Monte Carlo option model
Encyclopedia
In mathematical finance
, a Monte Carlo option model uses Monte Carlo method
s to calculate the value of an option
with multiple sources of uncertainty or with complicated features.
The term 'Monte Carlo method' was coined by Stanislaw Ulam in the 1940s. The first application to option pricing was by Phelim Boyle
in 1977 (for European options). In 1996, M. Broadie and P. Glasserman showed how to price Asian option
s by Monte Carlo. In 2001 F. A. Longstaff
and E. S. Schwartz
developed a practical Monte Carlo method for pricing American-style options.
, Monte Carlo valuation relies on risk neutral valuation. Here the price of the option is its discounted
expected value
; see risk neutrality and Rational pricing: Risk Neutral Valuation. The technique applied then, is (1) to generate several thousand possible (but random) price paths for the underlying
(or underlyings) via simulation
, and (2) to then calculate the associated exercise
value (i.e. "payoff") of the option for each path. (3) These payoffs are then averaged and (4) discounted to today. This result is the value of the option.
This approach, although relatively straightforward, allows for increasing complexity:
and Asian option
s and in real options analysis
. Additionally, as above, the modeller is not limited as to the probability distribution assumed.
Conversely, however, if an analytical technique
for valuing the option exists—or even a numeric technique, such as a (modified) pricing tree
—Monte Carlo methods will usually be too slow to be competitive. They are, in a sense, a method of last resort; see further under Monte Carlo methods in finance
. With faster computing capability this computational constraint is less of a concern.
Mathematical finance
Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...
, a Monte Carlo option model uses Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
s to calculate the value of an option
Option (finance)
In finance, an option is a derivative financial instrument that specifies a contract between two parties for a future transaction on an asset at a reference price. The buyer of the option gains the right, but not the obligation, to engage in that transaction, while the seller incurs the...
with multiple sources of uncertainty or with complicated features.
The term 'Monte Carlo method' was coined by Stanislaw Ulam in the 1940s. The first application to option pricing was by Phelim Boyle
Phelim Boyle
Phelim Boyle , a distinguished professor and actuary, is a professor of finance in the Laurier School of Business & Economics at Wilfrid Laurier University in Canada and is a pioneer of quantitative finance. He is best known for initiating the use of Monte Carlo methods in option pricing...
in 1977 (for European options). In 1996, M. Broadie and P. Glasserman showed how to price Asian option
Asian option
An Asian option is a special type of option contract. For Asian options the payoff is determined by the average underlying price over some pre-set period of time...
s by Monte Carlo. In 2001 F. A. Longstaff
Francis Longstaff
Francis A. Longstaff is the Allstate Professor of Insurance and Finance at the Anderson School of Management, University of California, Los Angeles, and the current Finance Area Chair....
and E. S. Schwartz
Eduardo Schwartz
Eduardo Saul Schwartz is a professor of finance at the Anderson School of Management, University of California, Los Angeles, where he holds the California Chair in Real Estate & Land Economics...
developed a practical Monte Carlo method for pricing American-style options.
Methodology
In terms of theoryFinancial economics
Financial Economics is the branch of economics concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment"....
, Monte Carlo valuation relies on risk neutral valuation. Here the price of the option is its discounted
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
; see risk neutrality and Rational pricing: Risk Neutral Valuation. The technique applied then, is (1) to generate several thousand possible (but random) price paths for the underlying
Underlying
In finance, the underlying of a derivative is an asset, basket of assets, index, or even another derivative, such that the cash flows of the derivative depend on the value of this underlying...
(or underlyings) via simulation
Simulation
Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....
, and (2) to then calculate the associated exercise
Exercise (options)
The owner of an option contract may exercise it, indicating that the financial transaction specified by the contract is to be enacted immediately between the two parties, and the contract itself is terminated...
value (i.e. "payoff") of the option for each path. (3) These payoffs are then averaged and (4) discounted to today. This result is the value of the option.
This approach, although relatively straightforward, allows for increasing complexity:
- An option on equityOption (finance)In finance, an option is a derivative financial instrument that specifies a contract between two parties for a future transaction on an asset at a reference price. The buyer of the option gains the right, but not the obligation, to engage in that transaction, while the seller incurs the...
may be modelled with one source of uncertainty: the price of the underlying stock in question. Here the price of the underlying instrument
is usually modelled such that it follows a geometric Brownian motionGeometric Brownian motionA geometric Brownian motion is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion, also called a Wiener process...
with constant drift
and volatilityVolatility (finance)In finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices...
. So: 
, where 
is found via a random sampling from a normal distribution; see further under Black–Scholes. (Since the underlying random process is the same, for enough price paths, the value of a european option here should be the same as under Black Scholes).
- In other cases, the source of uncertainty may be at a remove. For example, for bond optionBond optionIn finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC....
s the underlying is a bondBond (finance)In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
, but the source of uncertainty is the annualized interest rateInterest rateAn interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...
(i.e. the short rate). Here, for each randomly generated yield curveYield curveIn finance, the yield curve is the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S...
we observe a different resultant bond price on the option's exercise date; this bond price is then the input for the determination of the option's payoff. The same approach is used in valuing swaptionSwaptionA swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swaps....
s, where the value of the underlying swapSwap (finance)In finance, a swap is a derivative in which counterparties exchange certain benefits of one party's financial instrument for those of the other party's financial instrument. The benefits in question depend on the type of financial instruments involved...
is also a function of the evolving interest rate. (Whereas these options are more commonly valued using lattice based modelsLattice model (finance)In finance, a lattice model can be used to find the fair value of a stock option; variants also exist for interest rate derivatives.The model divides time between now and the option's expiration into N discrete periods...
, for path dependent interest rate derivativeInterest rate derivativeAn interest rate derivative is a derivative where the underlying asset is the right to pay or receive a notional amount of money at a given interest rate...
s - such as CMOsCollateralized mortgage obligationA collateralized mortgage obligation is a type of financial debt vehicle that was first created in 1983 by the investment banks Salomon Brothers and First Boston for U.S. mortgage lender Freddie Mac. A collateralized mortgage obligation (CMO) is a type of financial debt vehicle that was first...
- simulation is the primary technique employed.) For the models used to simulate the interest-rate see further under Short-rate model; note also that "to create realistic interest rate simulations" Multi-factor short-rate models are sometimes employed.
- Monte Carlo Methods allow for a compounding in the uncertainty. For example, where the underlying is denominated in a foreign currency, an additional source of uncertainty will be the exchange rateExchange rateIn finance, an exchange rate between two currencies is the rate at which one currency will be exchanged for another. It is also regarded as the value of one country’s currency in terms of another currency...
: the underlying price and the exchange rate must be separately simulated and then combined to determine the value of the underlying in the local currency. In all such models, correlationCorrelationIn statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
between the underlying sources of risk is also incorporated; see Cholesky decomposition: Monte Carlo simulation. Further complications, such as the impact of commodity pricesCommodity marketsCommodity markets are markets where raw or primary products are exchanged. These raw commodities are traded on regulated commodities exchanges, in which they are bought and sold in standardized contracts....
or inflationInflationIn economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
on the underlying, can also be introduced. Since simulation can accommodate complex problems of this sort, it is often used in analysing real options where management's decision at any point is a function of multiple underlying variables.
- Simulation can similarly be used to value options where the payoff depends on the value of multiple underlying assets such as a Basket optionBasket optionA basket option is a financial derivative, more specifically an exotic option, whose underlying is a sum or average of different assets. Examples are index options, options on a portfolio, etc....
or Rainbow optionRainbow optionRainbow option is a derivative exposed to two or more sources of uncertainty, as opposed to a simple option that is exposed to one source of uncertainty, such as the price of underlying asset. Rainbow options are usually calls or puts on the best or worst of n underlying assets, or options which...
. Here, correlation between assets is likewise incorporated.
- As required, Monte Carlo simulation can be used with any type of probability distributionProbability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
, including changing distributions: the modeller is not limited to normal or lognormal returns; see for example Datar-Mathews Method for Real Option ValuationDatar-Mathews Method for Real Option ValuationThe Datar-Mathews Method is a new method for Real options valuation. The DM Method can be understood as an extension of the net present value multi-scenario Monte Carlo model with an adjustment for risk-aversion and economic decision-making. The method uses information that arises naturally in a...
. Additionally, the stochastic processStochastic processIn probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
of the underlying(s) may be specified so as to exhibit jumpsJump processA jump process is a type of stochastic process that has discrete movements, called jumps, rather than small continuous movements.In physics, jump processes result in diffusion...
or mean reversion or both; this feature makes simulation the primary valuation method applicable to energy derivativeEnergy derivativeMajor players in energy derivatives include major trading houses, oil companies, utilities, financial institutions.-Definition:An energy derivative is a derivative contract based on an underlying energy asset, such as natural gas, crude oil, or electricity...
s. Further, some models even allow for (randomly) varying statisticalStatistical parameterA statistical parameter is a parameter that indexes a family of probability distributions. It can be regarded as a numerical characteristic of a population or a model....
(and other) parameterParameterParameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
s of the sources of uncertainty. For example, in models incorporating stochastic volatilityStochastic volatilityStochastic volatility models are used in the field of mathematical finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the...
, the volatilityVolatility (finance)In finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices...
of the underlying changes with time; see Heston modelHeston modelIn finance, the Heston model, named after Steven Heston, is a mathematical model describing the evolution of the volatility of an underlying asset...
.
Application
As can be seen, Monte Carlo Methods are particularly useful in the valuation of options with multiple sources of uncertainty or with complicated features, which would make them difficult to value through a straightforward Black–Scholes-style or lattice based computation. The technique is thus widely used in valuing path dependent structures like lookback-Lookback option
Lookback options are a type of exotic option with path dependency, among many other kind of options. The payoff depends on the optimal underlying asset's price occurring over the life of the option. The option allows the holder to "look back" over time to determine the payoff...
and Asian option
Asian option
An Asian option is a special type of option contract. For Asian options the payoff is determined by the average underlying price over some pre-set period of time...
s and in real options analysis
Real options analysis
Real options valuation, also often termed Real options analysis, applies option valuation techniques to capital budgeting decisions. A real option itself, is the right — but not the obligation — to undertake some business decision; typically the option to make, abandon, expand, or contract a...
. Additionally, as above, the modeller is not limited as to the probability distribution assumed.
Conversely, however, if an analytical technique
Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...
for valuing the option exists—or even a numeric technique, such as a (modified) pricing tree
Binomial options pricing model
In finance, the binomial options pricing model provides a generalizable numerical method for the valuation of options. The binomial model was first proposed by Cox, Ross and Rubinstein in 1979. Essentially, the model uses a “discrete-time” model of the varying price over time of the underlying...
—Monte Carlo methods will usually be too slow to be competitive. They are, in a sense, a method of last resort; see further under Monte Carlo methods in finance
Monte Carlo methods in finance
Monte Carlo methods are used in finance and mathematical finance to value and analyze instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining their average value over the range of resultant outcomes. This is usually done...
. With faster computing capability this computational constraint is less of a concern.
Articles
- Boyle, Phelim P., Options: A Monte Carlo Approach. Journal of Financial Economics 4, (1977) 323-338
- Broadie, M. and P. Glasserman, Estimating Security Price Derivatives Using Simulation, Management Science, 42, (1996) 269-285.
- Longstaff F.A. and E.S. Schwartz, Valuing American options by simulation: a simple least squares approach, Review of Financial Studies 14 (2001), 113-148
Software
- FairmatFairmatFairmat is a free-of-charge multi-platform software that allows to model financial contracts or projects with many contingencies by decomposing it into basic parts. Complex structures and dependencies are modeled using a graphical interface...
(freewareFreewareFreeware is computer software that is available for use at no cost or for an optional fee, but usually with one or more restricted usage rights. Freeware is in contrast to commercial software, which is typically sold for profit, but might be distributed for a business or commercial purpose in the...
) modeling and pricing complex options - MG Soft (freewareFreewareFreeware is computer software that is available for use at no cost or for an optional fee, but usually with one or more restricted usage rights. Freeware is in contrast to commercial software, which is typically sold for profit, but might be distributed for a business or commercial purpose in the...
) valuation and Greeks of vanilla and exotic options
External links
- Monte Carlo Simulation, Prof. Don M. Chance, Louisiana State UniversityLouisiana State UniversityLouisiana State University and Agricultural and Mechanical College, most often referred to as Louisiana State University, or LSU, is a public coeducational university located in Baton Rouge, Louisiana. The University was founded in 1853 in what is now known as Pineville, Louisiana, under the name...
- Pricing complex options using a simple Monte Carlo Simulation, Peter Fink (reprint at quantnotes.com)
- MonteCarlo Simulation in Finance, global-derivatives.com
- Monte Carlo Derivative valuation, contd., Timothy L. Krehbiel, Oklahoma State University–StillwaterOklahoma State University–StillwaterOklahoma State University–Stillwater is a land-grant, sun-grant, coeducational public research university located in Stillwater, Oklahoma, USA. OSU was founded in 1890 under the Morrill Act...
- Applications of Monte Carlo Methods in Finance: Option Pricing, Y. Lai and J. Spanier, Claremont Graduate UniversityClaremont Graduate UniversityClaremont Graduate University is a private, all-graduate research university located in Claremont, California, a city east of downtown Los Angeles...
- Option pricing by simulation, Bernt Arne Ødegaard, Norwegian School of ManagementNorwegian School of ManagementBI Norwegian Business School former name BI Norwegian School of Management is the largest business school in Norway and the second largest in all of Europe. BI has in total 6 campuses with the main one located in Oslo.-History:...
- Monte Carlo Method, riskglossary.com