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Valuation of financial nstruments from simple bonds to options

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- 1. FINANCIAL INSTRUMENTS valuation methodologies Dr A. A. Kotzé Financial Chaos Theory April 2007 Saggitarius A*: supermassive black hole at the Milky Way’s center
- 2. Before I came here I was confused about the subject. Having listened to your lecture I am still confused. But on a higher level. Enrico Fermi (1901-1954) Niels Bohr and Albert Einstein
- 3. What’s in the 1997 Nobel prize? Myron Scholes (1941 - ) Robert Merton (1944 - ) Fischer Black (1938 - 1995) Along the way, it changed the way investors and others place a value on risk, giving rise to the field of risk management, the increased marketing of derivatives, and widespread changes in the valuation of corporate liabilities. The theory "is absolutely crucial to the valuation of anything from a company to property rights“. In my view, financial economics deals with four main phenomena: time, uncertainty, options and information. William F. Sharp
- 4. Drivers of the markets • Only two emotions drive the market – GREED and FEAR • Two types of traders – SPECULATORS and HEDGERS
- 5. DERIVATIVES: why? Derivatives expanded the universe of instruments available for trading and hedging • Speculators use derivatives to expose portfolios to some market risk • Hedgers use derivatives to reduce the market risk they are exposed to
- 6. Benefits of derivatives • part of universe of instruments – bigger choice • basic instruments – used to construct others • participation in price moves – gearing or leverage • used as hedges • can buy or sell underlying at more favourable prices • can earn premium income • derivatives give options • tailor your position to your situation or risk tolerance • position yourself to your market expectation • limited influence on underlying market
- 7. Hedging • Transfer risk • Risks more precisely tailored to risk preferences and tolerances • Those willing to bear risk must be compensated • Most transactions still for speculating and not hedging • Speculators provide liquidity
- 8. Rapid growth • Rapid growth and development • Emergence of international financial institutions who intermediate bulk of international capital flows in the financial markets • Increase capacity of the financial system to bear and price risk and allocated capital •
- 9. Historical Developments
- 10. The Financial Market • Financial markets are mechanisms that link surplus and deficit investors • Participants: borrowers, lenders, financial intermediaries and brokers • Financial Market: interaction between the players in setting a price on financial instruments fulfilling demands and requirements
- 11. Time value of money • The notion that money has a time value is one of the basic concepts in the analysis of any financial instrument • Money has time value because of the opportunities for investing money at some interest rate • Interest is a fee for having the use of money
- 12. Zero coupon bond • A zero-coupon-bond is a bond that makes no interest payment over its lifetime • The face value is the amount paid at maturity • Most money market instrument are zero-coupon • Value Cash invested Cash + interest
- 13. Coupon bond These are bonds with regular interest payments e.g. R153, R157 The value of the bond is given by its net present value (NPV)
- 14. Pricing SA bonds Use Makeham’s formula: basically the NPV with a few twists to make it practical
- 15. Pricing SA bonds: the bond pricing formula
- 16. The bond pricing formula Settlement on coupon date; more than 6 months Settlement on coupon: less than 6 months Ex-coupon: more than 6 months Ex-coupon: less than 6 months Cum-interest: more than 6 months Cum-interest: less than 6 months
- 17. Bond Trading • Debt probably oldest financial product stemming from origins of mankind • Governments have been issuing public debt since 16th, 17th century? • First majour default: British Crown (Edward III) in 1345 • Bond trading also quite old – 1800’s • Derivatives on interest rate instruments are quite new
- 18. The zero coupon yield curve • There is no single interest rate for an economy • Factors: term of loan; credit • Government best credit – can borrow at best rates • A yield curve is a graphical representation of the term structure of interest rates for instruments of a similar credit rating
- 19. Yield curve is dynamic South African Yield Curves 5.500% 6.500% 7.500% 8.500% 9.500% 10.500% 11.500% 12.500% 13.500% 14.500% 0.01 0.08 0.25 0.75 1.25 1.75 3.00 5.00 7.00 9.00 11.00 13.00 15.00 17.00 19.00 21.00 23.01 25.00 27.00 29.00 Years YTM 06-Jan-03 30-Jun-03 31-Mar-04 30-Sep-04 05/05/2005 17/02/2006 28/02/2005 02-Apr-07
- 20. The forward rate • Can invest money today for a certain date • Question: if we receive money in the future, can we invest it at that point in time but fix the interest rate today? • Yes, by using forward rates • Forward rate determined by arbitrage argument
- 21. The forward rate: arbitrage
- 22. The Interest Rate Swap • A swap is an agreement between two parties to exchange cash flows in the future according to a prearranged formula • There is a fixed rate payer and a floating rate payer • Only the net interest differential is exchanged First swap done in 1981 between the World Bank and IBM Size of market: more than $150 trillion notional outstanding
- 23. The swap: mechanics
- 24. Swaps: an example • Two companies agree the following: • company B lends company A R100 million at the 3 month Jibar rate; • Company A lends company B R100 million at a fixed rate of 9.5%
- 25. Value of a swap Value of a swap using yield curve • Fixed leg: NPV • Floating leg:
- 26. Forwards/Futures A person holding a futures contract has the right to buy a certain underlying asset at a future date at a pre-determined price Payoff
- 27. What is an option? • Call option: gives the holder the right, BUT NOT the obligation to buy the underlying asset on a future date at a pre-determined price • Put option: gives the holder the right, BUT NOT the obligation to sell the underlying asset on a future date at a pre-determined price Payoff
- 28. The Black-Scholes formula If you buy an option, you buy a RIGHT without any OBLIGATION – when the option expires, you can decide to exercise or not. This costs money – the premium of the option
- 29. The Black-Scholes world • Stock prices follow a continuous random walk – Brownian motion • The efficient market hypothesis holds • Investors live in a risk-neutral world • Delta-hedging is done continuously In general, Black & Scholes assumed that the financial market is a system that is in equilibrium – without outside or exogenous influences, the system is at rest; everything balances out and supply equals demand. Any distortion or perturbation is thus quickly handled by the market players and equilibrium restored
- 30. Misconception A common misconception about option pricing models is that their typical use by option traders is to indicate the “right” price of an option. Options pricing models do not necessarily give the “right” price of an option. The “right” price is what someone is willing to pay for a particular option. An efficient market will give the best and truest prices for options. Many individuals trade warrants and some of them do not know what an option is nor do they have a pricing model.
- 31. Benefits of having a model • the concepts behind the formula provided the framework for thinking about option valuation and dynamics; • Due to research within the Black-Scholes framework many new insights into the market dynamics have come to light • such research led to an expanded universe of financial instruments encompassing more complex option structures; • quantitative risk management, stress testing and scenario analysis became possible; • hedging of derivatives were easier; • The market has progressed and expanded.
- 32. Types of options • European – can only exercise on expiry date; • American – can exercise any time on or before expiry • Exotics; – Barrier option: option either disappears or is created when certain market levels are breached – Asian: use averages to determine the final payoff value – Exchange one asset for another option – Cliquet – Ladder – Reset – Lookback New types of options are created nearly every day
- 33. Value of an option: risk management • The concept of delta-hedging: If we purchase an option we can trade in the cash instrument (called ``trading spot" or ``trading the cash") to hedge the option. By buying the option we pay the premium upfront. As the underlying's share price changes with time, the Delta will show us that we have to buy the cash at the lows and sell at the highs - we thus make money by delta hedging. In theory, if we hedge continuously we should make exactly what we paid for the option. On the other hand, if we sell the option, we earn the premium. The Delta will then show us to buy the cash at the highs and sell it at the lows - we thus loose money by delta hedging. By doing this continuously we should loose exactly what we earned with the premium. Hedging costs = value of derivative
- 34. Valuation: American options • No closed-form formula • Use numerical procedures like binomial trees • The discrete version of normal distribution is the binomial distribution • The binomial model models the time to expiration as a very large number of time intervals, or steps. Using probability theory a tree of stock prices is produced. • At each step it is assumed that the stock price will move up or down by an amount calculated using probabilities. This produces a binomial distribution, or recombining tree, of underlying stock prices. The tree represents all the possible paths that the stock price could take during the life of the option
- 35. The binomial distribution
- 36. Factors affecting option prices First 3 are known. Last 3 needs to be estimated • The spot price • The strike price • The time to expiry • The risk-free interest rate • The dividends/cash flows expected during the life of the option • The Volatility during the life of the option
- 37. Volatility • The most important of the three uncertain parameters, is the volatility. • The importance of the volatility parameter was highlighted by Black & Scholes through their model. Practitioners now needed to estimate only one parameter, the volatility, and input it into a relative simple formula to find the price of an option. Of the three uncertain parameters, changing volatility has the biggest impact on the price of an option. • Volatility measures variability, or dispersion about a central tendency — it is simply a measure of the degree of price movement in a stock, futures contract or any other market. • Standard Deviation
- 38. Volatility is not constant
- 39. New listed option • Safex listed a lookback option • Investors know it is near impossible to get the timing right. The floating strike lookback option comes to the rescue. • This path-dependent option is described as the portfolio manager’s “best friend”. This option will always let the investor have the best possible strike price thus maximising his payoff! • Very expensive! • To cheapen a lookback, one can limit the time in which the strike is set. One can even discretise the lookback feature to only include certain pre-determined dates.
- 40. New listed option Price • Continuous lookback: Black-Scholes type closed- form equation • Discrete lookback: value using numerical procedures like Monte Carlo simulation
- 41. Monte Carlo simulation • This is a difference equation • Through simulation of this equation we can obtain share prices at expiry and thus option prices by their payoff functions ttdSS ttt 2 exp 2 1 • If we assume stock prices follow Brownian motion we can describe their behaviour with
- 42. Derivative Strategies • If a fund manager wants to hedge a portfolio against market moves, is buying a put the only option? • NO • Options are very versatile and, used in combination with the underlying instruments give powerful strategies that can help you to hedge a portfolio, get synthetic exposure to the market or gear a position.
- 43. Payoff profiles • A payoff profile shows the payoff that would be received if the underlying is at its current level when the option expires • It highlights the risks associated with the strategy in a simple diagram: a future has unlimited profit potential, but such a diagram also shows the potential losses • It is easy to work with payoff profiles - they are additive meaning that we can add or subtract them from one another --- • useful in constructing more complex financial instruments or strategies
- 44. Derivatives as simple diagrams a future has unlimited profit potential, but such a diagram also shows the potential losses P/L Short Future K K’ Long Future
- 45. Payoff profiles: options K Long Put Payoff K Short Put Payoff K Short Call K Long Call
- 46. The put strategy • Buying a put is also a strategy • In general investors buy a put as a hedge when they are long the underlying stock • A bearish strategy
- 47. Put-call-parity SK Long put Long stock
- 48. Valuation of optionality in transactions • Most BEE deals have inherent optionality • SBSA: partial American option. Black shareholders obtain a put option on SBK to protect them against any downside. The European put turns into an American put on 1 January 2015. The strike is R40.50 put + long shares = synthetic call • Absa: variable strike partial American option where the American date is 2 July 2007 and the strike is 00.10000.69 00.10000.70,0maxint70.000.48 RSifR RSifRSRR K A AA Binomial Model
- 49. The zero cost collar strategy • Fund manager has a portfolio of shares he/she needs to hedge • Fund manager buys an ATM put from risk taker – this costs money (premium) • Risk taker buys an OTM call from the fund manager • Strike of OTM call is determined by ensuring that the call premium is equal to the premium of the ATM put Value = Put - Call
- 50. Advantages • Hedge down side market moves with zero upfront premium • Participates in upward moves • Tailor-made in terms of strike and expiry e.g. important for unit trust’s quarterly reports • Generally OTC and not Safex • Mostly done on the Alsi Top 40 index • Also suited to single stocks, Indi, Fini, Resi or a tailor- made index
- 51. The zero cost collar: graphically S Long stock K Short call Floor Ceiling Long put
- 52. Employee stock options • Companies issue these to employees • A particular structure: – Options are granted on certain dates. – 20% of the granted options vests 2 years after the grant date and 20% vests every year thereafter for the next 4 years. – No rights can be taken up until the vesting date of a particular tranche. – Employees have until 1 year after the last vesting date to take up the vested option rights. Vested options can be taken up on any date from the vesting date. – The option is a delayed delivery option. – Option strikes are set on the grant date.
- 53. Employee stock options Experts believe Black-Scholes is not appropriate: • There is usually a vesting period - options cannot be exercised. This vesting period can be as long as four to five years. • Options are very long dated – up to 10 years. • When employees leave their jobs during the vesting period they forfeit unvested options. • When employees leave after the vesting period they forfeit options that are out of the money and they have to exercise vested options that are in the money immediately. • Employees are not permitted to sell their employee stock options in the open market – this means there is no efficient discovery of prices. They must exercise the options and sell the underlying shares in order to realize a cash benefit or diversify their portfolios. This tends to lead to employee stock options being exercised earlier than similar regular options.
- 54. Employee stock options • A lattice structure, such as the binomial or trinomial model, incorporates assumptions about employee exercise behaviour over the life of each option grant. This results in more-accurate option values and compensation expense. • This is set out in IFRS 2
- 55. An equity linked note: guarantees • The fund manager now has cash and not a portfolio of shares • An interest rate play • Invested funds are guaranteed at expiry + there is potential upside if the share market performs well
- 56. Example • Investor invests R100 million with risk taker for 1 year • Risk taker guarantees R100 million after 1 year – credit risk • Risk taker invests R90,497,737.56 in money market – 10.5% NACA • This leaves R9,502,262.44 to buy exposure to market
- 57. Equity linked note structure Buy ATM call Cost = R13,040,447.90 Need R3,538,185.46 to make zero-cost Sell OTM call Value = Call1 – Call2 + money market
- 58. Contact Dr Antonie Kotzé Email: consultant@quantonline.co.za Phone: 082 924-7162 Disclaimer This article is published for general information and is not intended as advice of any nature. The viewpoints expressed are not necessarily that of Financial Chaos Theory Pty Ltd. As every situation depends on its own facts and circumstances, only specific advice should be relied upon.

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