Stratégies Options - Monte Carlo Simulation : A First Attempt

Several ways to price options that lead to the same results.....
How to price a european option using Monte Carlo

I - Theory:

Call put parity sows us that european style options are priced on the forward, that is, as if underlying has a drift equal to riskfree rate.
Black and Scholes model as well as binomial model start from a delta hedged portfolio that leads to risk neutral valuation.

An option is just the actualized value of an expectation under a risk neutral probability.

To get the risk neutral element in simulations, it's enough to make the asset drift equal to risk free rate.

II - The goal:
The main purpose of Monte Carlo simulation in option pricing is
- to simulate paths including asset volatility, maturity, dividends or revenues.
- to compute each final payoff
- to compute average payoff
- to 'present value" that outcome

III - Now let's go:

If the point is to price a 1 year call struck at 100 with interest rate set at 5% with a volatility at 30%,
- µ=5% ou 0.05
- annualized volatility σ = 30% ou 0.30

- In cells E21, "=5%"
- In cells G23, "=30%"
dt=1/365

Starting spot =: 100
- In cells B28, "0"
- In cells C28, "100"
- In cells B29, "1"
- In cells C29, "=C28*(1+$E$21*(1/365)+$G$23*sqrt(1/365)*LOI.NORMALE.STANDARD.INVERSE(ALEA()))"
- In cells B30, "2"
- In cells C30, "=C29*(1+$E$21*(1/365)+$G$23*sqrt(1/365)*LOI.NORMALE.STANDARD.INVERSE(ALEA()))"
- In cells B31, "3"
- In cells C31, "=C30*(1+$E$21*(1/365)+$G$23*sqrt(1/365)*LOI.NORMALE.STANDARD.INVERSE(ALEA()))"

....

- In cells B392, "364"
- In cells C392, "=C391*(1+$E$21*(1/365)+$G$23*sqrt(1/365)*LOI.NORMALE.STANDARD.INVERSE(ALEA()))"
- In cells B393, "365"
- In cells C393, "=C392*(1+$E$21*(1/365)+$G$23*sqrt(1/365)*LOI.NORMALE.STANDARD.INVERSE(ALEA()))"

Now the payoff :
- In cells C395, "=MAX($C$393-100;0)

Now it's just about to run more and more simulations....

Just doing the mean:
- In cells C399, "=MEAN(C395:HZ395)"
- In cells B400, "Call Value "
- In cells C400, "=EXP(-E21)*C399"

For example we find here 13.78205 for that simulation (the true value is 14.23 using Black & Scholes closed form).

F9 for another simulation.

Next : The Greeks : A First Attempt
Previous : Asset Path Simulation

Télécharger le pricer : ici

OPTIONS 101 - INDEX
OPTIONS 101 - CHAPTER I
OPTIONS 101 - CHAPTER II
OPTIONS 101 - CHAPTER III
OPTIONS 101 - CHAPTER IV
OPTIONS 101 - CHAPTER V
OPTIONS 101 - CHAPTER VI

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