Séminaire validation des modèles 27 mai 2010 Cost of hedging in illiquid markets Etienne KOEHLER, Barclays Capital Etienne Koehler Paris

Séminaire validation des modèles27 mai 2010Cost of hedging in illiquid markets

Etienne KOEHLER, Barclays Capital

Etienne Koehler Paris

Table of Contents

1. The two classical pricing approaches

2. Some alternative approaches

3. Impact of the crisis on options hedging

4. Some words about the crisis case and conclusion

The two classical pricing approaches



Two techniques, very often assumed to be equivalent:

One based on hedging (through a “replicating” portfolio of underlying)

One based (directly or not) on probabilities of a payoff to materialize

1


However, equivalence known to be not always true.

Recent real example: the sharp increase in the price of risk that started in August 2007 spread to all markets

Disruptions in option pricing, even in usually very liquid markets like Forex, have been huge

Big impact of discontinuity in option hedging and transaction costs, in an environment where classical assumptions hold no more


2


Some examples on the theoretical side:

Boyle and Emanuel (1980): the pricing error of an option is inversely proportional to the re-hedging frequency in discrete time intervals

Leland (1985) developed a modified Black-Scholes hedging strategy with a volatility adjusted by length of rebalance interval and rate of proportional transaction cost (or cost minimization)

Using Chaos decomposition, V. Lacoste (1996) deduced hedging strategies by a simultaneous minimization of the risk of the portfolio, the transaction costs and the tracking error of such strategies

V Lacoste and T Ané (2001): convex portfolios of options entail hedging costs inconsistent with the tight bid/ask spreads in the markets


3


Rarely put into question in usual trading life…

However Does the price at inception really provide enough money (for a given accepted level of risk) to hedge

the position? Hedging with the underlying or with other options? What if volatile phases? Pricing additivity?

Of the number of techniques, parameters, and their impact on the hedging, one is not easy to apprehend: the market costs of hedging

Hedging cost versus pay-off expectancy?

4


Usual modelling assumption: hedging is supposed to be cost free

However a short vega trader for example, with a positive theta, has to try to manage his position so as not to lose in the negative gamma management the money he gets from a positive carry

He then has to set himself strict guidelines on what moves to hedge

A look at hedging costs

5


But as the recent crises show: costs of hedging surge and can quickly put at risk part or all of the margins locked-in for a “medium to long term” product, quite likely that market conditions will vary widely during the life of

the product after a crisis the market stabilizes at a different level where the original assumptions are not true

anymore

Then models might actually dupe their users by making them believe that differentials (as HR) will fully hedge them

A look at hedging costs

6

Some alternative approaches


General framework anyway: to be able to keep on computing prices, we still need a space of random variables stable under the usual operations

Then: a priori Ito processes (drift + martingale part)

However, what about pricing additivity, or even sub or super additivity?? The price of a long short position may be quite different from zero Compared to the sum of individual prices, the price of a portfolio may be significantly less than (e.g.

long short) or more (risk concentration)


7


Jumps?

Levy processes and risk minimization?

Filtration extensions?


8


As seen in recent situations (or economic papers), three types of phases can be isolated: standard, high volatility, crises

A possible natural pricing: weighted average of pricings in each case.

VaR computation includes this idea, e.g. through historical scenarios, but rather to look at portfolio limits

Another possible approach

9


The probability of switching from one to the other of the 3 phases remains difficult to estimate.

It depends: On the maturity of the product (the longer, the more probable it will have to weather hard times) On the firm’s “appetite” for risk

The impreciseness can be narrowed through econometric but the percentage of time and probability that each product will live in each of the 3 phases remains a guess


10


Using historical data to propose these values and upgrade the pricing (in a way similar to a VaR computation) might be better but remains arbitrary

Another possibility is to include a provision on this model risk, which can be computed with the same line of idea as just mentioned


11

Impact of the crisis on options hedging

0.85

0.95

1.05

1.15

-30

10


Risk visualization = why pricing is not enough

Classical risk profile

Simple indicators

Ranges predicted by the market

Risk Aversion + utility function

Zero-cost switches

Immunization

Additional parameters

Inadequacy of mathematical indicators

Need to visualize and understand

Limits of the risk profile

Risk Profile

-30

-20

-10

0

10

20

30

40

1 2 3 4 5 6 7 8 9 10

P/L

Risk Surface

Vol delta

IR delta

12


Market probabilities might not match the risk aversion

Buyers and sellers = why is there a smile ?

The correlation problem

Case of a simple strategy

-3

-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8

P/L

Risk Surface

13


Disappearing of some market-makers

Widening bid/offer spreads

Huge jumps in market volatility, especially on the short-end

Gaps in the long term volatility liquidity

Effects on hedging : Difficulty to delta-hedge Higher cost of negative gamma Disruptions in smile propagation in the models


14


A standard P/C option delta-hedged

A complex IR / FX hybrid also hedged with vanilla products Long term PRDC structure Bearing FX, IR and correlations impacts Requiring constant hedging on its parameters during its lifetime

Real impact on 2 examples

15


(PRD) swap

Issue swap of a Callable Bond issued in JPY, with coupons linked to a Foreign Exchange rate (here USD/JPY).

Issuer pays JPY Libor – margin

Counterpart pays a coupon equal to a fixed amount in USD less a fixed amount in JPY, floored at 0:

Callable feature

Counterpart has the right to cancel the structure at each coupon date after a number of No Call periods

The PRD pay-off

0*** cpnJPYcpnUSD domNFXdualN

16


FX spot, JPY rates and USD rates all impact the call option embeded in the PRDC. The challenge is to value a « bermudan » option up to 30Y using a model which accounts for those three factors of risk (1FX+2IR).

Banks have developped sophisticated 3 factors models to monitor these products, usually based on Heath-Jarrow-Morton model with 3 Gaussian state variables:

Here, is the JPY short rate and is the instantaneous forward rate for time t read

on today’s JPY curve. The same notations apply for USD

Valuing the call option

)0(

)(log)(

),0()()(

),0()()(

FX

tFXtZ

tUSDft

USDrtX

tJPYft

JPYrtX

f

d First factor: JPY rate

Second factor: USD rate

Third factor: FX rate

)(trJPY ),0( tf JPY

17


Once we identify the main factors of risk, we need to specify the « risk neutral » dynamics of the model

with

And

describe the correlations USD rates / JPY rates, JPY rates / FX and USD rates / FX

3 factor model: Dynamics

)()()(2

1)()()(

)()())()()(()(

)()())()(()(

2 tdWtdtttrtrtdZ

tdWtdtttXttdX

tdWtdttXttdX

ZZXUSDJPY

ffZZffff

ddddd

t

fff

t

ddd dsststdsstst00

))(2exp()()(,))(2exp()()(

dtdWdWdtdWdWdtdWdW fZZfdZZddffd ,,,,,

18


If FX vols move down, the floor in the exotic coupon is worth less, and the bond is worth less.

Long term FX vols have more impact than short term FX vols because the vega increases with the maturity.

When the coupon is capped, the impact of the FX vols is lower.

PRDC is short of FX volatility but it is less exposed to FX volatility move than a simple PRD

PRDC: effect of FX vols

Scenario Variation MtM

Vol FX -1% -0,47%

Vol FX <= 10Y, -1% -0,12%

Vol FX > 10Y, -1% -0,35%

MtM change of the bond

19


If FX rate moves up, the PV of the exotic coupon is higher, and the mark-to-market of the bond moves up

If JPY rates move up, the PV of the notional moves down and the mark-to-market of the bond moves down (like in a classical bond)

If USD rates move up, the forward FX move down, the PV of the exotic coupon is lower and so is the mark-to-market of the bond

PRDC has the same direction of « rate » risk than a PRD but of a reduced amount

PRDC: effect of rates

Scenario Variation MtM

FX + 5 yens 1,43%

JPY rates + 10 bps -1,72%

USD rates + 10 bps -0,37%

MtM change of the bond

20


The correlations have an impact only on the callable feature

The impact of correlations should always be understood as after recalibration of the model to the IR volatilities and FX volatilities

Different effects are combined: the impact on the one-time callable feature, the impact on the switch option (difference between multi-callable and one-time callable) and the impact on the embedded floor

Such combinations are quite complex: the impact may even change sign depending on the particular structure and the market conditions

In what follows, we focus on the impacts on a « classical » 30Y PRDC with floor at 0, in current market conditions

Impact of correlations

21


Indication of hedging frequency = FX

Since 29/01/2001, in USD / JPY

Days move > 2 % 15

Days move > 1.5 % 47

Days move > 1 % 177

Days move > 0.9 % 256

Days move > 0.8 % 335

Days move > 0.7 % 442

Days move > 0.6 % 584

Days move > 0.5 % 720

22


Number of re-hedged for an initially ATM Forward, 5Y uds/jpy call option, depending on the precision required

Costs for the vanilla option

Tics between rehedgesYear 25 50 75 1002001 156 118 79 432002 185 119 74 382003 160 82 47 292004 170 108 62 362005 175 100 49 212006 164 94 51 272007 155 96 68 386M 2008 * 2 196 141 94 46

Total 01-05 846 527 311 167Total 02-06 854 503 283 151Total 03-07 824 480 277 151Total 04-08 860 539 324 168

Costs % premium 01-05 0,184 0,229 0,203 0,145Costs % premium 02-06 0,186 0,219 0,185 0,131Costs % premium 03-07 0,179 0,209 0,181 0,131Costs % premium 04-08 0,187 0,234 0,211 0,146

(Assumption = 50% of the rehedges are cost-free

Cost 10 tics = 0,0174% in spot

23


Costs for the PRDC structure

Number of re-hedged for a callable usd/jpy bond with floored coupon

Year Jpy rate Usd rate FX2001 27 72 432002 25 65 382003 34 81 292004 23 45 362005 27 22 212006 13 34 272007 24 55 382008 39 71 47

Total 01-06 149 319 194Total 02-07 146 302 189Total 03-08 160 308 198

Unitary cost 184,8 301,1 594Cost/sensitivity 6,36 2,23 8,49

Total cost 6 yearsRates and FX 17,08 bp flatCorrelations 5,12 bp flatTotal 22,20 bp flat

For 30 years: 111,0 bp flat

Cost IRS: 0,125% per annum per deal

Flat var / 0,125 bp rateUsd 10Y 0,9775Jpy 10Y 1,155

Rehedge = 10bp IR100 FX

Cost FX = Vanilla in bp

24


The price is higher than what the pricing models show

The Greeks are harder to monitor than what was expected

The cost of hedging is different in 3 phases : In a « standard market, models indications hold In a « volatile market », the increase in price can be linked to the volatility (Formula (A))

In a « crisis » market, some additional price / risk parameters have to be added, depending in part on the risk aversion of the hedger

Summary of back-testing on those 2 examples

25

Some words about the crisis case and conclusion

Some past crises

Year Crisis Markets

1974 Bank Herstatt Bank, Forex, Systemic risk

1979 Rise of the Fed Funds American monetary market

1980 Corner of silver metal Metals, energy, agricultural products

1982 Debt of Emerging Markets Bank, Interest rates, Systemic risk

1985 Bank of New York Bank, Systemic risk

1987 October 1978 krach Interest rates, Equity, Systemic risk

1989 Junk bonds Bank, Interest rates

1989 Japanese bubble Equity, Real estate, Banks

1990 Invasion of Kuwait Oil, Interest rates

1992 EMS crisis Forex, Interest rates

1993 EMS crisis: the return Forex, Interest rates

1994 Correction on bond market Interest rates

1994 Mexican economic crisis Forex, Interest rates, Systemic risk

1997 Asian economic crisis Forex, Bank

1997-1998 Brazil Forex

1998 Russian crisis (LTCM…) Interest rates, Systemic risk

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Some more past crises

Year Crisis Markets

2000 Internet bubble Equity

2000 Turkey Bank, Interest rates, Forex

2000 - ? Zimbabwe Hyperinflation

2001 11 September Systemic risk

2001 Junk bonds Interest rates

2001 Argentinean economic crisis Forex

2002 Brazil Bond market, Forex

2007-? Subprime crisis Real Estate, Bank, Equity, Systemic risk

2008-? Credit crisis Real Estate, Bank, Equity, Systemic risk

27


First, let us note that during a crisis only back-testing can give an estimate of the costs to hold a position

Then, if the latest market disruptions can give an idea of future ones, those hedging costs are quite expensive

Some words about the crisis case

28


Back-testing can provide a more accurate pricing for new transactions thanks to the analysis of past crisis, through the inclusion of formula A in the pricing function

A possibility might also be to buy deep out of the money options as a static hedge against these crisis scenarios

Some words about the crisis case

29


What is certain is that standard pricing undervalues the option

At the end of the day, the pricing needs to be adjusted by a function that we showed depends on volatility (from A)

A natural way to do it is to look at the pricing via hedging costs

Conclusion

30


Boyle, P. D. Emanuel, 1980, “Discretely Adjusted Option Hedges”, Journal of Financial Economics, Vol. 8, p.359-282

Leland, H. (1985). “Option Pricing and Replication with Transaction Costs”, Journal of Finance, 5, 1283–1301

"Wiener Chaos: A New Approach to Option Hedging" (, V. Lacoste).. Mathematical Finance, Special Issue on Market Imperfections, apr. 1996, Vol. 6, N° 2, p. 197‑213

"Understanding Bid-ask Spreads of Derivatives Under Uncertain Volatility and Transaction Costs" (T. Ane, V. Lacoste).. International Journal of Theoretical and Applied Finance (The), jan 2001, Vol. 4, N° 3, p. 467‑489

Zakamouline, Valeri ,Optimal Hedging of Option Portfolios with Transaction Costs (August 15, 2006). Available at SSRN: http://ssrn.com/abstract=938934

Toft, “on the mean-variance tradeoff in option replication with transaction costs”, Journal of Financial and Quantitative Analysis 31, 233–263

Kennedy, Forsith and Vetzal: “Dynamic Hedging under Jump Diffusion with Transaction Costs”, working paper, to be published in Operations Research, 2008

Cont, Rama and Tankov, Peter: “Calibration of jump-diffusion option pricing models: a robust non-parametric approach”, Journal of Computational Finance, Vol. 7, No. 3, 1-49 (2004)

Some references

31


Tankov, Peter: “Lévy processes in finance and risk management”, Wilmott magazine, Sept-Oct 2007

Minsky, Hyman P 1987. “Securitization,” Handout Econ 335A, Fall 1987. Mimeo, in The Levy Economics Institute

archives. 1986. Stabilizing an Unstable Economy. Yale University Press 1992. “The Financial Instability Hypothesis,” Working Paper No. 74. Annandale-on-Hudson, New York:

The Levy Economics Institute 1996. “Uncertainty and the Institutional Structure of Capitalist Economies,” Working Paper No. 155,

Annandale-on-Hudson: The Levy Economics Institute

N. E. KAROUI, M. QUENEZ Dynamic programming and pricing of a contingent claim in an incomplete market. in « SIAM Journal on

Control and optimization », numéro 1, volume 33, 1995, pages 29-66 Non-linear Pricing Theory and Backward Stochastic Differential Equations. Ed: W.J.RUNGGALDIER., in

« Financial Mathematics », Lectures Notes in Mathematics, volume 1656, Springer, 1997, note : Bressanone,1996

P. Jeanne, E. Koehler “The real costs of hedging options”, working paper

Some references

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Séminaire validation des modèles 27 mai 2010 Cost of hedging in illiquid markets Etienne KOEHLER, Barclays Capital Etienne Koehler Paris