Financial risk management course

Financial Risk Management 2010-11 Topics T1 Stock index futures Duration, Convexity, Immunization T2 Repo and reverse repo Futures on T-bills Futures on T-bonds Delta, Gamma, Vega hedgin

Financial Risk Management 2010-11 Topics

T1 Stock index futures

Duration, Convexity, Immunization

T2 Repo and reverse repo

Futures on T-bills Futures on T-bonds Delta, Gamma, Vega hedging

T3 Portfolio insurance

Implied volatility and volatility smiles

T4 Modelling stock prices using GBM

Interest rate derivatives (Bond options, Caps, Floors, Swaptions) T5 Value at Risk

T6 Value at Risk: statistical issues

Monte Carlo Simulations Principal Component Analysis Other VaR measures

T7 Parametric volatility models (GARCH type models)

Non-parametric volatility models (Range and high frequency models) Multivariate volatility models (Dynamic Conditional Correlation DCC models) T8 Credit Risk Measures (credit metrics, KMV, Credit Risk Plus, CPV)

T9 Credit derivatives (credit options, total return swaps, credit default swaps)

Asset Backed Securitization Collateralized Debt Obligations (CDO)

* This file provides you an indication of the range of topics that is planned to be covered in the

module However, please note that the topic plans might be subject to change

Financial Risk Management

Topic 1 Managing risk using Futures

Reading: CN(2001) chapter 3

Futures Contract:

Speculation, arbitrage, and hedging

Stock Index Futures Contract:

Futures Contract:

Speculation, arbitrage, and hedging

Stock Index Futures Contract:

Topics

Stock Index Futures Contract:

Hedging (minimum variance hedge ratio)

Hedging market risks

Stock Index Futures Contract:

Hedging (minimum variance hedge ratio)

Hedging market risks

Futures Contract

Agreement to buy or sell “something” in the future at

a price agreed today (It provides Leverage.)

Speculation with Futures: Buy low, sell high

Futures (unlike Forwards) can be closed anytime by taking

Speculation with Futures

Purchase at F0= 100

Hope to sell at higher price later F1= 110

Close-out position before delivery date.

Obtain Leverage (i.e initial margin is ‘low’)

Example: Example : : Nick Leeson: Feb 1995

Long 61,000 Nikkei-225 index futures (underlying

value = $7bn).

Nikkei fell and he lost money (lots of it)

- he was supposed to be doing riskless ‘index

arbitrage’ not speculating

$10

Speculation with Futures

Profit/Loss per contract

Speculation with Futures

Profit payoff (direction vectors)

-1

F increase

then profit increases

F increase then profit decrease

Arbitrage with Futures

Arbitrage with Futures

At expiry (T), FT = ST Else we can make riskless

profit (Arbitrage).

Forward price approaches spot price at maturity

Forward price ‘at a premium’ when : F > S (contango)

Forward price, F

Stock price, St

T Forward price ‘at a premium’ when : F > S (contango)

Forward price ‘at a discount’, when : F < S (backwardation)

0

At T, ST= FT

Arbitrage with Futures

General formula for non-income paying security:

F0= S0erT or F0 = S0(1+r)T

Futures price = spot price + cost of carry

For stock paying dividends, we reduce the ‘cost of

For stock paying dividends, we reduce the ‘cost of

carry’ by amount of dividend payments (d)

F 0 = S 0 e (r-d)T

For commodity futures, storage costs (v or V) is

negative income

F 0 = S 0 e (r+v)T or F 0 = (S 0 +V)e rT

Arbitrage with Futures

For currency futures, the ‘cost of carry’ will be

reduced by the riskless rate of the foreign currency

(rf)

F 0 = S 0 e (r-rf)T

Arbitrage with Futures

Arbitrage at t<T for a non-income paying security:

If F0> S0erTthen buy the asset and short the futures contract

If F0< S0erTthen short the asset and buy the futures contract

Hedging with Futures

Hedging with Futures

F and S are positively correlated

To hedge, we need a negative correlation So we

long one and short the other.

Hedge = long underlying + short Futures

Hedging with Futures

Simple Hedging Example:

You long a stock and you fear falling prices over the

next 2 months, when you want to sell Today (say

January), you observe S0=£100 and F0=£101 for

April delivery.

Today: you sell one futures contract so r is 4%

Today: you sell one futures contract

In March: say prices fell to £90 (S1=£90) So

F1=S1e0.04x(1/12)=£90.3 You close out on Futures.

Profit on Futures: 101 – 90.3 = £10.7

Loss on stock value: 100 – 90 =£10

Net Position is +0.7 profit Value of hedged portfolio

= S1+ (F0- F1) = 90 + 10.7 = 100.7

so r is 4%

Hedging with Futures

F1value would have been different if r had changed.

This is Basis Risk Basis Risk Basis Risk (b1= S1– F1)

Final Value = S1 + (F0 - F1) = £100.7

= (S1- F1 ) + F0

= b + F

= b1 + F0where “Final basis” b1= S1- F1

At maturity of the futures contract the basis is zero (since S1 = F1 ) In general, when contract is closed out prior to maturity b1= S1- F1 may not be zero However, b1will usually be small in relation to F0.

Stock Index Futures Contract

Hedging with SIFs

Stock Index Futures Contract

Stock Index Futures contract can be used to

eliminate market risk from a portfolio of stocks

If this equality does not hold then index arbitrage

(program trading) would generate riskless profits

( )

0 0

r d T

F = × S e −

(program trading) would generate riskless profits

Risk free rate is usually greater than dividend yield

(r>d) so F>S

Hedging with Stock Index Futures

Example: A portfolio manager wishes to hedge her

portfolio of $1.4m held in diversified equity and

Hedging with Stock Index Futures

The required number of Stock Index Futures contract

to short will be 3

In the above example, we have assumed that S and

0 0

Hedging with Stock Index Futures

Minimum Variance Hedge Ratio

∆∆∆∆ V = change in spot market position + change in Index Futures position

TVS f N

F F FVF

f N S S TVS

V

/,/.002

2/

2)0(

2)(

2/)

20(

2

∆

∆+

∆+

∆

=

σ

σ σ

σ

where, z = contract multiple for futures ($250 for S&P 500 Futures); ∆∆∆∆ S =

S 1 - S 0 , ∆∆∆∆ F = F 1 - F 0

The variance of the hedged portfolio is

Hedging with Stock Index Futures

To obtain minimum, we differentiate with respect to NNffff( ) and set to zero

FVF

TVS

N = − 0 β

implies f

FaceValue

Position Spot

of Value

p f

FVF

= −

Hedging with Stock Index Futures

Application: Changing beta of your portfolio: “Market Timing Strategy”

Example: βp (=say 0.8)is your current ‘spot/cash’ portfolio of stocks

) (

0

0

p h

•It’s ‘expensive’ to sell low-beta shares and purchase high-beta shares

•Instead ‘go long’ more Nf Stock Index Futures contracts

Note: If βh= 0, then Nf= - (TVS0/ FVF0) βp

Hedging with Stock Index Futures

If you hold stock portfolio, selling futures will place a

hedge and reduce the beta of your stock portfolio

If you want to increase your portfolio beta, go long

futures

Example: Suppose β= 0.8 and Nf= -6 contracts would

Example: Suppose β= 0.8 and Nf= -6 contracts would

make β= 0

If you short 3 (-3) contracts instead, then β= 0.4

If you long 3 (+3) contracts instead, then β= 0.8+0.4

= 1.2

Hedging with Stock Index Futures

Application: Stock Picking and hedging market risk

You hold (or purchase) 1000 undervalued shares of Sven plc

V(Sven) = $110 (e.g Using Gordon Growth model) P(Sven) = $100 (say)

Sven plc are underpriced by 10%.

Therefore you believe Sven will rise 10% more than the market over the next

3 months

But you also think that the market as a whole may fall by 3%

The beta of Sven plc (when regressed with the market return) is 2.0

Hedging with Stock Index Futures

Can you ‘protect’ yourself against the general fall in the market and hence any

‘knock on’ effect on Sven plc ?

Yes Sell Nfindex futures, using:

p f

FVF

TVS

N = − 0 β

FVF

Hedging with Stock Index Futures

Application: Future stock purchase and hedging market risk

You want to purchase 1000 stocks of takeover target with βp=2, in 1 month’s time when you will have the cash

You fear a general rise in stock prices

Go long Stock Index Futures (SIF) contracts, so that gain on the futures will

Financial Risk Management

Topic 2 Managing interest rate risks

Reference: Hull(2009), Luenberger (1997), and CN(2001)

Duration, immunization, convexity

Repo (Sale and Repurchase agreement) and Reverse Repo

Duration, immunization, convexity

Repo (Sale and Repurchase agreement) and Reverse Repo

Fooladi, I and Roberts, G (2000) “Risk Management with Duration Analysis”

Fooladi, I and Roberts, G (2000) “Risk Management with Duration Analysis”

Managerial Finance, Vol 25, no 3

Hedging Interest rate risks: Duration

Duration measures sensitivity of price changes (volatility) with

changes in interest rates

T

T t

T t

t

B C ParValue

P

r r

=

+ +

∑

Lower the coupons for a given time to maturity, greater change in price to change in interest

Duration

1

For a given percentage change in yield, the actual price increase is

greater than a price decrease

(1 + r )

T

T t

T t

t

B C ParValue

P

r r

=

+ +

∑

rates

Greater the time to maturity with a given coupon, greater change in price to change in interest rates

2

3

Duration (also called Macaulay Duration)

Duration of the bond is a measure that summarizes approximate response of bond prices to change in yields

A better approximation could be convexity of the bond

1

i

n

y t i i n

Duration is “how long” bondholder has to wait for cash flows

Modified Duration and Dollar Duration

For Macaulay Duration, y is expressed in continuous compounding

When we have discrete compounding, we have Modified Modified Duration

Duration (with these small modifications)

If yyyy is expressed as compounding mmm times a year, we divide DD

Example: Consider a trader who has $1 million in

bond with modified duration of 5 This means for

every 1 bp (i.e 0.01%) change in yield, the value of

the bond portfolio will change by $500.

A zero coupon bond with maturity of n years has a

Duration of a bond portfolio is weighted average of

the durations of individual bonds

Here, yield to maturity = 0.08, m = 2, y = 0.04, n = 6, Face value = 100.

Qualitative properties of duration

Duration of bonds with 5% yield as a function of

maturity and coupon rate.

Actually, where λ is the yield to maturity

per annum, and m is the number of coupon

payments per year.

λ

λ

m

D→1+

Properties of Duration

3 Durations are not quite sensitive to increase in

coupon rate (for bonds with fixed yield) They don’t

vary huge amount since yield is held constant and

it cancels out the influence of coupons.

4 When the coupon rate is lower than the yield, the

duration first increases with maturity to some

maximum value then decreases to the asymptotic

limit value.

5 Very long durations can be achieved by bonds with

very long maturities and very low coupons.

Changing Portfolio Duration

Changing Duration of your portfolio:

If prices are rising (yields are falling), a bond

trader might want to switch from shorter duration bonds to longer duration bonds as longer duration bonds have larger price changes

Alternatively, you can leverage shorter maturities Effective portfolio duration = ordinary duration x leverage ratio.

Immunization (or Duration matching)

This is widely implemented by Fixed Income Practitioners

You want to safeguard against interest rate increases

A few ideas:

time 0 time 1 time 2 time 3

0 pay $ pay $ pay $

A few ideas:

Immunization

Matching present values (PV) of portfolio and obligations

This means that you will meet your obligations with the cash from the portfolio

If yields don’t change, then you are fine.

If yields change, then the portfolio value and PV will both change

by varied amounts So we match also Duration (interest rate risk)

PV1+PV2 =PV obligation

PV +PV =PV

Example

Suppose Company A has an obligation to

pay $1 million in 10 years How to invest

in bonds now so as to meet the future

obligation?

• An obvious solution is the purchase of a

simple zero-coupon bond with maturity 10

years.

* This example is from Leunberger (1998) page 64-65 The numbers

are rounded up by the author so replication would give different

numbers

Immunization

Suppose only the following bonds are available for its choice.

coupon rate maturity price yield durationBond 1 6% 30 yr 69.04 9% 11.44Bond 2 11% 10 yr 113.01 9% 6.54Bond 3 9% 20 yr 100.00 9% 9.61

• Present value of obligation at 9% yield is $414,642.86.

• Present value of obligation at 9% yield is $414,642.86.

• Since Bonds 2 and 3 have durations shorter than 10 years, it is not possible to attain a portfolio with duration 10 years using these two bonds.

Suppose we use Bond 1 and Bond 2 of amounts V1& V2,

Observation: At different yields (8% and 10%), the value of the

portfolio almost agrees with that of the obligation.

Difficulties with immunization procedure

1 It is necessary to rebalance or re-immunize the portfolio from time to time since the duration depends

on yield

2 The immunization method assumes that all yields are equal (not quite realistic to have bonds with different maturities to have the same yield)

3 When the prevailing interest rate changes, it is unlikely that the yields on all bonds change by the same amount

Duration for term structure

We want to measure sensitivity to parallel shifts in the spot

rate curve

For continuous compounding, duration is called FisherFisherFisher Weil Weil

duration

duration

If x0, x1,…, xnis cash flow sequence and spot curve is stwhere

t = t0,…,tnthen present value of cash flow is

n

s t t i

Duration for term structure

Consider parallel shift in term structure:

n

s y t t

Duration applies to only small changes in y

Two bonds with same duration can have different

change in value of their portfolio (for large changes

in yields)

Convexity

Convexity for a bond is

Convexity is the weighted average of the ‘times squared’

2 2

2 1

2

11

REPO and REVERSE REPO

Short term risk management using Repo

Repois where a security is sold with agreement to buy it back at

a later date (at the price agreed now)

Difference in prices is the interest earned (called repo raterepo raterepo rate)

It is form of collateralized short term borrowing (mostly overnight)

Example: a trader buys a bond and repo it overnight The money from repo is used to pay for the bond The cost of this deal is repo rate but trader may earn increase in bond prices and any coupon payments on the bond

There is credit risk of the borrower Lender may ask for margin costs (called haircut) to provide default protection.

Example: A 1% haircut would mean only 99% of the value of collateral is lend in cash Additional ‘margin calls’ are made if market value of collateral falls below some level

Short term risk management using Repo

Hedge funds usually speculate on bond price differentials

using REPO and REVERSE REPO

Example: Assume two bonds A and B with different prices (say price(A)<price(B)) but

similar characteristics Hedge Fund (HF) would like to buy A and sell B

simultaneously This can be financed with repo as follows:

(Long position) Buy Bond A and repo it The cash obtained is used to pay for

(Long position) Buy Bond A and repo it The cash obtained is used to pay for

the bond At repo termination date, sell the bond and with the cash buy

bond back (simultaneously) HF would benefit from the price increase in

bond and low repo rate

(short position) Enter into reverse repo by borrowing the Bond B (as

collateral for money lend) and simultaneously sell Bond B in the market At

repo termination date, buy bond back and get your loan back (+ repo

rate) HF would benefit from the high repo rate and a decrease in price of

the bond

Interest Rate Futures

(Futures on T-Bills)

Interest Rate Futures

In this section we will look at how Futures contract written on a

Treasury Bill (T-Bill) help in hedging interest rate risks

Review - What is T-Bill?

T-Bills are issued by government, and quoted at a discount

Prices are quoted using a discount rate discount rate discount rate (interest earned as % of

face value)

Example: 90-day T-Bill is quoted at 0.080.080.08 This means annualized

return is 8% of FV So we can work out the price, as we know FV

Day Counts convention Day Counts convention (in US)

1. Actual/Actual (for treasury bonds)

2. 30/360 (for corporate and municipal bonds)

3. Actual/360 (for other instruments such as LIBOR)

9 01

Interest Rate Futures

So what is a 3-month T-Bill Futures contract?

At expiry, (T), which may be in say 2 months timethe (long) futures delivers a T-Bill which matures at T+90 days, with face value M=$100

As we shall see, this allows you to ‘lock in’ at t=0, the forward rate, f12

T-Bill Futures prices are quoted in terms of quoted index, quoted index, quoted index, Q Q (unlike discount rate for underlying)

Q = $100 – futures discount rate (df)

So we can work out the price as

9 01

1) You hold 3m T-Bills to sell in 1-month’s time ~ fear price fall

~ sell/short T-Bill futures

Cross Hedge: US T-Bill Futures

Example:

Today is May Funds of $1m will be available in August to invest for further 6 months in bank deposit (or commercial bills)

~ spot asset is a 6-month interest rate

Fear a fall in spot interest rates before August, so today BUY

T-bill futures

Cross Hedge: US T-Bill Futures

3 month exposure period

Desired investment/protection period = 6-months

Maturity date of Sept.

T-Bill futures contract

Maturity of ‘Underlying’

in Futures contract

Question: How many T-bill futures contract should I purchase?

Cross Hedge: US T-Bill Futures

We should take into account the fact that:

1. to hedge exposure of 3 months, we have used T-bill futures with 4 months time-to-maturity

2. the Futures and spot prices may not move one-to-one

We could use the minimum variance hedge ratio:

Question: How many T-bill futures contract should I purchase?

We could use the minimum variance hedge ratio:

However, we can link price changes to interest rate changes using Duration based hedge ratio Duration based hedge ratio

p f

Duration based hedge ratio

Using duration formulae for spot rates and futures:

So we can say volatility is proportional to Duration:

Duration based hedge ratio

Expressing Beta in terms of Duration:

( )

0

2 0

0

2 0

,

s F s

Cov

F FVF

0

∆ = + ∆ +

Duration based hedge ratio

Desired investment/protection period = 6-months

Maturity of ‘Underlying’

Example REVISITED

Purchase T-Bill future with Sept

delivery date

Known $1m cash receipts

Maturity date of Sept.

T-Bill futures contract

Maturity of ‘Underlying’

in Futures contract

Question: How many T-bill futures contract should I purchase?

Cross Hedge: US T-Bill Futures

May (Today) Funds of $1m accrue in August to be invested for 6- months

in bank deposit or commercial bills( Ds= 6 )

Use Sept ‘3m T-bill’ Futures ‘nearby’ contract ( DF= 3)

Cross-hedge

Cross Hedge: US T-Bill Futures

Suppose now we are in August:

3 month US T-Bill Futures : Sept Maturity Spot Market(May)

(T-Bill yields)

CME Index Quote Q f

Futures Price, F (per $100)

Face Value of $1m Contract, FVF

0 (6m) = 11% Q f,0 = 89.2 97.30 $973,000 August y (6m) = 9.6% Q = 90.3 97.58 $975,750 August y (6m) = 9.6% Q = 90.3 97.58 $975,750

Cross Hedge: US T-Bill Futures

Invest this profit of $5500 for 6 months (Aug-Feb) at y1=9.6%:

= $5500 + (0.096/2) = $5764

Loss of interest in 6-month spot market (y0=11%, y1=9.6%)

= $1m x [0.11 – 0.096] x (1/2) = $7000

Net Loss on hedged position $7000 - $5764 = $1236

(so the company lost $1236 than $7000 without the hedge)

Potential Problems with this hedge:

1 Margin calls may be required

2 Nearby contracts may be maturing before September So we may have to roll

over the hedge

3 Cross hedge instrument may have different driving factors of risk

Interest Rate Futures

(Futures on T-Bonds)

US T-Bond Futures

Contract specifications of US T-Bond Futures at CBOT:

Contract size $100,000 nominal, notional US Treasury bond with 8% coupon

Delivery months March, June, September, December

Quotation Per $100 nominal

Tick size (value) 1/32 ($31.25)

Last trading day 7 working days prior to last business day in expiry month

Delivery day Any business day in delivery month (seller’s choice)

Notional is 8% coupon bond However, Short can choose to

deliver any other bond So Conversion Factor Conversion Factor Conversion Factor adjusts “delivery

price” to reflect type of bond delivered

T-bond must have at least 15 years 15 years 15 years time-to-maturity

Quote ‘98‘98‘98 14’14’14’ means 98.(14/32)=$98.4375 per $100 nominal

Delivery day Any business day in delivery month (seller’s choice)

Settlement Any US Treasury bond maturing at least 15 years from the

contract month (or not callable for 15 years)

US T-Bond Futures

Conversion Factor (CF)Conversion Factor (CF): : : CF adjusts price of actual bond to be delivered by assuming it has a 8% yield (matching the bond to the notional bond specified in the futures contract)

Price = (most recent settlement price x CF) + accrued interest

Example: Example: Possible bond for delivery is a 10% coupon

(semi-
Example: Example: Possible bond for delivery is a 10% coupon annual) T-bond with maturity 20 years

(semi-
The theoretical price (say, r=8%):

Dividing by Face Value, CF = 119.794/100 = 1.19794 (per

$100 nominal)

40

40 1

5 100

119.794 1.04i 1.04

i

P

=

If Coupon rate > 8% then CF>1

If Coupon rate < 8% then CF<1

US T-Bond Futures

Cheapest to deliver::::Cheapest to deliver

In the maturity month, Short party can choose to deliver any

bond from the existing bonds with varying coupons and

maturity So the short party delivers the cheapest one

Short receives:

(most recent settlement price x CF) + accrued interest

Cost of purchasing the bond is:

Quoted bond price + accrued interest

The cheapest to deliver bond is the one with the smallest:

Quoted bond price - (most recent settlement price x CF)

Hedging using US T-Bond Futures

Hedging is the same as in the case of T-bill Futures (except Conversion Factor)

For long T-bond Futures, duration based hedge ratio is given by:

Financial Risk Management

Topic 3a Managing risk using Options

Readings: CN(2001) chapters 9, 13; Hull Chapter 17

Financial Engineering with Options

Black Scholes

Delta, Gamma, Vega Hedging

Financial Engineering with Options

Options Contract - Review

An option (not an obligation), American and European

-Put Premium

Financial Engineering with options

Synthetic call option

Put-Call Parity: P + S = C + Cash

Financial Engineering with options

Nick Leeson’s short straddle

You are initially credited with the call and put premia C + P (at t=0) but if at expiry there is either a large fall or a large rise in S (relative to the strike price K ) then you will make a loss

(.eg Leeson’s short straddle: Kobe Earthquake which led to a fall in S (S = “Nikkei-225”) and thus large losses).

Sensitivity of option prices Sensitivity of option prices (American/European non

Sensitivity of option prices (American/European dividend paying)

non
c = f ( K, S 0 , r, T, σ )

- + + + + dividend paying European options. This however can be negative for

Example: stock pays dividend in

p = f ( K, S 0 , r, T, σ )

Call premium increases as stock price increases (but less than

+ - - + +

Example: stock pays dividend in

2 weeks European call with 1 week to expiration will have more value than European call with 3 weeks to maturity.

Sensitivity of option prices

The Greek Letters

Delta, Delta, ∆ ∆ ∆ measures option price change when stock price increase by $1

Gamma, ΓΓΓΓ measures change in Delta when stock

price increase by $1

price increase by $1

Vega, υυυυ measures change in option price when there

is an increase in volatility of 1%

Theta, Theta, Θ measures change in option price when

there is a decrease in the time to maturity by 1 day

Rho, Rho, ρ measures change in option price when there

is an increase in interest rate of 1% (100 bp)

Sensitivity of option prices

Using Taylor series,

2 2

d f ≈ ∆ ⋅ d S + Γ ⋅ d S + Θ ⋅ d t + ρ ⋅ d r + υ ⋅ d σ

The rate of change of the option price with respect

to the share price

e.g Delta of a call option is 0.6

Stock price changes by a small amount, then the option price changes by about 60% of that

Option price

S

C

Slope = ∆ = ∂ c/ ∂ S

Stock price

∂

(for long positions)

If we have lots of options (on same underlying) then delta of portfolio is

S

∂

N

So if we use delta hedging for a short call position, we must keep a long position of N(d 1 ) shares

What about put options?

The higher the call’s delta, the more likely it is that the option ends up in the money:

respect to time

Also called the time decay of the option

For a European call on a non-dividend-paying stock,

Related to the square root of time, so the relationship is

Theta isn’t the same kind of parameter as delta

Theta isn’t the same kind of parameter as delta

The passage of time is certain, so it doesn’t make any sense to hedge against it!!!

Many traders still see theta as a useful descriptive statistic because in a delta-neutral portfolio it can proxy for Gamma

f S

Sometimes referred to as an option’s curvature

If delta changes slowly → gamma small → adjustments

to keep portfolio delta-neutral not often needed

0

S σ T

If delta changes quickly → gamma large → risky to

leave an originally delta-neutral portfolio unchanged for long periods:

S'

Making

Making a a Position Position Gamma Gamma Neutral Neutral

We must make a portfolio initially gamma-neutral as well as delta-neutral

if we want a lasting hedge

But a position in the underlying share can’t alter the portfolio gamma since the share has a gamma of zero

So we need to take out another position in an option that isn’t linearly dependent on the underlying share

If a delta-neutral portfolio starts with gamma Γ , and we buy w T options each with gamma Γ T , then the portfolio now has gamma

For any derivative dependent on a non-dividend-paying stock,

Δ , θ, and Г are related

The standard Black-Scholes differential equation is

where f is the call price, S is the price of the underlying

share and r is the risk-free rate

2

2 2

2

1 2

Θ + ∆ + Θ Γ =

NOT NOT a letter in the Greek alphabet!

Vega measures, the sensitivity of an option’s