m 3m 6m 1y 

25

Implied Vol 

8346 

9737 

11128 

12519 

13910 

15301 

16692 

18083 

19474 



• Observed market option prices are not consistent with the BS assumption of 

a single constant volatility, and not even with the assumption of a timedependent 

volatility. 

Market 

70% 

60% 

50% 

40% 

30% 

1Y 

2Y 

3Y 

3M 

Strike 

Any pricing algorithm needs to “re-price” the observed option prices 

(otherwise we would have internal arbitrage). 

26


Dupire’s Local Volatility 

• Classic Solution: Dupire’s Local Volatility [2] 

the second most used model in equity derivatives 

— Idea: find s as a function of spot and time such that 

dS 

S 

LV 

LV 

( t) 

( t) 

= ( 

t) 

dt s 

LV 

( t, 

S 

LV 

( t)) 

dW ( t) 

reprices all market prices, ie 

 

 

! 

 

S ( t) 

K MarketCallPrice( t, 

K) 

DF( t) 

E 

LV 

= 

27



• Key points 

— Dupire’s local volatility is unique within the class of diffusions 

— In this class it is well-defined and given as 

where 

s ( t, 

S): 

= ~ 

LV 

s 

~ s 

LV 

 

t, 

k 

 

2 

: = 

LV 

2k 

~ 

C( 

t, 

k) 

t 

2 

~ 

C( 

t, 

k) 

2 

k 

2 

 

t, 

 

S 

F( 

t) 

 

 

 

Normalized 

Call Prices 

~ 

C ( t, 

k) 

: = 

MarketCallPrice t,kF( 

t) 

DF( t) 

F( 

t) 

 

 

28



Implied Volatility SPX Dec 9 2005 

Local Volatility SPX 

80 

80 

70 

70 

60 

60 

50 

50 

40 

40 

30 

20 

10 

0 

09-Dec-18 

09-Jun-16 

09-Dec-13 

09-Jun-11 

09-Dec-08 

30 

20 

10 

0 

09-Jun-18 

09-Dec-15 

09-Jun-13 

09-Dec-10 

09-Jun-08 

09-Jun-06 

09-Dec-05 

Strike/Spot 

Strike/Spot 

29

Implied Volatility 

Implied Volatility 


Dupire’s Local Volatility – time dependency 

Implied Vol 

flattens out in the 

future inside the 

Local Vol Model 

Local Vol Calibration .STOXX50E 10/10/2005 

Local Vol Calibration .STOXX50E 10/10/2005 shifted forward by 2 years 

50.0 

50.0 

45.0 

45.0 

40.0 

40.0 

35.0 

35.0 

30.0 

30.0 

25.0 

20.0 

15.0 

10.0 

5.0 

0.0 

70% 80% 90% 100% 110% 120% 130% 

Strike/Spot 

1m 

4m 

1y 

25.0 

20.0 

15.0 

10.0 

5.0 

0.0 

70% 

80% 

90% 

100% 

Strike/Spot 

110% 

120% 

130% 

1m 

3m 

6m 

1y 

30



• Plus 

— Allows to consistently price all combinations of European options. 

— One-factor model fast to simulate 

— Break Even Vol formula applies 

• Cons 

— Dynamical behaviour is not consistent 

— Implied Vol needs to be sensible 

Widely used reference model for equity derivatives. 

31

Thank you very much for your attention. 

hans.x.buehler@jpmorgan.com 

32

Numerical Methods 

References 

— [1] Gatheral: The Volatility Surface, Wiley, 2006 

— [2] Dupire: Pricing with a Smile, Risk, 7 (1), pp. 18-20, 1996 

— [3] Black, Scholes: The Pricing of Options and Corporate Liabilities, Journal of Political 

Economy, 81, pp. 637-59, 1973 

— [4] Carr, Madan: Option Valuation Using the Fast Fourier Transform, Journal of Computational 

Finance, 2, 1998 

— [5] Ren, Madan, Qian: Calibrating and pricing with embedded local volatility models, Risk, 

September 2007 

— [6] Buehler, Volatility Markets, VDM Verlag, 2008 

— [7] Merton: Option Pricing When Underlying Stock Returns are Discontinuous. Journal of 

Financial Economics 3 (1976) pp. 125-144, 1976 

— [8] Cont, Tankov: ―Financial modeling with Jump Processes‖, Chapman & Hall / CRC Press, 

2003 

— [9] Buehler, Volatility and Dividends, Working paper, 2008, http://www.math.tuberlin.de/~buehler/ 

— [10] Glasserman, ―Monte Carlo Methods in Financial Engineering‖, Springer 2004 

— [11] Andersen, Piterbarg: Moment Explosions in Stochastic Volatility Models WP~2004, 

http://ssrn.com/abstract=559481 

— [12] Bermudez, Buehler, Ferraris, Jordinson, Lamnouar, Overhaus: ―Equity Hybrid 

Derivatives‖, Wiley, 2006 

33

Risk 1 - Hans Buehler

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