Financial Derivatives and Portfolio Management

This finance assignment delves into the intricacies of portfolio management and financial risk mitigation using derivatives. It examines a scenario where an institution seeks to create a neutral portfolio by strategically employing long and short positions in options contracts while considering key factors like delta, gamma, and vega. The assignment also analyzes the impact of interest rate fluctuations on profitability and explores factor analysis for understanding the relative importance of various market drivers.

Finance
Assignment

Table of Contents
Q1.................................................................................................................................................................. 3
Q2.................................................................................................................................................................. 3
Q3(6.14) AND 4(6.15).......................................................................................................................... 3
Q5.................................................................................................................................................................. 4
Q6 (7.18).................................................................................................................................................... 6
Q7.................................................................................................................................................................. 7
Q8.................................................................................................................................................................. 7

Q1
Suppose, at the maturity date, the oil price is ST and company has promised to pay
$1000 in addition to the price at the date of maturity. It can be presented here as under:
ST <$25: payoff = 0
$40 > ST > $25: payoff = 170(ST – 25)
ST > $40: payoff = $2,550
As per the stated payoff, it can be seen that if price at maturity is below $25, then
there will be zero return. However, price below $40 but above $25 will be require a pay
off on 170 call options at the difference between strike price and $25. In contrast, in the
last one, if price goes beyond $40 per barrel then the maximum payment will be $2,550.
Oil with the St of $25, the payoff is less from 170 options with a strike price of $40. It
indicates that bond is regular bond which depicts the long position on oil with having a
strike price of $25 per barrel whereas short position lies on oil price at $40 per barrel is
indicates bull spread strategy on the oil product. The strategy demonstrates that holder
can buy oil at lower prices and sell the oil barrel at high prices to take benefits from the
same.
Q2
As per the stated case, it is presented that arbitrageur can borrow money at an
interest rate of 10% annual to purchase 100 ounce and create a short futures contract for
its delivery in the following year. Buying $1500/ounce today and sell it in one year at
$1700/ounce will bring a return of 13.33% computed as follows:
= ($1700 - $1500)/ $1700 *100
The return rate is below the cost of borrowings of 10% per annum thus, it
becomes clear that arbitrageur can borrow money and earn a net return of (13.33%-10%)
3.33% on such proposal. It will be a profitable proposal for the arbitrageur and he can
buy ounce at the current price and create a short future contract for the bought quantity of
ounce and sell it into the future period.
Q3(6.14) AND 4(6.15)
As per the scenario stated, if senior tranches, Mezzanine tranches and Equity
tranches principle are assigned at 70%, 20% and 10% instead of 75%, 20% and 5%
respectively. In this, senior traches has been rated AAA, Mezzanine tranches has been
rated at BBB and Equity tranches are not rated. As per the figure 6.4, it can be said that
total AAA rated instrument senior tranche is founded at 90%, 75% + (20%*75%) = 90%
which is very high.

Senior tranche: 70%
Mezzanine tranche: 20%
Equity tranche: 10%
Losses to
subprime
portfolio
Losses to
mezzanine
tranche
Losses to equity
tranche of ABS
Cdo
losses to
mezzanine tranche
of ABS Cdo
Losses to senior
tranche of ABS
CDO
10% 33.30% 100% 93.30% 0%
13% 53.30% 100% 100% 28.20%
17% 80% 100% 100% 69.20%
20% 100% 100% 100% 100%
Q5
F0 = 30
Exercise price (K) = 30
Risk free rate = 5% OR 0.05
Volatility (σ) = 25% OR 0.25

As per the result founded in DerivaGem software, it is founded that delta, gamma,
Vega, theta and Rho is founded 3.7007997, 0.6274, 0.050, 0.1135, -0.0059 and 0.1512
respectively.
Suppose, if stock price increased to 30.1 then as per the new derived results, it is
founded that delta, gamma, Vega, theta and Rho is founded 3.7131, 0.627, 0.0502,
0.1139, -0.005 and 0.1517 respectively. Thus, the change in the option price is (3.7131-
3.7008) very less to 0.0123 whilst change in delta remains constant to 0.6274095.

Assuming, if volatility increased from 25% to 26% then price will goes up from
3.7008 to 3.8144 by 0.1136 which is in line with the Vega % to 0.1139. Thus, it indicates
that result founded is accurate and reliable.
Q6 (7.18)
7.17
Delta:
Gamma
Vega
On 4000 option, the gamma-neutral portfolio at the long position will be 4000*1.5
= 6000 whilst the delta will be as follows:
(4000*0.6)-450 = 1,950
As per the results, in addition to the 4,000, a short position of 1950 is essential for neutral
gamma & delta.

(A) Long position, 5000 options will create neutral portfolio in vega at where vega
portfolio 5000*0.8 = 4000 and the delta will be as follows:
5000*0.6 – 450 = 2,550
Thus, derived results reflect that apart from 5000 trading option, short-position of
2,550 sterling is required to create neutral delta and vega.
7.18
Let us assumption, W1 in the first option and W2 for the second one can be
illustrated as follows:
Let us take W1 = 3200
W2 = 2400
-450 + (3200*0.6) + (2400*0.1) = 1,710
Thus, founded results demonstrates that by taking long position of 3200 & 2400
option in the first and second option & short-position in 1,710 sterling which will create
neutral portfolio in the respect of delta, gamma and vega.
Q7
In the stated case, the mismatch between assets and liabilities has been founded to
$25million and return after taxation is ($2bn*12%) equal to $0.24billion. Thus, profit
before taxation will be $0.24 billion/(1-30%) came to $342,857. In the given case,
assuming that cash low remains constant for the five year instrument. Changes in the rate
of interest rate on the profitability is totally depends upon the fluctuations in the rates on
the end. For instance, if interest rate goes up by 5% on the last day then it will have no
impact on the return. If it changes at the starting date then interest payment on one-year
deposit subtracting the amount of interest payment on yearly loan of $25bn*Interest rate.
In such case, ROE will be zero, in contrast, if rate remains constant till the middle of the
year and in the mid it goes up by X% then before-tax return will be decline by
$25bn*half of interest rate.
$25bn * X/2 = $342,857
= 2.74%
On the other hand, if rates rose by 2.74% at the beginning date and remain same
for the half year and again go back to the employee at a mid at zero return. In each case,
on an average, rise in interest is 1.37 % which will present nil rate of return.

Q8
Delta of the portfolio for the first factors:
(5*0.216) + (-3*0.331) + (-1*0.372) + (2*0.392) + (5*0.404) + (7*0.394) + (8*0.376) +
(1*0.305)
= 8.590
Delta for the second factor:
(5*-0.501) + (-3*-0.429) + (-1*-0.267) + (2*-0.110) + (5*0.019) + (7*0.194) + (8*0.371)
+ (1*0.554)
= 3.804
Delta for the third factor
(5*0.627) + (-3*0.129) + (-1*-0.157) + (2*-0.256) + (5*-0.355) + (7*-0.195) + (8*0.068)
+ (1*0.575)
= 0.372
The relative importance of all the factors PC1, PC2 and PC3 can be seen through
multiplying the factorial exposure with the standard deviation. In the stated case, second
factor (3.804*4.77)/(8.590*17.55) = 12% which is really important as a first factor and
the third factor is (0.472*2.08)/(3.804*4.77) quantified to 5.4% which is the second
factor.