Risk Management and Governance...by selling 1000 ounces of gold. This is known as making the...
Transcript of Risk Management and Governance...by selling 1000 ounces of gold. This is known as making the...
Risk Management and Governance Hedging with Derivatives
Prof. Hugues Pirotte
Several slides based on Risk Management and Financial Institutions, 2e, Chapter 6,
Copyright © John C. Hull 2009
Why Manage Risks?
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Why hedging?...and using derivatives... Focus on core activity
Prevent shocks from propagating throughout the institution
Competitive power in a cyclical environment
Survivorship
Tax argument
Counterexample: may be dangerous to be non-herding!
» In some industries fluctuations in raw material costs are passed on to the purchasers of the end product
» In this case ``hedging” raw material costs actually increases risks!
» Ex: gold jewellery
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How do we manage or « hedge » risks? Natural Hedges
» Management of supply chain
» Cash management (multinational companies)
Hedging » Forwards & Futures
» Swaps
Insurance or « protection » » Options
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Hedging Examples
» A US company will pay £10 million for imports from Britain in 3 months and decides to hedge using a long position in a forward contract
» An investor owns 1,000 Microsoft shares currently worth $28 per share. A two-month put with a strike price of $27.50 costs $1. The investor decides to hedge by buying 10 contracts
Options vs. Forwards/Futures » A futures/forward contract gives the holder the obligation to buy or sell at
a certain price
» An option gives the holder the right to buy or sell at a certain price
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Reminder > Use of derivatives
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Reminder > Payoff profiles
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Reminder > Payoff profiles (2)
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The activity risk of the firm
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Derivatives’ Mapping
RISK SOURCE INSTRUMENT
Commodities Commodity Forwards
Commodity Futures
Commodity Options
Commodity Swaps
Stock Market/
Indices
Stock Index Futures
Stock Options
Stock Index Options
Volatility swaps
Convertibles
Equity Swaps
Interest-rates Forwards
FRAs
Interest-rate Futures
Treasury Bond Futures
Options on Bond Futures
IRS (plain vanilla, LIBOR-in-arrears, CMS, CMT, differential
swap, accrual swaps, cancelable, cancelable compounding, index
amortizing rate swap, forward starting)
Swaptions
Cross-
currency
swaps
Exchange rates FX Forward
Currency Futures
FX Options
CS
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Hedging Types
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Hedging with Linear Products
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Static or dynamic hedging? Reasons for dynamic hedging
» Basis risk
Timing
Risks of quality or of imperfect correlation (different underlying)
» Imperfections related to standardisation inherent to futures
» Uncertainties on treasury and carrying costs
» Optimal hedge vis-à-vis the payoff at maturity
Need to periodically (re-)assess the hedge » Pro: reallocating continuously (dynamically) a hedging strategy with
options is equivalent to taking a forward contract!
» Con: transaction costs
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Reminder (Futures) > Basis risk Initial strategy: to be “long” or to be “short”
Basis risk
» Basis (b) = Spot price to be hedged (S) – Futures price (F)
If S : strengthening of the basis
If S : weakening of the basis
» Case 1: different maturities
S1 = 2.50, F1 = 2.20, S2 = 2.00, F2= 1.90 b1 = 0.30, b2= 0.10
Suppose the hedger knows that asset will be sold at t2 and takes a futures position at time t1: S2 + (F1 – F2) = F1 + b2 = 2.30
Basis risk: Hedging risk because S2 is unknown at t1 no perfect hedge
» Case 2: different assets
S2* = price of asset underlying futures contract at t2
S2 = price of asset to be hedged
By hedging, a company ensures that the price paid (received) will be: S2 + (F1 – F2)
In this case, we can rewrite this as:
S
F
Time t1 t2
* *
1 2 2 2 2
basis if same asset basis between the two assets
F S F S S
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Futures > Optimal Hedge Ratio Use of the « minimum variance-hedge ratio »
Demonstration » The total profit of a hedged portfolio can be written as
» where is the quantity of contracts defined ex-ante and is the value to be hedged ex-ante. The long underlying position is thus hedged by a short position in the futures. Examining the unit profit, i.e dividing by , we have that :
)(
),(
t
tt
FVar
FSCovh
hQFFVV tTtTTtTpf )()( ,,
tQ
tQ
tV
pfTtTTtTTtTTQ
VVhFFSShFF
t
tT
)()( ,,,,
)(
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Futures > Optimal Hedge Ratio (cont’d)
» Thus,
» And
» Which means that the optimal ratio corresponds to the Beta of S with respect to F, in absolute terms.
FSFSpf hhFhSVarVar ,
222 2)(
,2
,( ) S S F
pf F S F
F
Min h h
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Futures > Use of mvh ratio
Purpose... » But if futures on asset asset to be hedged: hedge ratio should not be 1!
Use the minimum-variance hedge ratio (cf previous slides)
Typical case: the stock index futures! » Minimum-variance hedge ratio = the Beta!
Hedging amount needs to be recalculated every period! (beware of transaction costs)
if same and same S F S F T S
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Hedging with non-linear products > Options Protective put
Covered call
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Other Strategies Straddle
= + put + call (same expiration dates and strikes)
Strangle = + put + call (same expiration dates, strikes out-of-the-money each)
Bull spread = + call (low strike) – call (high strike) = + put (low strike) – put (high strike)
Bear spread: reverse
Butterfly spread = Bull+Bear spreads = + 2 call options at high and low strikes – 2 options at the middle strike price
Condor = Similar to butterfly spread but – 2 options at two different mid strikes
Cap/Floors
Collars = Cap + Floor
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Design > some strategies can be “unbundled”
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“Dynamic hedging” with options Until now…
» When the price of a product is linearly dependent on the price of an underlying asset a ``hedge and forget’’ strategy can be used
» Except if there is some basis risk.
Options can be used » To get a particular payoff profile at maturity (all the cases considered before) » By traders, brokers, etc.. who have portfolios of long and short positions given
their activity as an intermediary but they do not want to keep “open profiles”, only a “flat position”, also called delta-neutral.
» Or by hedgers who want simply to “flatten” their open profile given the analysis of their position and the market conditions.
Given the asymmetry of these products, to produce a flat profile means also to rebalance continuously, i.e. dynamically hedging.
Traders usually ensure that their portfolios are delta-neutral at least once a day
» Whenever the opportunity arises, they improve gamma and vega » As portfolio becomes larger hedging becomes less expensive
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Option List Example
Example Some portfolio
Example » Suppose that a $0.1 increase in the price of gold leads to the gold portfolio
increasing in value by $100 » The delta of the portfolio is 1000 » The portfolio could be hedged against short-term changes in the price of gold
by selling 1000 ounces of gold. This is known as making the portfolio delta neutral.
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Position Value
Spot gold 180’000
Forward contracts -60’000
Future contracts 2’000
Swaps 80’000
Options -110’000
Exotics 25’000
Total 117’000
Delta When examining a whole portfolio of exposures on 1 underlying,
we can be interested by the variability of that portfolio to the underlying’s price » That’s the delta » We know the delta for some traditional cases in finance
Beta for stocks against the index Duration for bonds against the variation of r
Delta of a portfolio is the partial derivative of a portfolio with respect to the price of the underlying asset (gold in this case)
Example » A bank has sold for $300,000 a European call option on 100,000 shares of
a non-dividend paying stock » S0 = 49, K = 50, r = 5%, = 20%, T = 20 weeks, = 13% » The Black-Scholes value of the option is $240,000 » How does the bank hedge its risk to lock in a $60,000 profit?
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Delta of an option
Option price
A
B Slope =
Stock price
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Delta Hedging Initially the delta of the option is 0.522
The delta of the position is -52,200
This means that 52,200 shares must purchased to create a delta neutral position
But, if a week later delta falls to 0.458, 6,400 shares must be sold to maintain delta neutrality
Tables 6.2 and 6.3 (pages 118 and 119) provide examples of how delta hedging might work for the option.
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Table 6.2: Option closes in the money
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Week Stock Price Delta Shares Purchased
0 49.00 0.522 52,200
1 48.12 0.458 (6,400)
2 47.37 0.400 (5,800)
3 50.25 0.596 19,600
…. ….. …. …..
19 55.87 1.000 1,000
20 57.25 1.000 0
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Risk Management and Financial Institutions, 2e, Chapter 6, Copyright © John C. Hull 2009
Table 6.3: Option closes out of the money
Week Stock Price Delta Shares Purchased
0 49.00 0.522 52,200
1 49.75 0.568 4,600
2 52.00 0.705 13,700
3 50.00 0.579 (12,600)
…. ….. …. …..
19 46.63 0.007 (17,600)
20 48.12 0.000 (700)
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Risk Management and Financial Institutions, 2e, Chapter 6, Copyright © John C. Hull 2009
Gamma of an option Gamma (G) is the rate of change of delta () with respect to the
price of the underlying asset
Gamma is greatest for options that are close to the money
S
C
Stock price
S'
Call price
C'' C'
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When gamma changes… The New Delta Old Delta + Old Gamma
For some practitioners: The New Delta Old Delta + Average Gamma ((Old+New)/2)
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Volatility Surface
Vega of an option Vega (n) is the rate of change of the value of a derivatives portfolio with
respect to volatility
Vega tends to be greatest for options that are close to the money
In practice a trader responsible for all trading involving a particular asset must keep gamma and vega within limits set by risk management
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Theta of an option Theta (Q) of a derivative (or portfolio of derivatives) is the rate of
change of the value with respect to the passage of time
The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of the option declines
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Rho of an option Rho is the partial derivative with respect to to a parallel shift in all
interest rates in a particular country.
The greater the underlying asset price and days to expiration, the greater the rho.
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Taylor expansion Standard
When volatility is uncertain
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tSt
tt
SS
tt
SS
2
22
2
2
2
2
2
)(2
1
)(2
1
2
2
2
2
2
2
)(2
1
)(2
1
P
SS
Pt
t
PPS
S
PP
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Option Greeks (call example)
Interpretation of gamma For a delta neutral portfolio,
Q t + ½GS 2
Negative Gamma Positive Gamma
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Dynamic hedging > Managing delta, gamma & vega
Delta–hedging () » Buy/Sell a delta number of underlying that will compensate the sensitivity on the
option side
(Delta-)Gamma (G) neutrality » What is the gamma of a Linear product?
» Otherwise: suppose a delta-neutral pf has a gamma of G, and a traded option has a gamma of GT. If the number of traded options added to the pf is wT, then the gamma of the pf is
» To make it gamma-neutral
STEPS:
(1) make the new portfolio gamma-neutral (by taking another position in the option)
(2) make it then delta-neutral (with the underlying)
(Delta-Gamma-)Vega (n neutrality (1) Same principle, but we need to solve a system of two linear equations to find the
weights in two different options on the same underlying
(2) And then again, make it delta-neutral.
T Tw G GT
Tw
G
G
0S C S Cw w w w
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Static option replication This involves approximately replicating an exotic option with a
portfolio of vanilla options
Underlying principle: if we match the value of an exotic option at a number of points on some boundary, we have matched it at all interior points of the boundary
Static options replication can be contrasted with dynamic options replication where we have to trade continuously to match the option
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Dynamic hedging and greeks Some option greeks...(from Black-Scholes)
1
qTCe N d
S
0 1
0 1
2
'
2
( )
( )
qT
qT
rT
S N d e
T
qS N d e
rKe N d
Q
2
1
2
0
'( ) qTP N d e
S S S T
G
Call Put
Delta
Gamma
Theta (per year)
Vega (per %)
Rho (per %)
2
1
2
0
'( ) qTC N d e
S S S T
G
1 1qTPe N d
S
2 / 21'( )
2
xN x e
0 1
0 1
2
'
2
( )
( )
qT
qT
rT
S N d e
T
qS N d e
rKe N d
Q
0 1'( ),
100
qTS T N d e
0 1'( ),
100
qTS T N d e
2( )
100
rTKTe N d
2( )
100
rTKTe N d
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References Books
» RMH: Chap. 6
» FRM: Instruments: Ch. 510
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