Session 11 - Binomial Trees (II) & Delta

During this session we extended the binomial tree methodology to the pricing of European Puts and American Options. Remember that it is never optimal to exercise before expiry an American Call (w/o dividends). However, it may be optimal to exercise an American put (w/o dividends).

We also reviewed Delta (how the premium of the option changes when the price of the underlying asset changes). We saw how to form a Delta Neutral position by hedging with options and what is the role of Delta when calculating the hedging ratio.

Have a look at the presentation here and to an example of Delta Neutral portfolio here.

Session 10 - Binomial trees

Remember: there are 3 basic ways to price options: closed formulas (like Black-Scholes), some kind of trees or Montecarlo simulations. During this class, we reviewed how to price a vanilla call with a binomial tree.

We assume a very simple world where the price of the underlying asset can only have two potential scenarios: up or down.

- First we expand the price of the underlying asset in the tree assuming a certain percentage for upward and downward movement.
- Then we price the option at the end of the tree. Given that we are at maturity, we can basically apply the formula for the payoff of the option (the outcome of exercising the option).
- We form a riskless portfolio by buying Delta shares and selling an option (a replicating portfolio). We calculate Delta making the portfolio riskless: it does not matter where the share goes, the value of my portfolio will always be the same.
- If the portfolio is riskless, we can bring it to present value using a risk-free rate. Now we know the value of the portfolio today and the value of the Delta shares today, so we can solve for the price of the option.

One important feature is that it does not matter what the real world probabilities are as we are working in a risk neutral world. In fact, a different way to price the option would be to solve for the expected value of the option using risk-neutral probabilities. The expected price for the share using this risk-neutral probabilities will be the forward.

We extended this methodology by increasing the number of branches in the tree.

Have a look at the presentation here and at the examples we saw in class here.

Session 9 - Strategies with options

We had a look at different combinations of options. The presentation can be found here. An Excel file with some of the strategies can be found here.

Session 8 - Basics on Options

At the very beginning of the session we had a look at Markets and we determined that EURUSD was moving a lot due to the recent activity in short-term interest rates in EUR and USD. This relationship is called the interest rate parity. I have prepared a chart with the following data: x = difference in implied yield for 2 year bond futures (US and EUR); y = EURUSD.

As you can see from R^2, the relationship is quite strong. The chart includes data from Jan-2011 until today.

According to the model, EURUSD should be trading at around 1.2961 given the current level of interest rates in EUR and US.

During this session we reviewed some basics on options. Remember, options provide a right to those who buy them and an obligation to those who sell them. They can be traded both in Exchanges and in OTC Markets.

We saw in class that options can be vanilla or exotic and we also saw that the premium of an option can be decomposed into Intrinsic Value (=payoff of the option) and Time Value (depends on volatility and other parameters).

We also had a look at put-call parity, a way to value option applying no-arbitrage assumptions.

The presentation can be found here.

Session 7 - Derivatives - FRAs & Eurodollar Futures

During this class we learned what a zero-coupon rate is and how to calculate discount factors (have a look at the Excel file).

We also reviewed how to calculate forward discount factors and forward rates avoiding potential arbitrage opportunities. A forward rate is a rate to be applied to a forward starting loan/deposit. The idea behind this concept is that if I do a 1) 12m deposit or 2) a 6m deposit and I reinvest the proceeds in a new forward starting 6m deposit, the outcome should be the same.

Additionally, we had a look at how FRAs (Forward Rate Agreements) and Eurodollar Futures work.

The presentation can be found here.

Session 6 - Derivatives - 2014

During this session we reviewed how to price forwards on currencies. Please, have a look at the Excel file. The idea is exactly the same as with other asset classes: if the forward is too expensive (if the forward price should be 1.3062 and it is trading at 1.28 - I can buy less USD for the same amount of EUR), I will sell USD forward; if the forward is too cheap (if the forward price should be 1.3062 and it is trading at 1.32 - I can buy more USD for the same amount of EUR), I will buy USD forward. There is only one no-arbitrage possibility: F = S * e ^ (rd - rf).

Additionally, we had a look at how we should price forwards on consumption assets. In this case, we must take into account any potential convenience yield (for instance, to avoid shortages of the product that could affect our production line) and, also, any storage cost.

Remember the logic behind of the formula:

F = S * e ^ (+ any potential cost - any potential income)

The first day we buy/sell a Forward, the MTM of the position (value) is equal to zero. Remember we do not have to pay anything when we buy/sell a Forward. However, as time goes by and as the underlying asset and interest rates move, the MTM of the Forward will change. Basically, we will compare the price at which we can buy/sell at Maturity with the current Forward price and we will bring the difference to present value.

Finally, in spite of the fact that we will assume both Futures prices and Forward prices to be the same during the course, we reviewed why they are not (correlation between interest rates and price of the underlying asset, different interest rates, credit risk). You can find the presentation here.

Session 5 - Derivatives - 2014

We started this session reviewing how simple interest rates and compounded interest rates work. Remember that for this course we will use continous compounding (discount factor = e ^ (-r*T); future value = e ^ (r*T)). It is interesting to remember also how to work with Natural Logs. Remember that Ln e ^(r*T) = r*T.

When we value Futures, we consider an Eonia curve (Eonia is a riskless interest rate and Futures are riskless due to the margining process). When we value forwards, we consider a Euribor curve. Additionally, we would have to charge an additional spread that depends on the creditworthiness of the counterparty (CVA).

We valued Forward for an asset producing no income assuming a no-arbitrage hypothesis. If the forward is too expensive, we can sell (short) the forward and ask for a loan to buy the underlying asset. At maturity, we give the asset to the person who bought the forward from us and we make a riskless profit. So, if the forward is expensive, many arbitrageurs will enter into this strategy, taking the price of the forward down.

If the forward is too cheap, we buy the forwards and short the underlying asset. We will make a deposit with the amount that we obtain by shorting the asset. At maturity, we buy the asset from the person who sold us the forward and we give it back through the short contract. We would make a riskless profit. So, if the forward is cheap, many arbitrageurs will enter into this strategy, taking the price of the forward up.

The only possible value to avoid any potential arbitrage opportunity would be F = S*e ^ (r*T). Please review the Excel file. Khan Academy explains this potential arbitrage opportunity here and here.

We applied the same logic to assets producing a discrete income and to assets producing a yield. You can find the presentation here.