Description

Answer all the questions provided on the PDF, providing an explanation when asked. Provide proof and formulas used for each question. You MUST be knowledgeable in Corporate Finance to complete this.

Econ 372/572 & Fintech 522,

Asset Pricing & Risk Management

Homework 4

**Due 5pm Duke time on Thursday April 19**

** Scan or compile your homework answer sheet into a single PDF file and

upload it to the course Sakai site **

Instructions on how to submit your completed homework on Sakai can be found at:

https://sakai-duke.screenstepslive.com/s/sakai_support/m/71158/l/1091053-how-do-studentssubmit-an-assignment

Fixed Income

1. A zero coupon bond with 2.5 years to maturity has a yield to maturity of 25% per annum. A 3year maturity annual-pay coupon bond has a face value of $1000 and a 25% coupon rate. The

coupon bond also has a yield to maturity of 25%.

a) Does the longer maturity bond have a larger interest rate sensitivity? Why or why not?

b) Calculate for each bond the percentage price change associated with a change of yield to

maturity from 25% to 26%.

Options

2. Construct profit diagrams or profit tables on expiration to show what position in IBM puts, calls

and/or underlying stock best expresses the investor’s objectives described below. Assume IBM

currently sells for $150 and for the profit diagrams/tables use share prices between $100 and

$200 (in $10 increments). Also assume that “at the money” puts and calls cost $15 each. (As

usual, the profit calculations ignore dividends and interest.)

a) An investor wants upside potential if IBM increases but wants (net) losses no greater than

$15 if prices decline.

b) An investor wants to capture profits if IBM declines in price but wants a guaranteed limited

loss if prices increase.

c) An investor wants to profit if IBM’s upcoming earnings announcement is either

unexpectedly good or disappointingly bad.

3. Stock XYZ is trading at $100 and pays no dividends. The interest rate is 5% per year continuously

compounded. A one-year European call option on XYZ has a strike $104. The volatility of XYZ is

20%.

a) What is the intrinsic value of the call?

b) What is the adjusted intrinsic value of the call?

c) Find the price of the European call, using the Black-Scholes model. What is the price of an

American call option with the same characteristics?

d) How many European calls do you need to buy/sell to hedge one share of XYZ?

4. Excel Question. Use the Black and Scholes file posted on Sakai. We want to explore the effect of

changing the time to expiration T from 1 to 2 years on the value of an out-of-the-money put

option. Throughout this exercise we keep the stock price at SÃ¢â€šâ‚¬=70, the strike price at X=40, the

dividend rate at ÃŽÂ´=0, and the stock price volatility at ÃÆ’=35%. For the interest rate r, consider

various values (from very low to very high).

a) Show how the effect on the put price for T=1Ã¢â€ â€™2 depends on the interest rate and explain

intuitively why this happens.

b) Confirm that the call price is always increasing in the time to expiration T.

Fixed-Income Securities:

Forward Rates & Yield Curve

Prof. Aino Levonmaa

1

Overview

Ã¯â€šÂ§ Forward rate: an interest rate on a future loan that is fixed

today

Ã¯Æ’Ëœ Inferred from the term structure of interest rates

y02

0

y01

1

f12

2

Ã¯â€šÂ§ Forward rate (f12) vs. spot rate (y01, y02)

Ã¯â€šÂ§ Yield curve (the term structure of interest rates)

Ã¢â‚¬â€œ the relationship between yield and maturity

Ã¯â€šÂ§ Theories for explaining the term structure of interest rates

Examples of Forward Rates

Ã¯â€šÂ§ A forward rate is an interest rate on a future loan

that is fixed today.

Ã¯â€šÂ§ The forward rate for 1-year lending starting t years

from now is denoted ft.

Ã¯â€šÂ§ A company will receive a payment next year and must

make a payment two years from now. The company

is worried about the re-investment risk related to the

incoming payment. Can the company lock in a lending

rate, starting one year from now?

Ã¯â€šÂ§ A firm foresees the need for short-term funds one

year from now but is worried about the interest rate

rising. Can they Ã¢â‚¬Å“lock inÃ¢â‚¬Â a rate for a one-year loan,

starting one year from now?

3

Engineering Forward Rates

Ã¯â€šÂ§ Suppose that the yields and prices of 1- and 2-year

zero coupon bonds are

P1 Ã¯â‚¬Â½

100

1 Ã¯â‚¬Â« y1

Ã¯Æ’â€º

100

P2 Ã¯â‚¬Â½

(1 Ã¯â‚¬Â« y2 ) 2

y1 Ã¯â‚¬Â½

Ã¯Æ’â€º

100

Ã¯â‚¬Â1

P1

Ã¯Æ’Â¦ 100 Ã¯Æ’Â¶

Ã¯Æ’Â·Ã¯Æ’Â·

y2 Ã¯â‚¬Â½ Ã¯Æ’Â§Ã¯Æ’Â§

Ã¯Æ’Â¨ P2 Ã¯Æ’Â¸

1

2

Ã¯â‚¬Â1

Ã¯â€šÂ§ Then, under the law of one price, what is the interest

rate specified today on a contract to buy a 1 year ZCB

1 year from now, i.e. what is the 1-year forward

rate?

4

Engineering Forward Rates

Ã¯â€šÂ§ The cash flows of the three instruments are as follows:

0

1

1-year zero

-P1

100

2-year zero

-P2

Forward contract

2

100

-Pf

100

Ã¯â€šÂ§ To find the price Pf and the forward rate from time 1 to

time 2

Ã¯â€šÂ§ Consider a strategy of:

Ã¢â‚¬â€œ Buying 1 unit of the 2 year zero and

Ã¢â‚¬â€œ Short selling P2/P1 units of the 1 year zero.

5

Engineering Forward Rates

Ã¯â€šÂ§ Cash flows to the strategy are:

Trade

t=0

t=1

t=2

Sell short

ZCB(1)

+(P2/P1)*P1=

P2

-(P2/P1)*100

0

Buy

ZCB(2)

-P2

0

100

Net CFs

0

-(P2/P1)*100

100

Ã¯â€šÂ§ Then, under no arbitrage, the price of the ZCB one year

from now must be:

Ã°Ââ€˜Â·Ã°ÂÅ¸Â

Ã°Ââ€˜Â·Ã°Ââ€™â€¡ = Ã°ÂÅ¸ÂÃ°ÂÅ¸Å½Ã°ÂÅ¸Å½

Ã°Ââ€˜Â·Ã°ÂÅ¸Â

What about the corresponding forward rate?

6

Engineering Forward Rates

Ã¯â€šÂ§ The forward rate is:

100

f1, 2

100

P1

(1 Ã¯â‚¬Â« y2 ) 2

(1 Ã¯â‚¬Â« y1 )

Ã¯â‚¬Â½

Ã¯â‚¬Â1 Ã¯â‚¬Â½ Ã¯â‚¬Â1 Ã¯â‚¬Â½

Ã¯â‚¬Â1 Ã¯â‚¬Â½

Ã¯â‚¬Â1

100

Pf

P2

1 Ã¯â‚¬Â« y1

2

(1 Ã¯â‚¬Â« y2 )

f1, 2

(1 Ã¯â‚¬Â« y2 ) 2

Ã¯â‚¬Â½

Ã¯â‚¬Â1

1 Ã¯â‚¬Â« y1

Ã¯â€šÂ§ This result can be generalized to calculate the

forward rate at any maturity:

(1 Ã¯â‚¬Â« yn Ã¯â‚¬Â«1 ) n Ã¯â‚¬Â«1

f n ,n Ã¯â‚¬Â«1 Ã¯â‚¬Â½

Ã¯â‚¬Â1

n

(1 Ã¯â‚¬Â« yn )

7

Engineering Forward Rates:

Example

Ã¯â€šÂ§ Suppose that

Ã¢â‚¬â€œ A 1-year zero has a YTM of 2%

Ã¢â‚¬â€œ A 2-year zero has a YTM of 3%

1) What are the prices of these bonds (assume FV =

100)?

2) How can you trade these bonds to replicate a loan

between year 1 and year 2 ? (Ã¢â‚¬Å“syntheticÃ¢â‚¬Â loan.)

3) What is the implicit interest rate in the second year?

Ã¢â‚¬â€œ This interest rate is the forward rate, f.

8

Forward Rates

Ã¯â€šÂ§ The forward rate is determined by no-arbitrage

(1 Ã¯â‚¬Â« yn Ã¯â‚¬Â«1 ) n Ã¯â‚¬Â«1

f n ,n Ã¯â‚¬Â«1 Ã¯â‚¬Â½

Ã¯â‚¬Â1

n

(1 Ã¯â‚¬Â« yn )

Ã¯â€šÂ§ Example with n=2 (and rewriting)

(1 Ã¯â‚¬Â« y2 ) Ã¯â‚¬Â½ Ã¯Ââ€ºÃ¯â‚¬Â¨1 Ã¯â‚¬Â« y1 Ã¯â‚¬Â©Ã¯â‚¬Â¨1 Ã¯â‚¬Â« f1, 2 Ã¯â‚¬Â©Ã¯ÂÂ

1/ 2

Ã¯â€šÂ§ Forward rate = interest rate that would need to

prevail in second year to make the long- and shortterm investments equally attractive.

9

Forward contracts and Futures

Ã¯â€šÂ§ Forward rates are also traded directly:

Ã¢â‚¬â€œ FRAÃ¢â‚¬â„¢s : Forward Rate Agreements

Ã¢â‚¬â€œ Interest Rate Futures

Ã¯â€šÂ§ We will come back to this in our class on Futures

10

The Yield Curve or the

Term-Structure of Interest

Ã¯â€šÂ§ The collection of YTM of zero-coupon bonds has many

names:

Ã¢â‚¬â€œ The term structure of zero-coupon bond yields

Ã¢â‚¬â€œ The term structure of (spot) interest rates

Ã¢â‚¬â€œ The yield curve

Ã¯â€šÂ§ Typical shapes of the term structure of interest:

Ã¢â‚¬â€œ Upward sloping (most common)

Ã¢â‚¬â€œ Downward sloping

Ã¢â‚¬â€œ Flat

Ã¢â‚¬â€œ Hump-shaped

11

The Yield Curve

Ã¯â€šÂ§ The yield curve can take many shapes; for example flat, upward

sloping, inverted, or humped.

15

The yield curve Jan 2020 vs. Jan 2021

13

1. The Expectations Hypothesis

Ã¯â€šÂ§ Assumptions:

Ã¢â‚¬â€œ No transaction costs.

Ã¢â‚¬â€œ No default.

Ã¢â‚¬â€œ Investors maximize profits; they are riskneutral.

Ã¢â‚¬â€œ All bonds are zero-coupon bonds.

Ã¯â€šÂ§ Implication of risk-neutrality: Investors choose

the maturity of their bonds to maximize holding

period returns.

14

Expectation Hypothesis

Ã¯â€šÂ§ Consider the choice between:

Ã¢â‚¬â€œ Buying 1-year zero coupon bond and holding it to maturity

Ã¢â‚¬â€œ Buying a 2-year zero coupon bond and then selling it after 1 year

when it will be a 1-year zero coupon bond.

Ã¯â€šÂ§ What are the expected holding period returns for these two

Ã¢â‚¬Å“one yearÃ¢â‚¬Â strategies?

Ã¯â€šÂ§ Alternatively, consider the choice between:

Ã¢â‚¬â€œ Buying 2-year zero coupon bond and holding it to maturity

Ã¢â‚¬â€œ Buying a 1-year zero coupon bond and then rolling the proceeds

into a second 1-year zero coupon bond at time 1.

Ã¯â€šÂ§ What are the expected holding period returns for these two

Ã¢â‚¬Å“two yearÃ¢â‚¬Â strategies?

15

E[HPR] One-year Strategies

Ã¯â€šÂ§ Expected HPR on the 1-year bond (known with certainty)

V1

100

E[ HPR ] Ã¯â‚¬Â½ HPR Ã¯â‚¬Â½

Ã¯â‚¬Â1 Ã¯â‚¬Â½

Ã¯â‚¬Â 1 Ã¯â‚¬Â½ y 0,1

V0

P0,1

Ã¯â€šÂ§ Expected HPR on buying a 2-year zero coupon bond at t=0 and

selling it at t=1

100

(1 Ã¯â‚¬Â« E[ y1, 2 ])

(1 Ã¯â‚¬Â« y0, 2 ) 2

E[V1 ]

E[ P1, 2]

E[ HPR] Ã¯â‚¬Â½

Ã¯â‚¬Â1 Ã¯â‚¬Â½

Ã¯â‚¬Â1 Ã¯â€šÂ»

Ã¯â‚¬Â1 Ã¯â‚¬Â½

Ã¯â‚¬Â1

100

V0

P0, 2

1 Ã¯â‚¬Â« E[ y1, 2 ]

2

(1 Ã¯â‚¬Â« y0, 2 )

16

E[HPR] One-year Strategies

Ã¯â€šÂ§ Under the expectations hypothesis, this HPR has to be equal to

the return from holding the 1-year bonds:

(1 Ã¯â‚¬Â« y0, 2 ) 2

1 Ã¯â‚¬Â« E[ y1, 2 ]

Ã¯â‚¬Â 1 Ã¯â‚¬Â½ y0,1 Ã¯Æ’â€º (1 Ã¯â‚¬Â« y0, 2 ) 2 Ã¯â‚¬Â½ (1 Ã¯â‚¬Â« y0,1 )(1 Ã¯â‚¬Â« E[ y1, 2 ])

Ã¯â€šÂ§ Or equivalently:

(1 Ã¯â‚¬Â« y0, 2 ) Ã¯â‚¬Â½ [(1 Ã¯â‚¬Â« y0,1 )(1 Ã¯â‚¬Â« E[ y1, 2 ])]

1

2

Ã¯â€šÂ§ The 2-year rate is the geometric average of the expected 1year rates

17

E[HPR] Two-year Strategies

Ã¯â€šÂ§ Consider buying the 2-year bond versus buying the 1-year

bond and rolling the proceeds over into a second 1-year bond

at maturity (in 1 year)

Ã¯â€šÂ§ Buy and hold 2-year bond:

Ã¯Æ’Â¦ V2

E[ HPR ] Ã¯â‚¬Â½ Ã¯Æ’Â§Ã¯Æ’Â§

Ã¯Æ’Â¨ V0

Ã¯Æ’Â¶

Ã¯Æ’Â·Ã¯Æ’Â·

Ã¯Æ’Â¸

1

2

Ã¯Æ’Â¦ 100 Ã¯Æ’Â¶

Ã¯Æ’Â·

Ã¯â‚¬Â 1 Ã¯â‚¬Â½ Ã¯Æ’Â§Ã¯Æ’Â§

Ã¯Æ’Â·

P

0

,

2

Ã¯Æ’Â¨

Ã¯Æ’Â¸

1

2

Ã¯Æ’Â¦

Ã¯Æ’Â§

100

Ã¯â‚¬Â1 Ã¯â‚¬Â½ Ã¯Æ’Â§

Ã¯Æ’Â§Ã¯Æ’Â§ 100

2

(

1

Ã¯â‚¬Â«

y

)

0, 2

Ã¯Æ’Â¨

Ã¯Æ’Â¶

Ã¯Æ’Â·

Ã¯Æ’Â·

Ã¯Æ’Â·Ã¯Æ’Â·

Ã¯Æ’Â¸

1

2

Ã¯â‚¬Â 1 Ã¯â‚¬Â½ y 0, 2

18

E[HPR] Two-year Strategies

Ã¯â€šÂ§ Rolling over the 1-year bond generates:

Ã¯Æ’Â¦

Ã¯Æ’Â¦100

Ã¯Æ’Â¶Ã¯Æ’Â¶

1

Ã¯Æ’Â§ 100Ã¯Æ’Â§

Ã¯Æ’Â·Ã¯Æ’Â·

E

[

P

]

Ã¯Æ’Â¦ E[V2 ] Ã¯Æ’Â¶ 2

1, 2 Ã¯Æ’Â¸ Ã¯Æ’Â·

Ã¯Æ’Â¨

Ã¯Æ’Â·Ã¯Æ’Â· Ã¯â‚¬Â 1 Ã¯â€šÂ» Ã¯Æ’Â§

E[ HPR] Ã¯â€šÂ» Ã¯Æ’Â§Ã¯Æ’Â§

Ã¯Æ’Â§

Ã¯Æ’Â·

P0,1

Ã¯Æ’Â¨ V0 Ã¯Æ’Â¸

Ã¯Æ’Â§

Ã¯Æ’Â·

Ã¯Æ’Â¨

Ã¯Æ’Â¸

1

2

Ã¯Æ’Â¦

Ã¯Æ’Â¶

Ã¯Æ’Â§ 100Ã¯â‚¬Â¨1 Ã¯â‚¬Â« E[ y1, 2 ]Ã¯â‚¬Â© Ã¯Æ’Â·

Ã¯â‚¬Â1 Ã¯â€šÂ» Ã¯Æ’Â§

Ã¯Æ’Â·

100

Ã¯Æ’Â§

Ã¯Æ’Â·

(

1

Ã¯â‚¬Â«

y

)

0

,

1

Ã¯Æ’Â¨

Ã¯Æ’Â¸

1

2

Ã¯â‚¬Â1

Ã¯â‚¬Â½ Ã¯Ââ€º(1 Ã¯â‚¬Â« y0,1 )Ã¯â‚¬Â¨1 Ã¯â‚¬Â« E[ y1, 2 ]Ã¯â‚¬Â©Ã¯ÂÂ 2 Ã¯â‚¬Â 1

1

19

Main Insights

Ã¯â€šÂ§ The general result, under the expectations hypothesis and its

risk neutrality assumption is that the yield on a long-term

bond is equal to the geometric average of expected short

term yields.

Ã¯â‚¬Â¨1 Ã¯â‚¬Â« yt (2)Ã¯â‚¬Â© Ã¯â‚¬Â½ Ã¯Ââ€ºÃ¯â‚¬Â¨1 Ã¯â‚¬Â« yt (1)Ã¯â‚¬Â©Ã¯â‚¬Â¨1 Ã¯â‚¬Â« Et Ã¯Ââ€ºyt Ã¯â‚¬Â«1 (1)Ã¯ÂÂÃ¯â‚¬Â©Ã¯ÂÂ

1

2

Ã¯â€šÂ§ All expected HPR are equalized in equilibrium.

Ã¯â€šÂ§ Typical shape of yield curve is flat.

Ã¯â€šÂ§ Short-term interest rates are more volatile than long-term

interest rates.

Ã¯â€šÂ§ What does an upward-sloping yield curve, i.e. y(2) > y(1), imply

about the expected future short-term interest rate?

20

Expectations Hypothesis: Example

Ã¯â€šÂ§ Suppose:

yt(1) = y01 = 2%

Et[yt+1(1)]= Et(y12) = 4%

Ã¯â€šÂ§ What should yt(2) (=y02) be under the expectations hypothesis?

Ã¯â€šÂ§ Investigate if short-term (1 year) investors are indifferent

between:

1) buying & holding a 1-year ZCB

2) buying a 2-year ZCB and selling it after 1 year

Ã¯â€šÂ§ Investigate if long-term (2 year) investors are indifferent

between:

1) buying & holding a 2-year ZCB

2) buying a 1-year ZCB and rolling the proceeds into

another 1-year ZCB at time 1

21

Link with forward rate

Ã¯â€šÂ§ Recall the definition of the forward rate:

Ã¯â‚¬Â¨1 Ã¯â‚¬Â« yt (2)Ã¯â‚¬Â© Ã¯â‚¬Â½ Ã¯Ââ€ºÃ¯â‚¬Â¨1 Ã¯â‚¬Â« yt (1)Ã¯â‚¬Â©Ã¯â‚¬Â¨1 Ã¯â‚¬Â« ft (1)Ã¯â‚¬Â©Ã¯ÂÂ

1

2

Ã¯â€šÂ§ Compare with prediction of EH theory

Ã¯â‚¬Â¨1 Ã¯â‚¬Â« yt (2)Ã¯â‚¬Â© Ã¯â‚¬Â½ Ã¯Ââ€ºÃ¯â‚¬Â¨1 Ã¯â‚¬Â« yt (1)Ã¯â‚¬Â©Ã¯â‚¬Â¨1 Ã¯â‚¬Â« Et Ã¯Ââ€ºyt Ã¯â‚¬Â«1 (1)Ã¯ÂÂÃ¯â‚¬Â©Ã¯ÂÂ

1

2

Ã¯â€šÂ§ Therefore, under the EH theory, the forward rate

equals the expected future 1-year interest rate :

f t (1) Ã¯â‚¬Â½ Et [ yt Ã¯â‚¬Â«1 (1)]

22

2. Liquidity Preference Theory

Ã¯â€šÂ§ Problem with the EH theory: yield curve is flat on average while

in the data it is upward sloping 90% of the time.

Ã¯â€šÂ§ Source of problem: risk-neutrality assumption in EH theory.

Ã¯â€šÂ§ short rate next period is not known, it is risky!

Ã¯â€šÂ§ When investors are risk averse, they care not only about the

expected short rate but also about its volatility.

Ã¯â€šÂ§ Investors in long-term bonds want to be compensated

Ã¢â‚¬â€œ For Ã¢â‚¬Å“tying upÃ¢â‚¬Â money for a long time.

Ã¢â‚¬â€œ For facing price risk if they need to sell before maturity.

Ã¯â€šÂ§ Conversely, issuers of bonds are willing to pay a higher interest

rate on long-term bonds because

Ã¢â‚¬â€œ They can lock in an interest rate for many years

23

Liquidity Preference Theory

Ã¯â€šÂ§ The associated risk premium is denoted the

liquidity premium (LP)

Ã¯â‚¬Â¨1 Ã¯â‚¬Â« yt (2)Ã¯â‚¬Â© Ã¯â‚¬Â½ Ã¯Ââ€ºÃ¯â‚¬Â¨1 Ã¯â‚¬Â« yt (1)Ã¯â‚¬Â©Ã¯â‚¬Â¨1 Ã¯â‚¬Â« Et Ã¯Ââ€º yt Ã¯â‚¬Â«1 (1)Ã¯ÂÂÃ¯â‚¬Â©Ã¯ÂÂ

1

2

Ã¯â‚¬Â« LP

Ã¯â€šÂ§ Based on this theory,

Ã¢â‚¬â€œ What is the typical shape of the yield curve?

Ã¢â‚¬â€œ Is the forward rate still equal to the expected

future short rate?

24

2. Liquidity Preference Theory: Examples

LP (+)

E (+)

YC (+)

Yield

Yield

YC (+)

LP (+)

E (-)

Maturity

YC (-)

1y

1y

10y

Yield

Yield

1y

10y

Maturity

LP(+)

YC (0)

E (-)

LP (+)

10y

1y

Maturity

Maturity

LP is yield spread due to liquidity premium; E is yield spread due to

expected future changes in short rate; YC is the total yield spread.

E (-)

10y

25

3. Segmented Markets Theory

Ã¯â€šÂ§ Some investors trade short-term bonds:

Ã¢â‚¬â€œ Short-term interest rates are determined by

supply and demand among these investors.

Ã¯â€šÂ§ Other investors trade long-term bonds:

Ã¢â‚¬â€œ Long-term interest rates are determined by supply

and demand among these investors.

Ã¯â€šÂ§ Each maturity sector is then a Ã¢â‚¬Ëœsegmented

marketÃ¢â‚¬â„¢ and the yield in that market is

determined independently.

26

4. Preferred Habitat Theory

Ã¯â€šÂ§ Borrowers and lenders have strong preferences

for particular maturities, but yields are not

determined independently.

Ã¯â€šÂ§ If expected returns are large enough, investors

will deviate from their preferred maturities.

Ã¯Æ’Ëœ Investors will accept additional risk in return for

additional expected returns.

27

Fixed Income Securities:

Interest Rate Management

Professor Aino Levonmaa

1

Outline

Ã¯â€šÂ§ Interest rate sensitivity of a bond price

Ã¢â‚¬â€œ What happens to prices of fixed income

securities as interest rates change?

Ã¯â€šÂ§ Duration

Ã¯â€šÂ§ Convexity

Ã¯â€šÂ§ Immunization: duration matching

2

Interest-Rate Sensitivity

Ã¢â‚¬Â¢ First order effect: Bond prices and interest

rates are negatively related.

Ã¢â‚¬Â¢ Maturity matters: Prices of long-term

bonds are more sensitive to interest-rate

changes than short-term bonds.

Ã¢â‚¬Â¢ Convexity: An increase in a bondÃ¢â‚¬â„¢s YTM

results in a smaller price decline than the

price gain associated with a decrease of

equal magnitude in the YTM.

3

Example: Interest-Rate Sensitivity

Price

90

89

88

P = 100 / (1+y)^T

5%

Ã¢â‚¬Â¢

Ã¢â‚¬Â¢

6%

7%

YTM

Price

1-Year ZCB

Price

2-Year ZCB

5%

95.24

90.70

6%

94.34

89.00

7%

93.46

87.34

YTM

Negative convex relation YTM and price:

Ã¢â‚¬â€œ Price change 2-year ZCB 6 Ã¢â€ â€™ 5%:

Ã¢â‚¬â€œ Price change 2-year ZCB 6 Ã¢â€ â€™ 7%:

Maturity Matters:

Ã¢â‚¬â€œ % change 1y 6 Ã¢â€ â€™ 7%:

Ã¢â‚¬â€œ % change 2y 6 Ã¢â€ â€™ 7%:

90.70-89.0 = 1.70

87.34-89.0 = -1.66

(93.46-94.34)/94.34 = -.93%

(87.34-89.00)/89.00 = -1.87%

4

Price Sensitivity of BondsÃ¢â‚¬Â¦

Ã¢â‚¬Â¢ Ã¢â‚¬Â¦ as interest rates change

Ã¢â‚¬Â¢ Maturity clearly plays a role

Ã¢â‚¬Â¢ This class introduce the precise concept that

matters: DURATION

5

Duration: The Concept

Ã¢â‚¬Â¢ Duration (or Macaulay duration)

Ã¢â‚¬â€œ Is defined as Ã¢â‚¬Å“the averageÃ¢â‚¬Â time you have to wait for

your payments

Ã¢â‚¬â€œ Is used to determine how sensitive the price of a bond

is to changes in the yield

(maturity matters Ã¢â€ â€™ duration matters)

Ã¢â‚¬Â¢ How long on average do you have to wait for your

payments in the following two situations?

-P

A:

B:

1000

t=0

t=1

t=2

t=3

t=4

-P

80

80

80

1080

t=0

t=1

t=2

t=3

t=4

6

Duration: The Derivation

The price sensitivity is linked to Ã¢â‚¬Å“the average timeÃ¢â‚¬Â you

have to wait for your payments, provided the weights

are the contributions to price of the bond:

CF1

CF2

CFT

PÃ¯â‚¬Â½

Ã¯â‚¬Â«

Ã¯â‚¬Â« … Ã¯â‚¬Â«

1

2

Ã¯â‚¬Â¨1 Ã¯â‚¬Â« y Ã¯â‚¬Â© Ã¯â‚¬Â¨1 Ã¯â‚¬Â« y Ã¯â‚¬Â©

Ã¯â‚¬Â¨1 Ã¯â‚¬Â« y Ã¯â‚¬Â©T

dP

CF1

CF2

CFT

Ã¯â‚¬Â½ Ã¯â‚¬Â1

Ã¯â‚¬Â2

Ã¯â‚¬Â … Ã¯â‚¬Â T

2

3

dy

Ã¯â‚¬Â¨1 Ã¯â‚¬Â« y Ã¯â‚¬Â©

Ã¯â‚¬Â¨1 Ã¯â‚¬Â« y Ã¯â‚¬Â©

Ã¯â‚¬Â¨1 Ã¯â‚¬Â« y Ã¯â‚¬Â©T Ã¯â‚¬Â«1

dP 1 Ã¯â‚¬Â« y

CF1

CF2

CFT

Ã¯â‚¬Â

Ã¯â‚¬Â½1

Ã¯â‚¬Â«2

Ã¯â‚¬Â« … Ã¯â‚¬Â« T

1

2

T

dy P

PÃ¯â‚¬Â¨1 Ã¯â‚¬Â« y Ã¯â‚¬Â©

PÃ¯â‚¬Â¨1 Ã¯â‚¬Â« y Ã¯â‚¬Â©

PÃ¯â‚¬Â¨1 Ã¯â‚¬Â« y Ã¯â‚¬Â©

7

Duration: Summary

Ã¢â‚¬Â¢ The duration (D) of a bond is defined as minus the

elasticity of its price (P) with respect to (1 plus) its YTM (y):

dP 1 Ã¯â‚¬Â« y T

Cashflow (t )

DÃ¯â‚¬Â½Ã¯â‚¬Â

Ã¯â‚¬Â½ Ã¯Æ’Â¥ wt t , where wt Ã¯â‚¬Â½

t

dy P

(

1

Ã¯â‚¬Â«

y

)

P

1

Ã¢â‚¬Â¢ We see that the duration D is equal to the average of the

cash-flow times, weighted by their contribution to the

present value of the bond!

Ã¢â‚¬Â¢ The relative price-response to a yield change is therefore:

Ã¯Ââ€žP

D

Ã¯Ââ‚¬Ã¯â‚¬Â

Ã¯Ââ€žy

P

1

Ã¯â‚¬Â«

y

Ã¯ÂÂ»

Modified

Duration

8

Duration: Example

Ã¢â‚¬Â¢ What is the duration of a 4-year coupon bond

with a face value of $1,000, a coupon rate of

8% and a YTM of 10%?

80

80

80

1080

PÃ¯â‚¬Â½

Ã¯â‚¬Â«

Ã¯â‚¬Â«

Ã¯â‚¬Â«

Ã¯â‚¬Â½ 936.6

1

2

3

4

Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10Ã¯â‚¬Â© Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10Ã¯â‚¬Â© Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10Ã¯â‚¬Â© Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10Ã¯â‚¬Â©

80

80

w1 Ã¯â‚¬Â½

Ã¯â‚¬Â½

0

.

0777

,

w

2 Ã¯â‚¬Â½

Ã¯â‚¬Â½ 0.0706

1

2

936.6 Ã¯â€šÂ´ Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10 Ã¯â‚¬Â©

936.6 Ã¯â€šÂ´ Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10 Ã¯â‚¬Â©

80

1080

w3 Ã¯â‚¬Â½

Ã¯â‚¬Â½ 0.0642 , w4 Ã¯â‚¬Â½

Ã¯â‚¬Â½ 0.7876

3

4

936.6 Ã¯â€šÂ´ Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10 Ã¯â‚¬Â©

936.6 Ã¯â€šÂ´ Ã¯â‚¬Â¨1 Ã¯â‚¬Â« .10 Ã¯â‚¬Â©

D Ã¯â‚¬Â½ w1 Ã¯â€šÂ´1 Ã¯â‚¬Â« w2 Ã¯â€šÂ´ 2 Ã¯â‚¬Â« w3 Ã¯â€šÂ´ 3 Ã¯â‚¬Â« w4 Ã¯â€šÂ´ 4 Ã¯â‚¬Â½ 3.56

9

Duration: Example

Ã¢â‚¬Â¢ If the YTM changes from y = 10% to 10.1%,

what would be the (relative) change in price?

Ã¯Ââ€žP

D

3.56

Ã¯Ââ‚¬Ã¯â‚¬Â

Ã¯Ââ€žy Ã¯â‚¬Â½ Ã¯â‚¬Â

0.001 Ã¯â‚¬Â½ Ã¯â‚¬Â0.32%

P

1

Ã¯â‚¬Â«y

1

Ã¯â‚¬Â«Ã¯â‚¬Â²

.10

Ã¯â‚¬Â±

Ã¯â‚¬Â³

Ã¯ÂÂ»

Modified

Duration

Modified

Duration

Ã¢â‚¬Â¢ Does the duration formula imply that longterm bonds are more risky (bigger price

changes) than short-term bonds?

Yes for same change; BUT: Long-term interest rates less

volatile

10

Duration: Useful Facts

Ã¢â‚¬Â¢ The duration of a portfolio is the weighted average of the

durations of the constituents: D p Ã¯â‚¬Â½ Ã¯Æ’Â¥ Ã¯ÂÂ· i Di

i

Ã¢â‚¬Â¢ What is the duration of zero-coupon bond?

Ã¢â‚¬Â¢ What must be true for the duration of a coupon bond?

Ã¢â‚¬Â¢ What happens to the duration of a coupon bond if (all else

equal) the coupon rate increases?

Ã¢â‚¬Â¢ What happens to the duration of the bond if (all else

equal) the YTM increases?

Ã¢â‚¬Â¢ The duration of a perpetuity is: (1+y)/y

11

Going back to our previous 8% coupon bond.

When YTM increases to 11%

P becomes 906.927 (decreased)

W1 = 0.0797 (Ã¢â€ â€˜)

W2 = 0.0716 (Ã¢â€ â€˜)

W3 = 0.0645 (Ã¢â€ â€˜)

W4 = 0.07844 (Ã¢â€ â€œ)

Ã¯Æ’Â¨ New D = 3.554 (Ã¢â€ â€œ)

12

Portfolio Duration

Ã¢â‚¬Â¢ How to compute the duration of a bond portfolio?

Ã¢â‚¬Â¢ -> Simply the weighted average duration of the bonds in the

portfolio.

Ã¢â‚¬Â¢ The weight for each bond duration is calculated as the market

value of the bond divided by the total market value of the bond

Bond Market Value ($m) Weight Mod.

portfolio.

Duration

A

10

0.1

4

B

40

0.4

7

C

30

0.3

6

D

20

0.2

2

Ã¢â‚¬Â¢ Then the portfolio duration is given by 0.1*4+0.4*7+0.3*6+0.2*2 =

5.4

Ã¢â‚¬Â¢ If the yields affecting all four bonds change by 100bps, the

portfolioÃ¢â‚¬â„¢s value will change by approximately 5.4%.

Duration: How good is the

approximation? (y=8%, T=30)

14

Limitations of Duration

Ã¢â‚¬Â¢ Duration is an approximation for the change in bond prices due to

the change in yields.

Ã¢â‚¬Â¢ Another key limitation: duration is based on the bond pricing

equation:

Ã¢â‚¬Â¢ It assumes that all cash flows are discounted at the same interest

rate, then duration assumes that yields for all maturities change

by an equal amount.

Ã¢â‚¬â€œ This means we assume a parallel shift in the yield curve.

Ã¢â‚¬Â¢ Improve the approximation by also using convexity

measure

Convexity

Ã¢â‚¬Â¢ The sensitivity of price with respect to yield is

approximated by a linear function when using duration.

Ã¢â‚¬Â¢ The relation is really non-linear, specifically, itÃ¢â‚¬â„¢s convex.

Ã¢â‚¬Â¢ The convexity of a bond is the curvature of its price-yield

relationship:

d 2P 1 T

(t 2 Ã¯â‚¬Â« t )

Cashflow (t )

Convexity Ã¯â‚¬Â½ 2 Ã¯â‚¬Â½ Ã¯Æ’Â¥ wt

,

where

w

Ã¯â‚¬Â½

t

dy P 1

(1 Ã¯â‚¬Â« y)2

(1 Ã¯â‚¬Â« y )t P

Ã¢â‚¬Â¢ The relative price response to a yield change can be

better approximated using convexity:

Ã¯Ââ€žP

D

1

Ã¯Ââ‚¬Ã¯â‚¬Â

Ã¯Ââ€žy Ã¯â‚¬Â« Convexity Ã¯â€šÂ´ (Ã¯Ââ€žy ) 2

P

1Ã¯â‚¬Â« y

2

16

DIY: Example Convexity

Ã¢â‚¬Â¢ Same 4-year coupon bond with F = $1,000,

coupon rate of 8% and YTM of 10%?

Ã¢â‚¬Â¢ Recall earlier calculation: w1=.0777,

w2=.0706, w3=.0642, w4=.7876, D = 3.5617.

Ã¢â‚¬Â¢ Calculate convexity = 14.133 =

=

1+12

Ã°Ââ€˜Â¤

1+0.1 2 1

+

2+22

3+32

4+42

Ã°Ââ€˜Â¤ +

Ã°Ââ€˜Â¤ +

Ã°Ââ€˜Â¤

1+0.1 2 2 1+0.1 2 3 1+0.1 2 4

Ã¢â‚¬Â¢ What happens to the bond price if YTM

increases to 11%?

17

Convexity: Summary

Ã¢â‚¬Â¢ When yields decline, the price increase in the

bond is underestimated by the simple duration

formula. A convexity term corrects the problem.

Ã¢â‚¬Â¢ The more convex a bond, the greater the expected

price increase for a given decrease in yield and the

smaller the expected price decrease.

Ã¢â‚¬Â¢ If interest rates are volatile, this is an attractive

asymmetry.

18

Interest-Rate Management

Ã¢â‚¬Â¢

Ã¢â‚¬Â¢

Ã¢â‚¬Â¢

Investors and financial institutions are subject to interest-rate

risk, for instance:

Ã¢â‚¬â€œ Homeowner: mortgage payments (ARM)

Ã¢â‚¬â€œ Bank: short-term deposits and long-term loans

Ã¢â‚¬â€œ Pension fund: owns bonds and must pay retirees

A change in the interest rate results in:

1. Price risk

2. Re-investment risk

Want to construct a portfolio which is insensitive to interestrate changes.

Ã¯Æ’Ëœ We want to Ã¢â‚¬Å“immunizeÃ¢â‚¬Â the portfolio of assets against ÃŽâ€y

19

Duration Matching: Immunization

Ã¢â‚¬Â¢ Duration matching means to make the duration

of assets and liabilities equal.

Ã¢â‚¬Â¢ Then, the sensitivity to interest-rate changes is:

D Assets Assets

D Liabilitie s Liabilitie s

Ã¯Ââ€žP Ã¯Ââ‚¬

P

Ã¯Ââ€žy Ã¯â‚¬Â

P

Ã¯Ââ€žy Ã¯â‚¬Â½ 0

1Ã¯â‚¬Â« y

1Ã¯â‚¬Â« y

Ã¢â‚¬Â¢ Interest rate changes makes the values of assets

and liabilities change by the same amount: The

portfolio is immunized.

20

Markets: GMÃ¢â‚¬â„¢s Pension Fund

Ã¢â‚¬Â¢ General MotorsÃ¢â‚¬â„¢ pension fund has

Ã¢â‚¬â€œ Liabilities with duration of about 15 years

Ã¢â‚¬â€œ Assets (bonds) with duration of about 5 years

Ã¢â‚¬â€œ Problem: Duration mismatch!

Ã¢â‚¬Â¢ Price risk: When the interest rate falls:

Ã¢â‚¬â€œ The value of the bonds increases, but

Ã¢â‚¬â€œ The present value of the liabilities increases more!

Ã¢â‚¬Â¢ Reinvestment risk:

Ã¢â‚¬â€œ At the new interest rate, the assets could not be

reinvested to make the future payments.

21

Solution: Immunization

Ã¢â‚¬Â¢ Suppose:

Ã¢â‚¬â€œ GMÃ¢â‚¬â„¢s pension fund must pay $100M in 15 years

Ã¢â‚¬Â¢ Assume no payment before year 15. Liability duration?

Ã¢â‚¬â€œ The current market interest rate is 6% at all maturities

Ã¢â‚¬Â¢ Assume the fund can only invest in 1-year and 30-year zero-coupon

bonds

Ã¢â‚¬â€œ How many securities should the fund buy to be immunized?

Ã¢â‚¬â€œ What are the prices of these bonds (face value is $100)?

Ã¢â‚¬Â¢ Right after the fund buys the bonds, the interest rate rises to 7%

Ã¢â‚¬â€œ What is the new value of the fundÃ¢â‚¬â„¢s bond position?

Ã¢â‚¬â€œ What is the new present value of the fundÃ¢â‚¬â„¢s liabilities?

22

Solution: Immunization

1

2

Mat.

3

4

5

6

7

8

9

10

Portfolio

weight

PV

holding

@ y=6%

Bond

price @

y=6%

No. of

bonds

(mill.)

FV

15yrs

@

y=6%

Bond

price

@

y=7%

PV

bonds

@

y=7%

FV

15yrs

@

y=7%

41.73

41.73

1.00

100

36.25

36.25

100

L

15

A

1

0.517

21.58

94.34

0.23

51.72

93.46

21.38

58.99

A

30

0.483

20.14

17.41

1.16

48.27

13.14

15.20

41.93

Total A

1.000

41.73

100

23

36.58 100.90

Problems with Immunization

Ã¢â‚¬Â¢ Our example: immunization against once-andfor-all change

Ã¢â‚¬Â¢ Reality: rates change frequently;

immunization requires rebalancing.

Ã¢â‚¬Â¢ It is an approximation that assumes:

Ã¢â‚¬â€œ Only risk of changes in the level of interest rates;

not in the slope of the term-structure.

Ã¢â‚¬â€œ Small interest rate changes Ã¢â‚¬â€œ improve duration

matching by also matching convexity.

24

Concepts to Know

Ã¢â‚¬Â¢ Duration

Ã¢â‚¬Â¢ Convexity

Ã¢â‚¬Â¢ Immunization

25

Options:

Basics and Strategies

Professor Aino Levonmaa

1

Overview

Ã¢â‚¬Â¢ Option basics

Ã¢â‚¬â€œ Option valuation on expiration date

Ã¢â‚¬â€œ Option strategies

Ã¢â‚¬Â¢ Next two classes: Option valuation prior to

expiration date

Ã¢â‚¬â€œ No-arbitrage bounds on option prices

Ã¢â‚¬â€œ Put Call Parity

Ã¢â‚¬â€œ Black-Scholes Pricing model

2

Option Basics

TODAY:

Ã¢â‚¬Â¢ Derivatives

Ã¢â‚¬Â¢ Option characteristics

Ã¢â‚¬Â¢ Value of options at expiration

Ã¢â‚¬Â¢ Option strategies

3

Derivatives

Ã¢â‚¬Â¢ A derivative is a security with a payoff that depends on

the price of another security

Ã¢â‚¬Â¢ The other security is called the underlying (security)

Ã¢â‚¬Â¢ Examples: options, futures, swaps.

Ã¢â‚¬Â¢ Derivatives are used for

Ã¢â‚¬â€œ Risk management, hedging

Ã¢â‚¬â€œ Executive compensation

Ã¢â‚¬â€œ Portfolio insurance

Ã¢â‚¬â€œ Speculation

4

Options Characteristics

Ã¢â‚¬Â¢ Option types

Ã¢â‚¬â€œ Call option: right to Buy underlying at

Ã¢â‚¬â€œ Put option: right to Sell underlying at

Ã¢â‚¬Â¢ Exercise price / strike price (X)

Ã¢â‚¬Â¢ Can exercise at or before expiration (T)

Ã¢â‚¬â€œ European: exercise only at expiration

Ã¢â‚¬â€œ American: exercise any time at or before expiration

Ã¢â‚¬Â¢ Pay option price or premium

Ã¢â‚¬Â¢ Call option: In-the-money, out-of-the-money, at-the-money

S0 > X

S0 < X
S0 = X
Ã¢â‚¬Â¢ Net profit includes cost of option
5
Value of Options at Expiration
Ã¢â‚¬Â¢ At expiration, if the stock price is ST, a Call
option with strike price X is worth:
Ã¯Æ’Â¬ ST Ã¯â‚¬Â X
CT Ã¯â‚¬Â½ Ã¯Æ’Â
Ã¯Æ’Â®0
if ST Ã¯â‚¬Â¾ X
if ST Ã¯â€šÂ£ X
Ã¢â‚¬Â¢ At expiration, if the stock price is ST, a Put
option with strike price X is worth:
Ã¯Æ’Â¬0
PT Ã¯â‚¬Â½ Ã¯Æ’Â
Ã¯Æ’Â® X Ã¯â‚¬Â ST
if ST Ã¯â€šÂ³ X
if ST Ã¯â‚¬Â¼ X
6
Stock Option Logic
Ã¢â‚¬Â¢ Q: Why do you buy a call?
Ã¢â‚¬Â¢ A: you expect the stock price to go up.
Ã¢â‚¬Â¢ Q But you could buy the stock outright?
Ã¢â‚¬Â¢ A: A call is sometimes a better vehicle because it gives
you downside protection. You think the stock will go up,
but you are worried it may go down. You protect against
the downside. The option payoff is not symmetric, the
stock payoff is symmetric.
Ã¢â‚¬Â¢ Q: Can you replicate the call option payoff with a
position in the stock only?
7
The Value of a Call at Expiration
Ã¢â‚¬Â¢ Payoff and net profit for a call option with a strike/
exercise price of X=$100 and premium of $10.
20
10
0
80
90
100
110
120
130
ST
-10
ST
Payoff
Profit
80
0
-10
90
0
-10
100
0
-10
110
10
0
120
20
10
130
30
20
8
The Value of a Put at Expiration
Ã¢â‚¬Â¢ Payoff and net profit for a put option with a strike/
exercise price of X=$100 and premium of $10
20
10
0
80
90
100
110
120
130
ST
-10
ST
80
90
100
110
120
130
Payoff
Profit
20
10
10
0
0
-10
0
-10
0
-10
0
-10
9
DIY: Value of a Short Call at Expiration
Ã¢â‚¬Â¢ Determine the payoff and net profit for a short a
call with a strike of X=$100 and premium of $10
20
10
0
80
90
100
110
120
130
ST
-10
ST
Payoff
Profit
80
90
100
110
120
130
10
DIY: Value of a Short Call at Expiration
Ã¢â‚¬Â¢ Determine the payoff and net profit for a short a
call with a strike of X=$100 and premium of $10
20
10
0
80
90
100
110
120
130
ST
-10
ST
Payoff
80
0
90
0
100
0
110
-10
120
-20
130
-30
Profit
10
10
10
0
-10
-20
11
Option Strategies
1.
2.
3.
4.
5.
Using calls for leverage
Protective put
Covered calls
Straddle/Strangle
Collars
12
DIY: 1. Call Options for Leverage
Ã¢â‚¬Â¢
Ã¢â‚¬Â¢
Example
Ã¢â‚¬â€œ Microsoft share price is S0=$70
Ã¢â‚¬â€œ A call option with X=$70 and 6-month maturity costs C0=$10
What is the payoff investing $7,000 in
A. Buy 100 shares of Microsoft
B. Buy 700 call options with X=$70
C. Buy 100 call options and invest $6,000 at the risk-free rate
(2% per 6 months).
ST
Payoff A:
Payoff B:
Payoff C:
60
__
__
__
65
__
__
__
70
__
__
__
75
__
__
__
80
__
__
__
85
__
__
__
90
__
__
__
13
DIY: 1. Call Options for Leverage
Ã¯â€šÂ§
Example
Ã¢â‚¬â€œ
Ã¢â‚¬â€œ
Ã¯â€šÂ§
Microsoft share price is S0=$70
A call option with X=$70 and 6-month maturity costs
C0=$10
What is the payoff investing $7000 in
A. Buy 100 shares of Microsoft
B. Buy 700 call options with X=$70
C. Buy 100 call options and invest $6000 at the risk-free
rate (2% per half year).
ST
Payoff A:
Payoff B:
Payoff C:
60
6000
__
0
__
6120
__
65
6500
__
0
__
6120
__
70
7000
__
0
__
6120
__
14
75
7500
__
3500
__
6620
__
80
8000
__
7000
__
7120
__
85
90
8500
9000
__
__
10500
__ 14000
__
7620
8120
__
__
DIY: 1. Call Options for Leverage
Return = Payoff/7000 -1
ST
Return A:
Return B:
Return C:
60
-14.3%
__
- 100%
__
-12.6%
__
65
70
75
-7.1%
__
__
0 % 7.1%
__
-100%
-100%
-50%
__
__
__
-12.6%
__
-12.6%
__
-5.4%
__
15
80
85
90
14.3%
__
21.4%
__
28.6%
__
0%
50%
100%
__
__
__
1.7%
__
8.9%
__
16%
__
2. Protective Put
Ã¢â‚¬Â¢ You own a share of Microsoft with current
price S0=$70. You are afraid that the stock
price will drop. How do you limit your possible
losses by trading options?
ST
Payoff Stock:
Payoff Put,X=70:
Payoff Total:
40
40
30
70
50
50
20
70
60
60
10
70
70
70
0
70
80
80
0
80
90
90
0
90
100
100
0
100
Ã¢â‚¬Â¢ This payoff is the same as that of a long call
with X=70 + a bond with a face value of 70!
Next week: put-call parity.
16
Protective
Put
Combined
payoff
Net profit on
combo
Put payoff
Premium
for buying
put
Stock payoff
ST Ã¢â€°Â¤ X
ST > X

Long stock

ST

ST

Long put

X-ST

0

Net

X

ST

3. Covered Call

Ã¢â‚¬Â¢ Suppose that you own a share of Microsoft

for S0=$70. You think that at-the-money call

options trading at $10 seem excessively

expensive, and you want to profit from this.

ST

40

Payoff Stock:

40

Payoff Short Call: 0

Payoff Total:

40

Profit Total:

50

50

50

0

50

60

60

60

0

60

70

70

70

0

70

80

80

80

-10

70

80

90

90

-20

70

80

100

100

-30

70

80

Ã¢â‚¬Â¢ You sold your upside.

Ã¢â‚¬Â¢ Compare: Ã¢â‚¬Å“naked option writingÃ¢â‚¬Â

18

Covered Call

Stock payoff

Combined profit

on combo

Combined payoff

Premium for

writing call

Short Call payoff

ST > X

ST Ã¢â€°Â¤ X

Long stock

ST

ST

Short call

-(ST-X)

0

Net

X

ST

19

4. Straddle and Strangle

Ã¢â‚¬Â¢ You have private information that a particular

stockÃ¢â‚¬â„¢s price will change dramatically soon, but

you do not know if it will go up or down. So you

buy

Ã¢â‚¬â€œ Straddle: Put (X=X1) + Call (X=X2), X1=X2

Ã¢â‚¬â€œ Strangle: Put (X=X1) + Call (X=X2), X1

Purchase answer to see full

attachment