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We Know that the Value of a Call Option Depends on the Values of:
– Stock Price– Time to Expiration– Volatility– Riskless interest rate (generally assumed to be constant)– Dividend yield (generally assumed to be constant)– Strike Price (never changes)
Delta () is the rate of change of the option price with respect to the underlying. (I.e., How does the option price change as the underlying price changes?)
Aim of Delta Hedging: Keep Total Wealth Unchanged. Let’s look at the end of Week 9.
• Value of written option at start: $240,000 (per B-S.)• Value of option at week 9 (=20%): $414,500.• Option Position Loss: $(174,500).• Cash Position Change, (Measured by Cum. Cost: $2,557,800 –
4,000,500 = $(1,442,700).• Value of Shares held:
– At start: 49 * 52,200 = $2,557,800
– At end of week 9: 53 * (67,400 + 11,300) = $4,171,100.
• The position delta determines how much a portfolio changes in value if the price of the underlying stock changes by a small amount.
• The portfolio might consist of several puts and calls on the same stock, with different strikes and expiration dates, and also long and short positions in the stock itself.
• The delta of one share of stock that is owned equals +1.0. The delta of a share of stock that is sold short is –1.0. Why?
• In this example, the position delta of 178.96 is positive.
• This means that if the stock price were to increase by one dollar, the value of this portfolio would rise by $178.96.
• If the stock price were to decline by one dollar, then the value of this portfolio would fall by $178.96. (Remember that this example assumes that the underlying asset of an option is one share of stock.)
• Knowing your portfolio's position delta is as essential to intelligent option trading as knowing the profit diagrams of your portfolio.
• The maximum profit on a time spread occurs when the stock price equals the strike price on the expiration date of the nearby call.
• The short-term written option expires worthless, and the longer term option can be sold with some time value remaining. – An important risk to be kept in mind is that if the written,
short-term option is in-the-money as its expiration date nears, it could be exercised early.
– Thus, in-the-money time spreads using American options will often have to be closed out early in order to preclude this possibility.
• Of course, there are real risks and costs that must be included.• For example, we have ignored transactions costs.
– In practice, the position will have to be closed out early if the options are in the money as September 21 nears, because the September call could be exercised early.
– If they are exercised, the trader will bear additional transactions costs of having to purchase the stock and making delivery. In addition, there are rebalancing costs of trading options to maintain delta neutrality. For this reason, neutral hedgers prefer low gamma positions, if possible.
• We assumed there are no ex-dividend dates between August 1 and September 21. Dividends would affect the computed deltas and also introduce early exercise risk.
• We assumed constant interest rates and constant volatility. These also affect the computed deltas and indeed, may be the source of the apparent mispricing.
• A cap is a purchased call option in which the underlying asset is an interest rate.
• Note that a call on an interest rate is equivalent to a put on a debt instrument.
• The interest rate cap is actually established when the interest rate call is combined with an existing floating rate loan. The call caps the interest that can be paid on the floating rate liability.
• An interest rate floor is a put on an interest rate ( = a call on a debt instrument).
• Lenders, or institutions that own floating rate debt (an asset), purchase interest rate floors.
• The floor is actually created when the interest rate put is combined with the floating rate asset. The put places a floor (a minimum) on how low the institution’s interest income can be.
• A collar combines the purchase of a cap at a high strike price with the sale of a floor at a low strike price.
• Frequently, the cost of the cap equals the selling price of the floor, so that a zero cost collar is “purchased”.
• A collar allows a firm with a floating rate liability to insure itself from the undesired impact of an interest rate rise, but pays for it by giving up the benefits of a rate decline.
• Corridors are purchased by a firm with a floating rate liability. The firm buys an interest rate call with a (relatively) low strike price, and sells a cheaper one with a higher strike price.
• Now, suppose a trader is considering using either calls or puts to maintain a delta neutral portfolio. This can be accomplished with the purchase of puts or by writing calls. Using Puts:
Delta Neutral Strategy: Long 1 put at $6.44; long 0.3849 shares at $100/share.Positive Position Gamma: (1* 0.0181) + (0.3849* 0) = 0.0181.
Value ofStock Put 0.3849 Port. Pct. Chg.Price Price Shares Value (From S=100)
Delta Neutral Strategy: Short 1 call at $10.30; long 0.6151 shares at $100/share. Note that this results in a Negative Position Gamma: (-1* 0.0181) + (0.6151* 0) = -0.0181.
• Note that when the delta hedger constructs a portfolio using puts, the value of the portfolio increases, regardless of the direction of the change in the underlying asset price.
• However, in the case where calls are used, the portfolio value falls whether the underlying asset price increases or decreases.
• From a cost standpoint, a delta neutral hedger would like to have a portfolio with a low, but positive gamma. Recall that gamma measures changes in delta, and that deltas change as S, T, , and/or r change.
• Thus, a portfolio that is delta neutral portfolio today may not be delta neutral tomorrow.
• A low position gamma will mean that the delta neutral investor will conserve on the transactions costs of readjusting the portfolio delta back to zero if S changes.
• But a positive gamma at least compensates a trader for bearing the risk of fluctuating delta, because the value of the portfolio will increase if the underlying asset’s price changes, all else equal.
• Traders can create portfolios that are neutral (insensitive) to any single “Greek” or combination of “Greeks.”
• From the examples above, you can see that a trader needs one option to remove one effect.
• In the example above, the trader made the portfolio insensitive to changes in the stock price using a delta neutral strategy (i.e., recall that traders can use either calls or puts in a delta neutral strategy).
• To neutralize additional effects, traders must add additional securities to their portfolios.
• Suppose the trader decides to sell 200 call options with the strike of 100, each on 100 shares of stock.
• Because the trader wants to create a delta-gamma hedged portfolio, the trader must simultaneously solve for the number of shares to purchase, S, and the number of K=110 calls to purchase, K110.
• This is straightforward because the gamma of the stock is zero, and the portfolio gamma is a weighted sum of the constituent gammas. The trader wants a gamma of zero. That is,
• Assuming the trader buys 194 of the calls having a strike price of 110.
• Solving for the number of shares is accomplished by noting that the delta of a share equals one and that the trader also wants to hold a delta neutral portfolio. Accordingly,
0 = (-200*100*0.6151) + (S*1) + (194*100*0.4365)
• Therefore S = 3,834 makes the portfolio delta-gamma neutral, given the other positions.
• To examine the performance of this delta-gamma hedge, suppose the stock price were to change to either $101 or to $99.