Part I
Provide answers to questions 2-4 of Quiz 3 in a separate pdf Öle. The questions
are repeated here.
2. For the lognormal model with mean return on the stock given by and
a volatility of 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%; = 7%; = 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:
3. Consider now a call option on R with strike K and call payo§
(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.
4. Derive the put call parity relation for options on R and a formula for the
put option on R:
Part II
Answer the following four questions in a separate Öle from that used for Part 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%:
1(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 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 -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% percentage
points accompanied by drop in the stock price of 2 dollars.
The position should be delta, gamma, vega and volga neutral as well.
2(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.
(c) Use the data in the Öle TGVVV.xls to estimate the risk neutral
correlation between stock returns and changes in volatility.
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
Önal 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.
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