Docstoc

FX Options Trading and Risk Mana

Document Sample
FX Options Trading and Risk Mana Powered By Docstoc
					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

				
DOCUMENT INFO
Shared By:
Categories:
Stats:
views:28
posted:4/12/2010
language:English
pages:40