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elvis.sob.tulane

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elvis.sob.tulane

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    3. Bond Pricing Recall that a bond with an embedded call option should be priced such that

    4. Non-callable Bond Pricing Traditional yield spread method Based on YTM Ignores differences in cash flow characteristics Static spread analysis Based on term structure Allows for differences in cash flow characteristics

    5. Static Spread Analysis The benchmark is the Treasury term structure

    6. Static Spread Analysis The term structure implies a set of zero prices that can be used to price any cash flow

    7. Static Spread Analysis The benchmark is the Treasury term structure.

    8. Static Spread Analysis We use the term structure to price the cash flow of a bond we wish to price.

    9. Static Spread Analysis The difference between the bond’s market price and its “theoretical” risk-free price reflects the yield premium investors demand for bearing risk. For example, the 12% 15-year corporate was priced at 86 15/16 recently. We calculated a theoretical price of 139.695 for this bond, which implies a price differential of $52.758 per $100 of face value.

    10. Static Spread Analysis The “static spread” is the yield pickup (over and above the term structure) needed to equate the present value of a bond’s cash flow with its price. That is, instead of using the term structure to value a bond’s cash flow, we use

    11. Static Spread Analysis For example:

    12. Static Spread Analysis Cut to EXCEL spreadsheet example

    13. Option Value Static spreads are not useful for pricing callable bonds because they do not account for interest rate volatility, which affects the value of the embedded option. Common practice in pricing callable bonds is to compute the “Option Adjusted Spread,” or OAS. Interest rate volatility is modeled as a binomial process.

    14. Binomial Option Pricing The binomial model is based on the simplifying idea that asset prices (or interest rates) can move up by u% or down by d% from period to period.

    15. For example Consider the stock price dynamics:

    16. For example Suppose a call option on this stock has a strike price of $45

    17. A replicating portfolio Consider a portfolio containing D shares of the stock and $B invested in risk-free bonds. The present value (price) of this portfolio is

    18. Value of the portfolio

    19. A replicating portfolio This portfolio will replicate the option if we can find a D and a B such that

    20. The replicating portfolio Solution: D = 1/3 B = -35/(3(1+r/12)). Eg, if r = 6%, then the portfolio contains 1/3 share of stock (current value $40/3 = $13.33) partially financed by borrowing $35/(3x1.005) = $11.61

    21. The replicating portfolio Payoffs at maturity

    22. The replicating portfolio Since the the replicating portfolio has the same payoff as the option in all states, the two must also have the same price. The present value (price) of the replicating portfolio is $13.33 - $11.61 = $1.72. Therefore, c = $1.72

    23. A general (1-period) formula

    24. Another look at it Recall the stock price dynamics

    25. Option Value Which implies the option value (for a $45 strike price):

    26. Covered Calls Suppose you write H covered call options:

    27. Writing H Covered Calls

    28. Writing H Covered Calls

    29. Risk-Free Investing Since an investment of $40 - (3 x Call Price) will produce $35 for certain, it must equal the present value of $35 discounted at the risk-free interest rate

    30. Price of a Call This will be true if and only if the call is priced at

    31. For Example If the risk-free interest rate is 6% per annum, then

    32. The One-Period Formula The one-period pricing formula for a European call paying no dividends in the standard binomial model is

    33. The One-Period Formula

    34. Example

    35. Two Periods The assumption that the underlier price changes only once during the life of the option is obviously unrealistic Suppose two price changes are possible during the life of the option At each change point, the stock may go up by Ru% or down by Rd%

    36. Two-Period Stock Price Dynamics For example, suppose that in each of two periods, a stocks price may rise by 3.25% or fall by 2.5% The stock is currently trading at $47 At the end of two periods it may be worth as much as $50.10 or as little as $44.68

    37. Two-Period Stock Price Dynamics

    38. Terminal Call Values

    39. Two Periods The two-period Binomial model formula for a European call is

    40. Example

    41. Binomial Trees Cut to EXCEL example

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