Option Delta Research Articles

Tighter ‘uniform bounds for Black–Scholes implied volatility’ and the applications to root-finding

Using the option delta systematically, we derive tighter lower and upper bounds of the Black–Scholes implied volatility than those in Tehranchi (2016) [11]. As an application, we propose a Newton–Raphson algorithm on the log price that converges rapidly for all price ranges when using a new lower bound as an initial guess. Our new algorithm is a better alternative to the widely used naive Newton–Raphson algorithm, whose convergence is slow for extreme option prices.

Valuation and hedging of cryptocurrency inverse options

Currently, the most liquidly traded options on the crypto underlying are the so-called inverse options. An inverse option contract is quoted and traded in the units of the underlying cryptocurrency. The main economic reason for the popularity of inverse contracts in the crypto exchanges (such as Deribit) is that inverse contracts enable traders to operate without maintaining fiat cash accounts. For the theoretical part, we show that inverse options are just regular vanilla options considered under the martingale measure using the forward of the underlying as the numéraire. This measure requires an adjustment to option delta. For the empirical part, we use Deribit options data of past five years to backtest delta-hedged option strategies. We introduce USD and Coin accounting of trading Profit&Loss (P&L) which is important for designing strategies in crypto options. We show empirically that USD and Coin accounting rules are equivalent when performance is measured is Coin and USD units, respectively. We establish that the risk-premia observed in options on Deribit is negative and significant so that strategies selling volatility are expected to generate positive risk-adjusted performance in the long-term.

Long-term option pricing with a lower reflecting barrier

AbstractThis paper considers the pricing of long-term options on assets such as housing, where either government intervention or the economic nature of the asset limits large falls in prices. The observed asset price is modelled by a geometric Brownian motion (“the notional price”) reflected at a lower barrier. The resulting observed price has standard dynamics but with localised intervention at the barrier, which allows arbitrage with interim losses; this is funded by the government’s unlimited powers of intervention, and its exploitation is subject to credit constraints. Despite the lack of an equivalent martingale measure for the observed price, options on this price can be expressed as compound options on the arbitrage-free notional price, to which standard risk-neutral arguments can be applied. Because option deltas tend to zero when the observed price approaches the barrier, hedging with the observed price gives the same results as hedging with the notional price and so exactly replicates option payoffs. Hedging schemes are not unique, with the cheapest scheme for any derivative being the one which best exploits the interventions at the barrier. The price of a put is clear: direct replication has a lower initial cost than synthetic replication, and the replication portfolio always has positive value. The price of a call is ambiguous: synthetic replication has a lower initial cost than direct replication, but the replication portfolio may give interim losses. So the preferred replication strategy (and hence price) of a call depends on what margin payments need to be made on these losses.

Arbitrage-free smile construction on FX option markets using Garman-Kohlhagen deltas and implied volatilities

This paper addresses arbitrage-free FX smile construction from near-term implied volatility dynamics proposed by Carr (J Financ Econ, 120(1), 1–20, 2016). The approach is directly applicable to FX option market conventions. Prices of market benchmark contracts (risk reversals and butterflies) are identified as the roots of a cubic polynomial and ATM-volatility can be matched by construction. Implied volatilities are computed with respect to (non-premium adjusted) option deltas. The approach is compared to the Vanna Volga Approach, which does not guarantee arbitrage-free prices. An empirical application to a normal and a stress scenario demonstrates that arbitrage-free implied volatilities coincide with those from the Vanna Volga Approach when prices are interpolated between the Delta25-call and Delta25-put options. Differences are observed when implied volatilities are extrapolated to the wings. Empirically, these differences are particularly relevant in a stress scenario during the Coronavirus crises (2020).

Contract rescission in the real estate presale market

AbstractThis study documents that over 10% of the presale contracts in the Hong Kong housing market between 1996 and 2014 were rescinded, resulting in a loss of HKD 436.67 million per year. We then investigate potential determinants of contracts rescission from a novel perspective of option theory. We find out‐of‐the‐money presale contracts (with market price being lower than the outstanding payment at settlement) have a 12.2% higher rescission rate. The rescission rate is also higher when presale homebuyers bear more of the price risk as proxied by option delta and time‐induced risk as proxied by time to maturity. Moreover, we find rescission rates drop significantly after the Hong Kong government's housing market macroprudential measures. Our findings shed light on understanding the mechanism of presale contracts rescission, homebuyers’ strategic default behavior, and the role of housing market regulation in mitigating rescissions.

What Incentivizes Managers to Engage in Financial Misreporting? An Analysis of Stock Delta, Option Delta, and Vega

Previous studies have provided mixed evidence on the relation between managers’ equity incentives and financial misstatements. These studies often measure equity incentives with the sensitivity of a manager’s stock and option portfolio to changes in equity price (i.e., portfolio delta) and/or the sensitivity of her option portfolio to changes in equity risk (i.e., vega). We hypothesize and find that the mixed results are attributable to the measurement error in portfolio delta because option compensation causes a manager’s incentive structure to be convex while stock compensation makes it linear. Splitting the portfolio delta into stock delta and option delta, we provide evidence that option delta has a dominating effect over stock delta and vega in explaining the likelihood and extent of financial misstatements. We also find that the positive association between option delta and misreporting variables is mainly driven by the effect of CEO’s option delta rather than that of CFO’s option delta. Our findings contribute to the literature by revealing the differential effects that stock delta, option delta, and vega have on providing managers with incentives for financial misreporting.

Speed-up credit exposure calculations for pricing and risk management

We introduce a new method to calculate the credit exposure of European and path-dependent options. The proposed method is able to calculate accurate expected exposure and potential future exposure profiles under the risk-neutral and the real-world measure. A key advantage is that it delivers an accuracy comparable to a full re-evaluation and at the same time it is faster than a regression-based method. The core of the approach is solving a dynamic programming problem by function approximation. This yields a closed-form approximation along the paths together with the option's delta and gamma. The simple structure allows for highly efficient evaluation of the exposures, even for a large number of simulated paths. The approach is flexible in the model choice, payoff profiles and asset classes. We validate the accuracy of the method numerically for three different equity products and a Bermudan interest rate swaption. Benchmarking against the popular least-squares Monte Carlo approach shows that our method is able to deliver a higher accuracy in a faster runtime.

Analytically Deriving Risk-Neutral Densities from Volatility Smiles in Delta

This article develops a method for analytically deriving RNDs (risk-neutral densities) of future asset prices from volatility smiles. It extends an existing analytical method, which is for volatility smiles with respect to the strike price, to cover smiles with respect to option delta. A worked-out example on currency options shows that the analytically derived RNDs are free of approximation errors that would arise with numerical methods. The proposed method should be useful to practitioners. <b>TOPICS:</b>Options, futures and forward contracts <b>Key Findings</b> • This article develops an analytical method for deriving RNDs (risk-neutral probability densities) of future asset prices from volatility smiles with respect to option delta. • A worked-out example shows that exact RNDs computed by the method are free of approximation errors associated with numerical derivatives. • The proposed method should be useful to practitioners.

Contract Rescission in the Real Estate Presale Market

Presale is a popular method for selling residential properties and an important risk-sharing instrument for both homebuyers and developers. However, in Hong Kong’s presale housing market from 1996 to 2014, over 10% of the presale contracts were rescinded, resulting in an aggregate loss of HKD 436.67 million per year. In this study, we examine the rescission of presale contracts from a novel perspective with respect to option theory. We find that the moneyness of presale contracts, defined as the market price at settlement over the contract price, is negatively associated with the contract rescission rate. Specifically, contracts that are out of the money (i.e., the market price is less than the remaining payment) at settlement have a 12.2% higher rescission rate than the average. The presale call option delta and time to maturity at the purchase time positively predict rescission, which indicates a higher rescission rate when the buyer shares more of the price and time-induced risk. The Hong Kong government’s housing market macroprudential measures lead to significantly lower rescission rates. In addition, we find that lower moneyness at settlement leads to longer holding periods after settlement, which indicates that presale option features continue to influence the buyers’ behaviour after settlement due to the disposition effect. Overall, our findings shed light on understanding the presale market mechanism, the buyers’ strategic default behaviour, and the government’s role in curbing speculation and reducing the costs induced by presale rescission.

The Black-Scholes Merton Model&amp;lt;br/&amp;gt;—Implications for the Option Delta and the Probability of Exercise

This paper analyzes the implications of the Black-Scholes-Merton model of option pricing, for the deltas of call and put options and their respective probabilities of exercise at expiration. It derives a threshold value of the stock price and shows that in certain cases the options will have a delta in excess of 0.50, and will also have more than a 50% probability of exercise, while other options will have a delta that is lower than 0.50 and a probability of exercise that is lower than 50%. Similar results are obtained for the Garman-Kohlhagen model, which is an extension of the Black-Scholes Merton model, for valuing foreign currency options.

Dynamics of foreign exchange implied volatility and implied correlation surfaces

The prices of currency options expressed in terms of their implied volatilities and the implied correlations between foreign exchange rates at a given point in time depend on option delta and time to maturity. Implied volatilities and implied correlations likewise may thus be represented as a surface. It is well known that these surfaces exhibit both skew/smile features and term structure effects and their shapes fluctuate substantially over time. Using implied volatilities on three currency pairs as well as historical implied correlation values between them, we study the nature of these fluctuations by applying a Karhunen-Loève decomposition that is a generalization of a principal component analysis. We demonstrate that the largest share in the dynamics of these surfaces' fluctuations may be explained by exactly the same three factors, providing evidence of strong interdependences between implied correlation and implied volatility of global currency pairs.

Pricing American Options With Jumps in Asset and Volatility

Jump risk plays an important role in current financial markets, yet it is a risk that cannot be easily measured and hedged. We numerically evaluate American call options under stochastic volatility, stochastic interest rates and jumps in both the asset price and volatility. By employing the Method of Lines (Meyer (2015)), the option price, the early exercise boundary and the Greeks are computed as part of the solution, which makes the numerical implementation time efficient. We conduct a numerical study to gauge the impact of jumps and stochastic interest rates on American call option prices and on their free boundaries. Jumps tend to increase the values of OTM and ATM options while decreasing the value of ITM options. The option delta is affected in a similar way. The impact of jumps on the free boundary is substantial and depends on the time to maturity. Near expiry, including asset jumps lowers the free boundary and the option holder is more likely to exercise the option, whilst including asset-volatility jumps elevates the free boundary and the option holder is less likely to exercise the option. This relation reverses at the beginning of the options life. The volatility, interest rates and their volatilities have a positive impact on the free boundaries and the option holder is less likely to exercise as these parameters increase.

The effect of executive stock option delta and vega on the spin-off decision

We investigate the role of CEO incentives around asset restructuring known as corporate spin-off. More specifically, we focus on executive stock option delta and vega vis-à-vis changes in firm value and firm riskiness in response to the corporate spin-off. Controlling for self-selection of the spin-off decision, we find that executive stock option vega is positively related to changes in firm value as well as changes in firm risk. Conversely, we find that executive stock option delta is negatively related to changes in firm value and firm risk. Finally, we estimate the Fazzari et al. (1988) investment model and show that at the business segment level, CEO incentives are positively linked to capital spending. Overall, our study extends the current literature by documenting the role of executive stock option delta and vega in the context of corporate spin-offs.

Finite Element Based Monte Carlo Simulation of Option Prices on LLvy Driven Assets

This paper extends the simulation algorithm by Andreasen & Huge (2011) to the simulation of option prices and deltas on Levy driven assets. The simulation is performed relying on the inverse transition matrix of the discretized PDE. Each row describes the evolution of option prices and can thus be seen as the underlying’s transition probability distribution. We demonstrate how one can get accurate prices and deltas of European options on VG and CGMY via Monte Carlo simulations.

Informed Trading in the Index Option Market

We propose a structural vector autoregressive (VAR) model of informed trading in option markets to analyze whether investors use options to trade on private information about the underlying price and/or the underlying's volatility. We decompose option order flow into exposures to the underlying asset (through the option delta) and its volatility (through the option vega). Our proposed methodological framework facilitates meaningfully aggregation of option order flows for different strike prices and maturities, and increases statistical power to identify informed trading. A fitted model confirms that S&P500 option trades are indeed informative about changes in both the underlying and volatility.

Stochastic idiosyncratic cash flow risk and real options: Implications for stock returns

Stocks with high idiosyncratic volatility perform poorly relative to low idiosyncratic volatility stocks. We offer a novel explanation of this anomaly based on real options, which is consistent with earlier findings on idiosyncratic volatility (the positive contemporaneous relation between firm-level stock returns and idiosyncratic volatility). Our approach is based on introducing stochastic idiosyncratic cash flow risk into an equity valuation model of firms with growth options. Within our model, a firm's systematic risk depends on the delta of its growth option. The growth option's delta is lower when idiosyncratic volatility rises, driving down the firm's systematic risk and hence its expected return – firms with higher idiosyncratic volatility therefore have lower expected returns. Our model additionally offers the following novel empirical predictions: (i) returns correlate positively with idiosyncratic volatility during intervals between large changes in idiosyncratic volatility (the switch effect), and (ii) the anomalies and the switch effect are stronger for firms with more real options and which undergo larger changes in idiosyncratic volatility. Empirical results support the predictions of our model.

Differential Risk Taking Implications of Performance Incentives from Stock and Stock Option Holdings

In this paper, we study the risk taking implications of managerial pay-for-performance incentives (delta). The extant empirical literature is built on the presumption that each unit of delta has an equal risk inducing effect regardless of its source. Instead, following the predictions of the principal-agent models of executive compensation, we differentiate between performance incentives from stock and option holdings. We show that while option delta is positively associated with firm riskiness, stock delta does not have a significant effect on risk taking. Consequently, the relationship between the total value of pay-for-performance incentives and firm risk strengthens as the relative contribution from option holdings increases. Our findings contribute to the debate on the executive pay reforms, stressing the need to consider the composition of stock-based pay when designing compensation packages to provide appropriate performance and risk incentives to the executives.

A Strange Disposition? Option Trading Based on Reference Prices

Using individual stock option data, we document a classic disposition effect in addition to an effect whereby highly unfavorable positions are more likely to be liquidated. Both behaviors are more prevalent for less sophisticated investors. However, for a given gain or loss, option traders are less likely to sell winners and extreme losers as volatility increases. The disposition effect also induces return predictability. When underlying stock prices increase relative to reference prices, next day option delta hedged call returns are significantly higher and put returns are significantly lower.

BOUNDS ON PRICES FOR ASIAN OPTIONS VIA FOURIER METHODS

The problem of pricing arithmetic Asian options is nontrivial, and has attracted much interest over the last two decades. This paper provides a method for calculating bounds on option prices and approximations to option deltas in a market where the underlying asset follows a geometric Lévy process. The core idea is to find a highly correlated, yet more tractable proxy to the event that the option finishes in-the-money. The paper provides a means for calculating the joint characteristic function of the underlying asset and proxy processes, and relies on Fourier methods to compute prices and deltas. Numerical studies show that the lower bound provides accurate approximations to prices and deltas, while the upper bound provides good though less accurate results.

Dynamic Factor Models for the Volatility Surface

The implied volatility surface is the collection of volatilities implied by option contracts for different strike prices and time-to-maturity. We study factor models to capture the dynamics of this three-dimensional implied volatility surface. Three model types are considered to examine desirable features for representing the surface and its dynamics: a general dynamic factor model, restricted factor models designed to capture the key features of the surface along the moneyness and maturity dimensions, and in-between spline-based methods. Key findings are that: (i) the restricted and spline-based models are both rejected against the general dynamic factor model, (ii) the factors driving the surface are highly persistent, (iii) for the restricted models option Delta is preferred over the more often used strike relative to spot price as measure for moneyness.