International Journal of Financial Engineering (IJFE)
Flexible Forward pricing through Leisen-Reimer Trees: implementation and performance comparison with traditional Markov chains Pier Giuseppe Giribone, Simone Ligato CARIGE Bank Group, Genoa (Italy) 15 Cassa di Risparmio, 16123 Genoa +39 010 5792548
[email protected],
[email protected] Abstract:
This article aims to estimate the fair-value of flexi-forwards, popular financial instruments on currencies, through Leisen–Reimer trees. The first part of paper deals with Markov chains suitable for pricing American options: Cox–Ross–Rubinstein, Jarrow–Rudd, Tian, Leisen–Reimer Trees. The correctness of the implementation in Matlab® has been tested by comparing their prices with those obtained through approximated closed-formulas. The second part highlights the better performance of Leisen–Reimer trees in terms of convergence speed and sensitivity. Finally, flexi-forward contracts have been priced by using the numerical methodologies which have outperformed in the previous parts. Keywords:
Alternative Stochastic Trees, Leisen Reimer (LR) Markov Chains, Flexible Forward pricing, Numerical schemes for American Option valuation, Numerical Techniques for American Option greeks Pricing a Flexi Forward contract
A flexi-forward, also known as time-option forward contract, is a financial product which provides an exchange of an agreed notional in a foreign currency in any moment between two fixed dates. If the right has not been exercised during option life, the holder has the duty to deliver the amount at maturity. In financial terms, a contract with these features can be decomposed in an American call option and a European down-and-out put barrier option, in which the barrier level is dynamically fixed in function of the probability state for which the American option goes in-the-money. The pricing of the put option can be obtained using the set of closed-formulas proposed by Rubinstein, but the determination of the barrier level must be necessarily quantified using a numerical method. Applying the discretization schemes of Leisen–Reimer to modeling the future states of underlying on which the American call is written, has been showed to be suitable for pricing a flexi-forward on currencies.
International Journal of Financial Engineering (IJFE)
This paper can be divided into three parts: The first part deals with Markov chains suitable for pricing American options: Cox–Ross–Rubinstein (CRR Tree), Cox-Ross–Rubinstein with Drift (CRRd Tree), Jarrow–Rudd (JR Tree, also known as EQP Tree), Tian Tree, Leisen–Reimer (LR 1,2 Tree). The correct implementation in an environment for numerical computing, such as Matlab® , has been tested by comparing their convergence prices with those obtained using the approximated closed formulas (also known as quasi-closed formulae) of Barone–Adesi–Whaley (BAW) and Bjerksund– Stensland (BS 1993 – BS 2002). The second part of the paper highlights the better performance of LR trees in terms of convergence speed. The measures of sensitivity for the American option are also estimated. The last section deals with the evaluation of flexi-forward instruments: the contracts have been priced by using numerical methodologies which have outperformed in the previous parts. Conclusion
This paper deals with the pricing of a Flexi Forward contract on currency: a popular instrument in financial markets, but it is hard to find an exhaustive scientific literature about this topic. The authors approach this problem by decomposing the product in two options: an American call and a European down-and-out put, with the barrier level computed dynamically in function of the probability for which the American call becomes in-the-money. For this purpose, a numerical valuation is needed and the implementation of stochastic trees can be considered a good idea to approach the evaluation. Both the American call value and the determination of the critical level of price for the barrier has been made through Markov chains. From the obtained results, among the applied models, it can be deducted that Leisen-Reimer tree is more suitable for approaching the problem of pricing a Flexi Forward contract. It is shown that the approach proposed by the authors has a better performance both for the convergence speed to fair-value and for the smoothness for the estimation of greeks.
International Journal of Financial Engineering (IJFE) References
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