# Fixed Income

If 10-year interest rates are at 3%…and its volatility is at 100 bps…how many standard deviations away from a yield of 1% are you?  If you increase volatility to 200 bps does it become more or less likely to reach 1%?

Does this change in volatility make the bond more or less risky?  Explain why.

Please try to create a two-branch binomial tree for a stock price.

Assume a 1-year option expiry.

Using a volatility of 25%

What is the probability up and down for each step?  What are your up and down amounts?

What would the price of a call option struck at 110% of the forward cost?  (Hint: Recall risk neutral means you care only about expected payoffs.  So just calculate a payoff and probability at the end of the tree.)

Now let’s make this a one-branch normal model (so two outcomes at the end).  Suppose forward interest rates are 2% and volatility is 1%.  Time is 1 year.  What is the up move and down move?  Is the probability of 1% higher, the same, or more than the probability of 3%?  What is the value of a 2% floor (payoff = max(2% – rate, 0)?

Suppose there is a crew race with following people.  Below the name is the average time they have for that type of race.  There is also a standard deviation.  Create a race using a monte carlo.  Give me the probabilities of winning the race for each.  Give me the probabilities for last place for each.  What is the probability that Paul is first and Frank last?

 Boat 1 2 3 4 5 6 7 8 Steph Caroline AB Deb Paul Angie SlowPoke Frank Average 390 395 400 405 420 420 450 490 St Dev 15 15 20 5 70 20 25 30

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