Risk Management and Governance Hedging with Derivatives. Prof. Hugues Pirotte

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1 Risk Management and Governance Hedging with Derivatives Prof. Hugues Pirotte Several slides based on Risk Management and Financial Institutions, e, Chapter 6, Copyright John C. Hull 009

2 Why Manage Risks? Prof. Hugues Pirotte

3 3 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

4 4 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

5 5 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 $8 per share. A two-month put with a strike price of $7.50 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

6 Reminder > Use of derivatives Prof. Hugues Pirotte 6

7 Reminder > Payoff profiles Prof. Hugues Pirotte 7

8 Reminder > Payoff profiles () Prof. Hugues Pirotte 8

9 9

10 The activity risk of the firm Prof. Hugues Pirotte 10

11 11

12 1 Derivatives Mapping RISK SOURCE Commodities Stock Market/ Indices Interest-rates Exchange rates INSTRUMENT Commodity Forwards Commodity Futures Commodity Options Commodity Swaps Stock Index Futures Stock Options Stock Index Options Volatility swaps 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 FX Forward Currency Futures FX Options CS Crosscurrency swaps Convertibles Equity Swaps

13 13 Hedging Types

14 14 Hedging with Linear Products

15 15 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

16 16 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 F» Case 1: different maturities S 1 =.50, F 1 =.0, S =.00, F = 1.90 b 1 = 0.30, b = 0.10 t 1 t Suppose the hedger knows that asset will be sold at t and takes a futures position at time t 1 : S + (F 1 F ) = F 1 + b =.30 Basis risk: Hedging risk because S is unknown at t 1 no perfect hedge» Case : different assets S * = price of asset underlying futures contract at t S = price of asset to be hedged By hedging, a company ensures that the price paid (received) will be: S + (F 1 F ) In this case, we can rewrite this as: * * F S F S S 1 basis if same asset basis between the two assets S Time

17 17 Futures > Optimal Hedge Ratio Use of the «minimum variance-hedge ratio» h Cov( St, Ft ) Var ( F ) t Demonstration» The total profit of a hedged portfolio can be written as Q t pf ( VT Vt ) ( FT, T Ft, T ) Qt h» 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 Q t, we have that : V t ( VT Vt ) Q t ( FT, T Ft, T ) h ST St ( FT, T Ft, T ) h pf

18 18 Futures > Optimal Hedge Ratio (cont d)» Thus, S hf S h F h S F Var ( pf ) Var,» And Min( ) h h pf F S, F S S, F F» Which means that the optimal ratio corresponds to the Beta of S with respect to F, in absolute terms.

19 19 Futures > Use of mvh ratio S F S F if same T and same S 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)

20 0 Hedging with non-linear products > Options Protective put Covered call

21 1 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 = + call options at high and low strikes options at the middle strike price Condor = Similar to butterfly spread but options at two different mid strikes Cap/Floors Collars = Cap + Floor

22 Design > some strategies can be unbundled

23 3 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

24 4

25 Option List Example Prof. Hugues Pirotte 5

26 6 Some portfolio Example Position Value Spot gold Forward contracts Future contracts 000 Swaps Options Exotics Total 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.

27 7 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» S 0 = 49, K = 50, r = 5%, = 0%, T = 0 weeks, = 13%» The Black-Scholes value of the option is $40,000» How does the bank hedge its risk to lock in a $60,000 profit?

28 8 Delta of an option Option price B Slope = A Stock price

29 Delta Hedging Initially the delta of the option is 0.5 The delta of the position is -5,00 This means that 5,00 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. and 6.3 (pages 118 and 119) provide examples of how delta hedging might work for the option. 9

30 30 Table 6.: Option closes in the money Week Stock Price Delta Shares Purchased , (6,400) (5,800) , , Risk Management and Financial Institutions, e, Chapter 6, Copyright John C. Hull 009

31 31 Table 6.3: Option closes out of the money Week Stock Price Delta Shares Purchased , , , (1,600) (17,600) (700) Risk Management and Financial Institutions, e, Chapter 6, Copyright John C. Hull 009

32 3 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 Call price C'' C' C S S' Stock price

33 33 When gamma changes The New Delta Old Delta + Old Gamma For some practitioners: The New Delta Old Delta + Average Gamma ((Old+New)/)

34 Volatility Surface Prof. Hugues Pirotte 34

35 35 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

36 36 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

37 37 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.

38 Taylor expansion Standard When volatility is uncertain 38 Prof. Hugues Pirotte t S t t t S S t t S S ) ( 1 ) ( 1 ) ( 1 ) ( 1 P S S P t t P P S S P P

39 Option Greeks (call example) Prof. Hugues Pirotte 39

40 40 For a delta neutral portfolio, Interpretation of gamma Q t + ½GS Positive Gamma Negative Gamma

41 41 Dynamic hedging > Managing delta, gamma & vega Delta hedging ()» Buy/Sell a delta number of underlying that will compensate the sensitivity on the option side S C S C w w 0 w w (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 G T. If the number of traded options added to the pf is w T, then the gamma T T of the pf is w G G T G» To make it gamma-neutral w G T STEPS: (1) make the new portfolio gamma-neutral (by taking another position in the option) () 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 () And then again, make it delta-neutral.

42 4 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

43 43 Dynamic hedging and greeks Some option greeks...(from Black-Scholes) N '( x) 1 e x / Delta Gamma Theta (per year) Call S C e qt N d 1 C G Q qt 1 S S S0 T 0 1 N '( d ) e S0N ' d1 e T qs N( d ) e rke rt qt qt N( d ) Put S P G Q P e qt N d 1 1 qt 1 S S S0 T S0N ' d1 e T qs N( d ) e 0 1 N '( d ) e qt qt rt rke N( d ) Vega (per %) Rho (per %), S T N '( d ) e 100 T N '( d ) e 100 qt , KTe N( d ) 100 rt rt S KTe N( d ) 100 qt

44 44 Books» RMH: Chap. 6» FRM: Instruments: Ch. 510 References

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