Tian-Shyr Dai

The Bino-Trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing

A model with a closed-form solution is the Holy Grail of derivatives valuation, because as computers have become increasingly powerful, exact answers to even very complicated formulas can typically be obtained almost instantaneously. Unfortunately, as derivative instruments have become increasingly complicated, closed-form valuation has become increasingly rare. Researchers are now trying to develop more efficient and accurate approximation techniques. Lattice models, such as the classic binomial model of Cox, Ross, and Rubinstein, are among the workhorses of this effort. Lattice models converge to accurate values as the number of node calculations increases, but convergence is often erratic and slow. The biggest problem is that the option payoff between two price nodes can be highly nonlinear, for example when the critical price barrier for a knock-out option falls between two layers of nodes, and a large jump in the computed option value occurs when a small change in the number of time steps causes the critical price to hop from one side of a node to the other. A variety of tricks have been proposed to deal with this problem by adapting the geometry of the lattice to make the nodes land directly on top of the critical prices. Generally this has required the additional flexibility afforded by a trinomial structure rather than the binomial, but that is costly in terms of efficiency. In this article, Dai and Lyuu introduce a new approach that achieves remarkable improvement in efficiency by combining binomial and trinomial structures. Here the trick is to construct a binomial lattice with nodes that land on top of the key prices, but to use a very small amount of trinomial lattice to connect the initial price—the model option value—to the binomial structure. This allows both the best placement of the tree relative to the critical areas and also great efficiency gains because binomial lattice probabilities for the terminal nodes can be computed directly using combinatorial results, skipping over calculations for all of the intermediate time steps.

Efficient option pricing on stocks paying discrete or path-dependent dividends with the stair tree

Pricing options on a stock that pays discrete dividends has not been satisfactorily settled because of the conflicting demands of computational tractability and realistic modelling of the stock price process. Many papers assume that the stock price minus the present value of future dividends or the stock price plus the forward value of future dividends follows a lognormal diffusion process; however, these assumptions might produce unreasonable prices for some exotic options and American options. It is more realistic to assume that the stock price decreases by the amount of the dividend payout at the ex-dividend date and follows a lognormal diffusion process between adjacent ex-dividend dates, but analytical pricing formulas and efficient numerical methods are hard to develop. This paper introduces a new tree, the stair tree, that faithfully implements the aforementioned dividend model without approximations. The stair tree uses extra nodes only when it needs to simulate the price jumps due to dividend payouts and return to a more economical, simple structure at all other times. Thus it is simple to construct, easy to understand, and efficient. Numerous numerical calculations confirm the stair trees superior performance to existing methods in terms of accuracy, speed, and/or generality. Besides, the stair tree can be extended to more general cases when future dividends are completely determined by past stock prices and dividends, making the stair tree able to model sophisticated dividend processes.

An efficient convergent lattice algorithm for European Asian options

Financial options whose payoff depends critically on historical prices are called path-dependent options. Their prices are usually harder to calculate than options whose prices do not depend on past histories. Asian options are popular path-dependent derivatives, and it has been a long-standing problem to price them efficiently and accurately. No known exact pricing formulas are available to price them under the continuous-time Black-Scholes model. Although approximate pricing formulas exist, they lack accuracy guarantees. Asian options can be priced numerically on the lattice. A lattice divides the time to maturity into n equal-length time steps. The option price computed by the lattice converges to the option value under the Black-Scholes model as n->~. Unfortunately, only subexponential-time algorithms are available if Asian options are to be priced on the lattice without approximations. Efficient approximation algorithms are available for the lattice. The fastest lattice algorithm published in the literature runs in O(n^3^.^5)-time, whereas for the related PDE method, the fastest one runs in O(n^3) time. This paper presents a new lattice algorithm that runs in O(n^2^.^5) time, the best in the literature for such methods. Our algorithm exploits the method of Lagrange multipliers to minimize the approximation error. Numerical results verify its accuracy and the excellent performance.

Accurate approximation formulas for stock options with discrete dividends

Pricing options on a stock that pays discrete dividends has not been satisfactorily settled in the literature. Frishling (2002) shows that there are three different models to model stock price with discrete dividends, but only one of these models is close to reality and generates consistent option prices. We follow Frishling (2002) by calling this model Model 3. Unfortunately, there is no analytical option pricing formula for Model 3, and many popular numerical methods such as trees are inefficient when used to implement Model 3. A new stock price model is proposed in this article. To guarantee that the option prices generated by this new model are close to those generated by Model 3, the distributions of the new model at exdividend dates and maturity approximate the distributions of Model 3 at those dates. To achieve this, a discrete dividend in Model 3 is replaced by a continuous dividend yield that can be represented as a function of discrete dividends and stock returns in the new model. Thus, the new model follows a lognormal diffusion process and the analytical option pricing formulas can be easily derived. Numerical experiments show that our analytical pricing formulas provide accurate pricing results.

An exact subexponential-time lattice algorithm for Asian options

Asian options are popular financial derivative securities. Unfortunately, no exact pricing formulas exist for their price under continuous-time models. Asian options can also be priced on the lattice, which is a discretized version of the continuous- time model. But only exponential-time algorithms exist if the options are priced on the lattice without approximations. Although efficient approximation methods are available, they lack accuracy guarantees in general. This paper proposes a novel lattice structure for pricing Asian options. The resulting pricing algorithm is exact (i.e., without approximations), converges to the value under the continuous-time model, and runs in subexponential time. This is the first exact, convergent lattice algorithm to break the long-standing exponential-time barrier.

Linear-time option pricing algorithms by combinatorics

Options are popular financial derivatives that play essential roles in financial markets. How to price them efficiently and accurately is very important both in theory and practice. Options are often priced by the lattice model. Although the prices computed by the lattice converge to the theoretical option value under the continuous-time model, they may converge slowly. Worse, for some options like barrier options, the prices can even oscillate wildly. For such options, huge amounts of computational time are required to achieve acceptable accuracy. Combinatorial techniques have been used to improve the performance in pricing a wide variety of options. This paper uses vanilla options, power options, single-barrier options, double-barrier options, and lookback options as examples to show how combinatorics can help us to derive linear-time pricing algorithms. These algorithms compare favorably against popular lattice methods, which take at least quadratic time.

Analytics for geometric average trigger reset options

The geometric average trigger reset option resets the strike price based on the geometric average of the underlying assets prices over a monitoring window. Similar contracts have been traded on exchanges in Asia. This paper derives an analytic formula for pricing this option with multiple monitoring windows. The analytic formula in fact is a corollary of a general formula that holds for a large class of path-dependent options: It prices any option whose value can be written as a linear combination of , where X is a multinormal random vector and b is some constant vector. Numerical experiments suggest that the pricing formula approximates the values of arithmetic average trigger reset options accurately. Thus pricing the arithmetic average trigger reset option can benefit from using this formula as the control variate in Monte Carlo simulation. Numerical results also suggest that the geometric average trigger reset option does not have significant delta jump as the standard reset option, and this useful property reduces the hedging risk dramatically.

Efficient, exact algorithms for asian options with multiresolution lattices

Asian options are a kind of path-dependent derivative. How to price such derivatives efficiently and accurately has been a long-standing research and practical problem. This paper proposes a novel multiresolution (MR) trinomial lattice for pricing European- and American-style arithmetic Asian options. Extensive experimental work suggests that this new approach is both efficient and more accurate than existing methods. It also computes the numerical delta accurately. The MR algorithm is exact as no errors are introduced during backward induction. In fact, it may be the first exact discrete-time algorithm to break the exponential-time barrier. The MR algorithm is guaranteed to converge to the continuous-time value.

Accurate and efficient lattice algorithms for American-style Asian options with range bounds

Abstract Asian options are popular path-dependent options and it has been a long-standing problem to price them efficiently and accurately. Since there is no known exact pricing formula for Asian options, numerical pricing formulas like lattice models must be employed. A lattice divides a certain time interval into n time steps and the pricing results generated by the lattice (called desired option values for convenience) converge to the theoretical option value as n → ∞ . Since a brute-force lattice pricing algorithm runs in subexponential time in n, some heuristics, like interpolation method, are used to strike the balance between the efficiency and the accuracy. But the pricing results might not converge due to the accumulation of interpolation errors. For pricing European-style Asian options, the evaluation on the major part of the lattice can be done by a simple formula, and the interpolation method is only required on the minor part of the lattice. Thus polynomial time algorithms with convergence guarantee for European-style Asian options can be derived. However, such a simple formula does not exist for American-style Asian options. This paper suggests an efficient range-bound algorithm that bracket the desired option value. By taking advantages of the early exercise property of American-style options, we show that part of the lattice can be evaluated by a simple formula. The interpolation method is required on the remaining part of the lattice and the upper and the lower bounds option values produced by the proposed algorithm are essentially numerically identical. Thus the theoretical option value is said to be obtained practically when the range bound algorithm runs on a lattice with large number of time steps.

Pricing barrier stock options with discrete dividends by approximating analytical formulae

Deriving accurate analytical formulas for pricing stock options with discrete dividend payouts is a hard problem even for the simplest vanilla options. This is because the falls in the stock price process due to discrete dividend payouts will significantly increase the mathematical difficulty in pricing the option. On the other hand, much literature uses other dividend settings to simplify the difficulty, but these settings may produce inconsistent pricing results. This paper derives accurate approximating formulae for pricing a popular path-dependent option, the barrier stock option, with discrete dividend payouts. The fall in stock price due to dividend payout at an exdividend date is approximated by an accumulated price decrement due to a continuous dividend yield up to time . Thus, the stock price process prior to time and after time can be separately modelled by two different lognormal-diffusive stock processes which help us to easily derive analytical pricing formulae. Numerical experiments suggest that our formulae provide more accurate and coherent pricing results than other approximation formulae. Our formulae are also robust under extreme cases, like the high volatility (of the stock price) case. Besides, our formulae also extend the applicability of the first-passage model (a type of structural credit risk model) to measure how the firm’s payout influences its financial status and the credit qualities of other outstanding debts.