# Project #51327 - Futures & Options

Please see attached word document for an outline of the assignment. Most of it is copied and pasted here as well.

1.     1. For the lognormal model with mean return on the stock given by m and volatility of s for an interest rate of r determine expressions for the physical and risk neutral probabilities that A < S < B. Evaluate these probabilities for r=3%, m=7%, s =20% and t=.0833 for A = 85, B = 90, and S0 = 100. What is the rate of return on the security that pays a dollar if in a month the stock is between 85 and 90 given that it starts at 100.

2.     2. Consider now a call option on R with strike K and call payoff: (R-K)+. In the Black Merton Scholes model derive the formula for pricing this call option. For an interest rate of 2% and a volatility of 15% with a three-month maturity and a strike at -3% what is the price of this call option.

3.     3. Derive the put call parity relation for options on R and a formula for the put option on R.

Part II

1.     1. The mean rate of return on a stock is estimated at 20% while the volatility is 40%. The risk free interest rate is 5%.

a.     What is the mean for the log price relative?

b.     Construct the final stock prices for a 10 period one year tree.

c.      Construct the statistical probabilities for these stock prices

d.     Construct the associated risk neutral probabilities

e.     Graph the statistical and risk neutral probabilities against the stock prices on the same graph.

f.      For the two strikes of 80 and 120 construct the final cash flows to call and put options at these strikes.

g.     Price the puts and calls using the statistical probabilities

h.     Price the puts and calls using the risk neutral probabilities

i.       Identify an arbitrage you would use against a counterparty quoting on statistical probabilities

j.       Show that this arbitrage fails against the counterparty quoting on the risk neutral probabilities

2.     2. A stock trades in the US market for \$98. The dividend yield on the stock is 4.15%. The volatility of the stock is 35%. The US interest rate, continuously compounded, is 6.5%. We wish to quote on quantoing the stock into a foreign currency that has a continuously compounded interest rate of 8.14%. The volatility of the exchange rate measured in units of foreign currency per US dollar is 12% while the correlation between the stock and the exchange rate is 0.45. Prepare a quote on a six month call option struck at \$110 and quantoed into the foreign currency.

3.     3. Suppose the spot price on the underlying asset is \$100 with a continuously compounded interest rate of 2% and a zero dividend yield. A one and three month put struck at 90 and a call struck at 110 have the following information.

 One month 90 put One month 110 call 90 3 month put 110 3 month call Price 0.5337 0.0381 1.9051 0.7788 Delta -0.1141 0.0225 -0.2088 0.1689 Gamma 0.0209 0.0116 0.0191 0.0280 Vega 5.5709 1.5435 14.3599 12.6010 Volga 23.3412 39.6638 25.6412 70.3471 Vanna -0.6711 0.6855 -.6325 1.4679 IV 0.32 0.16 0.30 0.18

a.     Design a self financed position for a prospective investor who would like to benefit by 5 dollars from an increase in volatility of 2% percentage points accompanied by drop in the stock price of 2 dollars. The position should be delta, gamma, vega and Volga neutral as well.

b.     Construct a spot slide in the spot range 70 to 130 for the designed position. Use flat or constant implied volatilities as the spot is moved.

c.      Use the data in the file TGVVV.xls to estimate the risk neutral correlation between stock returns and changes in volatility.

4.     4. The data for this question is provided in vswap.xls.

a.     For all the maturities provided determine the quote on the variance swap contract.

b.     Graph the variance swap quote against the maturity.

c.      Prepare a quote on the forward variance swap between the tenth and final maturities.

d.     Prepare quotes on variance swaps on all maturities conditional on the spot being in the range of 80% to 120% of the initial value.

 Subject Business Due By (Pacific Time) 12/12/2014 11:00 pm
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