Option Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values

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1 Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X S T Time value the difference between the option price and the intrinsic value Call Option Value before Expiration Determinants of Call Option Values The volatility value is higher near X 1

2 Restrictions on Option Value: Call Figure 18.2 Range of Possible Call Option Values Volatility value is positive Figure 18.3 Call Option Value as a Function of the Current Stock Price Early Exercise: Calls The right to exercise an American call early is valueless as long as the stock pays no dividends until the option expires. (For dividend paying stock it may still make sense to exercise a deep in the money call option early in order to collect the dividend) There is no reason to exercise early" means "you are at least as well off owning the option as you are having exercised the option and owning the stock". The value of American and European calls is therefore identical. The call gains value as the stock price rises. Since the price can rise infinitely, the call option is worth more alive than dead. due to positive volatility value 2

3 Early Exercise: Puts American puts are worth more than European puts, all else equal. Figure 18.4 Put Option Values as a Function of the Current Stock Price The possibility of early exercise has value because: for deep in the money put options, it may make sense to exercise the option early in order to obtain the intrinsic value (X S) earlier so that it can start to earn interest immediately Once the firm is bankrupt, it is optimal to exercise the American put immediately because of the time value of money. Binomial Option Pricing: Text Example Binomial Option Pricing: Text Example U = 1.2, d= 0.9 Consider borrowing interest rate 10% => the pay off after 1Y are as above replicate payoff concept 3

4 Binomial Option Pricing: Text Example Replication of Payoffs and Option Values Hedge Ratio Expanding to Consider Three Intervals Assume that we can break the year into three intervals. For each interval the stock could increase by 20% or decrease by 10%. Assume the stock is initially selling at $100. For every three call option written, 1 share of stock must be held in order to hedge away risk 4

5 Expanding to Consider Three Intervals Possible Outcomes with Three Intervals What will happen if there are infinite intervals and the interest rates and volatility are constant over the period? Black Scholes Option Valuation Black Scholes Option Valuation 5

6 Figure 18.6 A Standard Normal Curve Example : Black Scholes Valuation Probabilities from Normal Distribution Call Option Value Implied Volatility Implied volatility is volatility for the stock implied by the option price. Using Black Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock? Investor compare implied volatility with observed stock volatility to make trading rule. 6

7 Black Scholes Model with Dividends Example : Black Scholes Put Valuation Put Option Valuation: Using Put Call Parity Using the Black Scholes Formula 7

8 Call Option Value and Hedge Ratio Portfolio Insurance Buying Puts results in downside protection with unlimited upside potential Limitations i i Tracking errors if indexes are used for the puts Maturity of puts may be too short Hedge ratios or deltas change as stock values change Profit on a Protective Put Strategy Hedge Ratios Change as the Stock Price Fluctuates 8

9 Hedging On Mispriced Options Hedging and Delta Option value is positively related to volatility. If an investor believes that the volatility that is implied in an option s price is too low, a profitable trade is possible. Profit must be hedged against a decline in the value of the stock. Performance depends on option price relative to the implied volatility. Example 18.6 Speculating on Mispriced Options Implied volatility = 33% Investor s estimate of true volatility = 35% Option maturity = 60 days Put price P = $4.495 Exercise price and stock price = $90 Risk free rate = 4% Delta =.453 Table 18.3 Profit on a Hedged Put Portfolio 9

10 Example 18.6 Conclusions As the stock price changes, so do the deltas used to calculate the hedge ratio. Gamma = sensitivity of the delta to the stock price. Gamma is similar to bond convexity. The hedge ratio will change with market conditions. Rebalancing is necessary. Delta Neutral When you establish a position in stocks and options that is hedged with respect to fluctuations in the price of the underlying asset, your portfolio is said to be delta neutral. The portfolio does not change value when the stock price fluctuates. Table 18.4 Profits on Delta Neutral Options Portfolio Empirical Evidence on Option Pricing The Black Scholes formula performs worst for options on stocks with high dividend payouts. The implied volatility of all options on a given stock with the same expiration date should be equal. Empirical test show that implied volatility actually falls as exercise price increases. This may be due to fears of a market crash. 10