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Derivative Securities (Fach) / Options (Lektion)
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- What is an option? An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. the holder can choose not to exercise the right and allow the option to expire.
- Call option Sell If value of the underlying asset S > Strike Price X The value of a call increases as the value of the underlying asset increases Exercise : S-X
- Put option Sell If value of the underlying asset X > value of underlying S The value of a put increases as the value of the underlying asset decreases Exercise : X-S
- Increase in Stock Price Call Value : increases Put Value: decreases
- Increase in Strike Price Call Value : decreases Put Value: increases
- Increase in variance of underlying asset Call Value : Increases Put Value: Increases
- Increase in time to expiration Call Value: Increases Put Value: Increases Life of the Option; both calls and puts benefit from a longer life.
- Increase in interest rates = Lower price, d.h. X decreases Call Value: Increases Put Value: Decreases
- Increase in dividends paid are likely to reduce the price appreciation component of the asset meaning reduced price of X Call Value: Decreases Put Value: Increases
- The Limiting Distributions If as t -> 0, price changes become smaller, the limiting distribution is the normal distribution and the price process is a continuous one If as t->0, price changes remain large, the limiting distribution is the poisson distribution, i.e., a distribution that allows for price jumps. The Black-Scholes model applies when the limiting distribution is the normal distribution .It explicitly assumes that the price process is continuous and that there are no jumps in asset prices.
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- The Black-Scholes Model Designed to value European options, which were dividend-protected Value of a call option in the Black-Scholes model can be written as a function of the following variables: Value of call = S N (d1) - X e-rt N(d2) S = Current value of the underlying assetX = Strike price of the option t = Life to expiration of the option r = Riskless interest rate corresponding to the life of the option σ2 = Variance in the ln(value) of the underlying asset
- Binomial Option Pricing model Cox, Ross and Rubinstein (1979) Not only a simple approximation of a complex problem: is a powerful tool for valuing quite general derivative securities can be used when no analytical closed form solution of the continuous time models exists In the limit, when the discrete steps converge to a continuum, the model converges to an exact formula in continuous time similar to State-Preference Theory, bit more specific -> uses discrete time and discrete variables. Time modelled as a series of points in time at which the uncertainty over the previous period is resolved and new decisions are made - in between ’nothing happens’ Uncertainty in variables modelled by distinguishing only 2 different future states of the world
- Hedge ratio Delta, ∆, is often used in finance is the number (fraction) of shares needed to replicate the option called the ”hedge ratio” or the ”option delta” is measured as spread in option values divided by spread in stock values
- hedging portfolio or the option equivalent portfolio. Equivalent to option ⇒ generates same payoffs as option Means they must have same current price to avoid arbitrage opportunities. So: O = ∆S + D
- What does not appear in the formula O = pOu +(1−p)Od / r What does not appear in the formula: Investors’ attitudes toward risk Other securities or portfolio’s (market portfolio) Real probabilities q and 1 − q Option pricing models do not ’explain’ prices of existing securities, like CAPM and APT They ’translate’ prices of existing securities into option prices
- Hedging portfolio for put ? Short position in the stock and long position in the risk free debt
- Option Characteristics 1. A long option is a limited liability investment -> gives the right, not the obligation to buy/sell 2. Economically options represent flexibility: possibility to choose best alternative walk away from bad outcomes 3. Also found in real investments (real options) flexibility to change cash flows profiting from opportunities, cutting off losses DCF assumes passive, not flexible, position: accept cash flows as they come DCF cannot handle flexibility well 4. Options are almost always riskier than underlying values
- Reasons to use options to insure against to profit from to speculate
- Intrinsic value of an option Amount by which option is in the money
- Options Hedge ratio BS N(d1)
- time value Premium - intrinsic value
- European Call Option Sell the option in the open market prior to expiration and buy the underlying asset at the exercise price on the expiration date.
- American Put-Option sell the underlying asset at the striking price on or before the expiration date and potentially benefit from a stock price decrease
- to adjust for stock splits.. the exercise price of the option is reduced by the factor of the split and the number of options held is increased by that factor.
- The current market price of a share of JNJ stock is $60. If a put option on this stock has a strike price of $55, the put is out of the money and sells for a lower price than if the market price of JNJ stock is $50. right
- Implied Volatitily As expectations rise, or as the demand for an option increases, implied volatility will rise. Options that have high levels of implied volatility will result in high-priced option premiums. Conversely, as the market's expectations decrease, or demand for an option diminishes, implied volatility will decrease Read more: Implied Volatility: Buy Low And Sell High http://www.investopedia.com/articles/optioninvestor/08/implied-volatility.asp#ixzz3u6g1hABF Follow us: Investopedia on Facebook
- Exercise Prices and Δ Δ The higher the X, the higher Delta for puts -Δ Δ The lower X, the higher Delta for calls Δ
- Underpriced assets Actual Call < Synthetic Call Buy the Call, sell the Synthetic Call