financial risk amangement

Size of the OTC vs Exchange Traded
derivatives maket

Convergence of Spot and Forward

Bid vs Ask
• Spot and Forward quotes for USD/GBP
exchange rate
Bid Ask
Spot 1.5541 1.5545
1 month forward 1.5538 1.5543
•From the perspective of a dealer
•If you wanna look from your perspective then never forget : world is your
enemy

Relationship between F & S
• Consider a stock paying 0 dividend worth $60.
One can borrow or lend money for 1 year at
5%. What should the 1 year forward price of
the stock be ?
Ans: 63

• What would you do if Forward is priced at
$67.
Profit - 4

• What would you do if Forward is priced at
$58.
Profit - 5

Options
• Call
• Put
• Exercise Price/ Strike
• Expiry / Maturity
• American
• European

Prices of call option on Google; Stock
Price: bid $871.23, offer 871.37

• Cost to enter a forward vs options
• Price of call option decreases as the strike
prices increase
• Price of put option increases as the strike
prices inrease

• Long Call
• Short Call
• Long Put
• Short Put
Possible Positions

Questions
• An investor enters into a short forward
contract to sell Rs100,000 for US dollars at an
exchange rate of Rs60.25 per dollar. How
much does the investor gain or lose if the
exchange rate at the end of the contract is
• A) Rs.60.12
• B) Rs.60.35

Questions
• A trader enters into a short cotton futures
contract when the price is 50 cents per pound.
The contract is for the delivery of 50,000
pounds. How much does the trader gain or
lose if the cotton price at the end of the
contract is
• A) 48.20 cents per pound
• B) 51.30 cents per pound

Questions
• Suppose that a march call option to buy a
share for $50 costs $2.5 and is held until
March. Under what circumstances will the
holder of the options make a profit? Let’s also
draw a diagram illustrating how the profit
from along position in the option depends on
the stock price at the maturity.

Questions
• A trader writes a december put option with a
strike price of $30.The price of the option is
$4. Under what circumstances does the trader
make a gain?

Questions
• A company knows that it is due to receive a
certain amount of foreign currency in 4
months. What type of option contract is
appropriate for hedging?

Questions
• A US company expects to have to pay 1 million
CAD in 6 months. Explain how the exchange
rate can be hedged using
• A) Forward Contract
• B) Option contract

Questions
• Suppose the USD/Sterling spot and forward
exchange rates as follows:
What opportunities are open to an arbitrageur in the following situations?
a) 180 day european call option to buy Pound 1 for $1.42 costs 2 cents.
b) 90 day european put option to sell Pound 1 for $1.49 costs cents.
Spot 1.5580
90 day forward 1.5556
180 day forward 1.5518

Questions
• The price of gold is currently $1400 per ounce.
The forward price for delivery in 1 year is
$1500 per ounce. An arbitrageur can borrow
money at 4% per annum. What should the
arbitrageur do? As the cost of storing gold is
zero and that gold provides no income.

Questions
• A trader buys a european call and sells a
european put option. The options have the
same underlying asset, strike price, and
maturity. Describe the trader’s position. Under
what circumstances does the price of the call
equals the price of put.

Hedging
• Short Hedge
• Long Hedge

Basis risk
• Hedged and underlying asset are different
• Exact date to buy and sell is not known
• Hedge may be required to close before its
delivery
Basis = S - F

Explanation
• S1 = 2.5 ; F1 = 2.2
• S2 = 2 ; F2 = 1.9
• Effective price => S2+F1-F2

• Cross hedge
• Hedge ratio = Size of the position taken in
futures contract to the size of the exposure
• Minimum Variance Hedge Ratio
h* = ρ (σs/σf)
Required Contract = Size of Position * h/Size of
future

Hedging Equity Portfolio
• Formula
• (B2-B1)/Bf * Sp/Sf

Types of Rates
• TreasuryRates
• LIBOR
• Federal Fund Rate
• Repo Rate
• Risk FreeRate
• Call Rate, MIBOR

Treasury Rate
• Rate an investor earns on Treasury bills and
Treasury bonds
• Instruments used by govt to borrow money

LIBOR
• Unsecured short term borrowing rates
between banks
• Cal each business day for 10 currencies and 15
borrowing priods
• One popular derivative that uses libor as
reference is interest rate swap.

Call Rate
• Overnight borrowing rates in India
• Banks and Corporate entities can borrow
• However, only banks can be lenders

Zero Rate
• N day zero rate is the Interest rate earned on
an investment that starts today and lasts for n
days.
• 1 year Zero Coupon bonds continuously
compounded quotes at 97.

Bond Pricing
• PV cash flows
• Same discount rate or ?

Bond Yeild
• Single discount rate applied to all cashflows
that gives the market price

Forward rate
• Future rates implied by current zero rates

• Payoff for a long forward
• Payoff for short forward
• Strip
• Pricing of future
F = Se^(rT)
If F > Se^(rT), arbitrageurs do what? [Cash and carry]

• Security XYZ Ltd trades in the spot market at Rs. 1150.
Money can be invested at 11% p.a. The fair value of a
one-month futures contract on XYZ is what?
• A two-month futures contract trades on the NSE. The
cost of financing is 10% and the dividend yield on Nifty
is 2% annualized. The spot value of Nifty 4000. What is
the fair value of the futures contract ?
• Current Price of the bond is 930. 4 month risk free rate
(CC) is 6% p.a. Value of future ?

• When asset provides income :
• F=(S-I)e^rT
• Q: Consider a 10 month forward contract on a
stock when the stock price is $50. Assume risk
free rate to be 8% p.a. for all maturities. We
also assume that dividendsof $0.75 are
expected after 3, 6, 9 and 12 months.
• Ans:51.14

• When asset provides known yield, rather than
cash income.
• F = Se^(r-q)T
Q. Consider a 6 month stock forward thatis
expected to provide income equal to 2% of the
asset price once during the 6 month period. Risk
free rate of interest is 10% p.a. The asset price is
$25.
Continuous rate = ln (1+ annual return)

Value of Fwd contract today
• At time 0 for time T, forward contract price = K
• At time t, between today and T, Fwd contract
price = F0

Which contract will u buy, Forward or
future?

• A long forward contract on a non dividend
paying stock was entered sometime ago. It
currently has 6 months to maturity. The risk
free rate of interest (with continuous
compounding) is 10%p.a., the stockprice is
$25 and delivery price is $24. Calculate the
current value of the forward.

Forward on currencies
Underlying asset is one unit of foreign currency.
However, for major exchange rates other than pound, australian dollar and NZ
dollar, a spot or forward exchange rate is normally quoted as the number of units of
the currency that are equivalent to one US dollar
Q. Suppose the year interest rate in Ausand the United States are 3% and 1%
respectively. The spot exchange rate is 0.9800 USD per AUD. The 2 year forward
exchange rate should be ?

Future pricing and storage cost
Q. Consider a 1 year future contract on an investment asset that provides no income. It
costs $2 per unit to store the asset, with the payment being made at the end of the
year. Assume that the spot price is $450 per unit and the risk free rate is 7% pe annum
for all maturities.

Backwardation and Contango
• When the future price is below the expected
future spot price, it is known as
backwardation.
• When future price is above the expected
future spot, it is known as contango.

Option Terminology
• Buyer of an option
• Writer of an option
• Call option
• Put option
• Option premium
• Expiration date
• Strike Price
• AmericanOption
• In the money option
• At the money option
• Out of money option
• Intrinsic value of option
• Time value of an option

Application option
• Have underlying, do what ?
• Bullish on security, do what?
• Bearish security, do what ?

• Equity Analysts Inc. (EQA) is an equity portfolio
management firm. One of its clients has decided to be
more aggressive for a short period of time. It would like
EQA to move the beta on its $65 million portfolio from
0.85 to 1.05. EQA can use a futures contract priced at
$188,500, which has a beta of 0.92, to implement this
change in risk.
A Determine the number of futures contracts EQA
should use and whether it should buy or sell futures.
B At the horizon date, the equity market is down 2
percent. The stock portfolio falls 1.65 percent, and
the futures price falls to $185,000. Determine the
overall value of the position and the effective beta

• Global Asset Advisory Group (GAAG) is a pension fund
management firm. One of its funds consists of $300 million
allocated 80 percent to stock and 20 percent to bonds.
The stock portion has a beta of 1.10 and the bond portion
has a duration of 6.5. GAAG would like to temporarily
adjust the asset allocation to 50 percent stock and 50
percent bonds. It will use stock index futures and bond
futures to achieve this objective. The stock index futures
contract has a price of $200,000 (after accounting for the
multiplier) and a beta of 0.96. The bond futures contract
has an implied modified duration of 7.2 and a price of
$105,250. The yield beta is 1. The transaction will be put in
place on 15 November, and the horizon date for
termination is 10 January.

• Quantitative Mutual Funds Advisors (QMFA) uses modern analytical
techniques to manage money for a number of mutual funds. QMFA
is not necessarily an aggressive investor, but it does not like to be
out of the market. QMFA has learned that it will receive an
additional $10 million to invest. Although QMFA would like to
receive the money now, the money is not available for three
months. If it had the money now, QMFA would invest $6 million in
stocks at an average beta of 1.08 and $4 million in bonds at a
modified duration of 5.25. It believes the market outlook over the
next three months is highly attractive. Therefore, QMFA would like
to invest now, which it can do by trading stock and bond futures. An
appropriate stock index futures contract is selling at $210,500 and
has a beta of 0.97. An appropriate bond futures contract is selling
for $115,750 and has an implied modified duration of 6.05. The
current date is 28 February, and the money will be available on 31
May.

• Total Asset Strategies (TAST) specializes in a
variety of risk management strategies, one of
which is to enable investors to take positions in
markets in anticipation of future transactions in
securities. One of its popular strategies is to have
the client invest when it does not have the
money but will be receiving it later. One client
interested in this strategy will receive $6 million
at a later date but wants to proceed and take a
position of $3 million in stock and $3 million.

• FCA Managers (FCAM) is a U.S. asset management firm. Among its
asset classes is a portfolio of Swiss stocks worth SF10 million, which
has a beta of 1.00. The spot exchange rate is $0.75, the Swiss
interest rate is 5 percent, and the U.S. interest rate is 6 percent.
Both of these interest rates are compounded in the LIBOR manner:
Rate × (Days/360). These rates are consistent with a six-month
forward rate of $0.7537. FCAM is considering hedging the local
market return on the portfolio and possibly hedging the exchange
rate risk for a six-month period. A futures contract on the Swiss
market is priced at SF300,000 and has a beta of 0.90.
A What futures position should FCAM take to hedge the Swiss
market return? What return could it expect?
B Assuming that it hedges the Swiss market return, how could it
hedge the exchange rate risk as well, and what return could it
expect?

FRA
• forward rate agreement (FRA) is an over-the-counter
transaction designed to fix the interest rate that will apply
to either borrowing or lending a certain principal during a
specified future period of time. The usual assumption
underlying the contract is that the borrowing or lending
would normally be done at LIBOR. If the agreed fixed rate is
greater than the actual LIBOR rate for the period, the
borrower pays the lender the difference between the two
applied to the principal. If the reverse is true, the lender
pays the borrower the difference applied to the principal.
Because interest is paid in arrears, the payment of the
interest rate differential is due at the end of the specified
period of time. Usually, however, the present value of the
payment is made at the beginning of the specified period

•
• Q.5 A portfolio manager needs to hedge a possible decrease in interest rates by shorting a 3 X 6 FRA. The current term structure for the LIBOR is
given below:
•
• Term (Days)
• Interest Rate (Annualised)
• F(30)
• 5.83
• F(90)
• 6.0
• F(180)
• 6.14
• F(360)
• 6.51
•
• a) What is the forward rate portfolio manager would receive on the FRA?
•
• It is now 30 days since the Portfolio Manager took the short position in FRA. Interest rates have fallen and the new term structure for
LIBOR is given below :
•
•
• Term (Days)
• Interest Rate (Annualized)
• F(60)
• 5.5
• F(150)
• 5.62
•
• What is the market value of the FRA based on notional principal of $15,000,000?

Swaps
• Interest rate swaps
• Floating rate – LIBOR
Rate of interest which at which bank with a AA
credit rating is able to borrow from other
banks.

• Consider a hypothetical 3-year swap initiated
on March 5, 2014, between Microsoft and
Intel. We suppose Microsoft agrees to pay
Intel an interest rate of 5% per annum on a
principal of $100 million, and in return Intel
agrees to pay Microsoft the 6-month LIBOR
rate on the same principal. Microsoft is the
fixed-rate payer; Intel is the floating-rate payer
. LIBOR on March 5 is 4.2%.

VaR
• I am X percent certain there will not be a loss
of more than V dollars in th next N days.

• VaR = portfolio size * z score of confidence
interval * std deviation

• Microsoft shares worth 10M, daily std
deviation 2%, 99% confidence interval, 10 day
var?
• AT&T 5 Million ; p=0.3, std deviation1%
Z 2.326

Could someone help me calculate this?
• 3 year spot rate = 4%
• 5 year spot rate = 5%
• 4 year forward rate 3 years from today = 6%
• 3 year forward rate 7 years from today = 7%
• What is the 2 year forward rate 5 years from
today?
• ~5.5

• Consider a $1 million 90-day forward rate
agreement based on 60-day London Interbank
Offered Rate (LIBOR) with a contract rate of
5%. If, at contract expiration, 60-day LIBOR is
6%, the short must pay:
• A) $1,650.17.
• B) $1,652.89.
• C) $1,572.33.
• D) $1,666.67.

• BACKGROUND: Client A has a $20 million technology equity
portfolio. At the beginning of the last quarter, Allison forecasted a
weak equity market and recommended adjusting the risk of the
portfolio by lowering the portfolio’s beta from 1.20 to 1.05. To
lower the beta, Allison sold 25 December NASDAQ 100 futures
contracts at $124,450. During the quarter, the market decreased by
3.5 percent, the value of the equity portfolio decreased by
5.1 percent, and the NASDAQ futures contract price fell from
$124,450 to $119,347. Client A has questioned the effectiveness of
the futures transaction used to adjust the portfolio beta.
• QUESTION: With respect to Client A, Allison’s most appropriate
conclusion is the futures transaction used to adjust the beta of the
portfolio was: A effective. B ineffective because the effective beta
on the portfolio was 1.64. C ineffective because the effective beta
on the portfolio was 1.27

• Security XYZ Ltd trades in the spot market at
Rs. 1150. Money can be invested at 11%
p.a. The fair value of a one-month futures
contract on XYZ is calculated as follows:

• A two-month futures contract trades on the
NSE. The cost of financing is 10% and the
dividend yield on Nifty is 2% annualized. The
spot value of Nifty 4000. What is the fair value
of the futures
• contract ?

Covered call
Mildly bullish
Max Profit: c+ X -s
Max Loss: S-p
Breakeven: S-p

Butterfly
• The strategy can be done by selling 2 ATM
Calls, buying 1 ITM Call, and buying 1 OTM
Call options (there should be equidistance
between the strike prices)

collar
• Similar to covered call

• Call options on a stock are available with strike
prices of $15, $17½, and $20, and expiration
dates in 3 months. Their prices are $4, $2, and
$½, respectively. Explain how the options can
be used to create a butterfly spread. Construct
a table showing how profit varies with stock
price for the butterfly spread.

• Suppose that put options on a stock with a
strike price of INR 100 and INR 110 cost INR 6
and INR 9, respectively. How can the following
be created. a bull spread a bear spread

Binomial Tree
• U = size of up move =1.33
• D = 1/U = 1/1.33 = 0.75
• Π (u) = probability of up move = 0.55
• Π(d) = probability of down move =1 – 0.55 =
0.45

• Use the info in the previous example to
calculate the value of a put option on the
stock with an exercise price of $30.

One period binomial tree
30
30 * 1.33 = 40
30 * 0.75 = 22.5
Π (u) = (1 +R – D)/ U-D
Π(d) = 1 – π (u)

• Cal value of call option at the start of the
period . X =30,R = 7

• Suppose you have a stock currently priced at
50 and a two period european call option with
strike price of 45. The size of an up move is
1.25. The risk free rate of interest is 7%.
Compute the value of call option using a two
period binomial model.
• 12.51

BSM
• Option value is the value of holding a position in an option
at a given point in time during the life of the option. Let’s
explore equations through which one can calculate an
option’s value. The option valuation equations that we will
explore can be used to determine the option value at any
point in time during an option’s life. They can also be used
to determine the fair premium at initiation. The academics
that initially derived these equations are Fischer Black,
Myron Scholes, and Robert Merton.1 These equations are
therefore known as the “Black-Scholes-Merton model” or
the “Black-Scholes model.” In the subsequent chapter we
will learn to understand why this model represents option
value. The value of a long call option and long put option
are

Assumptions of bsm
• ■ Options are European-style and not American-
style.
• ■ The underlying asset pays no income.
• ■ There are no “frictions” such as transaction
costs or taxes.
• ■ The risk-free interest rate is known and
constant.
• ■ The underlying asset volatility is known and
constant.
• ■ Returns are normally distributed.

Option on interest rate: caps and
floors

• Delta : describes the relationship between
asset price and option value.
Call option - > +ve
Put option -> -ve

• Vega : measures the sensitivity of the option
price to changes in the volatility of returns on
the underlying asset.
• Call - ?
• Put - ?

• Rho
Option price to risk free rate

• Theta : option price sensitivity to the passage
of time.
• As time passes, value of call option decreases (
all else equal) - “time decay”
• True for put options

Delta neutral portfolio
• Long position in stock offset by selling a call.
• Options needed to delta hedge = no. of shares
/ delta of call

• Effect of increase in stock price by $1?

• Gamma : rate of change of delta to change in
spot prices.
Call and put have same gamma.
Long position in call and put will have same
gamma.

• What does gamma of 0.04 imply
• * close to expiry

CDS
• Cds creates short position in the reference
obligation for the party buyin swap

Pricing vs valuation
Plain vanilla swap
- One party agrees to pay floating and receive
fixed.
- At the initiation, fixed rate is selected so that
present value of floating = present value of
fixed.
- This fixed rate is “swap rate”.
- Determining the swap rate is pricing the swap.

• We can price the swap if by using the insight
that the swap is equivalent to issuing a fixed
rate bond and buying an identical floating rate
bond.
•

Market Value of Swap
• At any payment date, the market value of
swap is the difference between the value of
the fixed rate bond and floatin rate bond.
• Fixed rate payer is long a floating rate bond
and short a fixed rate bond, the position will
have positive value when fixed rate bond is
trading at discount.
Don’t get confused why are v including principal
when in swap principal is not exchanged.

SIP PEPSi = Be Cool
• Put X = 100, Maturity 1 Year
• Call X = 100, Maturity 1 Year
• Put Call PArity

The expected annual return for a $100,000,000
portfolio is 6.0% and the historical standard
deviation is 12%. Calculate VaR at 5%
probability.

Interest rate parity
• Suppose 6 month interest rate in India is 5%
(or 10% per annum) and in USA are 1% (2%
per annum). The current USDINR spot rate is
50. What is the likely 6 month USDINR futures
price?

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