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Using Option Contracts to Hedge Risk

3/24/2012. Fin 355-Hedging w Options | Dr. Menahem Rosenberg. 2. Options Examples. Traditional US mortgages give the householder the right to call (refinance) the mortgage at a strike equal to the outstanding principleIf interest rates have fallen below the note's rate, when the home owner will co

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Using Option Contracts to Hedge Risk

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    1. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 1 Using Option Contracts to Hedge Risk Definition: An American call (put) option is the right, but not the obligation to buy (sell) an asset at a specified price any time until its expiration date. A European option can be exercised only at a specific time.

    2. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 2 Options Examples Traditional US mortgages give the householder the right to call (refinance) the mortgage at a strike equal to the outstanding principle If interest rates have fallen below the note’s rate, when the home owner will consider refinancing the mortgage The owners of a limited liability corporation have the right, but not the obligation, to ‘put’ the company to the corporation’s creditors and bondholders Limited liability is, in effect, a put option

    3. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 3 Option Terminology Call: an option to buy Put: an option to sell Strike or Exercise Price: the fixed price specified in an option contract Expiration or Maturity Date: The date after which an option can’t be exercised American Option: an option that can be exercised at any time up to and including maturity date European Option: an option that can only be exercised on the maturity date

    4. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 4 Option Terminology Intrinsic (Tangible) Value: The hypothetical value of an option if it were exercised immediately At-the-Money: an option with a strike price equal to the value of the underlying asset Out-of-the-Money: an option that’s not at-the-money, but has no intrinsic value In-the-Money: an option with a intrinsic value Writer the option seller

    5. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 5 Quote Example

    6. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 6 Quote Example

    7. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 7 Option Payoff C , P - The value of a Call and Put option, respectively, at maturity. K - Exercise (Strike) price. S - Stock Price at option Maturity. (1) C = Max {0, S - K} (2) P = Max {0, K - S}

    8. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 8 Option Payoff Diagrams Call Option: Put Option:

    9. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 9 Option Payoff Diagrams Call Option (including premium): Put Option (including premium):

    10. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 10 Call Value Value prior to maturity: a - Zero stock value – zero Call value b - Deep in the money, call value converges to the Stock value minus strike. c – Probability of stock end value above strike, while zero value otherwise, generates the call positive value.

    11. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 11 Factor affecting Option Prices K - Strike T - Time to Maturity S - Stock Price V - Stock volatility D - Stock Dividend r - risk free rate.

    12. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 12 The Black-Scholes Model The most widely used model for pricing options is the Black-Scholes model This model is completely consistent with the binary model as the interval between stock prices decreases to zero The model provides theoretical insights into option behavior The assumptions are elegant, simple, and quite realistic We will work with the generalized form of the model because the small additional complexity results in considerable additional power and flexibility First, notation:

    13. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 13 The Black-Scholes Model: Notation C = price of call P = price of put S = price of stock E = exercise price T = time to maturity ln(.) = natural logarithm e = 2.71828... N(.) = cum. norm. dist’n The following are annual, compounded continuously: r = domestic risk free rate of interest d = foreign risk free rate or constant dividend yield s = volatility

    14. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 14 The Black-Scholes Model: Equations

    15. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 15 Observable Variables All the variables are is directly observable, excepting the volatility, and possibly, d We do not have to delve into the psyche of investors to evaluate options We do not forecast future prices to obtain option values There are often many options trading on a single underlying security: puts, calls different maturates different strikes American, European, exotic

    16. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 16 Many Options, One Volatility We have assumed that s is a parameter and is fixed, and is associated with the underlying stock, and not the option While we can’t observe s directly, we could use one option to find the implied volatility, and then apply this to the other options on the same stock to look for pricing aberrations

    17. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 17 Put-Call Parity S0, S1 – Stock price today and at option maturity. Portfolio A - Buy a Call (strike K) and lend the present value of the strike Portfolio B - Buy a stock and buy a Put (strike K)

    18. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 18 Put-Call Parity Portfolio A terminal value equals portfolio B terminal value, therefore, their present value must equal too. c - call premium p - put premium

    19. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 19 Synthetic Securities The put-call parity relationship may be solved for any of the four security variables to create synthetic securities: c=S+p-K S=c-p+K p=c-S+K K=S+p-c

    20. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 20 Protective Put The purchase of a put contract to hedge against a decline in the portfolio assets. Strategy cost: Put premium, difference between current asset value and the strike. Portfolio value decline is limited to the the premium cost less (plus) the in (out) the money intrinsic value.

    21. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 21 Hedging by Selling Calls Covered call sell. Create a portfolio V, with S stock value and c call value. h - ratio of required stock per call contract (hedge ratio) For hedging purpose, make V a constant.

    22. 3/25/2012 Fin 355-Hedging w Options | Dr. Menahem Rosenberg 22 Covered Call Long Stock Short a Call

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