Exotic Options: An Introduction to Pricing with Monte Carlo Simulations

Question 1

What are the two key conditions that should hold in a risk neutral world?

Answer 1

1. The expected return of any investment (including risky ones such as stocks) must equal the risk-free rate
2. The discount rate used for expected payoffs in the future, even if the payoff is uncertain, is the risk-free rate

Question 2

What are the key steps to using Monte Carlo simulation for option pricing?

Answer 2

1. Define the random process for the underlying asset price (consistent with Risk Neutral Valuation)
2. Simulate a large number of price paths and calculate the corresponding payoff
3. Calculate the expected (average) payoff and discount it

Question 3

How to simulate S_t+Δt

Answer 3

S_t+Δt = S_t * e ^{[r-q-((σ^2)/2)]*Δt + σ * ε * sqrt(Δt)}

Where the power of e is the return over period Δt, where r is the risk-free rate, q is the continuous dividend yield
For calculating S₂ you start from S₁, to calculate S₃ you start from S₂ etc.

Question 4

What are the steps to simulate a price path in an Excel worksheet for MC simulation?

Answer 4

1. Extract ε from a standard normal distribution
2. Calculate the return over Δt
   R_Δt = lnS_t+Δt - lnS_t = [r-q-0.5 * σ²] * Δt + σ * ε * sqrt(Δt)
3. Calculate S_t+Δt = S_t * e^R

Question 5

How to extract ε from a standard normal distribution in an Excel worksheet?

Answer 5

1. Extract a random number from a [0;1] uniform distribution -> =RAND()
2. Use the inverse function of the cumulative standard normal distribution to obtain ε

For a non-normal distribution just use the inverse of the appropriate cumulative function

Question 6

What happens after we simulate a price path of the underlying for Monte Carlo valuation?

Answer 6

• After having simulated a sufficient number of paths, the average payoff can be calculated based on the price of the underlying.
• The payoff can then be discounted at the risk-free rate according to risk-neutral valutaion (assuming the option premium is paid upfront)

Question 7

How do we determine how many simulation is enough for Monte Carlo valuations?

Answer 7

Convergence! Whenever the estimated price doesn’t change much if I add other simulations.

Question 8

How to obtain Greeks with Monte Carlo simulations?

Answer 8

Greeks can be obtained numerically by running multiple parallel simulations (with the same vector for ε values) with a slightly altered initial pricing input.