Options and Derivatives

An option is a financial contract that gives the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price (the strike) on or before a specified date (the expiry). A call option gives the right to buy; a put option gives the right to sell. Options are a subset of the broader class of derivatives, which includes forwards, futures, swaps, and any other contract whose value derives from an underlying asset. The global derivatives market is measured in hundreds of trillions of dollars of notional outstanding, with options accounting for a small fraction of the notional but a disproportionate fraction of the trading volume and the practitioner employment (Hull, 2017).

The modern theory of derivative pricing begins with the Black-Scholes-Merton model (Black & Scholes, 1973; Merton, 1973), for which Myron Scholes and Robert Merton received the 1997 Nobel Memorial Prize in Economic Sciences. The central result is the no-arbitrage pricing formula: in a continuous-time frictionless market, the price of a European option is determined by the current price of the underlying, the strike, the time to expiry, the risk-free rate, and the volatility of the underlying but not by the expected return of the underlying. The result is the famous risk-neutral pricing principle: derivatives can be priced by computing the expected payoff under the risk-neutral measure (the measure under which the discounted underlying is a martingale) and discounting at the risk-free rate.

The options market is divided by exchange-traded options (vanilla calls and puts, listed on CBOE, CME, Eurex, and other exchanges; standardised strikes and expiries; cleared through a central counterparty) and over-the-counter (OTC) options (customised strikes, expiries, and underlyings; traded bilaterally between counterparties; cleared through swap execution facilities or bilateral credit support annexes). The two markets have different pricing conventions, different liquidity profiles, and different standard practices. The exchange-traded market is dominated by retail and institutional users of vanilla options; the OTC market is dominated by dealer pricing of exotic options and structured products (Hull, 2017).

The volatility surface is the central object of modern options pricing. It is the function σ(K, T) that gives the Black-Scholes implied volatility for each strike K and maturity T. In the Black-Scholes model, volatility is constant across strikes and maturities; in the real market, the implied volatility varies with both. The pattern is characteristic: implied volatility is higher for low-strike puts (downside protection is in demand) than for high-strike calls, and the pattern is more pronounced for short maturities than for long. The graph of implied volatility against strike for a fixed maturity is called the volatility smile (or smirk, for the asymmetric pattern observed in equity index options); the graph against maturity is called the term structure of volatility.

The dominant models for the volatility surface are: local volatility (Dupire, 1994; Derman & Kani, 1994), in which the instantaneous volatility σ(S, t) is a deterministic function of the underlying price and time, calibrated to fit the observed market smile exactly; stochastic volatility (Heston, 1993), in which the volatility itself follows a stochastic process, producing a smile endogenously; and SABR (Hagan, Kumar, Lesniewski & Woodward, 2002), in which the volatility follows a CEV-like process with explicit correlation to the underlying, producing a closed-form asymptotic approximation to the smile that has become the industry standard for interest rate derivatives.

The volatility surface is recalibrated daily at every major dealer, with the calibration feeding both the pricing of new options and the risk management of the existing options book. The principal risk in a volatility-surface model is the model risk: the smile produced by the model is a fit to the observed market, but the dynamics of the smile (how the smile moves as the underlying moves) are determined by the model's assumptions about the underlying dynamics. The dominant practitioner reference for volatility surface calibration is Gatheral (2006), which provides the standard treatment of the surface and the standard critique of the local volatility model's dynamics (Gatheral, 2006).

The Greeks are the first- and second-order sensitivities of the option price to the underlying parameters. The dominant Greeks are: delta (∂V/∂S, the sensitivity to the underlying), gamma (∂²V/∂S², the convexity), vega (∂V/∂σ, the sensitivity to implied volatility), theta (∂V/∂t, the time decay), and rho (∂V/∂r, the sensitivity to the risk-free rate). For multi-asset or path-dependent options, the Greeks are extended to include the cross-greeks (the sensitivity of one option's price to another asset's price or parameter).

The Greeks are the operational language of options risk management. The delta is the position in the underlying that the options book is equivalent to; the gamma is the change in that position as the underlying moves; the vega is the change in the position as the implied volatility changes. A typical options desk at a major dealer manages a book with a delta-neutral position, a gamma limit, a vega limit, and a theta budget, and the daily risk report is organised by the Greeks. The standard reference is Hull (2017), which provides the complete treatment of the Greeks for the standard option types.

The Greeks are also the basis of the most common options hedging strategy: delta hedging, in which the options position is hedged by a position in the underlying that exactly offsets the delta. The result of delta hedging is a position whose value is approximately independent of small moves in the underlying, but whose value depends on the realised volatility. The latter dependence is the source of the famous “p-vol vs. r-vol” distinction: the option is sold at the implied volatility p, and the hedger realises the actual volatility r over the hedging period. The hedger profits if r < p, and loses if r > p. The practice of delta hedging, and the design of optimal delta-hedging strategies in the presence of transaction costs, is one of the dominant research areas in derivatives (Hull, 2017).

Exotic options are options with payoffs that depend on the path of the underlying, not just its terminal value. The dominant exotic option types are: barrier options (which are knocked in or out if the underlying crosses a barrier), Asian options (with payoffs depending on the average price of the underlying over the option life), lookback options (with payoffs depending on the maximum or minimum of the underlying), basket options (with payoffs depending on a basket of underlyings), and Bermudan options (with early-exercise features at multiple dates). Each of these is a major product category with its own pricing literature and its own standard market practice.

The dominant pricing methods for exotic options are: closed-form solutions where they exist (for a small class of payoffs, mostly in the Black-Scholes model), numerical solutions of the pricing PDE (finite difference methods), Monte Carlo simulation (the dominant method for high-dimensional and path-dependent payoffs), and approximate analytical methods (the dominant method for fast pricing in real time, where the closed-form solution does not exist but an approximation is acceptable). The reference for finite-difference methods is Wilmott, Howison & Dewynne (1995); the reference for Monte Carlo methods in finance is Glasserman (2003).

The dominant risk in exotic options pricing is the correlation between the underlying factors, particularly for multi-asset or path-dependent payoffs. The correlation risk is typically the largest single source of pricing error, and the calibration of the correlation surface is one of the principal quantitative challenges in a multi-asset options book. The dominant practitioner reference for exotic options pricing and risk is the combination of Gatheral (2006) for the volatility surface, Glasserman (2003) for the Monte Carlo methods, and Wilmott, Howison & Dewynne (1995) for the finite-difference methods.

Interest rate derivatives are derivatives on interest rate-sensitive underlyings: forward rate agreements, interest rate swaps, swaptions, caps and floors, bond options, and the entire class of structured interest rate products. The interest rate derivatives market is the largest segment of the global derivatives market, with interest rate swaps alone accounting for over $300 trillion of notional outstanding. The dominant pricing model is the swap rate curve plus the swaption volatility cube; the dominant risk model is the sensitivity-based delta-gamma-vega framework with the addition of key rate durations and partial durations for the underlying rates.

The dominant volatility model for interest rate derivatives is the swaption volatility cube: a three-dimensional surface giving the Black implied volatility for each swaption as a function of the swap tenor, the option expiry, and the strike relative to the forward swap rate. The standard practitioner reference is Hagan, Kumar, Lesniewski & Woodward (2002) the SABR model which provides a closed-form asymptotic approximation to the volatility cube that is the industry standard for swaption pricing, calibration, and risk management. The standard reference for the broader interest rate derivatives market is Brigo & Mercurio (2006).

The dominant risk in interest rate derivatives is the yield curve risk: the sensitivity of the derivatives book to changes in the level, slope, and curvature of the yield curve. The standard decomposition is key rate durations (the sensitivity of the book to a 1bp move in each of the key rates on the yield curve) and partial durations (the sensitivity to a 1bp move in any specific maturity on the curve). The principal practitioner risk is the non-parallel yield curve shift: a parallel shift is rare in practice, and the typical interest rate move is a steepening, a flattening, or a butterfly. The risk model must capture these non-parallel shifts explicitly (Hull, 2017).

Credit derivatives are derivatives whose value depends on the credit quality of a reference entity. The dominant credit derivative is the credit default swap (CDS), which is insurance on the default of the reference entity. The buyer of the CDS pays a periodic premium; the seller pays the notional on default. The CDS market is the standard market for credit risk transfer, with single-name CDS, index CDS (the most important is the CDX.NA.IG in the US and the iTraxx in Europe), and tranche CDS all trading actively. The dominant pricing reference is O'Kane (2008).

The dominant credit pricing models are: the reduced-form model (Jarrow & Turnbull, 1995), in which default is a Poisson process with intensity calibrated to the CDS spread; the structural model (Merton, 1974), in which default is the first passage of the firm's assets through a default boundary inferred from the firm's capital structure; and the copula model (Li, 2000), which is the dominant model for the pricing of portfolio credit derivatives such as CDO tranches. The copula model was the subject of intense criticism after the 2008 financial crisis, in which the copula's tail dependence assumption was shown to be a poor model of actual default clustering.

The dominant credit risk is the correlation between the defaults of different reference entities, which is the principal input to the copula model for portfolio credit derivatives. The post-2008 literature has substantially revised the standard models: the base correlation model has been replaced by the base correlation skew model, the implied correlation model has been developed, and the regulatory framework for the standardised approach to counterparty credit risk (SA-CCR) has been introduced. The current state of the field is one of cautious use of the copula model for pricing, with substantial reliance on the standardised approach for capital (Hull, 2017).

Hedging in practice is the daily operational discipline of managing the Greeks of an options book. The standard hedging cycle is: at the start of the day, the desk runs the Greeks report (delta, gamma, vega, theta, rho) on the existing book; the desk identifies the Greeks that are out of policy (e.g., the book has a long gamma position that exceeds the gamma limit); the desk executes the hedge (typically by trading the underlying, by trading related options, or by trading a correlated instrument); at the end of the day, the Greeks report is re-run to confirm that the hedge has brought the Greeks within policy.

The standard hedge instruments are: the underlying (for delta and gamma), options on the same underlying (for vega and gamma), and correlated instruments (for cross-gamma and cross-vega). The choice of hedge instrument is a trade-off between the cost of the hedge (the bid-ask spread of the hedge instrument, the market impact of the hedge trade) and the precision of the hedge (how well the hedge instrument matches the desired Greek). The dominant practitioner reference is Hull (2017), which provides the complete treatment of the hedging strategies for the standard option types.

The standard risk in a hedging operation is the slippage of the hedge. A delta hedge that is supposed to be neutral at the underlying price at the time of the hedge may not be neutral at the underlying price a few minutes later, due to the time elapsed in the hedging operation. The slippage is the source of the “p-vol vs. r-vol” distinction: the hedger is selling p-volatility (the implied) but realising r-volatility (the actual), and the difference is the source of the hedger's P&L. The dominant research area in hedging is the design of optimal hedging strategies in the presence of transaction costs, market impact, and the discrete-time nature of the hedging operation. The standard reference is Almgren & Chriss (2000) (Hull, 2017).

The honest limits of options pricing are the same as the honest limits of any model-based discipline. Every options pricing model is a model, with explicit assumptions, and the assumptions are sometimes wrong. The principal assumption failure is the dynamics of the underlying: the Black-Scholes model assumes geometric Brownian motion, the Heston model assumes mean-reverting volatility, the SABR model assumes a CEV-like process, and each of these is an approximation. The standard response is to use a model-implied volatility, but the model-implied volatility is itself a model fit, and the model's dynamics are not guaranteed to match the realised dynamics.

The second limit is the volatility surface. The volatility surface is the central object of options pricing, and the standard models for the volatility surface are: local volatility, which fits the surface exactly but produces implausible dynamics; stochastic volatility, which produces plausible dynamics but cannot fit the surface exactly; and SABR, which is a closed-form approximation that fits the surface approximately. Each of these is a trade-off, and each has characteristic failure modes. The dominant research area in volatility surface modelling is the design of models that combine the fitting accuracy of local volatility with the dynamic plausibility of stochastic volatility (Gatheral, 2006).

The third limit is the human. Options pricing is a discipline of judgement. The pricing model's output is the input to the trader's decision, not the decision itself. The most common single source of major losses in the options book is a known model risk that was not hedged, a known position that was not monitored, or a known model that was overridden. The honest practitioner treats the pricing model as a tool, not as an oracle, and uses the model to make better decisions, not to replace the decision (Hull, 2017).