2. A contract that specifies the rights and obligations between two parties to receive or deliver future cash flows (or exchange of other securities or assets) based on some future event.
3. Historically, all derivative valuation was performed assuming a single standard discount curve (LIBOR). This methodology was based on the belief all market participants had equal credit risk. However during the crisis, the assumption that each institution had equal credit risk was clearly invalidated.
PURPOSE OF DERIVATIVES
1. Hedging
2. Speculating
3. Arbitrage
4. Accessing remote markets
5. Distribution of risk as securities
TYPES OF DERIVATIVES
1. Linear
(i) Forwards
(ii) Futures
(iii) Swaps
2. Non-Linear
(i) Vanilla Options, Warrants
(ii) Exotic” Options (e.g. Barrier Options, Cliquet/Forward starts)
(iii) Credit default “swaps”
3. Derivative Markets
(i) Interest Rates - Swaps, Futures, Options, FRAs, Exotics MATURE MARKETS
(ii) Foreign Exchange – Forwards, Swaps, Options, Exotics
(iii) Equities (Shares) - Forwards, Swaps, Options, Exotics
(iv) Credit – Vanilla Credit Default Swaps, CDOs (Tranched Default Swaps)
(v) Commodities- Forwards, Futures, Options
(vi) Weather – Swaps
(vii) Freight Routes – Forwards, Options
(viii) Energy – Forwards, Options
(ix) Emissions (New Market) - Forwards
(x) Property (New Market) - Swaps
YIELD CURVE
1. Discounting cash flows at different times is a routine but key part of derivatives valuations.
2. Discount Rates for varying time periods can be implied from pure interest rate instruments such as Govt Bonds, Interest Rate Futures, Interest Rate Swaps, Treasury Bills, Money Market Deposits.
3. Each could give slightly different 6 month interest rate for example.
4. Market Convention is to use Money Market Deposits for early maturities (say 3 months), then Interest Rate Futures (say up to 18 mths) and then Interest Rate Swaps thereafter.
5. Reasons: these are standard interest rate instruments used for hedging by the banks/market makers over these maturities.
6. Yield curves usually have upward sloping term structure; the longer the maturity, the higher the yield.
7. For longer maturities, more catastrophic events might occur that may impact the investment, hence the need for a risk premium. Distant future more uncertain than the near future, and risk of future adverse events (e.g. default) being higher, hence liquidity premium.
MATURE MARKETS
1. Characteristics
(i) Commoditisation/standardization
(ii) Low transaction costs : narrow bid-offer spreads
(iii). Very active trading by specialized traders
(iv). Warehousing of risk/transactions by banks acting as “market makers”
(v). Derivatives trades exceed the trading in the underlying (“physical”)
(iv). The derivatives market often leads the underlying (“physical market”) in terms of pricing.
2. Valuation - Linear Instruments(Example: Equity (Share) Swap)
(i) Party A pays Party B the one year increase in Boots Plc shares Party B pays Party A the market rate of interest + Y %
(ii) Party A can hedge by buying the Boots Plc shares
(iii) Pricing is based on the assumption that Party A will buy the share now and sell when the swap matures.
(iv) Assume current price of Boots = S, dividends received over the year = D and interest rate = r% p.a.
(v) The cost of doing this trade is a function of the ciost of buying and holding the share (net of dividends received). Y= S *r - D
(vi) KEY POINT : Minimum price for the trade (ignoring party A’s profit charge) is the cost of the RISK FREE REPLICATING PORTFOLIO
3. Valuation - Non-Linear Instruments (Example: Equity (Share) Call Option)
(i) Party A pays 15p (option premium) to Party B for the right to “call” (buy) Boots Plc shares at a predetermined price of 100p
(ii) Party A can hedge by buying the Boots Plc shares in advance but what happens if Party B does not exercise the option? Party A will be left with shares and associated price risk.
(iii) Utilize the concept of a DYNAMIC HEDGE in order to produce the risk free replicating portfolio.
NEW DERIVATIVES MARKETS
1.Characteristics
(i) Non standard deals/contracts
(ii) Little or no “market making” with banks matching buy and sell trades exactly
(iii) Low turnover
(iv) Derivatives take pricing lead from physical market
2. Valuation in Underdeveloped Market (Example: Property Index Swap)
(i) Party A pays Party B the one year increase in the UK IPD Index
Party B pays Party A the market rate of interest + X %
(ii) Hedging by Party A is inconvenient/expensive
(iii) Party A “looks to” the expected performance/forecast of the UK property before deciding what X % should be.
BINOMIAL MODEL FOR PRICING CALL OPTION
1. Current share price = 100 today, Suppose next day price will be 115 or 95 but we do not know probability Pre-agreed exercise price of the call option is 100
2. Suppose we buy 0.75 share to hedge the call sold option
3. Portfolio valuation next day:
(i) If share price rises = 0.75 (115) - 15 = 71.25
(ii) If share price falls = 0.75 (95) - 0 = 71.25
Note: Call option value will be 0 if no exercise next day
4. Since next day portfolio value is the same regardless of whether price rises or falls, we can say that we are hedged. Hedge ratio is 0.75 shares for every call option sold. We have succeeded in creating a RISK FREE PORTFOLIO.
5. if portfolio is risk free, then it will pay risk-free rate of interest (r).Calculation of premium from call option sale as follow:
Note: Assuming risk free rate at 5%
(i) Next day Portfolio Valuation / Initial Investment = 1+r
(ii) 71.25 / [(.75)(100) - Premium Received from call option sale] = 1+0.05
(iii) c = 7.143
6. The minimum price of the call option is 7.143 p
BLACK-SCHOLES EQUATION
1. It is an analytical solution (I.e. one-step) and therefore more elegant and computationally much more efficient. B-S makes one additional assumption that returns on assets are normally distributed.
2. Stock price behaviour is assumed/modelled as geometric Brownian motion
(i) % change stock price = constant return on stock or “drift” + “noise” or Weiner process (Brownian motion)
3. Black Scholes price of call option
c = S. N (d,) - ke- rT
N (d2)
4. Black Scholes price of put option
p = ke-rT N (d2)-
S N (-d1)
5. Black-Scholes Model assumptions
(i) No transaction costs and market allows short selling
(ii) Trading and prices are continuous
(iii) Underlying asset prices follow geometric Brownian Motion
(iv) Interest rate during life of option is known and constant
(v) The option can only be exercised on expiry
RISK MANAGEMENT MEASUREMENTS
1. Delta - measure of how option value changes with changes in underlying asset e.g. share price
2. Theta - measure of change in option value with change in time to maturity
3. Vega - change in option value with change volatility of underlying asset
4. Rho - change in option with respect to interest rates (in the case of a share option where delta relates to stock)
5. Gamma - Delta is not static. Gamma is a measure of how the delta itself changes with changes in
ADOPTING MULTIPLE PRICING CURVES FOR OTC DERIVATIVES VALUATION
1. Historically, financial institutions used a single standard curve to value derivatives. However following the collapse of Lehman Brothers, The basis spread between LIBOR rates of different maturities widened around 365 basis points.
2. Market participants have started to move away from a single curve for both discounting and forecasting. Instead, they are using multiple curves, each playing a specific role in valuation. Forecast curves continue to be based on LIBOR, but are built specifically for different tenors.
3. Market participants are adopting a two-curve framework for valuing derivatives. One curve is used for discounting and a curve that matches the maturity of the underlying floating rate is selected for projection.
4. To adapt to the issue that LIBOR no longer reflects equal credit risk, many market participants are now using a discount curve built from OIS. The underlying reasons for the move from a LIBOR-based curve to an OIS-based (overnight Indexed Swaps) standard curve are:
(i) The intense focus on collateral led the market to understand that the discounting methodology used to value derivatives must match the calculation of interest paid on collateral.
(ii) During the credit crisis, banks refused to lend to each other because of counterparty credit risk. This observation resulted in a perceived breakdown of the reliability of LIBOR as a benchmark, as it is a consensus composite.
(iii) When a derivative is in-the-money, the counterparty with positive mark-to-market collects collateral from the other counterparty. Interest is paid on posted collateral, including both bilateral International Swaps and Derivatives Association (ISDA) Credit Support Annexes (CSAs) and through centrally cleared LCH.Clearnet SwapClear margin accounts. The rate used is a standard overnight rate, such as Fed Funds, EONIA, or Sterling OverNight Index Average (SONIA).
(iv) hese rates are considered as close to ‘risk free’ as possible since the rates exist only for a single day. This process protects the positive counterparty in case of default. Since the in-the-money counterparty is paying interest on posted cash collateral, the counterparty is essentially funding the position with the overnight rate.
THOUGHTS
1. In order to apply binomial model to obtain realistic option prices, we need to divide time up into many small steps. As steps in time become shorter and shorter, discrete time will tend towards continuous time and the jumps or falls in share prices in each period will become infinitesimally small. For reasonably accurate results, the time to maturity should be divided into at least 50 steps.
2. For reasonably accurate results, the time to maturity should be divided into at least 50 steps.
3. IAS 39 requires that all derivatives are marked at fair value (mark to market) IAS sets out a hierarchy for the determination of fair value:
(i) For instruments traded in active markets, use a quoted price.
(ii) For instruments for which there is not an active market, use a recent market transaction.
(iii) For instruments for which there is neither an active market nor a recent market transaction, use a valuation technique.
4. A large section of the derivatives market use valuation techniques with parameters implied from known points. If valuation technique cannot be calibrated back to market (e.g. complex products) then fair value is the price at which the trade was done
5. In an active, efficient market, valuation is based on the RISK FREE, REPLICATING PORTFOLIO. Traders are employed to manage the exposure to the underlying exposure to the derivatives portfolio
6. For LINEAR Derivatives (Swaps, Futures) hedging is usually STATIC since future exchange is certain. For NON- LINEAR Derivatives (Options), exercise is not certain so hedging is dynamic.
7. Opinions vary on how far to take the new OIS-based methodology. Some participants believe that an OIS curve should be used to discount all trades. Others believe that it should only be used for collateralized trades with the bank’s unsecured cost of funding used for uncollateralized trades.
8. Some still continue to use the “legacy” way, the old standby LIBOR curve—using LIBOR deposits, futures, and swap rates. These banks claim that this method still captures the most efficient and liquid market. While the first and third options are attractive for their simplicity, the second option is probably the most accurate, but is much more subjective and complex.
(Source: ABN-AMRO, numerix)
