CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options

Transcription

1 CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options Paul Obour Supervisor: Dr. Antony Ware University of Calgary PRMIA Luncheon - Bankers Hall, Calgary May 8, 2012

2 Outline 1 Introductory Section Hedging 2 Currency Translated Options, CTOs What are CTOs Exotic Option (CTO) Pricing 3 CTO Hedging Strategies Hedging CTOs CTO Hedging Challenges 4 Future Work What next? References

3 Introductory Section Hedging Instruments & Purposes Mechanisms available for hedging: Insurance An organized futures markets Crude Oil Trades: Two organized exchanges; NYMEX & ICE In a bilateral over-the-counter (OTC) market Why hedge crude oil? To reduce volatility of earnings (or costs)

4 Introductory Section Hedging Oil markets Source: BP Statistical Review of World Energy June 2011

5 Introductory Section Hedging USD-CAD & Crude Oil Price Path May April 2012

6 Introductory Section Hedging Five Hedging Strategies 1 Pure Futures Strategies Always selling 12-months forward Selectively selling 3-months forward 2 Pure Options Strategies Using plain puts to obtain insurance or lock in a budget price 3 Options Combinations Strategies Spreads, Straddles, Strangles, Collars In-, out-of-, and near-the-money options 4 Exotic (Asian) Options Strategies 5 Insurance Instruments Blending risks with insurance programs

7 Currency Translated Options, CTOs What are CTOs Definition and Features of CTOs CTOs are options on foreign assets where the payoff is exchanged into domestic currency at expiration. Flexos Foreign commodity option struck in foreign currency Compos 1 Foreign commodity option struck in domestic currency 2 Commodity linked foreign exchange option Quantos Fixed exchange rate foreign commodity option

8 Currency Translated Options, CTOs Exotic Option (CTO) Pricing Pricing CTOs Evaluate the option without the currency impact since CTOs can be seen as standard European call or put options: 1 Black Scholes (1973) - European options on non-dividend paying stock 2 Merton (1973) - extended to continuous dividend paying shares Once the standard option valuation is completed, adjustments must be made due to the currency effects : 1 The correlation between the underlying stock and F/X 2 The difference in the interest rates in the two currencies Consider all impacts of interest rates on option valuation.

9 Currency Translated Options, CTOs Exotic Option (CTO) Pricing Pricing the Flexos The terminal payoff for a call is: C (0) d (F,T ) = X.max(F T K f,0) The fair value of a Flexo option in domestic currency is: C (0) K f,t (t,f,x) = X.e r f τ [FN(d 1 ) K f N(d 2 )] (1) where F = F(t) is the commodity futures, τ = T t, X = X(t) is the F/X futures and d 1 = ln F K f + σ2 F 2 τ, d 2 = d 1 σ F τ σ F τ

10 Currency Translated Options, CTOs Exotic Option (CTO) Pricing Pricing the Compos The terminal payoff for a call on the first version of Compos is: C (1) d (F T,X T,T ) = max(x T F T K d,0) = max(z T K d,0) The corresponding price formula is seen to be: where C (1) K d,t (t,z,x) = Ze δτ N( ˆ d 1 ) K d e r dτ N( ˆ d 2 ) (2) d 1,2 ˆ = ln(z/k d) ± 1 2 σ2 Z τ, σ 2 σ Z τ Z = σ 2 F + σ 2 X + 2ρσ F σ X

11 Currency Translated Options, CTOs Exotic Option (CTO) Pricing Pricing the Compos The second Compos version has a terminal payoff on a call as: C (2) d (X T,T ) = F(T )max(x T K,0) where K is the strike of the exchange rate expressed in domestic currency. The fair value of a commodity linked foreign exchange option in the domestic currency is given by: where C (2) K,T (t,f,x) = e r dτ F(XN( d 1 ) e ρσ X σ F τ KN( d 2 ) (3) d 1 = ln(x/k d) + (ρσ X σ F σ2 X )τ, d 2 = d 1 σ X τ σ X τ

12 Currency Translated Options, CTOs Exotic Option (CTO) Pricing Pricing the Quanto Option The payoff for a call is: C (3) d (F T,X T,T ) = X 0.max(0,F T K f ) = max(0,[f T.X 0 ] K d ) Equivalently, the payoff can be defined in domestic currency. Then, the no-arbitrage price is defined using the domestic martingale measure by: C Kf,T (t,f,x 0 ) = e r dτ X 0 E P [ (F(T ) Kf ) + F t ] The fair value of a Quanto in domestic currency is given by: where C Kf,T (t,f,x 0 ) = e r dτ X 0 (e ρσ X σ F τ FN(d 1 ) K f N(d 2 )) (4) d 1 = ln(f/k f ) + ( ρσ X σ F σ2 X )τ σ F τ, d 2 = d 1 σ F τ

13 CTO Hedging Strategies Hedging CTOs CTOs Hedging Approach Hedge structure can be Linear & Non-linear. We consider the former to the BS dynamic delta hedge implemented for CTOs on equities. Flexo Structure Flexos are hedged in exactly the same manner as standard European options. Compos Structure Margrabe (1978) proposed a similar hedging strategy to the Black & Scholes approach since we have the option to exchange one risky asset for the other. There are two Deltas that must be estimated and both risky assets must be used in the dynamic hedging.

14 CTO Hedging Strategies Hedging CTOs Hedging Flexos Reiner proposed using the BS replicating strategy in the foreign market. In domestic currency, this would be the cost of the hedge multiplied by the current spot F/X rate. From (1), the replicating portfolio on a call would require: F = e r f τ N(d 1 ) positions in the futures, denominated in the foreign currency, and units of foreign cash. B f = K f e r f τ K d N(d 2 ) The equivalent cost in domestic currency is not calculated by multiplying by the spot F/X rate, but rather the futures F/X.

15 CTO Hedging Strategies Hedging CTOs Hedging Compos For the first type of Compo call, the N(d 1 ) term indicates the quantity of the equity that must be held and N(d 2 ) indicates the quantity of foreign exchange to be sold. From (2), we can see that replicating portfolio requires: positions in the actual futures and units of domestic currency. Z = e r dτ N(d 1 ) B d = e r dτ K d N(d 2 ) The domestic cash position is B d = C Z FX B f X

16 CTO Hedging Strategies Hedging CTOs Hedging Quantos For a long European call, the delta hedge can be determined by looking at (4), given by F,X = X 0 e (r d+ρσ X σ F )τ N(d 1 ) where the proportions have been translated into the domestic currency at X 0. Then in the foreign market, F = X 0 X(t) e (r d+ρσ X σ F )τ N(d 1 ) positions must be taken in the underlying futures, with the domestic currency holding of B d = C and B f = F F held in foreign cash.

17 CTO Hedging Strategies Hedging CTOs CTO Option Prices & An Overview of Hedge Perfomance CTO Type Option Price FlexoCall ComposCall (1) ComposCall (2) QuantoCall (*) - same strike price was used in computation

18 CTO Hedging Strategies CTO Hedging Challenges CTO Hedging Difficulties Estimation of correlation. Most practitioners choose to assume that the correlation between the asset and the FX is zero. There would be a perfect hedge if the assumptions of the Black s model were precisely true. Even if one were to hedge as much as finitely possible, transaction costs would quickly offset any saving the hedge provided. The model relies on estimated parameters which changes over the life of the option.

19 Future Work What next? Going Forward! To consider using non-linear hedging strategies to identify the sources of variation not replicated by the linear hedge. Use copulas to capture the correlations in data more efficiently and to implement the hedge for bivariate options in an arbitrage-free setting. Overcome the difficulties in tail behaviours by trying to get a usable skew data. Work around different sets of data to try and eliminate the risk due to model selection. Price Flexos without the normality assumption - Lévy processes.

20 Future Work What next? Pricing Flexos under Lévy Processes Consider the market with one exchange rate and one foreign asset (X,F): X t = X 0 e L1 t F t = F 0 e L2 t Using this new measure, the evaluation formula becomes C Kf,T (X,F) = e r dt X T (F T K f )dq C [ ] = e r dt X 0 E Q e L1 T (F T K f )d Q C = e rdt e βt X 0 (F T K f )d Q = e (r d+β)t X 0 E Q [(F T K f );C] = e (r d+β)t X 0 E Q [ (F T K f ) +]. Unlike the Gaussian case, the discounting rate in the non-gaussian model above is (r d + β), where β = (r d r f ). C

21 Future Work References References Cont, R and Tankov, P (2004) Financial Modelling with Jump Processes; Chapman & Hall David Applebaum (2009) Lévy Processes and Stochastic Calculus; Cambridge University Press, 2nd edition, pp: William Margrabe (1978) The Value of an Option to Exchange One Asset for Another, The Journal of Finance, VOL. XXXIII, NO. I S. Huang and M. Hung (2005) Pricing Foreign Equity Options under Lévy processes, Journal of Futures Markets, 25(10),pp Robert G. Tompkins and José C. Wong (2001) Exotic Options, European Banking Study, pp:

22 Future Work References Thanks for Coming!