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 .nal 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 .nal cash .ows 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 financed position for a prospective investor who would
like to benefit 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 flat or constant implied volatilities as the spot is moved.
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