# Financial Derivatives- Futures and Options Project cyni09

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.