# Bermuda Option - Explained

What is a Bermuda Option?

# What is a Bermuda Option?

A Bermuda option refers to an exotic option type which is only exercisable to predetermined dates, usually on a day per month. Bermuda options are a blend of European and American options. Bermuda is can be exercised at the expiration date, and on some specified dates which occur between the date of purchase and the expiration date.

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## Academic Research on Bermuda Option

• Properties of American option prices, Ekstrm, E. (2004). Properties of American option prices. Stochastic Processes and their Applications114(2), 265-278. We investigate some properties of American option prices in the setting of time- and level-dependent volatility. The properties under consideration are convexity in the underlying stock price, monotonicity and continuity in the volatility and time decay. Some properties are direct consequences of the corresponding properties of European option prices that are already known, and some follow by writing solutions of different stochastic differential equations as time changes of the same Brownian motion.
• On bermudan options, Schweizer, M. (2002). On bermudan options. In Advances in Finance and Stochastics (pp. 257-270). Springer, Berlin, Heidelberg. A Bermudan option is an American-style option with a restricted set of possible exercise dates. We show how to price and hedge such options by superreplication and use these results for a systematic analysis of the rollover option.
• A simple derivation of and improvements to Jamshidian's and Rogers' upper bound methods for Bermudan options, Joshi, M. S. (2007). A simple derivation of and improvements to Jamshidian's and Rogers' upper bound methods for Bermudan options. Applied Mathematical Finance14(3), 197-205. The additive method for upper bounds for Bermudan options is rephrased in terms of buyer's and seller's prices. It is shown how to deduce Jamshidian's upper bound result in a simple fashion from the additive method, including the case of possibly zero final payoff. Both methods are improved by ruling out exercise at suboptimal points. It is also shown that it is possible to use subMonte Carlo simulations to estimate the value of the hedging portfolio at intermediate points in the Jamshidian method without jeopardizing its status as upper bound.
• Bermudan option pricing with Monte-Carlo methods, Douady, R. (2001). Bermudan option pricing with Monte-Carlo methods. In Quantitative Analysis In Financial Markets: Collected Papers of the New York University Mathematical Finance Seminar (Volume III) (pp. 314-328). We explain, compare and improve two algorithms to compute American or Bermudan options by Monte-Carlo. The first one is based on threshold optimisation in the exercise strategy (Andersen, 1999). The notion of fuzzy threshold is introduced to ease optimisation. The second one uses a linear regression to get an estimate of the option price at intermediary dates and determine the exercise strategy (Carriere, 1997; LongstaffSchwartz, 1999). We thoroughly study the convergence of these two approaches, including a mixture of both.
• Pricing of perpetual Bermudan options, Boyarchenko, S. I., & Levendorskii, S. Z. (2002). Pricing of perpetual Bermudan options. Quantitative Finance2(6), 432-442. We consider perpetual Bermudan options and more general perpetual American options in discrete time. For wide classes of processes and payoffs, we obtain exact analytical pricing formulae in terms of the factors in the WienerHopf factorization formulae. Under additional conditions on the process, we derive simpler approximate formulae.
• Dual valuation and hedging of Bermudan options, Rogers, L. C. G. (2010). Dual valuation and hedging of Bermudan options. SIAM Journal on Financial Mathematics1(1), 604-608. Some years ago, a different characterization of the value of a Bermudan option was discovered which can be thought of as the viewpoint of the seller of the option, in contrast to the conventional characterization which took the viewpoint of the buyer. Since then, there has been a lot of interest in finding numerical methods which exploit this dual characterization. This paper presents a pure dual algorithm for pricing and hedging Bermudan options.
• An efficient algorithm for Bermudan barrier option pricing, Ding, D., Huang, N. Y., & Zhao, J. Y. (2012). An efficient algorithm for Bermudan barrier option pricing. Applied Mathematics-A Journal of Chinese Universities27(1), 49-58. An efficient option pricing method based on Fourier-cosine expansions was presented by Fang and Oosterlee for European options in 2008, and later, this method was also used by them to price early-exercise options and barrier options respectively, in 2009. In this paper, this method is applied to price discretely American barrier options in which the monitored dates are many times more than the exercise dates. The corresponding algorithm is presented to practical option pricing. Numerical experiments show that this algorithm works very well and efficiently for different exponential Lvy asset models.
• A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options, Egloff, D., Kohler, M., & Todorovic, N. (2007). A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options. The Annals of Applied Probability17(4), 1138-1171. Under the assumption of no-arbitrage, the pricing of American and Bermudan options can be casted into optimal stopping problems. We propose a new adaptive simulation based algorithm for the numerical solution of optimal stopping problems in discrete time. Our approach is to recursively compute the so-called continuation values. They are defined as regression functions of the cash flow, which would occur over a series of subsequent time periods, if the approximated optimal exercise strategy is applied. We use nonparametric least squares regression estimates to approximate the continuation values from a set of sample paths which we simulate from the underlying stochastic process. The parameters of the regression estimates and the regression problems are chosen in a data-dependent manner. We present results concerning the consistency and rate of convergence of the new algorithm. Finally, we illustrate its performance by pricing high-dimensional Bermudan basket options with strangle-spread payoff based on the average of the underlying assets.
• Application of the fast Gauss transform to option pricing, Broadie, M., & Yamamoto, Y. (2003). Application of the fast Gauss transform to option pricing. Management science49(8), 1071-1088. In many of the numerical methods for pricing American options based on the dynamic programming approach, the most computationally intensive part can be formulated as the summation of Gaussians. Though this operation usually requiresO(NN') work when there areN' summations to compute and the number of terms appearing in each summation isN, we can reduce the amount of work toO(N+N') by using a technique called the fast Gauss transform. In this paper, we apply this technique to the multinomial method and the stochastic mesh method, and show by numerical experiments how it can speed up these methods dramatically, both for the Black-Scholes model and Merton's lognormal jump-diffusion model. We also propose extensions of the fast Gauss transform method to models with non-Gaussian densities.
• Pricing American exchange options in a jumpdiffusion model, Lindset, S. (2007). Pricing American exchange options in a jumpdiffusion model. Journal of Futures Markets: Futures, Options, and Other Derivative Products27(3), 257-273. A way to estimate the value of an American exchange option when the underlying assets follow jumpdiffusion processes is presented. The estimate is based on combining a European exchange option and a Bermudan exchange option with two exercise dates by using Richardson extrapolation as proposed by R. Geske and H. Johnson (1984). Closedform solutions for the values of European and Bermudan exchange options are derived. Several numerical examples are presented, illustrating that the early exercise feature may have a significant economic value. The results presented should have potential for pricing overthecounter options and in particular for pricing real options.
• Primal-dual simulation algorithm for pricing multidimensional American options, Andersen, L., & Broadie, M. (2004). Primal-dual simulation algorithm for pricing multidimensional American options. Management Science50(9), 1222-1234. This paper describes a practical algorithm based on Monte Carlo simulation for the pricing of multidimensional American (i.e., continuously exercisable) and Bermudan (i.e., discretely exercisable) options. The method generates both lower and upper bounds for the Bermudan option price and hence gives valid confidence intervals for the true value. Lower bounds can be generated using any number of primal algorithms. Upper bounds are generated using a new Monte Carlo algorithm based on the duality representation of the Bermudan value function suggested independently in Haugh and Kogan (2004) and Rogers (2002). Our proposed algorithm can handle virtually any type of process dynamics, factor structure, and payout specification. Computational results for a variety of multifactor equity and interest-rate options demonstrate the simplicity and efficiency of the proposed algorithm. In particular, we use the proposed method to examine and verify the tightness of frequently used exercise rules in Bermudan swaption markets.
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