Black Scholes Option Valuation. Option Valuation Part III. Put Call Parity. Example 18.3 Black Scholes Put Valuation

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1 Black Scholes Option Valuation Option Valuation Part III Example 18.3 Black Scholes Put Valuation Put Call Parity 1

2 Put Call Parity Another way to look at Put Call parity is Hedge Ratio C P = D (S F X) D = discount factor usually expressed as 1 / (1+r f ) T The RHS is the present value of long call + write put The RHS is the present value of forward contract (the pay off of the forward contract S f X is paid in the future ; ie, S F future value of underlying asset and X strike price at expiry) the spot price is given by D * F = S (spot price is present value, forward price is future value, discount factor relates these). For every 3 call option written, 1 share of stock must be held in order to hedge away risk Using the Black Scholes Formula Hedge Ratio For every 3 call option written, 1 share of stock must be held in order to perfectly hedge away risk Hedge ratio = 1/3 2

3 Call Option Value and Hedge Ratio Portfolio Insurance Buying Puts results in downside protection with unlimited upside potential Limitations Tracking errors if indexes are used for the puts Hedge ratios or deltas change as stock values change, maturity of puts may be too short The constant updating of the hedge ratio is called dynamic hedging (aka delta hedging) For small change in S, the Hedge ratio is the slope Hedge Ratios Change as the Stock Price Fluctuates Hedging and Delta 3

4 Portfolio Insurance : Example Suppose a portfolio is currently valued at $100 and at the money put option on the portfolio have a hedge ratio of 0.6. (meaning the option value swing by 0.6 for every random changein portfolio value, in theopposite direction. recall the pay ff diagram of put option) If stock portfolio value fall by 2% Portfolio Insurance : Example For 2% reduction in stock portfolio. The net loss in protective put position is Stock Loss 2% of 100 = $2 Gain on Put Option 0.6*2 = +$1.2 Net Loss = $0.8 (delta= option value change / stock price change) Equivalently, we can create a synthetic put option by selling 60% (delta) of share and proceed in risk free T bills. The rationale is that the synthetic put option must be able to offset 60% of change in stock value. One way to do so is to directly reduce portfolio risk by selling 60% of shares and put the proceed amount in risk free asset. Hence, the return is Stock Loss 2% of 40 = $0.8 Gain/Loss on risk free asset = 0 Net Loss = $0.8 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. Empirical Test on Option Pricing In general, the model reasonably track the actual value of the options. However, the Black Scholes model performs worst for options on stocks with high dividend payouts. But some empirical test reject Black Scholes : The implied But some empirical test reject Black Scholes : The implied volatility of all options on a given stock with the same expiration date should be equal. Empirical test by Robinstien (1994) show that implied volatility actually falls as exercise price increases. Clearly, Black Scholes is missing something This effect do not exist prior to market crash in So it may be due to fears of a market crash. 4

5 Conclusions As the stock price changes, so do the deltas used to calculate the hedge ratio. The hedge ratio will change with market conditions. Rebalancing is necessary. The hedge ratio (H) is the number of shares required to hedge the price risk involved in writing one option. H is near 0 for deep out of the money call and approach 1 for deep in the money. Portfolio insurance can be obtained by purchasing a protective put option on an equity position. When appropriate put option is not available, portfolio insurance entails a dynamic hedging where a fraction of the equity portfolio equal to the desired put option s delta is sold, with proceed in R f 5