Monte carlo option pricing final v3

Monte Carlo Simulation: Option Pricing

Acedo Fabia Reyes Sorbito Vidamo

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

• to determine quantities such as expected returns, risk, possible downsides, probabilities of making certain profits or losses, etc.

Exploring portfolio or cash flow statistics

• to calculate the present value of the expected payoff of an option under a risk-neutral random walk

Pricing Derivatives

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

• Flexible• Can generally be easily

extended and developed as required

• Easily understood

Advantages

• May be computationally intensive

• Calculations can take much longer than analytical models

• Solutions are not exact, but depend on the number of repeated runs

Disadvantages

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 Index

Given:

𝑆 𝑡 = 1,000 ( )𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑝𝑟𝑖𝑐𝑒 𝑋 = 1,000 ( )𝑠𝑡𝑟𝑖𝑘𝑒 𝑝𝑟𝑖𝑐𝑒

𝑟 = 6% . ., 𝑝 𝑎 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑙𝑦 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑒𝑑( − )𝑟𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑟𝑎𝑡𝑒 𝑇 = 1 250 ( 𝑦𝑟 𝑜𝑟 𝑡𝑟𝑎𝑑𝑖𝑛𝑔 𝑑𝑎𝑦𝑠 𝑒𝑥𝑝𝑖𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓

)𝑡ℎ𝑒 𝑜𝑝𝑡𝑖𝑜𝑛 𝜎 = 15.9% ( )𝑎𝑛𝑛𝑢𝑎𝑙 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦

Application: Option PricingExample 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)

0

50

100

150

200

250

Relative frequency distribution : FTSE Index returns (1984-1992)

0.00

0.20

0.40

0.60

0.80

1.00

Cumulative frequency distribution : FTSE Index returns (1984-1992)


Stage 2: Imitate the input variables

• Generate a large number of uniformly distributed random numbers ranging from 0 to 1

• Each random variable is located on the y-axis of the cumulative density function and the corresponding return on the x-axis is taken

0.00

0.20

0.40

0.60

0.80

1.00

Cumulative frequency distribution : FTSE Index returns (1984-1992)

For example, random number 0.472


Stage 3: Simulate the underlying variable

Adjust 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 rate

000189.0

250060201006940 2

25006022

r

..r

ee .σr

where σ2 is the variance of FTSE index daily returns

We need to solve for r, such that


Stage 3: Simulate the underlying variable

Compound 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.

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

2502125021 10001000 rrrrrT

reeeeS


Stage 4: Repeat the process

Repeat Stages 1 to 3 many times (say, 125,000 times) and compute the value of the call option at time T, (CT).

1

•Simulate future values (ST)

2

•Calculate CT = max [ST - X, 0]

3

•The average of all 125,000 CT , when discounted is the option value

Reduce the variability of the mean to be compatible with your accuracy requirements:


Stage 5: Increase accuracy

2

SE

Sn t

Rule of thumb for determining the number of runs:

It follows that to reduce the standard error by a factor of 10, the number of simulations must be increased by a one-hundred fold

n

jjT S

nS

1

1 2

1

1var

n

jTjS SS

nj nSE j

T

S

S

var


Stage 5: Increase Accuracy

Each time random variable r is drawn, its complement 1-r calculated and used to drive a parallel run of the simulation.

This tends to lead to negatively correlated output values; hence lower variance

Antithetic variate technique

Find an option that is both highly correlated to the one you’re trying to estimate and has a definite value.

Control variate technique

Variance Reduction Techniques:

Examples: Path Dependent Options

Asset: S = 100 Strike price : X = 105

Step: 0.01 Drift: 5% Volatility: σ = 20%

Simulation:δS = rS δt + σS δt φ

then transformed into lognormal S(t + δt) = S(t) + δS

S(t + δt) = S(t) exp((r − 1/2σ2) δt + σsqrt(δt) φ)

XLS formula: Sim = So*EXP((rate-0.5*σ*σ)*step +σ*SQRT(step)*NORMSINV(RAND()))

Option Value:option value = e−r(T−t) E [payoff(S)]

XLS formula: PV = Mean*EXP(rate*ST)

Notes:

Taking relatively small number of paths

250 simulations for a one year pay off

Pay off:Call Payoff = MAX(ST - X,0)Put Payoff = MAX(X -ST ,0)

Examples: Path Dependent Options Time Sim1 Sim2 Sim3 Sim4 … Sim249 Sim250 0 100.00 100.00 100.00 100.00 … 100.00 100.00 0.01 100.71 100.36 99.84 99.26 … 97.41 98.32 0.02 101.31 100.66 98.61 101.81 … 97.17 94.99 0.03 105.43 99.23 97.72 105.42 … 95.68 96.04 0.04 108.38 97.93 96.81 108.78 … 95.18 96.79 0.05 107.00 93.64 97.27 109.51 … 97.23 96.57 0.06 106.35 94.31 98.53 107.92 … 95.21 96.16 0.07 108.95 95.88 99.76 108.67 … 95.22 96.12 . . . . . . . . 0.08 109.61 96.85 98.34 109.81 … 92.74 101.18 0.98 111.24 122.08 124.02 139.08 … 127.40 89.89 0.99 110.41 118.50 122.21 141.00 … 128.76 86.23 1.00 109.82 120.11 119.42 139.42 … 129.31 87.60

CALL Payoff 4.82 15.11 14.42 34.42 24.31 0.00Mean 8.08

PV 8.50 PUT Payoff 0.00 0.00 0.00 0.00 0.00 17.40

Mean 8.90 PV 9.35

Examples: Exotic Options

ST

average over entire run

Optimum value for a callA knock in an option could

become active only when the stock crosses

this line

Asian Lookback Barrier

Asian option payoffs are calculated using the

average over the entire run rather than the

value of the underlying at time t.

Fixed Lookback option payoffs are calculated using

the optimal value (maximum for a call, minimum for a put)

Barrier options either knock in or knock out

above or below a certain barrier.

Simulation at run one:

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%20Defence%20Systems%20(GR-77)/p4c04.pdf

Thank You!

Monte carlo option pricing final v3

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