c6 solution

Case 6 - Solution / expected answers Note: For extreme precision, all intermediate computations have been carried out with exact values. Figures of intermediate results in this solution may be rounded for simplicity of notation only. Final results are the only ones to be rounded according to the rules of the game.

Part 1:

Question 1: a. The correct answers are presented in Table 1:

Table 1

Traded on an organized market Future contract

Expiration date is standardized Future contract

Traded on an over-the-counter market Forward contract

The investor can choose the exact quantity of underlying asset Forward contract

Prices are quoted on the exchange Future contract

These answers are based on the definitions of Forward and Future contracts as stated in technical document 1.

b. The correct answer is presented in Table 2:

Table 2

This contract totally eliminates the risk for a given maturity. Forward

This contract eliminates the downside risk, but offers the possibility to gain from advantage fluctuations.

Option

This contract partially eliminates the risk since it may differ in the maturity and the asset to be hedged.

Future

Since Forward contract is built to meet the exact needs of an investor, it has the size and maturity fixed by it; hence, it eliminates favorable and unfavorable risk as well.

From technical document 2, an Option give the holder the right but not the obligation to exercise it. If market conditions are unfavorable, the investor exercise the Option, else he simply abandon the Option to gain full advantage of price fluctuation.

From technical document 1, a Future contract’s size and maturity are standard, so they may differ from what the investor needs; hence, a Future partially eliminates the risk.

c. The correct answer is presented in Table 3: Table 3

To construct complex strategies using Options, the investor needs an estimation of two parameters: the expected volatility and the future market condition.

Yes

Hedging with Options eliminates good and bad risk, even when the spot price at maturity of the underlying asset is in favor of the investor.

No

Hedging with Future contract is more attractive than hedging with options. No

It is possible to construct complex strategies using Future except than simply buy and sell a contract.

No

An investor's risk aversion does affect the construction of binomial trees. No

Future price depends on the volatility of the underlying asset, same as Options. No

Complex strategies involve more than one Option to be established. Future expected volatility affects the price of an Option, as documented in technical document 3. Moreover, future market condition helps the investor to choose the side to be hedged, i.e., upside risk, downside risk, or both.

Hedging with Options eliminates only bad risk (unfavorable price movement), and let the investor to gain profit from favorable price movement.

Following the previous point, it’s clear that hedging with Options is more attractive than Futures.

There are only two possible strategies using Future contracts: buy or sell.

Following technical document 2, the construction of binomial trees is independent from the investor’s risk aversion.

Future prices are independent from the volatility, as documented in technical document 1.

Question 2: The correct answer is presented in Table 4: Table 4

Annual index return

21.16%

Forward price 1627.702

Hedging strategy Sell

Explanation:

The annual index return of 2014 is computed as follow (we use the discrete formula given in technical document 3):

2/10/14 2/01/14

2/01/14

S S

S

, but this return is for the period between January 2, 2014 and October 2, 2014

(29+28+31+30+31+30+31+31+30+2 = 273 days), we must than annualize it:

2/10/14 2/01/14

2/01/14

1693.87 1277.06

1277.06 21.16%273/365

S S

Sr

T t

An index is a security with a continuous dividend rate, which is the annual return. So, following technical document 1, we have the Forward price on October 2, 2014 maturing on December 31, 2014 (29+30+31 = 90 days) is:

90

0.05 .02116365

0 0 1693.87 1627.702r q t

F S e e

Felix holds the portfolio, so he wants to set the future price at which he could sell his portfolio. The strategy consists in selling a Forward contract.


Maturity 21-Jan-15 Future price 1612.639 Basis risk 81.231

Explanation:

Since a Future contract’s maturity is standard, an investor should choose the maturity which comes always after the desired date, and not before, to avoid a naked (unhedged) position. So the correct maturity is January 21, 2015.

The Future price in this case is: 90 21

0.05 .02116365

0 0 1693.87 1612.639r q t

F S e e

The basis risk is the spot price minus the Future price at time when the hedge is performed: basis = 1693.87 – 1612.639 = 81.231

1462.42

1462.42


Buy Call No

Sell Call No

Buy Put Yes

Sell Put No

Explanation: A Call option gives its holder the opportunity to buy the underlying asset at maturity (European Option), and since Felix wants to sell his portfolio, he has then to buy a Put Option, which gives him the right but not the obligation to sell the underlying asset.

Question 5: The correct answers are shown in Table 7: Table 7

TimeToMaturity 0.2329

Spot price 1693.87

Future price 1676.65

Interest rate 5%

Perceived return 9.39%

Using the formula given in technical document 1: 0 0

r q tF S e

, we must compute the real index rate of return “q”, given that on

October 2, 2014 the Future price is 1676.65 (from data – quotes, Future price column, last observation), and the spot price is 1693.84. The given Future prices are for contracts maturing on December 26, 2014; hence, the time to maturity (from October 2,

2014) is 29+30+26 = 85 days or 85/365 = 0.2329 fraction of year. Finally,

2/10/14

2/10/14

1676.65ln ln

1693.840.05 9.39%

0.2329

F

Sq r

t

.

Question 6: The correct answer is presented in table 8: Table 8

Number of Futures

2

Best strategy Sell Forward

Because it's cheaper to perform. No

Because it covers the whole position of JMK. Yes

Because it minimizes the risk better than the other strategy. Yes

Because it's easily accessible on the market and can be closed up by an inverse transaction. No

Explanation:

A Future contract’s size is 700 indices; however, Felix has 1000 indices to cover, he should then sell 2 Future contracts to avoid letting a part of his position uncovered (naked position).

A Forward contract is always preferred to a Future contract, because it fits the exact needs of Felix (in terms of maturity and size), so the best strategy is to sell a Forward, in this case he can fix the needed size of 1000 indices.

A Future or Forward has no value, so it’s not about the price. A Forward contract may cover his whole position as it is flexible. Moreover, it reduces the risk compared to a Future contract. However, it is not easily accessible on the market, because it is traded over-the-counter and it cannot be closed up by an inverse transaction (the underlying asset must be delivered).

Part 2:

Question 1: On the October 2, 2014, the price of the underlying asset is 1693.87. Following technical document 2 (binomial trees), we must construct the binomial tree of this Put over 3 periods (Table 9). Each period has its own up “u” and down “d” coefficients, and the length of periods must be in fractions of years. Table 9

02-Oct 04-Nov 18-Dec 21-Jan

Period 0.0904 0.1205 0.0932

u (up coef) 1.02 1.03 1.05

S (index price) 1693.87 K (Exercise) 1693.9

d (down coef) 0.98 0.97 0.95

Risk-free rate 0.05 Index dividend yield 0.21

For the first period starting on October 2, 2014 and ending on November 4, 2014,

For the second period starting on November 4, 2014 and ending on December 18, 2014,

For the third period starting on December 18, 2014 and ending on January 21, 2015,

To find the length of these periods in years, it’s convenient to use the function YEARFRAC in Excel, but remember to use the actual/actual day count basis. The stock price and Put values as well as the risk-neutral probability at each node of the tree are presented in

Table 10. First, the value of the underlying asset S is computed at each node starting from time 0 (October 2, 2014) where S = 1693.87. For the first period, we have S(u) = S * u1 = 1693.87 * 1.02 = 1727.75; the value of S(u,d) = S(u) * d2 = 1727.75 * 0.97 = 1675.91, and so on. The Put values on the tree are solved backward, i.e., starting from maturity when the value of the Put is P = Max[K – S, 0]. For example, the value of P(u,u,d) is Max[K – S(u,u,d), 0] = Max[1693.87 – 1690.60, 0] = 3.27. Next, let’s take for example the calculation steps of P(u,d): we need to compute the risk neutral probability Prob(u,d). Following

technical document 2, ( )

( )

, the value of P(u,d) is then

[

( ) ] [ ( ) ] Continue computing the risk neutral probability at each period and Put values the same way for all nodes until you reach the first one at October 2, 2014.

Table 10

S(u,u,u) 1868.56

P(u,u,u) 0.00

S(u,u) 1779.58

Prob 0.35

P(u,u) 2.11

S(u,u,d) 1690.60

S(u) 1727.75

P(u,u,d) 3.27

Prob 0.18

P(u) 54.08

S(u,d,u) 1759.71

P(u,d,u) 0.00

S(u,d) 1675.91

Prob 0.35

P(u,d) 65.77

S(u,d,d) 1592.12

S 1693.87

P(u,d,d) 101.75

Risk neutral prob 0.14

Option price "P" 89.44

S(d,u,u) 1795.28

P(d,u,u) 0.00

S(d,u) 1709.79

Prob 0.35

P(d,u) 44.97

S(d,u,d) 1624.30

S(d) 1659.99

P(d,u,d) 69.57

Prob 0.18

P(d) 95.54

S(d,d,u) 1690.70

P(d,d,u) 3.17

S(d,d) 1610.19

Prob 0.35

P(d,d) 107.23

S(d,d,d) 1529.68

P(d,d,d) 164.19

Question 2: The correct answer is: Explicit volatility = 0.11297 Explanation: The annual historical volatility of 2014 is the standard deviation of daily price returns multiplied by the square root of 252 (technical

document 4). The daily returns are computed as follow 1

ln tt

t

Su

S

, and the daily volatility is

2

1

1

1

T

t

t

s u uT

, since our

data are on a daily basis. An estimate of the explicit volatility is then ˆ 252 0.11297s .

Question 3: The correct answers are presented in Table 11: Table 11

Spot Price 1693.87 Index return 0.21

Exercise price 1693.87 d1 -0.75768

Time To Maturity 0.30411 d2 -0.81998

sigma 0.11297

Risk-free rate 0.05 Option price (Put) 92.43426

Explanation: We use the Black & Scholes formula in presence of a continuous dividend yield “y”:

2 1

r T t y T tP Ke N d Se N d

Where:

2

1

2 1

ln2

Sr y T t

Kd

T t

d d T t

For Options on index, the dividend yield represents the annual index return y = 0.21, and the risk-free interest rate is r = 0.05 per

annum. The time to maturity is from October 2, 2014 till January 21, 2015: ( )

. The volatility is the

one found in part 2 – question 2, σ = 0.11297.


Because binomial tree is discrete, and Black & Scholes is for continuous time. No Because the volatility is different in both models. Yes Because both models do not give the same value at any time. No

Explanation: The binomial tree is an approximation of the Black & Scholes model, for the same data we should get approximately the same results. However, the difference in both values 89.44 (binomial tree) and 92.43 (B&S) is essentially due to the difference in price changes, i.e., in the binomial tree we used u and d coefficients for a price fluctuation of 2% (first period), 3% (second period) and 5% (third period); and for the Black & Scholes model the price change is almost 11% (the volatility).


Buy European Call No

Buy European Put No

Buy American Put Yes

Buy American Call No

Explanation: Since Felix is willing to sell his portfolio, he should buy a Put option. An American option is more flexible than a European one because it offers the investor the possibility to exercise the option at any time till maturity, contrary to the European option which gives the holder to exercise only at maturity.


Spot Price = 1693.87

Black & Scholes model is reliable No

Time To Maturity = 0.2329

Because the implied volatility is Variable

Dividend rate = 0.21 Risk-free rate = 0.05

Strike price Market Call

price Theoretical Call price

Implied Volatility

1400 301.7 301.7000 56.60%

1425 278.9 278.9000 53.96%

1450 233.15 233.1499 42.21%

1470 208.65 208.6500 37.78%

1475 157 157.0000 13.35%

1500 187.5 187.4999 37.23%

1525 187.55 187.5500 42.79%

1550 182.7 182.6999 45.97%

1575 111.85 111.8500 26.57%

1585 100.4 100.4000 24.63%

1595 115.7 115.7001 31.53%

1600 104.75 104.7501 28.78%

1620 74.15 74.1501 22.13%

1625 84.25 84.2500 26.19%

1630 108.6 108.6000 34.83%

1645 72.3 72.3001 25.32%

1650 76 76.0001 27.22%

1655 62.55 62.5500 23.57%

1665 49.5 49.5000 20.67%

1675 54.9 54.9000 23.70%

1680 48.5 48.5000 22.21%

1685 44 44.0000 21.32%

1695 39.25 39.2500 20.87%

1700 35.95 35.9500 20.29%

1705 34.05 34.0500 20.17%

1715 29.6 29.6000 19.62%

1725 23.75 23.7500 18.43%

1735 20.15 20.1500 17.94%

1745 16.45 16.4500 17.27%

1750 15.7 15.7000 17.36%

1800 5 5.0000 15.07%

Explanation: The strike prices and option market prices are given in Data - options. The implicit volatility is computed using the Black & Scholes formula (technical document 2). Unfortunately, the Black & Scholes formula does not provide a direct solution for the volatility. This means that the implicit volatility can only be derived using numerical procedures, such as the solver add-in for Excel. Let’s take the case when the strike price is 1400 (first line), from the given quotes, the Call value for this maturity is 301.7, so we

solve the following equation relative to σ:

1 2

y T t r T tC Se N d Ke N d

, where:

2

1

2 1

ln2

Sr y T t

Kd

T t

d d T t

The time to maturity is from October 2, 2014 till December 26, 2014: ( )

In Excel, we choose an arbitrary value of the volatility (say 0.2) and use the solver by setting the target cell (which gives the value of C) equal to 301.7 by changing the variable cell containing the implicit volatility. The implied volatility deduced from the Black & Scholes model is far from the explicit volatility. Moreover, it’s changing over time which is in conflict with the hypothesis of Black & Scholes, we conclude then that the B&S model is unreliable.

c6 solution

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