# delta neutral

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A Primer for the Mathematics of Financial Engineering by Dan Stefanica

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-hedging position to be taken in the asset is .6. 3.7. THE CONCEPT OF HEDGING . .6.- AND r-HEDGING keeps the portfolio close to Delta-neutral, and therefore reduces the risk of losses if the price of the underlying asset changes rapidly, but increases the trading costs), and rebalancing only when the hedge becomes inaccurate enough (which reduces the trading costs but increases the risk of having to manage a portfolio that is close to, but not exactly Delta-neutral, most of the time). To achieve a better hedge of the portfolio, i.e., to have a portfolio that is even less sensitive to small changes in the price of the underlying asset than a Delta-neutral portfolio, we look for a portfolio that is both Delta-neutral and Gamma-neutral, i.e., such that .6. (II) av = as' 107 all = as = ° and r(II) a2ll = aS2 = 0, where the Gamma of the portfolio is denoted by r(ll).

As the price of the underlying asset changes, the portfolio might become unbalanced, i.e., not Delta-neutral. In this case, the hedge needs to be rebalanced, i.e., units of the underlying asset must be bought or sold in order to make the portfolio Delta-neutral again. Deciding when to rebalance the portfolio is a compromise between rebalancing the hedge often (which Therefore, if a portfolio is Gamma-neutral, its Delta will be rather insensitive to small changes in the price of the underlying asset. In particular, if the portfolio is both Delta-neutral and Gamma-neutral, then the Delta of the portfolio will change significantly only if larger changes in the price of the underlying asset occur. The need to rebalance the hedge for such a portfolio occurs less often than in the case of a portfolio that is Delta-neutral but not Gamma-neutral. This saves trading costs and the portfolios are, generally speaking, better hedged most of the time.

will be insensitive to small changes in the price of the underlying asset. Such a portfolio is called Delta-neutral, since all .6.(ll) av = as = as - .6. = 0. Recall that the correct hedging position for a long call is short .6. shares. Therefore, the hedging position for a short call is long .6. shares. Similarly, a long put position is hedged by going long .6. shares: as the spot price of the underlying asset goes up, a long put position loses value while a long asset position will gain value, thus offsetting the loss on the long put. A short put is hedged by going short .6. shares. Option Position Hedge Position Sign of Option Delta Short Asset Positive Delta Long Call Long Asset Negative Delta Short Call Long Put Long Asset Negative Delta Short Asset Short Put Positive Delta Note that a portfolio is Delta-neutral only over a short period of time. As the price of the underlying asset changes, the portfolio might become unbalanced, i.e., not Delta-neutral.

Mathematics for Finance: An Introduction to Financial Engineering by Marek Capinski, Tomasz Zastawniak

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The value of the delta neutral portfolio consisting of the shored stock, invested cash and sold options will be −32, 332.29 + 33, 914.69 − 1, 582.40 = 0.00 dollars. 296 Mathematics for Finance One day later the shorted shares will be worth 17, 765 × 1.81 = 32, 154.64 dollars, whereas the cash investment will grow to 33, 914.69e0.05/365 ∼ = 33, 919.34 dollars. The put price will increase to 0.035182 dollars, so the price of 50, 000 puts will be 1, 759.11 dollars. The value of the delta neutral portfolio will be −32, 154.64 + 33, 919.34 − 1, 759.11 ∼ = 5.59 dollars. 9.4 The price of a single put after one day will now be 0.038885 dollars, the 50, 000 options sold will therefore be worth 1944.26 dollars, the stock and cash deposit positions remaining as in Solution 9.3. The delta neutral portfolio will bring a loss of 179.56 dollars. 9.5 If the stock price does not change, S(t) = S(0) = S, then the value of the portfolio after time t will be given by V (t) = SN (d1 ) − Xert e−rT N (d2 ) − C E (S, t), where C E (S, t) is given by the Black–Scholes formula and d1 , d2 by (8.9).

Exercise 9.2 Find the stock price on day one for which the hedging portfolio attains its maximum value. Exercise 9.3 Suppose that 50, 000 puts with exercise date in 90 days and strike price X = 1.80 dollars are written on a stock with current price S(0) = 1.82 dollars and volatility σ = 14%. The risk-free rate is r = 5%. Construct a delta neutral portfolio and compute its value after one day if the stock 1 ) = 1.81 dollars. price drops to S( 365 Going back to our example, let us collect the values V of the delta neutral portfolio for various stock prices after one day as compared to the values U of the unhedged position: S 58.00 58.50 59.00 59.50 60.00 60.50 61.00 61.50 62.00 V −71.35 −31.56 −3.26 13.69 19.45 14.22 −1.77 −28.24 −64.93 U 1, 100.22 849.03 586.35 312.32 27.10 −269.11 −576.07 −893.53 −1, 221.19 196 Mathematics for Finance Now, let us see what happens if the stock price changes are considerable: S 50 55 60 65 70 V −2, 233.19 −554.65 19.45 −481.60 −1, 765.15 U 3, 594.03 2, 362.79 27.10 −3, 383.73 −7, 577.06 If we fear that such large changes might happen, the above hedge is not a satisfactory solution.

The values of the portfolio are given below (for comparison we also recall the values of the delta neutral portfolio): 1 S( 365 ) 58.00 58.50 59.00 59.50 60.00 60.50 61.00 61.50 62.00 delta-gamma −2.04 0.30 1.07 0.81 0.02 −0.79 −1.11 −0.49 1.52 delta −71.35 −31.56 −3.26 13.69 19.45 14.22 −1.77 −28.24 −64.93 We can see that we are practically safe within the given range of stock prices. For larger changes we are also in a better position as compared with delta hedging: 1 S( 365 ) delta-gamma delta 50 −614.08 −2, 233.19 55 −78.22 −554.65 60 0.02 19.45 65 63, 13 −481.60 70 440.81 −1, 765.15 As predicted, a delta-gamma neutral portfolio oﬀers better protection against stock price changes than a delta neutral one. Delta-Vega Hedging. Next we shall hedge against an increase in volatility, while retaining cover against small changes in the stock price.

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Automate This: How Algorithms Came to Rule Our World by Christopher Steiner

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The program would then scan the warrens of stock and option data and route it through Peterffy’s algorithms. First and foremost, the algorithms searched for options that were egregiously mispriced. What Peterffy was especially interested in—and what he wrote his algorithms to search for—were what’s known as delta neutral trades. In these trades, an overpriced call option, which Peterffy would sell, could be coupled with buying an underpriced put option to create a position that wouldn’t be adversely affected by a spike or a dip in the market. For example, say shares of IBM were trading hands at \$75. A delta neutral trade for Peterffy could look something like this: He would sell 10,000 call options (the right to buy IBM at a strike price of \$75 during the next sixty days) for \$1 each (the overpriced option), taking in the \$10,000. Almost simultaneously, another Peterffy trader would buy 10,000 put options (the right to sell IBM at a strike price of \$75 during the next sixty days) for 85 cents each (the underpriced option), for \$8,500.

The result of the trade: a near-riskless \$1,500 profit. Because options on stocks of big companies such as IBM traded at dozens of different strike prices as well as multitudes of diverse expirations, the 1980s options market was rife with delta neutral trades for those who could find them. Some floor traders had caught on to the tactic—they scanned the tapes and tickers for mispriced options that could easily be hedged for riskless profits. But the power of a few men searching out trades in the pits was no match for Peterffy’s indefatigable machine. The computer would put together trades that made for delta neutral plays and print them out immediately. Peterffy, confined to a chair with a bum knee, would then place a call to his floor clerk, who would get the trades to his corps of women. And on it went. Four months after hacking the Quotron link and getting his computer system operational, Peterffy was making more money than ever.

I can do that.”4 He had learned math, he said, studying astronomy in the Netherlands and in the air force, but it’s most likely any math being done for his trades came straight out of Peterffy’s algorithm, which was gnawing on raw data from the hijacked Quotron line. Just as with Peterffy’s other traders, Van Peebles frequently ran sorties to a large bank of phones on the trading floor to communicate with Timber Hill’s headquarters. On the phone, he scribbled down a jumble of letters, numbers, and fractions—his instructions. From there, he careened back into one of the roiling pits and put up his hands, ordering fresh delta neutral trades. Van Peebles’s story accentuated the success of the most improbable trading squad roaming the pits of New York, perhaps to this day: three blonde women and one highly acclaimed black writer, director, and actor, all of them well-disguised proxies of an algorithm that dwelled inside a machine. THE IPAD’S FORERUNNER Peterffy’s strategy to play to the pits’ infamous chauvinism paid off as the options specialists continued to take his women’s orders above most others.

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Hedge Fund Market Wizards by Jack D. Schwager

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There are other systems that have a greater variety of values that may provide modest improvements in either strategy or bet sizing. 9In a static hedge, the warrant position is counterbalanced by an equal delta opposite position in the stock at time of trade implementation, resulting in a delta neutral combined position. (Delta neutral means that the value of the combined position will remain approximately unchanged for small changes in price.) In a dynamic delta hedge, the offsetting stock position is continually adjusted to maintain an approximately delta neutral total position. 10 Delta neutral hedging means that purchases and sales of the underlying stock are used to keep the combined position balanced so that it is approximately unaffected by small price changes in either direction. 11 http://edwardothorp.com/id9.html. 12If this question is intelligible, no explanation is necessary; if it isn’t, any adequate explanation would be far too lengthy and tangential. 13Bruce Kovner is the founder of Caxton Associates, a highly successful global macro fund, and was profiled in my book Market Wizards (New York: Institute of Finance, 1989).

Kassouf thought he could tell something about the direction of stock prices and would sometimes take a fundamental view on a stock, whereas I was afraid to do that because I didn’t believe I had any forecasting power. I thought we should always hedge to be delta neutral. 10 So we went our separate ways. He and his brothers started a managed money business, while I managed my individual accounts for a while. When did you develop your own option-pricing model? In 1967, I took some of the ideas about how to price warrants in the Random Character of Stock Prices by Paul Cootner and thought I could derive a formula if I made the simplifying assumption that all investments grew at the risk-free rate. Since the purchase or sale of warrants combined with delta neutral hedging led to a portfolio with very little risk, it seemed very plausible to me that the risk-free assumption would lead to the correct formula.

The Concepts and Practice of Mathematical Finance by Mark S. Joshi

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If we have sold a call option then we are short Gamma and the procedure of hedging will cost us money over the life of the option. If we are long Gamma then, over the lifetime of the option, our hedging makes us money realizing the option's value. To see why this is the case, we expand the Taylor series of the portfolio value, P(S, t), as a function of S, we have z P(S + AS, t) = P(S, t) + aS (S, t)AS + 2 aS2 (S, t)OS2 O(OS3). (4.5) If the portfolio is Delta-neutral then the main term is 1 a2p a as2 (s, t)AS2, and thus the stock's price variations up and down will cause money to be lost or gained according to the sign of the Gamma. We saw that as maturity approaches the Delta of a call option behaves more and more like a step function equal to 0 below the strike and 1 above it. This means that Practicalities 82 0.25 0.2 m 0.15 -0.01 E E -0.1 --.0.05 m -0.4 0.1 0.05 0 80 85 90 95 100 105 110 115 120 Spot Fig. 4.8.

Exercise 4.2 How does the graph of the Vega of a call option vary as a function of time to expiry? Exercise 4.3 If a digital call and a digital put have the same expiry and strike, what relations will their Greeks satisfy? Exercise 4.4 If a derivative has a negative Vega and volatility increases what happens to the price? Exercise 4.5 A portfolio consisting of a short position in a call option and a long position in a stock is Delta-neutral. Suppose the stock price jumps; how will the value of the portfolio change if the option is priced according to the Black-Scholes formula before and after the jump? Exercise 4.6 Derive simple approximations for at-the-money Vega and Theta. Exercise 4.7 Show that call and put options with the same strike and expiry have the same Vega. Do this without using the Black-Scholes formula. Practicalities 92 Exercise 4.8 In the Black-Scholes model, we have the following parameters: S 100 K 110 r 0.05 sigma 0.1 Ti For a call option, we have value Delta Vega Gamma 2.174 0.343 36.78 0.0367 Find the value, Delta, Vega and Gamma of a put option with the same strike.

To illustrate the issues involved in hedging using vanilla options in a jumpdiffusion world, let's consider a simple model with only a single possible jump amplitude. Suppose we use the vanilla call option, 0, struck at 100 as a hedging instrument and we want to hedge a call option, C, struck at 110. We can set up a hedging portfolio consisting of -1 units of C, and a units of 0 and A stocks, together with bonds to ensure the self-financing condition holds. We want the portfolio's value to be invariant under small changes in S, that is, to be Delta-neutral, and to be invariant under jumps. This means we require a and A to be such that ac a0 - - +a as +0=0, (15.18) and that -C(S, t) + aO(S, t) + AS = -C(JS, t) + aO(JS, t) + AJS, (15.19) 15.7 Matching the market 377 where J is the jump amplitude. We have two equations in two unknowns which we can solve for a and A. We are now perfectly hedged and the price of C is guaranteed. Suppose we have a larger finite number of jump amplitudes, then we can hedge in a similar fashion provided we add in another hedging option for each jump amplitude, we just have to solve a larger linear system.

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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If they own options then their exposure to the underlying is, to a first approximation, the same as if they own delta of the underlying. Those who are not speculating on direction of the underlying will hedge by buying or selling the underlying, or another option, so that the portfolio delta is zero. By doing this they eliminate market risk. Typically the delta changes as stock price and time change, so to maintain a delta-neutral position the number of assets held requires continual readjustment by purchase or sale of the stock. This is called rehedging or rebalancing the portfolio, and is an example of dynamic hedging. Sometimes going short the stock for hedging purposes requires the borrowing of the stock in the first place. (You then sell what you have borrowed, buying it back later.) This can be costly, you may have to pay a repo rate, the equivalent of an interest rate, on the amount borrowed.

This can be costly, you may have to pay a repo rate, the equivalent of an interest rate, on the amount borrowed. Gamma The gamma, Γ, of an option or a portfolio of options is the second derivative of the position with respect to the underlying: Since gamma is the sensitivity of the delta to the underlying it is a measure of by how much or how often a position must be rehedged in order to maintain a delta-neutral position. If there are costs associated with buying or selling stock, the bid-offer spread, for example, then the larger the gamma the larger the cost or friction caused by dynamic hedging. Because costs can be large and because one wants to reduce exposure to model error it is natural to try to minimize the need to rebalance the portfolio too frequently. Since gamma is a measure of sensitivity of the hedge ratio Δ to the movement in the underlying, the hedging requirement can be decreased by a gamma-neutral strategy.

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Expected Returns: An Investor's Guide to Harvesting Market Rewards by Antti Ilmanen

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CCW return volatility is mainly driven by equity returns (directional exposure), especially if done with OTM options. Absolute return seekers who look for less correlated return sources should prefer variance swaps that largely avoid directional exposure. Variance swap returns are mainly driven by the difference between implied and realized volatilities (squared). However, even though variance swaps are designed to be mathematically delta neutral, they exhibit empirical market directionality (a mild positive equity market beta) because falling equity markets tend to coincide with rising volatility. As noted above, delta-hedged straddles were the best (but noisy) way to buy or sell volatility until variance swap contracts were created and their OTC markets became liquid. Traditionally, CCW has been more common than variance selling.

Volatility-selling returns are based on the simulated Merrill Lynch Equity Volatility Arbitrage Index since 1989, which tries to gain from the typically positive gap between market-implied volatility and subsequent realized volatility of the S&P 500 index’s Bloomberg ticker MLHFEV1 index. Chapter 15 also analyzes the simulated covered-call-writing indices and put-writing indices (BXM, BXY, PUT in Bloomberg) on the S&P 500 index from the Chicago Board of Exchange, dating back to the 1980s. These indices are not delta neutral so their returns reflect market-directional exposures as much as volatility exposures. Data on volatility-sorted stock portfolios are from Andrew Ang (Columbia University), David Blitz (Robeco), and Giuliano De Rossi (UBS). Importantly, no trading costs have been subtracted from any of the strategy style returns. B.3 FACTOR PROXIES Growth. Monthly change in the consensus forecast of U.S. real GDP growth over the next year, based on a survey by Consensus Economics.

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Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues by Alain Ruttiens

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.); and, more fundamentally, an option replication problem: as a matter of fact (see Sec-tion 12.1.5), despite the credit derivatives are labeled “swaps”, they are actually conditional swaps, that is, options products. Option valuation is based on the ability to build a riskless portfolio, combining some short position in underlying and some long position in a corresponding option. In the case of a credit derivative, it is impossible to take a short position on an underlying default risk, hence, no delta neutral mechanism, and the option theory is not grounded in this case, because such option is not replicable. As a consequence, speculative trading on credit derivatives is actually very different from speculation on most other instruments, because of the absence of a grounded fair or theoretical value to be compared with observed market prices. Figure 13.9 is a graph of an iTraxx premium: how to objectively appreciate to what extent should these prices be over- or under-estimated?

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The Crisis of Crowding: Quant Copycats, Ugly Models, and the New Crash Normal by Ludwig B. Chincarini

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LTCM believed that this was due to investor demand for long-term insurance protection and a low supply of these options in the marketplace. At the time, many retail investors demanded longer-term insurance protection for their investment portfolios, and thus drove up prices. LTCM traders had many ways to take advantage of their view. Overall, they tried to sell expensive long-term options and then dynamically hedge the unwanted risk with futures and short-term options. In option lingo, they wanted to be delta-neutral: focused only on the idea that equity index volatility would be lower than the level market prices implied, with no exposure to equity market price shifts. If volatility turned out to be lower than implied during the five years after the option changed hands, LTCM made a good profit. LTCM sold options that were priced as if volatility would be as high as 20% per year in the coming years. Historical data, however, suggested that volatility would be only 10 to 13%.