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
The two classical pricing approaches
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
The two classical pricing approaches
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
The two classical pricing approaches
2
The two classical pricing approaches
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
The two classical pricing approaches
3
The two classical pricing approaches
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
The two classical pricing approaches
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
The two classical pricing approaches
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
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)
Some alternative approaches
7
Some alternative approaches
Jumps?
Levy processes and risk minimization?
Filtration extensions?
Some alternative approaches
8
Some alternative approaches
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
Some alternative approaches
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
Another possible approach
10
Some alternative approaches
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
Another possible approach
11
Impact of the crisis on options hedging
0.85
0.95
1.05
1.15
-30
10
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
14
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
(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
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
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
Impact of the crisis on options hedging
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
26
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
Some words about the crisis case and conclusion
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
Some words about the crisis case and conclusion
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
Some words about the crisis case and conclusion
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
Some words about the crisis case and conclusion
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
Some words about the crisis case and conclusion
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
32
DisclaimerThis document has been prepared by Barclays Capital, the investment banking division of Barclays Bank PLC ("Barclays"), for information purposes only. This document is an indicative summary of the terms and conditions of the securities/transaction described herein and may be amended, superseded or replaced by subsequent summaries. The final terms and conditions of the securities/transaction will be set out in full in the applicable offering document(s) or binding transaction document(s).
This document shall not constitute an underwriting commitment, an offer of financing, an offer to sell, or the solicitation of an offer to buy any securities described herein, which shall be subject to Barclays’ internal approvals. No transaction or services related thereto is contemplated without Barclays‘ subsequent formal agreement. Barclays is acting solely as principal and not as advisor or fiduciary. Accordingly you must independently determine, with your own advisors, the appropriateness for you of the securities/transaction before investing or transacting. Barclays accepts no liability whatsoever for any consequential losses arising from the use of this document or reliance on the information contained herein.
Barclays does not guarantee the accuracy or completeness of information which is contained in this document and which is stated to have been obtained from or is based upon trade and statistical services or other third party sources. Any data on past performance, modelling or back-testing contained herein is no indication as to future performance. No representation is made as to the reasonableness of the assumptions made within or the accuracy or completeness of any modelling or back-testing. All opinions and estimates are given as of the date hereof and are subject to change. The value of any investment may fluctuate as a result of market changes. The information in this document is not intended to predict actual results and no assurances are given with respect thereto.
Barclays, its affiliates and the individuals associated therewith may (in various capacities) have positions or deal in transactions or securities (or related derivatives) identical or similar to those described herein.
IRS Circular 230 Disclosure: Barclays Capital and its affiliates do not provide tax advice. Please note that (i) any discussion of U.S. tax matters contained in this communication (including any attachments) cannot be used by you for the purpose of avoiding tax penalties; (ii) this communication was written to support the promotion or marketing of the matters addressed herein; and (iii) you should seek advice based on your particular circumstances from an independent tax advisor.
BARCLAYS CAPITAL INC., THE UNITED STATES AFFILIATE OF BARCLAYS CAPITAL, THE INVESTMENT BANKING DIVISION OF BARCLAYS BANK PLC, ACCEPTS RESPONSIBILITY FOR THE DISTRIBUTION OF THIS DOCUMENT IN THE UNITED STATES. ANY TRANSACTIONS BY U.S. PERSONS IN ANY SECURITY DISCUSSED HEREIN MUST ONLY BE CARRIED OUT THROUGH BARCLAYS CAPITAL INC., 200 PARK AVENUE, NEW YORK, NY 10166.
NO ACTION HAS BEEN MADE OR WILL BE TAKEN THAT WOULD PERMIT A PUBLIC OFFERING OF THE SECURITIES DESCRIBED HEREIN IN ANY JURISDICTION IN WHICH ACTION FOR THAT PURPOSE IS REQUIRED. NO OFFERS, SALES, RESALES OR DELIVERY OF THE SECURITIES DESCRIBED HEREIN OR DISTRIBUTION OF ANY OFFERING MATERIAL RELATING TO SUCH SECURITIES MAY BE MADE IN OR FROM ANY JURISDICTION EXCEPT IN CIRCUMSTANCES WHICH WILL RESULT IN COMPLIANCE WITH ANY APPLICABLE LAWS AND REGULATIONS AND WHICH WILL NOT IMPOSE ANY OBLIGATION ON BARCLAYS OR ANY OF ITS AFFILIATES.
THIS DOCUMENT DOES NOT DISCLOSE ALL THE RISKS AND OTHER SIGNIFICANT ISSUES RELATED TO AN INVESTMENT IN THE SECURITIES/TRANSACTION. PRIOR TO TRANSACTING, POTENTIAL INVESTORS SHOULD ENSURE THAT THEY FULLY UNDERSTAND THE TERMS OF THE SECURITIES/TRANSACTION AND ANY APPLICABLE RISKS.
Barclays Bank PLC is registered in England No. 1026167. Registered Office: 1 Churchill Place, London E14 5HP. Copyright Barclays Bank PLC, 2009 (all rights reserved). This document is confidential, and no part of it may be reproduced, distributed or transmitted without the prior written permission of Barclays.
33