Alpha-Quantile Options are similar to lookback options in the sense that the payoff is based on a lookback period but differs in that the payoff is on the smallest level where the asset spends a portion of its time in during the option life and not just a specified level as in standard lookbacks.

Under Black-Scholes Framework || Jump Diffusion || Forward Shooting Grid || || Monte Carlo Simulation || Known Names & Variants || References || Advanced Readings || Pricing Models

A relatively new form of exotic options which have not appeared on trading floors as of yet is the so-called alpha-quantile option, an option based on the widely used lookback option. The alpha-quantile option is similar to lookback options in the sense that the payoff is based on a lookback period but differs in that the payoff is on the smallest level where the asset spends a portion of its time in during the option life and not just a specified level as in standard lookbacks. The aim of the alpha-quantile is to provide a payoff which is similar in magnitude as a standard lookback option, but at a fraction of the cost.

Pricing:

Several authors have considered the pricing of alpha-quantile options; namely beginning with Miura (1992), and then further extended by Akahori (1995), Dassios (1995, 1996) and more recently by Ballotta (2000). We focus on two general pricing methodology; an analytical formula under a Black-Scholes framework, and under Jump Diffusion processes.

One of the main questions concerning the pricing of an alpha-quantile option is the distribution of quantile itself, which numerous authors have concentrated on, and we will consider within this document.

Black-Scholes Framework

The pricing for alpha-quantiles can be structured within a Black-Scholes framework as shown by Dassios (1995) and Akahori (1995), in which analytical solutions have been found, beit in an integral form.

The alpha-quantile call and put on a non-dividend paying stock can then be expressed respectively in the form of an expected payoff function:

Where S is the asset price, is the quantile expressed as a positive fraction with a range of 0 and 1, T is the time to maturity, r is the risk free rate, X is the strike price, E is the expectated value under a risk neutral probability measure P and is the filtration process for the Brownian motion at time t - where t is in the range of 0 and T.

By making use of what is known as the Dassios-Port-Wendel identity (see Dassion 1995), the integral closed form solution for an alpha-quantile call can then be given as:

Where ' is given as:

The function (a, t) represents the occupation time of the Brownian motion with a drift u. For further details of the derivation and use, see Dassios (1995) and Ballotta & Kyprianou (2001). is a replication of the alpha-quantile that is independent of the filtration process .

Alpha-quantile put options can be represented in a similar fashion.

By considering the above integral, one can price an alpha-quantile by use of a straightforward Monte Carlo simulation implementation.

Jump Diffusion

Merton's (1976) jump diffusion framework became an important concept within option pricing because of its application to stock markets and the random price movements which can take place over a period of time.

In the case of alpha-quantiles, a similar Levy jump diffusion framework can be considered in which the jumps are modeled by a Poisson driven process. Ballotta (2001, 2002) considered jump processes under three different martingale measures (Merton, Esscher & minimal measure) and found slight variations in results with a standard error in the range of single digit percentages.

The three different martingale measures is not the context of this document and for further reading, refer to Ballotta (2001), for our purposes, we focus on the use of the jump diffusion framework given by Merton. Under a European option setting, Merton gives the process of the underlying stock as:

Where is the drift term, represents the number of jumps per year and is the jump size as a proportion of the asset price. Furthermore, dp is the Poisson counter which governs the jumps, z is a Wiener process.

For alpha-quantile options, a variation of the Merton process is given, and again, Ballotta (2001) provides the SDE of an alpha-quantile under jump process as:

Where L is the Levy jump process, is a martingale measure, is the volatility process, dz is a Wiener process and is given as 0 if there are no jumps during the life of the option and takes the value of if there is a jump. is normally distributed with mean and variance of .

The path of the underlying jump process can be found be solving:

Where j takes the value of 0 to n, n being the number of subintervals within the option period, y and z are independent normal random variables and I is given as 1 if a jump occurs between and 0 otherwise.

The value of depends on the choice of the risk neutral measure (Merton, Esscher or minimal) chosen. See Ballotta (2001) for the proof of each measure and formulation.

* Note that care has been made to preserve the original author's notation, but we have adjusted the notation in certain places for consistency..

Monte Carlo Simulation

Both the aforementioned techniques (Black-Scholes pricing, Jump Diffusion framework) require the use of simulation methods in order to find a price effectively. Although simulation methods for alpha-quantile options are not computationally extensive - Ballotta (2001) found that 100,000 paths on a PIII 64mb RAM machine takes a mere 7 seconds using a C++ program, variance reduction techniques can be employed to render pricing of these instruments even more rapid.

A natural choice for variance reduction is the use of a control variate in order to find a close approximation to the alpha-quantile. Based on similarities to standard lookback options, we can consider the use of lookback options as the control variate.

For those of you who are unfamiliar with variance reduction techniques, these are techniques such as antithetic variables, similarity reduction and control variate methods which can be used to reduce dimensionality in the option pricing problem and provide faster simulation results and greater accuracy.

By using lookback options as a control variate, one can improve on the estimation given for the alpha-quantile option - see Hull (2002) for a brief introduction to variance reduction techniques. Furthermore, perhaps the use of a more similar type of lookback such as a partial lookback would provide even better simulation results; but as Dr. Ballotta kindly pointed out, we must consider the tradeoff between simulation time and accuracy.

Forward Shooting Grid

Hull & White (1993) and Ritchken, Sankarasubramanian & Vijh (1993) propose the forward shooting grid as a method to price exotic Asian and lookback options with the intention for the method as a way of reducing the complexity of multi-dimensional pricing problems associated with highly path-dependent exotic options. The usefulness of this method is that the partial differential equation does not need to be solved at each node (as is the case with finite differences)

A forward shooting grid is a method used to price path-dependent options by expanding on a standard tree method. A state vector at each time node is used to simulate the process for the underlying asset. Barraquand & Pudet (1996) and Forsyth, Vetzal & Zvan (1999) highlight problems associated with the convergence of the grid which may lead to inconsistent computation for these types of options, particularly if the interpolation scheme chosen is incorrect, but at the same time, conditions are also given where the pricing problem is convergent under the method.

Kwok & Lau (2001) explore the pricing of alpha quantiles using a forward shooting grid and note the similarities in the computation between alpha quantiles, Parisians and reset options, even though the inherent features between each class of option is significantly different. The application of the grid can be done by applying the appropriate discrete evolution function to the alpha quantile feature and using a trinomial version of the forward shooting grid.

Other Known Names / Variants:

Cumulative Parisian Options
Edokko Options
Lookback Options
Parisian Options
Partial Lookback Options
Quantile Lookback Options
Quantile Options

Additional/Useful List of resources

Papers:

Akahori, J., "On Dassios' Formula of Brownian Quantiles", Working Paper (1995)
Akahori, J., "Some Formulae for a New Type of Path-Dependent Option", Annals of Applied Probability, 5, pp.383-8 (1995)
Andrianjakaherivola, E., & Russo, F., "The Quantile of a Diffusion: Pricing a Quantile Lookback Option", Universite de Paris Working Paper, 2001
Ballotta, L., "Lévy processes, Option Valuation and Pricing of the Alpha-Quantile Option" Monografia dell'Istituto di Econometria e Matematica, Università Cattolica Sacro Cuore Milano Thesis (2001)
Ballotta, L., & Kyprianou, A., "A Note on the Alpha-Quantile Option", Applied Mathematical Finance, 8, pp 137-144 (2001)
Ballotta, L., "Alpha-quantile option in a Jump-Diffusion Economy" Financial Engineering, E-Commerce and Supply Chain (2002)
Barraquand, J. & Pudet, T., "Pricing of American path-dependent contingent claims" Mathematical
Finance 6, pp. 17–51. (1996)
Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities", The Journal of Political Economy (May '73)
Conze, A., & Viswanathan, R., "Path Dependent Options: The Case of Lookback Options", Journal of Finance, S. 1893-1907 (1991)
Dassios, A., "The Distribution of the Quantile of a Brownian Motion with Drift & The Pricing of Related Path-Dependent Options", Annals of Applied Probability, 5, pp.389-98 (1995)
Fujita, T., & Miura, R., "Edokko Options: A New Framework of Barrier Options'', Asia-Pacific Financial Markets 9(2) pp. 141-15 (2003)
Fujita, T., "A Note on the Joint Distribution of a,ß-Percentiles and Its Application to the Option Pricing'', Asia-Pacific Financial Markets 7(4), pp. 339-344 (2000)
Kwok, Y.K., Lau, K.W., "Pricing Algorithms for Options with Exotic Path-Dependence", Journal of Derivatives, (2001)
Hull, J., "Options, Futures & Other Derivatives", Prentice-Hall, 2002, Chapter 18
Hull, J., & White, A., "Efficient Procedures for Valuing European and Ameircan Path-Dependent Options", Journal of Derivatives, 1, pp. 21-31 (1993)
Merton, R.C., "Option Pricing When Underlying Returns are Discontinuous" Journal of Financial Economics, 3, 125-144. (1976)
Miura, R., "A Note on Lookback Options Based on Order Statistics", Hitotsubashi Journal of Commerce & Management, 27, pp.15-28 (1992)
Yor, M., "The Distribution of Brownian Quantiles", The Journal of Applied Probability, 32, pp.405-416 (1995)

Advanced Readings:

Wendel, J.G., "Order Statistics of Partial Sums", Annals of Mathematical Statistics, 31, pp. 1034-44 (1960)