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?

Prof. Hugues Pirotte 2

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

3 Prof. Hugues Pirotte

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


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


Reminder > Use of derivatives


Reminder > Payoff profiles


Reminder > Payoff profiles (2)



The activity risk of the firm



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


Hedging Types


Hedging with Linear Products


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


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


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

)()( ,,,,

)(


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


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


Hedging with non-linear products > Options Protective put

Covered call


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


Design > some strategies can be “unbundled”


“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




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.


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?


Delta of an option

Option price

A

B Slope =

Stock price


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.

29


Table 6.2: Option closes in the money

30

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


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)


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'


When gamma changes… The New Delta Old Delta + Old Gamma

For some practitioners: The New Delta Old Delta + Average Gamma ((Old+New)/2)



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


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


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.


Taylor expansion Standard

When volatility is uncertain


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


Option Greeks (call example)

Interpretation of gamma For a delta neutral portfolio,

Q t + ½GS 2

Negative Gamma Positive Gamma


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


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


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


References Books

» RMH: Chap. 6

» FRM: Instruments: Ch. 510


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