# Fundamentals of Financial Mathematics Exam Questions

## Chapter 1
#### 1. Consider a non-dividing paying stock. Proof that if interest rates are zero (r = 0), the put-call parity for European vanillas is also valid for American vanillas. (Give + proof the put-call parity. Proof the validity for American vanillas.)

#### 2. Why do put and call have the same price as $K = S_0 e^rT$?

Trivial

#### 3. Prove that under the no-arbitrage assumption, the gamma is the same of both a European put and call with the same strike and time to maturity.

Proof: Under no-arbitrage assumption, we have the put call parity for European options:
$$S(0) - exp(-rT)K +EP(K,T) - EC(K,T) = 0 $$
Taking derivative of the option prices with respect to stock price:
$$ 1 + \frac{\partial{EP(K,T)}}{\partial S} -\frac{\partial{EC(K,T)}}{\partial S} = 0 $$
Taking second-order derivative:
$$\frac{\partial^2 EP(K,T)}{\partial S^2} = \frac{\partial^2 EC(K,T)}{\partial S^2}$$

#### 4. Give Calendar Spread and prove.

## Chapter 2

#### 1. Give a detailed explanation of the pricing of a 3-step binomial tree model of an American call/put option.

#### 2. Discuss pricing or 2-step trinomial American Put.

#### 3. Prove that exercising an American Call is never optimal, using non arbitrage argument.

## Chapter 3

#### 1. Explain the relationship of an equivalent martingale measure with an arbitrage free market and complete model. Give an example of an arbitrage free, incomplete model.

(1). No-Arbitrage Theorem: A market is arbitrage free if and only if at least one equivalent martingale measure exists.
(2). Completeness Theorem: An arbitrage-free market is complete (all contingent claims can be replicated) if and only if there exists a unique equivalent martingale measure.

Example: Trinomial tree pricing model is arbitrage free, incomplete model, because there exists more than one EMM.

#### 2. What is the difference between real world and risk neutral world? What does this mean for the binomial tree model and the black scholes model?

(1) In financial engineering there are essentially two different worlds. The so-called, real-world (sometimes also referred to as the historical or physical world or Pworld) and the pricing world (sometimes also referred to as the the risk-neutral world or Q-world or P∗-world).

Most of the times, the probability measuring how things happen in the realworld is denoted with a P; therefore also this real-world is often named the Pworld. This probability measure is measuring how things actually happen in reality and one refers often to it as the physical measure. Therefore, one is typically estimating distributions in this world on the basis of historically observed real data of the underlying asset, like daily log-returns of a stock. In contrast, the risk-neutral world is the pricing world, created by financial markets trading event risk and modelled by financial engineers. Most of the times, the probability measure of how things happen in the pricing world is denoted with a Q and one refers to it as the pricing measure. One can prove that in absence of arbitrage, under this measure, traded assets all behave in a ”risk-neutral” way, meaning that there expected return is equal to the return of the risk-free account.

In many applications the actual P is irrelevant, one tries to estimate directly the special pricing measure Q, via a so-called calibration procedure. The information to estimate Q comes from observable market prices of derivatives. Since, these market quotes are discounted expectations under one particular Q of the related payoffs of the derivatives, one can try to estimate this underlying probability measure Q from these market quotes.

(2) Binomial tree model and Black-Scholes model are complete models, which means there exists unique EMM in Binomial tree model and unique EMM in BS model.

The Black-Scholes model and the Binomial tree model is an example an arbitrage-free and complete model. However, all other more advanced models are not complete and there can be more (often infinitely many) measures under which the underlying asset behaves risk-neutrally. Important is however, that there is one such special Q under which all the derivatives are priced, i.e. the price of any derivative on the underlying asset is given by the discounted expected payoff of the derivative under Q. It is therefore why one refers to such a measure as the pricing measure. This pricing measure is as mentioned given/determined by the market.

## Chapter 4

#### 1. True or False: The price of an ODBC is always higher than IDBC if $H <K \leq S_0$.

## Chapter 5

#### 1. Give the Black-Scholes stochastic differential equation. Explain where it comes from (what it represents) and that the solution is a geometric Brownian motion (give the solution).

#### 2. Derive the Black-Scholes partial differential equations for the price O for options: $\frac {\partial O} {\partial t} + \frac {1}{2} \sigma^2 S^2 \frac{\partial^2 O} {\partial S^2} + r S \frac{\partial O} {\partial S} - r O = 0$. (Proof that this is true, using a lemma. Your answer should be a mixture or correct mathematical formulas and sentences.)

#### 3. Explain relationship between Itô's lemma, Feyman-Kac and Black & Scholes.

#### 4. True or False: the integral of $\int W_s dW_s = \frac{W_s}{2}$.
Answer: False.

#### 5. Give BS formula for EC.

#### 6. Given the BS SDE $dX = \mu dt + \sigma dW$, derive the BS PDE, using a portfolio hedging argument.

#### 7. Relate the BS SDE to statement "the total change in the value of a delta-hedged portfolio is equal to zero on average." (The last question on exam)

See P117