The first and second definition is quite synonymous.
3 important concepts:Find an algorithm for how the most basic investments evolve randomly. Equities: often the lognormal random walkcan be represented on a spreadsheet or in code as how a stock price changes from one period to the next by adding on a random return. Fixed-incomeBGM model in modeling how interest rates of various maturities evolveCredit A model that models the random bankruptcy of a company. Can represent any interrelationships between investments which can achieved through correlations.Understand the derivatives theory for after performing simulations of the basic investments, there is a need to have models for more complicated contracts that depend on them such as options/derivatives/contingent claims. May be able to use the results in the simulation of thousands future scenarios to examine portfolio statisticsIe. how classical Value at Risk can be estimated
Risk-neutrality assumption – We make the assumption that investors are risk neutral, i.e., investors do not increase the expected return they require from an investment to compensate for increased risk.Cox and Ross (1976) have shown that the assumption of risk neutrality can be used to obtain solutions of option valuation problems.’ This implies that the expected return on the underlying asset is the risk-free rate and that the discount rate used for the expected payoff on an option is the risk-free rate.
In this stage, we want to convert the Random Variable with Normal distribution to that of a Random Variable with an identical distribution to the empirical distribution. To do this:From a normal distribution, we transform it with a lognormal distribution.The mean of the lognormal distribution is assumed to be the mean daily return.This will be equated to the continuously compounded daily rate of return rate since we make an assumption that the option is priced within a risk neutral framework. Then we compute for new r or mean of the random variable. It will lead to:New random observations of daily returns with: Current asset price is compounded by the daily equivalent of 6%paProbability distribution maintains it shape but is shifted to the left as the mean is lower.
Monte carlo option pricing final v3
Monte Carlo Simulation• A method that numerically ‘imagines’ many possible scenarios to solve deterministic and probabilistic problems• A numerical method which enables the modeling of the future value of a variable by simulating its behavior over time.• It calculates statistical properties such as expectations, variances or probabilities of certain outcomes.• The method is usually quite simple to implement in basic form and so is extremely popular in practice.
Finance Application• represents the future behavior of equities, exchange rates, interest rates etc., for two reasons – study the possible future performance of a portfolio – price derivatives Exploring portfolio or Pricing Derivatives cash flow statistics • to determine quantities • to calculate the present such as expected value of the expected returns, risk, possible payoff of an option under downsides, probabilities a risk-neutral random of making certain profits walk or losses, etc.
Monte Carlo Simulation and Option Pricing• The Monte Carlo technique on option pricing, first proposed by Boyle (1977), simulates the process generating the returns on the underlying asset and invokes the risk-neutrality assumption.• The method is “simple and flexible … it can easily be modified to accommodate different processes governing the underlying asset returns.”• The accuracy of the results depends on the number of simulations
MCS to Option Pricing: Pros and Cons Advantages Disadvantages • May be computationally intensive• Flexible • Calculations can take much• Can generally be easily longer than analytical extended and developed as models required • Solutions are not exact, but• Easily understood depend on the number of repeated runs
Stages of a Monte Carlo Simulation• Identify the probability distribution of the input variables.• Imitate the movement of the input variables by repeatedly drawing random numbers, which are adjusted to have the same probability distribution as the underlying variables.• Simulate the underlying variable by combining the input variables together according to the logic of the system.• Repeat this process many times to get the simulated future value.• Increase accuracy by applying variance reduction techniques
Application: Option Pricing Suppose we want to value a 1-year European call option on the FTSE IndexGiven: = 1,000 ( ) = 1,000 ( ) = 6% . ., ( − ) =1 250 ( ℎ ) = 15.9% ( )
Application: Option Pricing Example Stage1 Stage2 Stage3 Stage4 Stage5 Stage 1: Identify the probability distribution Transform uniformly distributed random variables into a random variable with a probability distribution that matches the empirical distribution of the option’s underlying asset (FTSE index) Relative frequency distribution : FTSE Index returns (1984-1992) Cumulative frequency distribution : FTSE Index returns (1984-1992)250 1.00200 0.80150 0.60100 0.40 50 0.20 0 0.00
Application: Option Pricing Example Stage1 Stage2 Stage3 Stage4 Stage5 Stage 2: Imitate the input variables Cumulative frequency distribution : FTSE Index returns (1984-1992) • Generate a large number of1.00 uniformly distributed0.80 random numbers ranging0.60 from 0 to 10.400.20 • Each random variable is located on the y-axis of the0.00 cumulative density function and the correspondingFor example, random number 0.472 return on the x-axis is taken
Application: Option Pricing Example Stage1 Stage2 Stage3 Stage4 Stage5 Stage 3: Simulate the underlying variableAdjust the observed daily return given the assumption of a risk neutral framework, i.e., we assume that the return on the FTSE index is equivalent to the risk-free rateWe need to solve for r, such that σ2 2 er e 0.06 250 r 0.01006942 2 0.06 250 r 0.000189where σ2 is the variance of FTSE index daily returns
Application: Option Pricing Example Stage1 Stage2 Stage3 Stage4 Stage5 Stage 3: Simulate the underlying variableCompound the current asset price by the random daily returns for each of the trading days during the life of the option. This is equal to 1 simulation run. r250 ST 1000 e r1 e r2 e r250 1000 e r1 r2 where rn = random observation of the 1-day continuously compounded return drawn from the same empirical probability distribution as the underlying data n = number of trading days
Application: Option Pricing Example Stage1 Stage2 Stage3 Stage4 Stage5 Stage 4: Repeat the processRepeat Stages 1 to 3 many times (say, 125,000 times) and compute the value of the call option at time T, (CT). • Simulate future values (ST) 1 • Calculate CT = max [ST - X, 0] 2 • The average of all 125,000 CT , when discounted is the option 3 value
Application: Option Pricing Example Stage1 Stage2 Stage3 Stage4 Stage5 Stage 5: Increase accuracy Reduce the variability of the mean to be compatible with your accuracy requirements: 1 n 1 n 2 varS j ST Sj varS j Sj ST SEST n j 1 n j 1 n Rule of thumb for determining the number of runs: 2 St n SEIt follows that to reduce the standard error by a factor of 10, the number of simulations must be increased by a one-hundred fold
Application: Option Pricing Example Stage1 Stage2 Stage3 Stage4 Stage5 Stage 5: Increase Accuracy Variance Reduction Techniques:Antithetic variate technique Control variate techniqueEach time random variabler is drawn, its complement 1-rcalculated and used to drive a Find an option that is bothparallel run of the simulation. highly correlated to the one you’re trying to estimateThis tends to lead to negatively and has a definite value.correlated output values;hence lower variance
Examples: Path Dependent Options Asset: S = 100 Drift: 5% Step: 0.01 Strike price : X = 105 Volatility: σ = 20% Notes: Simulation: δS = rS δt + σS δt φ Taking relatively small then transformed into lognormal number of paths S(t + δt) = S(t) + δS S(t + δt) = S(t) exp((r − 1/2σ2) δt + σsqrt(δt) φ)250 simulations for a one year pay off XLS formula: Sim = So*EXP((rate-0.5*σ*σ)*step +σ*SQRT(step)*NORMSINV(RAND())) Pay off: Option Value:Call Payoff = MAX(ST - X,0) option value = e−r(T−t) E [payoff(S)]Put Payoff = MAX(X -ST ,0) XLS formula: PV = Mean*EXP(rate*ST)
Examples: Exotic Options Simulation at run one: Asian Lookback Barrier A knock in an Optimum value for a call option could ST become active only when the stock crosses this line average over entire runAsian option payoffs are calculated using the Fixed Lookback option Barrier options eitheraverage over the entire payoffs are calculated knock in or knock out run rather than the using the optimal value above or below a certainvalue of the underlying (maximum for a barrier. at time t. call, minimum for a put)
More on Options Pricing• Any option that has a payoff formula can be priced with a Monte Carlo method.• Very versatile - Multiple underlying, changing volatility, a different distribution assumption for the random walk, etc.• For some options, like the Asian option, variance reduction strategies can make the Monte Carlo method very accurate at low trial sizes – meaning that it is both accurate and quick.• Besides being used for these particular options, this method is also often used to double check other implementations.
Conclusion• The Monte Carlo method is primarily used for pricing of several kinds of exotic options for which there is no formula or the formula is difficult.• It is also good as a way to double check any implementation of option pricing.• Unfortunately, there aren’t really any programs that simply do Monte Carlo approximations of options you basically have to write an implementation yourself, or copy code from someone.
References:• Boyle, P (1977) Options: a Monte Carlo approach. Journal of Financial Economics 4 323–338• Glasserman, P (2003) Monte Carlo Methods in Financial Engineering. Springer Verlag• Jackel, P (2002) Monte Carlo Methods in Finance. John Wiley & Sons• Watsham and Parramore (1997) Quantitative Methods in Finance• Wilmott, P. Frequently asked Questions in Quantitative Finance• Wilmott,P. Paul Willmot Introduces Quantitative Finance. p581-604• Monte-carlo Simulation. http://www.sars.org.uk/old-site- archive/BOK/Applied%20R&M%20Manual%20for%20Defenc e%20Systems%20(GR-77)/p4c04.pdf