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.
Tuesday, October 21, 2014
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.
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.
Sunday, October 12, 2014
Session 9 - Strategies with options
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.
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.
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