Digital Options

2011 ◽  
pp. 231-234
Keyword(s):  
2008 ◽  
Vol 11 (08) ◽  
pp. 905-941 ◽  
Author(s):  
ERIC C. K. YU ◽  
WILLIAM T. SHAW

We propose a general approach that requires only a simple change of variable that keeps the valuation of call and put options (convertible bonds) with strike (conversion) price resets two-dimensional in the classical Black–Scholes setting. A link between reset derivatives, compound options and "discrete barrier" type options, when there is one reset is then discussed, from which we analyze the risk characteristics of reset derivatives, which can be significantly different from their vanilla counterparts. We also generalize the prototype reset structure and show that the delta and gamma of a convertible bond with reset can both be negative. Finally, we show that the "waviness" property found in the delta and gamma of some reset derivatives is due to the discontinuous nature of the reset structure, which is closely linked to digital options.


2018 ◽  
Vol 59 (3) ◽  
pp. 349-369
Author(s):  
ZIWIE KE ◽  
JOANNA GOARD ◽  
SONG-PING ZHU

We study the numerical Adomian decomposition method for the pricing of European options under the well-known Black–Scholes model. However, because of the nondifferentiability of the pay-off function for such options, applying the Adomian decomposition method to the Black–Scholes model is not straightforward. Previous works on this assume that the pay-off function is differentiable or is approximated by a continuous estimation. Upon showing that these approximations lead to incorrect results, we provide a proper approach, in which the singular point is relocated to infinity through a coordinate transformation. Further, we show that our technique can be extended to pricing digital options and European options under the Vasicek interest rate model, in both of which the pay-off functions are singular. Numerical results show that our approach overcomes the difficulty of directly dealing with the singularity within the Adomian decomposition method and gives very accurate results.


1999 ◽  
Vol 31 (02) ◽  
pp. 551-577 ◽  
Author(s):  
Sid Browne

We study a variety of optimal investment problems for objectives related to attaining goals by a fixed terminal time. We start by finding the policy that maximizes the probability of reaching a given wealth level by a given fixed terminal time, for the case where an investor can allocate his wealth at any time between n + 1 investment opportunities: n risky stocks, as well as a risk-free asset that has a positive return. This generalizes results recently obtained by Kulldorff and Heath for the case of a single investment opportunity. We then use this to solve related problems for cases where the investor has an external source of income, and where the investor is interested solely in beating the return of a given stochastic benchmark, as is sometimes the case in institutional money management. One of the benchmarks we consider for this last problem is that of the return of the optimal growth policy, for which the resulting controlled process is a supermartingale. Nevertheless, we still find an optimal strategy. For the general case, we provide a thorough analysis of the optimal strategy, and obtain new insights into the behavior of the optimal policy. For one special case, namely that of a single stock with constant coefficients, the optimal policy is independent of the underlying drift. We explain this by exhibiting a correspondence between the probability maximizing results and the pricing and hedging of a particular derivative security, known as a digital or binary option. In fact, we show that for this case, the optimal policy to maximize the probability of reaching a given value of wealth by a predetermined time is equivalent to simply buying a European digital option with a particular strike price and payoff. A similar result holds for the general case, but with the stock replaced by a particular (index) portfolio, namely the optimal growth or log-optimal portfolio.


2011 ◽  
Vol 14 (07) ◽  
pp. 1045-1090 ◽  
Author(s):  
MITYA BOYARCHENKO ◽  
MARCO DE INNOCENTIS ◽  
SERGEI LEVENDORSKIĬ

We calculate the leading term of asymptotics of the prices of barrier options and first-touch digitals near the barrier for wide classes of Lévy processes with exponential jump densities, including the Variance Gamma model, the KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In the case of processes of infinite activity and finite variation, with the drift pointing from the barrier, we prove that the price is discontinuous at the boundary. This observation can serve as the basis for a simple robust test of the type of processes observed in real financial markets. In many cases, we calculate the second term of asymptotics as well. By comparing the exact asymptotic results for prices with those of Carr's randomization approximation, we conclude that the latter is very accurate near the barrier. We illustrate this by including numerical results for several types of Lévy processes commonly used in option pricing.


10.2196/15733 ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. e15733 ◽  
Author(s):  
Jordan Glenn ◽  
Erica Nicole Madero ◽  
Michelle Gray ◽  
Nami Fuseya ◽  
Mari Ikeda ◽  
...  

Background As the global prevalence of dementia continues to rise, multidomain lifestyle interventions that address modifiable risk factors associated with pathological cognitive decline are increasing. Although some digital options have been developed to increase the reach and scalability of these programs, because of cultural differences, the efficacy of the programs in one population cannot easily be generalized to populations in other countries. Objective This investigation aimed to examine the usability and engagement of a digitally delivered multidomain cognitive lifestyle intervention developed in the United States for a Japanese population. Methods This feasibility investigation utilized a quasi-experimental, single-arm, nonrandomized, longitudinal design where participants engaged in the behavioral intervention on a smartphone. Of the 559 participants that initially enrolled (age: mean 51 years, SD 7.5 years; 51.7% female [289/559]), 242 completed the final testing trial. Participants enrolled in a multidomain lifestyle program that consisted of (1) psychoeducational material, (2) physical activity tracker, (3) nutrition tracker, (4) audio-based meditations, and (5) health coaching. Engagement with the program was assessed through the total number of app sessions and the use of the exercise, diet, and meditation tracking features within the app. The total number of minutes exercised was collected through subjective user inputs, and nutrition was quantified by the Mediterranean-DASH Intervention for Neurodegenerative Delay diet adherence score. Results Significant relationships existed between overall nutrition score and frequency of nutrition tracking (r=0.18), frequency of physical activity tracking (r=0.19), and the total number of minutes exercised (r=0.22). Total minutes exercised was significantly correlated with total app sessions (r=0.57), frequency of physical activity tracking (r=0.85), frequency of nutrition tracking (r=0.64), number of times participants meditated (r=0.46), and total lessons read (r=0.36). The number of completed lessons was significantly related to frequency of physical activity tracking (r=0.40), frequency of nutrition tracking (r=0.43), the total number of times participants meditated (r=0.35), and total minutes exercised (r=0.33). Dividing the cohort into two groups based on lesson completion (<10 lessons completed vs ≥10 lessons completed), significant differences were observed between the total minutes exercised, frequency of physical activity tracking, frequency of nutrition tracking, and total number of times participants meditated (all P values <.01). Conclusions Overall, this cross-cultural feasibility study in Japanese users demonstrated that the various engagement metrics were significantly correlated, and greater engagement was related to improved nutrition scores and increased time exercising. In addition, the relationships between lesson completion and other engagement metrics suggest that there may be value in exploring mechanisms that enhance lesson completion. Future research should examine the program in randomized control trials to more rigorously evaluate program efficacy.


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