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FX Options Trading and Risk Management Paiboon Peeraparp Feb. 2010 1 Risk Uncertainties for the good and worse scenarios Market Risk Operational Risk Counterparty Risk Financial Assets Stock , Bonds Currencies Commodities Non-Financial Assets Weather Inflation Earth Quake 2 Today Topics Hedging Instruments Risk Management Dynamic Hedging Volatilities Surface 3 Instruments Forwards Contracts to buy or sell financial assets at predetermined price and time Linear payout No initial cost Options Rights to buy or sell financial asset at predetermined price (strike price) and time Non-Linear payout Premium charged 4 Participants Hedgers Want to reduce risk Speculators Seek more risk for profit Brokers / Dealers Commission and Trading Regulators/ Exchanges Supervise and Control 5 FX (Foreign Exchange) Market Over the counter Trade 24 hours Active both spot/forwards/options Banks act as dealers 6 FX Banks Trade to accommodate clients Make profit by bid/offer spread Absorb the risk from clients Offer delivery service Other Commission Fees Trade on their own positions Trade on their views (buy low and sell high) 7 Forwards Valuation (1) An Electronic manufacturer needs to hedge gold price for their manufacturing in 1 years. A dealer will need to T= 0 1. borrow $ 1,000 at interest rate of 4% annually 2. buy gold spot at $ 1,000 T = 1 yr 1. repay loan 10,40 (principal + interest) 2. Charge this customer at $ 1,040 Valuation by replication , F = Sert In FX and commodities market, we call F-S swap points 8 Forwards Valuation (1) We call the last construction as the arbitrage pricing by replicate the cash flow of the forward. If F is not Sert but G G > F , sell G, borrow to buy gold spot cost = F G < F , buy G, sell gold spot and lend to receive = F The construction is working well for underlying that is economical to warehouse it. For the others, it typically follows the mean reverting process. Physical / Paper Hedging Physical Hedging Deliver goods against cash No basis risk Paper Hedging Cash settlement between contract rate and market rate at maturity Market rate reference has to be agreed on the contracted date. Some basis risk incurred Option Characteristics (1) P/L of Call Option Strike at 34.00 2.50 2.00 1.50 Time value P/L 1.00 0.50 Intrinsic Value 0.00 30.00 31.00 32.00 33.00 34.00 35.00 36.00 37.00 -0.50 C-Ke-rt > 0, we call the option is in the money C-Ke-rt = 0, we call the option is at the money C-Ke-rt < 0, we call the option is out of the money 11 Option Characteristics (2) For a plain vanilla option An option buyer needs to pay a premium. An option buyer has unlimited gain. An option seller has earned the premium but face unlimited risk. This is the zero-sum game. 12 P/L Diagram An importer needs to pay USD vs. THB for 1 year. Underlying Forward P/L P/L P/L + = Rate Rate Rate Underlying Option P/L P/L P/L + = Rate Rate Rate 13 Options Details Buyer/Seller Put/Call Notional Amount European/ American Strike Time to Maturity Premium 14 Option Premium (1) Normally charged in percentage of notional amount Paid on spot date Depends on (S,σ,r,t,K) can be represented by V= BS(S,σ,r,t,K) if the underlying follows BS model. Option Premium (2) BS(S,σ,r,t,K) σ1> σ2 then BS(S,σ1,r,t,K) > BS(S,σ2,r,t,K) t1> t2 then BS(S,σ,r,t1,K) > BS(S,σ,r,t2,K) r1> r2 then BS(S,σ,r1,t,K) > BS(S,σ,r2,t,K) In reality, the call and put are traded with the market demand supply. From the equation C,P = BS(S,σ,r,t,K), we solve for σ and call it implied volatility. The is another realized volatility ∑ is the actual realized volatility. Volatilities Put/Call Parity Call option for buyer Put option for seller P/L P/L P/L + = Rate Rate Rate K Call option for seller P/L F=C–P F = S-Ke-rt Rate C-P = S-Ke-rt 18 Options Path Independence Plain Vanilla European Digital Path Dependence Barriers American Digital Asian Etc. 19 Combination of Options (1) Risk Reversal Buy Call option Sell Put option + = 1. View that the market is going up (Strikes are not unique). 2. Can do it as the zero cost. 3. If do it conversely, the buyer of this structure view the market is going down. 20 Combination of Options (2) Straddle Buy Call option Buy Put option + = Butterfly Spread Buy Call & Put option Sell Straddle + = 21 Combination of Options (3) Create a suitable risk and reward profile Finance the premium Better spread for the banks 22 Risk Reward Analysis Combine your underlying with the options and see how much you get and how much you lose. Underlying + = Underlying More = + risk more return 23 Structuring Dual Currency Deposit is the most popular product that combine sale of option and a normal deposit . For example, the structure give the buyer of this deposit at normal deposit rate + r % annually. But in case the underlying asset has gone lower the strike, the buyer will receive underlying asset instead of deposit amount. This structure will work when the interest rates are low and volatilities are high. 24 FX Option Quotation in FX market (1) 1. Quotes are in terms of BS Model implied volatilities rather than on option price directly. 2. Quotes are provided at a fixed BS delta rather than a fixed strike. 3. However implied volatilities are not tradeable assets, we need to settle in structures. 25 FX Option Quotation in FX market (2) Standard Quotation in the FX markets 1 Straddle - A straddle is the sum of call and put at the same strike at the money forward 2 Risk Reversal (RR) - A RR is on the long call and short put at the same delta 3 Butterfly - A Butterfly is the half of the sum of the long call and put and short Straddle. 26 BBA FX Option Quotation Spot GBP/USD Rate Option Volatility 25 Delta Risk Reversal 25 Delta Strangle Date: 1 Month 3 Month 6 Month 1 Year 1 Month 3 Month 1 Year 1 Month 3 Month 1 Year 2-Jan-08 1.9795 9.80 9.80 9.43 9.25 -0.82 -0.79 -0.38 0.23 0.32 0.39 3-Jan-08 1.9732 9.73 9.73 9.48 9.25 -0.61 -0.59 -0.62 0.26 0.33 0.39 4-Jan-08 1.9754 9.45 9.45 9.38 9.20 -1.20 -1.22 -1.30 0.29 0.33 0.39 7-Jan-08 1.9725 9.55 9.55 9.23 9.18 -1.14 -1.18 -0.83 0.27 0.32 0.39 For 3 months (Vatm = 9.8) VC25d-VP25d = -0.79 ((VC25d+VP25d)/2)-Vatm = 0.32 Solve above equation VC25d = 9.725 , VP25d = 10.045 27 Volatility Smile Volatility Smile 10.10 10.00 9.90 Implied Vol 9.80 Volatility Smile 9.70 9.60 9.50 25d 50d 25d Strike 28 Volatility Surface (1) Volatility Surface (2) Vol. Stock Index Vol. FX Vol. K/S K/S Single Stock Vol. K/S Volatility Surface (3) • Implies volatilities are steepest for the shorter expirations and shallower for long expiration. • Lower strike and higher strikes has higher volatilities than the ATM. implied volatilities. • Implied volatilities tend to rise fast and decline slowly. • Implied volatility is usually greater than recent historical volatility. 31 Smile Modeling In the BS Model the stock’s volatilities are constant, independent of stock price and future time and in consequence ∑(S,t,K,T) = σ In local volatility models, the stock realized volatility is allowed to vary as a function of time and stock price. we may write the evolution of stock price as dS/S = µ(S,t)dt + σ(S,t)dZ We firstly match the σ(S,t) with ∑(S,t,K,T) and this can be done in principle. The problem is to calibrate the σ(S,t) to match with the characteristic of the pattern of the smile FX Option Formula 33 Bank Options Hedging A bank has a lot of fx options outstanding in the book. They manage overall risk by look into the change of option price given change in one parameter. Each dealer is limited by the total amount of risk in his book. The Greek A call option depends on many parameters: c( S , , r , t ) A Taylor Expansion: c ct t cS S c cr r cSS (S ) ... 1 2 2 theta delta vega rho gamma ct cS c cr c SS A dealer try to keep all parameter hedged except the one they want to take the view. Dynamic Hedging (1) Set C(S,t) be the option call price From Taylor series expansion Assume ∆S = ∑S√∆t (∆S)2 = ∑2S2∆t C(S+∆S,t+∆t) = C(S,t)+∂C/∂t ∆t+∂C/∂S ∆S + ∂2C/∂S2 (∆S)2/2 + … For a fixed t, and define Γ = ∂2C/∂S2 Consider C(S+∆S,t) = C(S,t)+∂C/∂S ∆S + Γ(∆S)2/2 + … 36 Dynamic Hedging (2) We like to create a hedged portfolio Define θ = ∂C/∂t C(S+∆S,t+∆t) = C(S,t)+θ∆t+∂C/∂S ∆S + Γ(∆S)2/2 dP&L = C(S+∆S,t+∆t) - C(S,t) - ∂C/∂S ∆S = θ∆t+ Γ(∆S)2/2 Suppose r=0, the hedge portfolio has the same return as riskless portfolio θ∆t + Γ(∆S)2/2 = 0 or θ∆t + Γ/2 ∑2S2∆t = 0 or θ + Γ/2 ∑2S2 = 0 Step by step hedging Time Option Stock Value Cash Value Net Position Value t C - ∂C/∂S S (∂C/∂S S)-C 0 t+dt C+dC - ∂C/∂S ((∂C/∂S S)-C) (1+rdt) C+dC - ∂C/∂S (S+dS) (S+dS)+ ((∂C/∂S S)- C) (1+rdt) 37 Dynamic Hedging (3) dP&L =[C+dC - ∂C/∂S (S+dS)]+ ((∂C/∂S S)-C) (1+rdt) =dC- ∂C/∂S dS –r(C- ∂C/∂S S)dt Using Ito’s Lemma for dC we obtain = θdt+ ∂C/∂S dS +1/2ΓS2σ2dt- ∂C/∂S dS –r(C-∂C/∂S S)dt = [θ+ 1/2ΓS2σ2-r∂C/∂S-rC]dt By Black-Scholes equation with σ = ∑ θ+ 1/2ΓS2∑2-r∂C/∂S-rC = 0 dP&L = 1/2 ΓS2(σ2-∑2)dt Real World Hedging A Taylor Expansion: c ct t cS S c cr r 1 cSS (S )2 ... 2 Daily P/L = Delta P/L + Gamma P/l + Theta P/L = ∂C/∂S (∆S) + 1/2Γ (∆S) 2 + θ (Δt) •The dealer job is to design a option book with the risk that he feel comfortable with. •For a delta hedged position Gamma and Theta have the opposite signs •For a long call or put, Gamma is positive and Theta is negative. •For a short call or put, the situation is reversed. Option Sensitivities (K=10, T=0.2, r=0.05, =0.2) European Call Option Price European Call Option Delta 2.5 1 0.9 2 0.8 0.7 1.5 0.6 Price Delta 0.5 1 0.4 0.3 0.5 0.2 0.1 0 0 8 8.5 9 9.5 10 10.5 11 11.5 12 8 8.5 9 9.5 10 10.5 11 11.5 12 S S European Call Option Gamma European Call Option Theta 0.5 0 0.45 -0.2 0.4 0.35 -0.4 0.3 -0.6 Gamma Theta 0.25 0.2 -0.8 0.15 -1 0.1 -1.2 0.05 0 8 8.5 9 9.5 10 10.5 11 11.5 12 -1.4 8 8.5 9 9.5 10 10.5 11 11.5 12 S S

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posted: | 4/12/2010 |

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