### Questions

**1)** What is Monte Carlo Simulation ?

This is a technique that uses statistical sampling to approximate solutions to quantitiative problems.

It simulates the underlying process and then calculates an average result.

**2)** How can you improve the precision of a Monte Carlo Simulation ?

Increase the number of trials.

**3)** What type of model would you use to price the following instruments ?

Option on an Equity - Black Scholes

Option on a Bond -

Option on a Currency - Black Scholes

Option on a Commodity - Black

Option on an Interest Rate Instrument -

Option on a Future or Forward - Black

**4)** What important assumptions does the Black Scholes model make ?

*) the returns on the underlying asset are normally distributed

*) the price of the underlying asset follows a geometric brownian motion

*) the options are european and can only be accessed at maturity

*) the risk free interest rate is known and is constant

*) the volatility of the underlying is known and is constant.

*) no dividends are paid on the underlying asset

**5)** Can you write down the Black Scholes Equation ?

This equation has infinitely many solutions so we need to impose boundary conditions.

**6)** What is the Blacks model ?

This is a variation of the Black Scholes model where the spot price of the underlying asset is replaced by a discounted futures price.

This formula can be derived from the Margrabes formula

This can be described as a log-normal forward model.

Can be used to price STIR Options

**7)** What is the difference between the Black Scholes model and the Blacks model ?

Black Scholes - focuses on the diffusion of the Spot price on the assumption of Geometric Brownian Motion. This model is used when the spot price is easily available and the drift term is just the risk free rate (Equivalent Martingale Measure)

In equity and FX markets,

Black - This model is used when the Spot price is not easily available and determining the drift term is quite complicated.

**8)** Can you give some examples of One-Factor Short Term Interest Rate Models ?

Merton (1973) -

Vasicek (1977) -

Rendleman-Bartler (1980) -

Cox-Ingersoll-Ross (1985) -

Ho-Lee (1986) -

Hull-White (1990) -

Black-Derman-Toy (1990) -

Black-Karasinski (1991) -

Kalotay-Williams-Fabozzi (1993) -

**9)** Can you describe an Arbitrage Free Option Pricing model ?

Also called yield curve option pricing model.

These models can accommodate different volatility assumptions along the yield curve.

**10)** What is the Vasiceks model ?

This is a model that can be used to price options on bonds.

The short rate is assumed to satisfy a stochastic differential equation.

This is a special version of the Ornstein-Uhlenbeck process with constant volatility.

Interest rates can be negative

it is mean reverting

uses a normal distribution ?

**11)** What is the Cox-Ingersoll-Ross model ?

Volatility depends also on the square root of the short rate ensuring the interest rate is not negative.

Use chi-squared distribution

**12)** What is the Black-Derman-Toy model ?

Uses discretely compounded interest rates

**13)** What is the Garman-Kohlhagen model ?

This is a model for pricing European Options on Currencies.

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