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.
Showing posts with label valuation. Show all posts
Showing posts with label valuation. Show all posts
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, September 29, 2013
Clase 6 curso 2013-2014 - Forwards
En las últimas dos clases hemos estado revisando como se valoran forwards y futuros.
La base que utilizamos en la valoración es la asunción de "no-arbitraje". Esta asunción nos permite determinar el precio de los forwards/futuros asegurando que, si está caro, el Mercado arrastrará el precio abajo, mientras que, si está barato, el mercado llevara el precio hacia arriba.
Como hemos visto durante estos días, cuando hay costes (el coste de oportunidad que representa el tipo de interés, el coste de almacenamiento en los commodities...), el precio Spot se capitaliza a esos costes; cuando hay rendimientos (dividendos, cupones de bonos, rendimientos de conveniencia, el tipo de interés en otra moneda...), el precio Spot se descuenta a esos rendimientos.
He incorporado una hoja Excel en la que podéis ver cómo valorar un forward de divisa asumiendo que no hay arbitraje posible.
La última presentación que hemos revisado en clase podéis encontrarla aquí.
La base que utilizamos en la valoración es la asunción de "no-arbitraje". Esta asunción nos permite determinar el precio de los forwards/futuros asegurando que, si está caro, el Mercado arrastrará el precio abajo, mientras que, si está barato, el mercado llevara el precio hacia arriba.
Como hemos visto durante estos días, cuando hay costes (el coste de oportunidad que representa el tipo de interés, el coste de almacenamiento en los commodities...), el precio Spot se capitaliza a esos costes; cuando hay rendimientos (dividendos, cupones de bonos, rendimientos de conveniencia, el tipo de interés en otra moneda...), el precio Spot se descuenta a esos rendimientos.
He incorporado una hoja Excel en la que podéis ver cómo valorar un forward de divisa asumiendo que no hay arbitraje posible.
La última presentación que hemos revisado en clase podéis encontrarla aquí.
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