Dynamic Hedging: Managing Vanilla and Exotic Options

Taleb is one arrogant dude who loves flooding his books with archaic words which were last employed in the English Language by Geoffrey Chauncer. But alas, Dynamic Hedging is a strong advanced text which goes through many nuanced topics. For example, he makes some good points on managing option greeks. Some chapters I really enjoyed which are hugely important in practice that you don't learn in any classroom: soft American options, discrete delta vs continuous delta, fungibility. Just a warning that you might have to read over sections multiple times before you digest ideas. For example, for american options, you can tend to think of the early exercise having some sensitivity to interest rates (as rates go higher, it becomes more optimal to exercise puts and less optimal to exercise calls), so in some circumstances, the early exercise provision of american option is actually an option on rates. Also, every mathematician teaches delta as a continuous derivative d[option value]/d[spot], but really what's important is to the know the delta at discrete intervals since no one hedges continuously and also since in a real options book the greek sensitivities might flip or go through extreme changes over discrete intervals. Just some great material which makes you think hard. The structure of the book jumps over the place, but mainly Taleb is focused on options, volatility, and exotics. So not exactly a good book on vanilla rates or commodities for example. This text is certainly one I keep as a reference guide on my desk. As a sign of its value, everytime I read it, I do learn something new. I rated it highly based solely on the excellent and juicy material but the writing style is really horrible. Not for beginners but a great read for anyone interested in the deep details of trading derivatives.

By dynamically hedging an asset, its drift is neutralized. This is why the underlying asset’s risk premium (drift) does not show up in Black Scholes. The value depends only on replication cost, which is a function of volatility and risk free rate. Dynamic hedging makes every option path dependent. The same vol, beginning price, and ending price can produce different P&L depending on hedging frequency. Only continuous hedging avoids path dependence, but means significant transaction costs. Diversification to capture edge over time is often not possible because of externally imposed stopping times (leverage, risk managers, P&L absorbing barriers). By put call parity, whatever raises the price of the call must also paradoxically raise the price of the put, so preferences like bullishness can’t possibly impact price. Delta-neutral ITM calls should have same time value as delta-neutral OOTM put. If a call trades at a premium to the roll rate (financing rate less carry) because of high expected return to underlying, then one could sell the call and replicate it at the roll rate (an arbitrage). This shows that the ability to replicate a payoff negates its expected return from impacting its value. Despite its many assumptions, traders get Black-Scholes. It is better to improve on a simple and seasoned model than to operate with a more advanced but newer model. Less parameters is better. Hedging options with options mitigates the imperfections of Black Scholes. Real world hedging cannot be continuous. There is a tradeoff between variance of returns and transaction costs. The more Greeks one wishes to immunize, the greater the cost. More volatile parameters require more allowance for variance or cost. Derivatives with more or less convex second-order sensitivities will therefore be more or less costly to manage, requiring markups or markdowns (Black Scholes doesn’t allow for this). With long gamma use limit orders, with short gamma use stop orders. It’s better to be roughly hedged against a broad set of eventualities than exactly hedged against a narrow parameter. A very narrow hedge might be ok for very short term and for liquid markets. Fungibility is degree of specificity required to satisfy delivery obligations. More fungible is less specific. Fungibility reflects the feasibility of risk-neutral replication of a derivative security. Low fungibility makes arbitrage difficult. High fungibility leads to larger market, more liquidity, and easier pricing. Currencies are highly fungible. Commodities with specific delivery location tend not to be because of shipping considerations. Derivatives on highly fungible commodities can usually be arbitraged both ways on the timeline. Derivatives with low fungibility can generally only be arbitraged one way (buy spot, sell forward, store…difficult to short spot so can’t arbitrage back month discounts). Long term contracts on perishable commodities cannot be arbitraged through physical storage. Option arbitrage can only be performed if a risk-neutral forward market exists. Stacking is a short-term approximate hedging technique that uses a few liquid instruments. Sacrifices precision for ease. It’s common to target minimum variance. Front month equivalent would be extreme example. Black Scholes Greeks often need to be discretized and considered together rather than in isolation, especially for portfolios. Otherwise hedge ratios will deviate from Greeks. When one factor changes, others generally do too. Shadow measure of a Greek is when one considers the sensitivity not in isolation, but includes potential impact in other Greek terms as well. Numerical methods (full reval) naturally achieves this. Using discrete shock (both up and down) with full revaluation for delta naturally incorporates gamma and higher order derivatives. Delta by itself is bad for risk management (imagine long call vs short call of same delta…one has infinite loss potential, the other has loss bounded by premium). Higher order Greeks can be material for a portfolio of options, which is a problem for value at risk. ATM option gamma is highest near expiration, while OOTM option gamma is highest when far from expiration. This time dependence suggests gamma hedges can range from immediate and precise to general and rough. Netting up-gamma and down-gamma (full reval) is misleading because of different signs. A risk reversal is when the up-gamma and down-gamma have different signs for some increment of price change (i.e. sign of gamma changes at some price, so third moment; price goes from convex function to concave function of underlying). With a calendar spread (long one maturity, short another), a gamma basis may exist because of variance differential between the maturities. Must adjust gammas for different maturities by finding betas between them. When discretizing the shock for portfolio Vega, one should not assume a parallel change in vol. Shorter maturities tend to be more volatile. Vega should be weighted by maturity (higher weighting for more volatile maturities). Vol exhibits strong mean reversion. Forward vol can be computed similar as forward rates, and Vega of options can then be expressed as a function of all forward vols leading to its expiration, rather than just the vol corresponding to its expiration time. Local vol generalizes forward vol. If interest rates are zero and forward vol equals expected future spot vol, then theta should be zero. If not, Theta should be modified to reflect change in vol for change in time (i.e., shadow theta). Interest cost of holding premium in theta is eliminated if trader funds themself. That is, the interest rate drives cost of borrowing but also discounting, so higher rate means more borrowing cost but lower price, offsetting. Rho should be split into components. Rhop is risk associated with financing of the option premium. Rho1 is risk associated with base rate (which would include rhop). Rho2 is risk associated with risk neutral curve (dividend payout for equity options, foreign rate for fx options, etc.). Rho assumes parallel shift. Modified rho applies a one-factor yield curve model to capture the relative variance of each maturity to get true sensitivity to rates. Weighting maturities based on variance improves rho. First and second order exposures of derivative prices to underlying variables are not comparable when the variables have different volatilities. OOTM options “bleed” delta and gamma as time passes. That is, the sensitivities fall. ITM is opposite. This effect is greater for shorter-dated options. Gamma bleed is third derivative with respect to time. Bleed is a function of spot, which makes fourth derivative important. Change in delta and gamma for change in vol can also cause bleed or reverse bleed. Odd orders of sensitivity are about symmetry, while even orders of sensitivity are about convexity. Lock delta is a measure of delta risk at extreme boundaries. Shocking to zero and infinity zeroes out all other Greeks and reveals potentially hidden behavior of positions. It ignores dynamic hedging, so it’s only what happens when a trader doesn’t have time to hedge. Nonparametric stress tests can be better than parametric. Hard deltas (cash or futures) don’t vanish with large moves in the underlying. Soft deltas (options) vanish after big moves. In self-financing strategies, trader funds by paying risk free rate on negative balances and earning on positive balances. There are users and manufacturers of derivatives. Users are concerned with terminal value. Manufacturers engage in dynamic hedging, which alters the product. First order hedges make calls and puts identical to manufacturer, meaning strike and expiration matter more. Sticky strike refers to stability of underlying spot price around a strike because of large open interest and resulting bid-ask accumulation. At expiration, non-delta-hedgers must buy their deltas below and sell above, creating stickiness. This is similar to government-imposed barriers to spot prices, such as currency pegging and commodity support. One expects volatility to drop as a price approaches a barrier, but in reality barriers lead to political tension and sometimes higher volatility of the price. To pass this through a model, some variable must adjust (i.e., stabilizing exchange rate leads to more rate vol). Expiration pin risk is the expiration variance for an options position due to the absence of timely information about the outcome of the option assignment lottery. Exercise decisions come after closing expiry price. When the price is near the strike at expiration, it is unclear how to hedge until final settlement is printed. Pin risk causes put call parity (C-P=F) not to hold for non-cash-settled listed options. Forward delta for European option is the equivalent cash position with the same delivery date as the underlying asset. It is the cash delta adjusted to be in terms of forwards (fit to the exact period in the future by transforming potential future exposure through delta of forward and discounting). Mirror image rule using put-call parity says P+xF=(C-F)+xF=C-(1-x)F where x is % and F is forward. So if put has forward delta 0.3, call has forward delta 0.7. Vol for OOTM put should equal that of corresponding ITM call with same strike and expiration. Omega is the duration or expected life of an American or barrier option. If long an American call, one must consider the financing and carry costs (opportunity cost from not exercising) against costs after exercising and replicating. It’s optimal to exercise American option when financing cost of premium (from reinvestment) exceeds time value or the cost of delta hedging. When no carry costs, the option is “soft American” and acts more like a European outside of deep ITM. With carry costs, option is “hard American” i.e., it acts like an American option. Owning the underlying asset provides collateral that relieves funding costs (ignoring carry). Options on futures fall under soft exercise rules; while options on cash assets fall under hard exercise rules. Primary risks drive bulk of variance of pnl e.g., for equity derivatives, underlying equity is primary risk while interest rates are secondary. Primary risks are focus of hedging. Secondary risks can be cleaned up with less frequent residual hedges. Proprietary trading worked better at funds than banks because of diversification effects at the latter. When a bank manager of prop desks has traders running correlated or offsetting strategies, one’s win is often another’s loss. On net, the manager cannot outperform. Such problems did not exist at funds where strategies were built to be independent. Bank prop trading underperformed and slowly disappeared. Market makers are needed because of imperfect synchronization between buyers and sellers. Bid-ask spreads are a function of competition between market makers. Risk from market movement dominates risk from mispricing, meaning hedging strategy is more important than pricing methods. Profitable strategies get mimicked, causing markets to become one-sided (liquidity dries up on one side of the trade). Adverse selection problems occur where informed traders are the ones trading with market makers. This is particularly acute during shocks. Market makers pick up the spread without much risk during normal times but tend to get hosed during shocks, making them effectively short vol. The competing trading dichotomy is value trading (inventory positions that have positive expected return) vs greater fool theory (inventory positions in greater demand). Volcker rule relegates banks to the latter. Trading pnl tends to follow arcsin law (long streaks of losses or gains) as opposed to just being 50/50 gains losses from month to month. Slippage (execution price vs mid-market price) is main reason funds limit assets under management. Lack of liquidity can thicken tails of return distribution. Trigger points (for barrier options or stop-losses, for example) can encourage one-directional bulk trades, breaking Markovian property of asset prices. Perfect hedges can still lead to ruin if one side requires intermediate funding while the other does not produce necessary cash flows. Deep pockets avoid such squeezes and allow for purchase of distressed inventory from others. Predilection for short-term profits makes the efficient frontier (modern portfolio theory) unenforceable, as returns have path-dependent restrictions (must survive to get terminal payoff). Arbitrage using purely modeled prices is often a mirage, as it then depends on proprietary pricing methods and no agreed upon benchmark. Failure to account for counterparty credit risk or funding costs can also give the illusion of arbitrage in accounting systems for bilateral derivatives. Numeraire has direct impact on hedging. If profits are in foreign currency, they must be hedged. If profits are in native currency, they don’t. If one expects 20% increase in one currency per another, this corresponds to a different decrease after inverting the exchange rate. An exchange rate approaching zero in terms of one currency is infinity when inverted. Flipping numeraire doesn’t change option price (call on one currency is put on other) but does change delta. Parametric VaR is useful for short term hedging and works ok with the body of the distribution. Critiques of parametric VaR. Markets are heteroskedastic, so vol doesn’t scale with square root of time. Correlations are stochastic and often approach 1 during times of stress, so joint tails are not easily modeled. No explicit consideration of liquidity outside parameterization, which can dominate market risk in many markets. If everyone uses VaR, it becomes procyclical, raising correlations and volatilities. Computation requires all positions in a single portfolio, and large portfolios have huge covariance matrices where rounding and estimation errors cumulate. Parametrization requires arbitrary choices around realized vs implied vol and time periods. Derivatives with a multitude of first and second order Greeks cannot be simplified to a single Greek. Stochastic dominance principle can be used to value options using just terms and conditions. Strike relationship (straddle is more valuable than a strangle). Long butterflies have nonnegative value. Time value increases with time. Higher exercise frequency is more valuable than less. No barrier is more valuable than barrier. Every covariance matrix must be positive definite (positive eigenvalues). Convexity with respect to some parameter commands more value than lack of convexity. Correlation describes behavior, but the magnitude of the relationship depends on relative volatilities. Heteroskedasticity implies stochastic correlations. Bachelier’s assumption of constant dollar volatility leads to possible negative values but might better represent volatility over medium-sized increments. Financial asset prices might be said to be arithmetic in the short term and geometric in the long run. Since returns compound, higher vol raises compounding, increasing divergence between arithmetic and geometric process. Long call on bond is equivalent to long put on yields. If one is lognormal, the other cannot be. Trinomial trees allow for changes in vol through the middle node, whereas binomial trees often fail to recombine when incorporating vol skew. Brownian motion has nonzero local time, which is how much time is spent repeatedly visiting a certain price level. Attempts to sell the option for the premium and then dynamically rebalance at the strike runs potentially unlimited losses no matter the price change threshold since smaller thresholds will just be triggered more often. There’s also an execution lag, meaning the rebalancing will never perfectly sync the price changes. This residual mismatch is the gamma risk and ultimately prevents hedging for cheaper than the premium. This motivates Ito calculus. If Brownian motion were smooth then hedging would be perfect. Quadratic variation produces the nonzero local time that prevents this. Trading long or short a derivative exposes one to direction. Long one tenor and short another exposes one to slope. Carry is the cost or benefit of holding an asset through time. Storage costs, insurance, financing, for commodities. Coupon less financing costs for bonds. A commodity in backwardation can have positive carry from roll yield. Total return = carry + capital gain/loss. Convergence is a similar concept but related to expected price movements. Splitting Greeks into horizon groups based on cash flow date only works when dates are well known. Vol skew is caused by vol of vol (kurtosis). If asset price and vol are negatively correlated, one would expect ITM options to have more skew than OOTM. Skew of return distribution expresses correlation between returns and squared returns (which is vol when expected return is zero). Lower vol means higher skew in a trending market. Infinite variance can disguise as high sample vol of vol. Getting density from second derivative of options price with respect to strike (Breeden Liechtenberger) is suspect because it ignores the path dependency that arises from changes in vol. When bond prices and equities go up, everyone is happy because issuers aren’t consciously short. No one thinks in terms of having to buy back one’s debt or equity at an inopportune time. Rather, issuers are also happy because their future issuance will be on better terms. A miracle of modern finance. Currencies dominated by commercial exchange tend to have high correlation between price and rates. Rates are raised to defend the currency or avoid capital flights. Speculative currencies exhibit independence between price and rates. A binary is long gamma (convex) and losing theta when OOTM and short gamma (concave) and earning theta when ITM. ATM binary acts like a forward. Approaching maturity, binary price approaches Dirac delta at strike (infinite delta). Shape of vol series for binary is similar to a narrow call spread, but different in scale. Binary price is lim h->0 (C(K+h)-C(K))/h which notice is derivative of call wrt strike. In continuous time finance replication would require infinite call spread quantity for infinitely small strike difference (h), but with discrete price ticks the replication is a finite quantity. Replicating call spread should narrow as approach expiration. If there is vol skew in the call spread (i.e. vol is function of h), it impacts the replication of the binary option. To be priced as a fair bet, skew to one side of distribution must be offset by more observations to the other. This changes value of binary option even though magnitude of return skew is irrelevant to payoff. Delta accounts for payoff and frequency, while binary only accounts for frequency. Gap delta is the delta to a discrete jump rather than an infinitesimal move. It shows up as the difference in deltas near a barrier for a binary option. Literally how much delta moves at a discontinuity. Hedging delta of an American binary requires growing short on underlying as price approaches strike, but everything must be purchased back if strike is hit. Unwinding this gap delta is challenging (pin risk). The same applies to barrier options. Avoid hedging discontinuous exposure with a continuous one. Unwinding a derivative position with jump risk means subsequent market movement as hedges are rebalanced and stop-losses triggered. A mined market is one with many barriers in place with much whipping around of price near these barriers. Execution costs are worse in such markets. A cleared marked has no such orders in it. Risk reports are often viewed as scenarios in which the portfolio is untouched. Such a perspective is unhelpful when positions with jumps are involved, as with binaries and barriers. E.g. there is execution risk with a stop-loss.

Excellent! This book is about how to avoid the absorbing barrier of bankruptcy! Taleb does an excellent job of covering the practical and mathematical (in a practical way) properties of options trading. There are no recommendations about volatility spreads but you will learn more about how vol, the Greeks, and time impact your P/L than you thought possible. The most interesting things discussed were the practical uses and impacts of unexpected market shifts and the consequences to the trader (liquidity holes, huge change in pricing and the impact on the traders portfolio, the limitations of Black scholes and delta hedges... when and how to use them). I bet you have never thought about the 7th derivative of an option contract and how it affects you P/L. Taleb discusses how to deal with all of these issues and much more. Build an anti-fragile portfolio. Excellent guide on the properties of options and their impact on your portfolio! Thanks Nassim!

Very academic, but extremely insightful. I would recommend this book to anyone looking to get past just the basics. This book assumes that you are already familiar with the basic options spreads (i.e., Iron Condors, Vertical Spreads, etc.) and provides many equations for creating Delta neutral hedges and managing your risk. He then identifies the short-comings of each equation. Most of his examples are accompanied by charts to help his explanations. As an intermediate trader, this book helped bring my options trading to the next level. Although it does not cover many of the derivatives that caused the collapse in 2008, such as Credit Default Swaps or Interest Rate Swaps. Overall, a very good book and I would recommend it to any beginning trader that is looking to advance.

This isn't so much a review as an inquiry - I'm still working through this book, but I've noticed that it is replete with algebraic errors. Taleb doesn't always go through every step of his calculations, so they're difficult to follow. Does anyone else notice these errors or am I getting caught in the algebra? An example is on page 38 towards the top: He says that 12,500,000/.985 is 1,269,000. Even with the rounding (shaving off the 355), I'd say he's off by a factor of 10. Shouldn't the answer by 12,690,000? To confirm this, in the next paragraph he writes that a 1 cent rise in the futures price will yield $125,000 in profits on the future (12,500,000*.01) or $126,900 on the forward hedge. Intuitively, it seems that to get $126,900, you could divide $125,000 by .985 or multiply 12,690,000 (the answer you should've previously arrived at) by .01. There are other errors like this throughout the book - anyone else notice them?

The book is a difficult read the first time around while you get used to Taleb's style. It is because his style is so refreshingly different from everything published on the subject before and 10 years later I don't see anything that comes close. If you are a beginner, get the basic option math down (Hull) then read Taleb, and be patient. I have the original edition and have read it several times from cover to cover and I don't see the "numerous" errors several reviewers are complaining about. There are a few very obvious typos towards the end of the book, but all the conclusions and intuition is right. If you think it's wrong, it's most likely you, not the book. Go reread it and think about it, you will get it and you will want to thank Taleb for making you think. I am a vol trader and I use this book to filter out idiots. If this book ticks you off, you don't have a makeup of a trader. Do yourself a favor and find something else to do. People who give this book a negative review are most frequently quants. His criticism towards guants has a purpose - it is to make you as an impressionable young trader aware that a person with a PhD does not have all the answers, and that you should not be intimidated and use your own head. If this ticks quants off then so be it. The book is not for them.

This is a fairly advanced Text on options. Taleb jumps right into option Greeks from the introduction. It is mostly oriented towards institutional traders and as a person with exposure just to equity options it goes right above me. So I had to go back and forth with Wiki to understand what he is talking about. You could see traces of Fooled by Randomness and Black Swan as he deals with various risk issues. Little bit pricey but goes to show even professional traders use broken models to invest millions of client money. My personal experience is Fundamentals of an equity affect its option pricing more than Vega/Theta/Row. Delta and Gama are somewhat in play. Taleb calls them liquidity holes but mostly it is Fundamental analysis. If a company has a massive debt restructuring coming up then options will behave differently before and after the event. Even Beta changes after the event.

A great book from the best professional in the options trading I know. If you finished reading books like "Options, futures and other derivatives" it's a good time to go up one level and read the Dynamic Hedging by N. Taleb. It requires more effort to understand the author's intentions, but it totally worth it.

Muito didático e claro. Oferece muita informação de diversos cenários. Vale cada centavo.

Still a very good reference for volatility trading 25 years after its publication. Perhaps it needs some updating on the statistical analysis of volatility, but you can find those updates in more recent publications of the author.

I All I can say is that this is the best Start for anybody wanting to trade Exotic Options At a retail Level, but also wishing to start a career in any global Financial Center. This Book could change your life I.e ' you will Win or you Will Lose' playing this Game. Good Luck Everybody the Game is On.

I did not enjoy that book as much when I was still unfamiliar with the field: maybe it is a bit disorganised. Some of it is however very accessible to beginners. When I go back to it I always appreciate it a bit more than the previous time. It deals with important subjects in derivatives trading that are rarely presented anywhere else.

That this is a pirate printed book selling on the market for 5and a half k bucks