Summary: Derivatives

Read the summary and the most important questions on Derivatives

• 1 Week 1

• 1.1.1 Binomial Model and Risk-Neutral pricing

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• In the pricing equation, what can we say about q and 1-q?

  Q and 1-q can be interpreted as probabilities:
  

  • Sum is equal to 1
  • positive if u > 1+r > d
  • if this is NOT the case -> arbitrage opportunity

• Risk neutral pricing: 

  if there are no arbitrage opportunities, the price of every redundant asset is equal to the expectation under the risk-neutral probability measure  of its discounted future payoff

• What are the risk neutral probabilities in a binomial tree model?

  .

• Why does risk neutral pricing NOT imply risk-neutral investors?

  • q and 1-q are artificial probabilities (and not real probabilities)
    
  • Risk neutral pricing is based on replication and no-arbitrage and so on non-satiated investors and NOT on risk-neutrality (no assumptions about risk attitude of investors are made)
    

• Risk-neutral pricing is relative pricing:

  Derivative price is determined relative to stock price, interest rate and dynamics of the stock price, these are given exogenously

• Discounted derivative prices are Q-martingales:

  The expected future value is equal to the value today, or, stated differently, the process does on average neither increase nor decrease

• Application of martingales in option theory: under the risk neutral measure Q:

  o Discounted stock prices grow on average with 0
  o Stock prices grow on average with r

• Normalizing derivative prices: 

  Normalizing derivative prices typically refers to the process of scaling the prices of different derivative contracts to a common basis, usually a standard unit or a percentage. This is done to compare the prices of different derivatives with each other, or to compare the prices of a single derivative at different points in time.

• Theorem: American call on underlying that pays no dividends:

  If the underlying pays no dividends and if the interest rate is positive, it is never optimal to exercise and American call before maturity

• Theorem (American call on a dividend paying stock): 

  if the stock pays dividends and interest rates are positive, it might be optimal to exercise an American call immediately before the dividend payment