In a market with transaction costs, generally, there is no nontrivial portfolio that dominates a contingent claim. Therefore, in such a market, preferences have to be introduced in order to evaluate the prices of options. The main goal of this article is to quantify this dependence on preferences in the specific example of a European call option. This is achieved by using the utility function approach of Hodges and Neuberger together with an asymptotic analysis of partial differential equations. We are led to a nonlinear Black-Scholes equation with an adjusted volatility which is a function of the second derivative of the price itself. In this model, our attitude towards risk is summarized in one free parameter a which appears in the nonlinear Black-Scholes equation : we provide an upper bound for the probability of missing the hedge in terms of a and the magnitude of the proportional transaction cost which shows the connections between this parameter a and the risk.

In the presence of transaction costs the perfect option replication is impossible which invalidates the celebrated Black and Scholes (1973) model. In this chapter we consider some approaches to option pricing and hedging in the presence of transaction costs. The distinguishing feature of all these approaches is that the solution for the option price and hedging strategy is given by a nonlinear partial differential equation (PDE). We start with a review of the Leland (1985) approach which yields a nonlinear parabolic PDE for the option price, one of the first such in finance. Since the Leland's approach to option pricing has been criticized on different grounds, we present a justification of this approach and show how the performance of the Leland's hedging strategy can be improved. We extend the Leland's approach to cover the pricing and hedging of options on commodity futures contracts, as well as path-dependent and basket options. We also present examples of finite-difference schemes to solve some nonlinear PDEs. Then we proceed to the review of the most successful approach to option hedging with transaction costs, the utility-based approach pioneered by Hodges and Neuberger (1989). Judging against the best possible tradeoff between the risk and the costs of a hedging strategy, this approach seems to achieve excellent empirical performance. The asymptotic analysis of the option pricing and hedging in this approach reveals that the solution is also given by a nonlinear PDE. However, this approach has one major drawback that prevents the broad application of this approach in practice, namely, the lack of a closed-form solution. The numerical computations are cumbersome to implement and the calculations of the optimal hedging strategy are time consuming. Using the results of asymptotic analysis we suggest a simplified parameterized functional form of the optimal hedging strategy for either a single option or a portfolio of options and a method for finding the optimal parameters.

[Note: eBook now available; see Amazon author page for details.] Dr. Crack studied PhD-level option pricing at MIT and Harvard Business School, taught undergrad and MBA option pricing at Indiana University (winning many teaching awards), was an independent consultant to the New York Stock Exchange, worked as an asset management practitioner in London, and has traded options for 20+ years. This unique mix of learning, teaching, consulting, practice, and trading is reflected in every page. This revised 6th edition gives clear explanations of Black-Scholes option pricing theory, and discusses direct applications of the theory to trading. The presentation does not go far beyond basic Black-Scholes for three reasons: First, a novice need not go far beyond Black-Scholes to make money in the options markets; Second, all high-level option pricing theory is simply an extension of Black-Scholes; and Third, there already exist many books that look far beyond Black-Scholes without first laying the firm foundation given here. The trading advice does not go far beyond elementary call and put positions because more complex trades are simply combinations of these. UNIQUE SELLING POINTS -The basic intuition you need to trade options for the first time, or interview for an options job. -Honest advice about trading: there is no simple way to beat the markets, but if you have skill this advice can help make you money, and if you have no skill but still choose to trade, this advice can reduce your losses. -Full immersion treatment of transactions costs (T-costs). -Lessons from trading stated in simple terms. -Stylized facts about the markets (e.g., how to profit from reversals, when are T-costs highest/lowest during the trading day, implications of the market for corporate control, etc.). -How to apply European-style Black-Scholes pricing to the trading of American-style options. -Leverage through margin trading compared to leverage through options, including worked spreadsheet examples. -Black-Scholes pricing code for HP17B, HP19B, and HP12C. -Five accompanying Excel sheets: forecast T-costs for options using simple models; explore option sensitivities including the Greeks; compare stock trading to option trading; GameStop example; and, explore P(ever ITM). -Practitioner Bloomberg Terminal screenshots to aid learning. -Simple discussion of continuously-compounded returns. -Introduction to "paratrading" (trading stocks side-by-side with options). -Unique "regrets" treatment of early exercise decisions and trade-offs for American-style calls and puts. -Unique discussion of put-call parity and option pricing. -How to calculate Black-Scholes in your head in 10 seconds (also in Heard on The Street: Quantitative Questions from Wall Street Job Interviews). -Special attention to arithmetic Brownian motion with general pricing formulae and comparisons of Bachelier (1900) with Black-Scholes. -Careful attention to the impact of dividends in analytical American option pricing. -Dimensional analysis and the adequation formula (relating FX call and FX put prices through transformed Black-Scholes formulae). -Intuitive review of risk-neutral pricing/probabilities and how and why these are related to physical pricing/probabilities. -Careful distinction between the early Merton (non-risk-neutral) hedging-type argument and later Cox-Ross/Harrison-Kreps risk-neutral pricing -Simple discussion of Monte-Carlo methods in science and option pricing. -Simple interpretations of the Black-Scholes formula and PDE and implications for trading. -Careful discussion of conditional probabilities as they relate to Black-Scholes. -Intuitive treatment of high-level topics e.g., bond-numeraire interpretation of Black-Scholes (where N(d2) is P(ITM)) versus the stock-numeraire interpretation (where N(d1) is P(ITM)). -Introduction and discussion of the risk-neutral probability that a European-style call or put option is ever in the money during its life.

New Tools to Solve Your Option Pricing ProblemsFor nonlinear PDEs encountered in quantitative finance, advanced probabilistic methods are needed to address dimensionality issues. Written by two leaders in quantitative research-including Risk magazine's 2013 Quant of the Year-Nonlinear Option Pricing compares various numerical methods for solving hi

[Note: eBook now available; see Amazon author page for details.] THE AUTHOR: Dr. Crack studied PhD-level option pricing at MIT and Harvard Business School, taught undergrad and MBA option pricing at Indiana University (winning many teaching awards), was an independent consultant to the New York Stock Exchange, worked as an asset management practitioner in London, and has traded options for over 20 years. This unique mix of learning, teaching, consulting, practice, and trading is reflected in every page. This revised 5th edition gives clear explanations of Black-Scholes option pricing theory, and discusses direct applications of the theory to trading. The presentation does not go far beyond basic Black-Scholes for three reasons: First, a novice need not go far beyond Black-Scholes to make money in the options markets; Second, all high-level option pricing theory is simply an extension of Black-Scholes; and Third, there already exist many books that look far beyond Black-Scholes without first laying the firm foundation given here. The trading advice does not go far beyond elementary call and put positions because more complex trades are simply combinations of these. UNIQUE SELLING POINTS -The basic intuition you need to trade options for the first time, or interview for an options job. -Honest advice about trading: there is no simple way to beat the markets, but if you have skill this advice can help make you money, and if you have no skill but still choose to trade, this advice can reduce your losses. -Full immersion treatment of transactions costs (T-costs). -Lessons from trading stated in simple terms. -Stylized facts about the markets (e.g., how to profit from reversals, when are T-costs highest/lowest during the trading day, implications of the market for corporate control, etc.). -How to apply European-style Black-Scholes pricing to the trading of American-style options. -Leverage through margin trading compared to leverage through options, including worked spreadsheet example. -Black-Scholes pricing code for the HP17B, HP19B, and HP12C. -Three downloadable spreadsheets. One allows the user to forecast T-costs for option positions using simple models. Another allows the user to explore option sensitivities including the Greeks. -Practitioner Bloomberg Terminal screenshots to aid learning. -Simple discussion of continuously-compounded returns. -Introduction to "paratrading" (trading stocks side-by-side with options to generate additional profit). -Unique "regrets" treatment of early exercise decisions and trade-offs for American-style calls and puts. -Unique discussion of put-call parity and option pricing. -How to calculate Black-Scholes in your head in 10 seconds (also in Heard on The Street: Quantitative Questions from Wall Street Job Interviews). -Special attention to arithmetic Brownian motion with general pricing formulae and comparisons to Bachelier (1900) and Black-Scholes. -Careful attention to the impact of dividends in analytical American option pricing. -Dimensional analysis and the adequation formula (relating FX call and FX put prices through transformed Black-Scholes formulae). -Intuitive review of risk-neutral pricing/probabilities and how and why these are related to physical pricing/probabilities. -Careful distinction between the early Merton (non-risk-neutral) hedging-type argument and later Cox-Ross/Harrison-Kreps risk-neutral pricing -Simple discussion of Monte-Carlo methods in science and option pricing. -Simple interpretations of the Black-Scholes formula and PDE and implications for trading. -Careful discussion of conditional probabilities as they relate to Black-Scholes. -Intuitive treatment of high-level topics e.g., bond-numeraire interpretation of Black-Scholes (where N(d2) is P(ITM)) versus the stock-numeraire interpretation (where N(d1) is P(ITM)). -Introduction and discussion of the risk-neutral probability that a European-style call or put option is ever in the money during its life.

The traditional Black-Scholes theory on pricing and hedging of European call options has long been criticized for its oversimplified and unrealistic model assumptions. This dissertation investigates several existing modifications and extensions of the Black-Scholes model and proposes new data-driven approaches to both option pricing and hedging for real data. The semiparametric pricing approach initially proposed by Lai and Wong (2004) provides a first attempt to bridge the gap between model and market option prices. However, its application to the S & P 500 futures options is not a success, when the original additive regression splines are used for the nonparametric part of the pricing formula. Having found a strong autocorrelation in the time-series of the Black-Scholes pricing residuals, we propose a lag-1 correction for the Black-Scholes price, which essentially is a time-series modeling of the nonparametric part in the semiparametric approach. This simple but efficient time-series approach gives an outstanding pricing performance for S & P 500 futures options, even compared with the commonly practiced and favored implied volatility approaches. A major type of approaches to option hedging with proportional transaction costs is based on singular stochastic control problems that seek an optimal balance between the cost and the risk of hedging an option. We propose a data-driven rule-based strategy to connect the theoretical approaches with real-world applications. Similar to the optimal strategies in theory, the rule-based strategy can be characterized by a pair of buy/sell boundaries and a no-transaction region in between. A two-stage iterative procedure is provided for tuning the boundaries to a long period of option data. Comparing the rule-based strategy with several other existing hedging strategies, we obtain favorable results in both the simulation studies and the empirical study using the S & P 500 futures and futures options. Making use of a reverting pattern of the S & P 500 futures price, we refine the rule-based strategy by allowing hedging suspension at large jumps in futures price.