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 Önal 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; 120 construct the Önal cash áows 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. 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. 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
|
3 month 90 put
|
3 month 110 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
|
-0.6325
|
1.4679
|
|
Iv
|
0.32
|
0.16
|
0.30
|
0.18
|
(a) Design a self Önanced position for a prospective investor who would like to beneÖt by 5 dollars from an increase in volatility of 2% per-
centage 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 áat or constant implied volatilities as the spot is moved.
4. The data for this question is provided in vswap.xls.
(a) For the two maturities closest to three and six months determine the quote on the variance swap contract.
(b) Determine the price on the forward variance swap starting at the Örst maturity of part a and Önishing at the second maturity of part a.
Attachment:- vswap.xls