Quantitative Script by Costea, Andrei by Andrei Costea

Quantitative Script © Andrei Costea, 2008

Table of Contents: 1. Mathematics 2. Statistics 3. Finance 3.1 Return and Risk 3.2 Equity 3.3 Fixed Income 3.4 Derivatives 4. Derivations

1 10 23 24 28 36 43 63

Detailed Table of Contents: 1. Mathematics 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Calculus Algebra Summations and Products Trigonometry Hyperbolic Trigonometry Geometry Complex Numbers The Mandelbrot Set Matrix Algebra

2. Statistics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12

3 4 6 7 7 8 8 8 9

11 11 12 13 13 13 14 15 16 19 20 21

24 25 26 27

28 28 29 30 32 33 34 34 34

36 36 37 37 37 39 39 40 41 42 42

Dividend Discount Models Discounted Cash Flow Valuation Econometric Equity valuation Portfolio Selection (MPT and PMPT) International Asset Allocation Estimation Risk and Asset Allocation The Capital Asset Pricing Model The Market Model Arbitrage Pricing Theory

Values Returns Term Structure of Interest Rates Forward Rates Single‐factor Duration Models under a Flat Term Structure Cash Flow Matching Duration Matching Single‐factor Duration Models under a Non‐flat Term Structure Key Rate‐Duration Methods for the Identification of Term Structures Dirty Price of a Coupon Bond

Forwards and Futures Hedging with Futures Options Option Pricing with other Underlyings Options Trading Strategies Constant Proportion Portfolio Insurance Exotic Options Swaps Swap Derivatives Credit Derivatives Interest Rate Derivatives (additional) Interest Rate Models

43 45 47 50 52 54 55 56 58 59 60 61

4. Derivations

Computation of Returns and Interest Rates Shortfall‐Risk Measures Value at Risk (VaR, CVaR, mVaR) Risk‐adjusted Performance Measures

Derivatives 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6 3.4.7 3.4.8 3.4.9 3.4.10 3.4.11 3.4.12

4.1 4.2 4.3 4.4

Fixed Income 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7 3.3.8 3.3.9 3.3.10 3.3.11

3.4

Equity 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.2.8 3.2.9

3.3

Return and Risk 3.1.1 3.1.2 3.1.3 3.1.4

3.2

Moment Computation Rules The Gini Coefficient Moments and Co‐Moments of the Distribution Chebyshev Inequality Estimation of Moments Important Theorems Law of Total Probability/Expectation/Variance/Skewness Normal and Lognormal Distributions Other Distributions Probability Theory Hypothesis Tests Distribution Tables

3. Finance 3.1

The Black/Scholes Formula Portfolio Selection (MPT) The Capital Asset Pricing Model The Gordon‐Growth Formula

64 66 70 71

Index of Symbols Α

alpha

beta

gamma

delta

epsilon

epsilon variant

zeta

eta

theta

theta variant

iota

kappa

lambda

omicron

pi variant

rho

rho variant

sigma

sigma variant

tau

upsilon

phi

phi variant

chi

psi

omega II

1. Mathematics

Calculus (differentiation and integration) ∙

∙

ln ∙

∙

;

∙

log

additionally: log

∙ log

Differentiation:

∘

1 ∘

Important Functions:

erf

erfc

Γ Γ Γ

√

additionally:

erf

√ erf

erf

erfc

erf

∗

erf ∗

∗

Integration by Parts:

Integration by Substitution:

Example:

⇒

1⇔

⇒

Leibniz Integral Rules: ,

, ,

∙

Interchange of the Order of Integration: Generally:

Specific:

Algebra Chains (concatenation): U

;

; ∙

∙

Chain Rule: °

a) b)

∙

Tangency: ∙

Tangent line: Tangent plane:

∙

or:

∙

Tangent at the level curve:

∙

slope of the tangent:

Fourier Series: cos

sin

Taylor Expansion: 1 ∙ 2!

∙

1 ∙ 3!

∙

1 ∙ 2!

1 ∙ !

…

∙

2∙

∙

…

Examples: 1

⋯,

∞

⋯

Elasticity: ∙

∙

⁄ ⁄

Binomial Computations: ∙ !

∙

!∙ !

1 1 1

1 1

0 1

Permutations: !

With distinction of the elements:

,…,

!∙…∙

Without distinction of the elements: Combinations:

Considering the order of the elements:

Without consideration of the order: Some other useful formulas: 2

10 ∙

∓

2∙

∙

Progressions: ∙

1 ∙

with ∑ ∙ with ∑

∙

Siegel’s Paradox: If a fixed fraction of a given amount is lost, and then the same fraction of the remaining amount is gained, the result is less than the original and equal to the final amount if a fraction is first gained, and then lost. ∙ 1

∙ 1

In the foreign exchange market, it can be observed that the loss of one currency does not fully offset the gain of the other currency, where is the price quotation of the exchange rate at time : 1 ≶

1 ⟹

, ,

Summations and Products ∙

1 2

∙

;

1 1 1

; | |

∙

1 ∈

∙

1 ∈

;

1 1

0,1

1 ∈

;

1 ∈

0,1

Trigonometry cos

sin

Trigonometric relations and identities: cos

cos

2 cos

sin

2 sin

cos

cos cos

∓ sin sin

cos sin

cos

sin

tan

sec

cos

sin

tan

√1

tan θ

√1

cos θ cos θ 1

cos

cot θ

√1 1 cot θ tan θ

sin

tan θ

√sec θ 1 sec θ

√csc θ csc θ

cot θ cot θ

1 csc θ 1

cos sin

∓

sec θ √1

2 sin sin cos

1 cos θ

sin θ

sin θ 1

√1

∓

tan θ

cos θ sin θ

√1

cos

√csc θ

Hyperbolic Trigonometry sinh

∙ sin

∙ tan

tanh

sec

sech

cosh 1

cosh

coth

csch

sinh

tanh

sinh

sech

∙ cosh

cos

∙ cot

∙ csc

tanh ∙ sech 1

cosh

∙ sinh

coth

coth ∙ csch sch

tanh

∙ ln cosh

coth

∙ ln sinh

Hyperbolic trigonometric relations and identities: cosh

sinh

1 cosh sinh cosh

cosh

0 ∀

tanh

cosh sinh

cosh

sinh

cosh

coth

tanh

sech

cosh ln ln

√

sinh

cosh cosh sinh

sinh sinh

sinh

cosh

sinh

| |

tanh

csch

∙

ln | | √ | |

cosh √

cosh 1

1 © Andrei Costea, 2008

Geometry 2

4 3

1 ∙√

Complex Numbers ∙ is a complex number with

1. The complex conjugate of

sin

cos

tan

∙

√

The radius of the complex unit circle is given by

∙ is ̅

∙ .

∙ , where:

The polar coordinate representation is

∙

∙ cos ∙

The representation from Euler’s formula is

The hyperbolic representation of Euler’s formula is

∙ cos cos

sin ∙ . sin ∙ sin ∙ and

∙

. cos

sin ∙ .

The Mandelbrot Set :

→

: ↦

given by

The iterated sequence is 0,

0 ,

, … where

denotes the

iterate of the

series. The Mandelbrot set is a subset of the complex plane and is given by: ∈

∶ sup| ∈

0 |

∞

Matrix Algebra det

det

‖ ‖ cos

‖ ‖

√

‖ ‖∙‖ ‖

The pairs of solutions for the characteristic equation are the characteristic roots and the characteristic vectors . 2

2. Statistics

Moment Computation Rules ∙

∙

⇒

∙

∙ ∙

∙

∙ , ∙ ∙

∙

∙ ,

∙

Special cases: ⋯

⋯ 1

⋯

The Gini Coefficient Discrete Variables: 1,2, … ,

⋯

∑

∑ ∑

0,0

,…,

1,1

The plotted points , result in the Lorenz curve. The difference between the diagonal and the Lorenz curve represents the concentration area. The larger the concentration area the higher the concentration of the variables. Continuous Variables: ∗

∗

∑

∗

⁄2

∑

1 ⁄

2∙∑ ∙∑

∙

Practical approximation for small : 0

∗

∙

1 11

Moments of the Distribution Population Mean (1st moment): U

∑ ̅

Population Variance (2nd centralized moment): U

∑

Skewness (3rd centralized moment): U

3 1

Kurtosis (4th centralized moment): U

4 1

1 2

1 1

Moment Generating Function: ∙

∙

Co‐moments of the Distribution ,

, 3 ,

, ,

Chebyshev Inequality |

⁄

Estimation of Moments For ̂ to be the estimator of

the following conditions have to be satisfied:

1) The estimator has to be unbiased: We call ̂ to be the bias of the estimator ̂ . ̂ If ̂ is an unbiased estimator of then: 2) The estimator has to be consistent: →

If ̂ is a consistent estimator of then: ̂ → which is the same as ̂ 0 3) The estimator has to be efficient: If is another estimator of and both ̂ and are unbiased, then ̂ is said to be more efficient than . if ̂

Important Theorems The Law of Large Numbers (LLN): U

The weak form of the law: |

→

0 ,

The strong form of the law: →

The Central Limit Theorem (CLT): U

∑

→

≔

√

≡

∑

∞

⁄√

≔ →

→

∞ 0

The CLT then states that, in the limit with → ∞, the distribution function towards the Standard Normal distribution function : →

(in standardized form) will tend

Law of Total Probability |

Law of Alternatives (discrete case): ∩

alternatively: |

Law of Total Expectation |

alternatively: |Ω

⇒

Law of Total Variance |

Law of Total Skewness This is a special case of the Law of Total Cumulance. |

3∙

Normal Distribution ~

⇒

∙

√

;

∙ 1

erf

⇒

0,1

√

3 ~

For returns: Observations:

Lognormal Distribution ~

⇒ ln

⇒

√

∙

, 0 ,

: ~

⇒

∙ 1

erf

√

0 ,

∙

1∙

For returns:

;

∙

2 ln 1

ln 1

~ ln 1

∙

ln 1

Other Distributions Discrete Uniform Distribution: ~

1 1 2

1,2, … , 1

Bernoulli Distribution (discrete): U

0,1 1

1 1

6 1 1

Binomial Distribution (discrete): U

It originates from the sum of i.i.d. Bernoulli variables ~ ~

0,1, … , 1

Poisson Distribution (discrete): U

0,1, … , 1

√

Geometric Distribution (discrete): U

1 1

0,1, … ,

Continuous Uniform Distribution: ~

1.8

1 Exponential Distribution (continuous): U

0 1

Double‐exponential Distribution (continuous): U

| |

2 2

0 0

Pareto Distribution (continuous): U

1 1

‐) Distribution (continuous):

Chi‐Square‐(

Student t‐Distribution (continuous): U

From ~

√

0,1 and the estimated standard deviation

1 ,

1,2, …

with ~

and for large :

∑

0,1 .

Gamma Distribution (continuous): U

From

and thus respectively

, 0

we obtain the gamma function

∞ , where Γ

and therefore:

Furthermore we have that ~Γ

∙Γ

, 0

∞,

Weibull (Two‐Parameter) Distribution (continuous): ~

∞

0 Γ 1

Γ 1

Probability Theory Set operations: ⊆

⇔

∈

| ∉ ∩

⇒

∈

| ∈

∩

∈

∅ ⇒

∪

| ∈

∖

| ∈

∈

∉

de Morgan’s Rule: ∩

̅ ∪ "none will occur"

∪

̅ ∩ "will not occur simultaneously" ∪

∩

"exactly one of both will occur"

Laplace Probability Definition: | | |Ω|

Axioms of Kolmogorov: U

1) 2) 3)

0 1

Ω ∪

Probabilities Computation Rules: a) b) c) d) e)

∅ ̅

0 1

⊆

∪ ∩ ∩

∩ ∪

∩

∖ ∩

∩

Conditional Probabilities: ∩

| ∩

∙

⇒

∙

∑

∙

Independency of Two Events (stochastic independent events): U

∩

∅⇒

∩

∙

Hypothesis Tests :

Null Hypothesis: The p‐value: for

Alternative Hypothesis:

2Φ ,

∑

The Standard Error: ,

The t‐Statistic: Reject

if |

√

as an estimator for

√

Distribution Tables 21

3. Finance

3.1 Return and Risk

Computation of Returns and Interest Rates Present Value Method:

∏

1 1 1

∙

∙ 1

Annuity Method: ∙

1 1

∙ 1

∙

Internal Rate of Return (IRR/Yield) Method: ≝ 0 The Simple (discrete) Annual Interest Rate: ,

∙ 1

∙

365

The Effective (compounded) Annual Interest Rate: 1

is the effective annual interest rate and the number of equidistant payment periods. The discrete effective annual interest rate (or discretely compounded annual interest rate): ∙ 1

The continuous effective annual interest rate (or continuously compounded annual interest rate): lim →

lim →

∙ 1

∙

∙ ∙

∙

Shortfall‐Risk Measures Shortfall Probability: 0 1

Shortfall Expectancy: ∙

∙

Mean Excess Loss: |

∙

Shortfall Variance: ∙

∙

2 1 ⇒

∙

Lower Partial Moments (LPM): ∙

∗

∙∑

∙

∗

∙∑

∙

Upper Partial Moments (UPM): U

∙

Co‐Lower/Upper Partial Moments (CoLPM/CoUPM): ,

max

,0 ,

max

∙

Value at Risk (VaR) Normal ~

∙

Lognormal ~

∙√

⇒ ln ~

For returns:

Normal ~

Lognormal ln 1

√

where:

∙ ~

Alternatively:

∙√ ∙

∙√

∙

⇒

∙√

∙

The Value at Risk of a portfolio of funds:

∙

Conditional Value at Risk (CVaR) |

Interpretation:

Normal ~

Lognormal ~

: ,

∙

, where Φ is the cumulative distribution function (cdf) and is the probability density function (pdf). :

For returns:

Normal ~

Lognormal ln 1

| ,

∙

Modified Value at Risk (mVaR) The modified VaR for returns using the Corner‐Fisher approximation: 1 6

3 24

∙ 26

∙

5 36

∙

Risk‐adjusted Performance Measures

∙

∑

∙∑

∙

Modigliani Modigliani (M2)/‐Muralidhar (M3) Leverage Performance Measures: The M2 measure leads to the same ranking as the Sharpe ratio, but with an improved economical interpretation. This is due to the fact that the slopes of the two market lines do not change. The market line of (the portfolio) whereas the market line of (the norm asset) has a slope of . It is only the volatility that is has a slope of adjusted. If the risk‐free rate is used as the comparable return, the measure underlies the assumption that has zero volatility. This assumption only holds if the maturity of is the same as the holding period of . 1

∙

∗

∙

The M3 performance measure corrects the M2 for the difference in correlations, but has never been implemented in practice. GH1 and GH2 (Graham Harvey) Performance Measures: U

GH1 adjusts the volatility by exhibiting an adjusted return leverages to match the volatility of . 1

∗

, leaving

and

∗

unchanged. It leverages or de‐

GH2 adjusts the volatility by exhibiting an adjusted return ∗ , leaving and unchanged. It leverages or de‐ leverages to match the volatility of . It is similar to M2 but it does not rely on the assumption of zero risk for the risk‐free rate. In the case of positive volatility, the market line will be replaced by a market curve due to non‐ zero correlation between the changes in interest rates and the changes in asset returns. Both measures should be positive if there is outperformance. © Andrei Costea, 2008 27

3.2 Equity

Dividend Discount Models (DDM) 1 1

⇔

⁄ ⁄

1 1

⁄

Discounted Cash Flow Valuation (DCF) &

∆

is the levered (equity) beta.

∆ ∙

∙ 1

∙

∙ 1

∆

∙ 1

∆

∙

∆ ∆ ∆

∙

∆

∙

∆

is the unlevered (asset) beta.

∙ %

. %

∙

∙ 1 ∙

∙

1 1

∙

⟶

∙

∙ 1

∙

∆

Growth is valuable when: ∙ 1

or when: ∙ 1

Consequently, the Economic Value Added is given by: ∙ 1

∙

Econometric Equity Valuation ,…,

;

,…,

;

∙

Whitbeck/Kisor Model: /

∙

Portfolio Selection (MPT and PMPT) Check the Derivations chapter for further details and definitions. Mean‐Variance Optimization (M/V): U

∑

Objective function: ∑

Side conditions:

∑

1 ,…,

1 ⁄2

Number of inputs: 2

→

⁄2

Index Model: ∙

, ≔ 0; , ,

1) 2) 3)

∙

0 ,

∙

Number of inputs: 3 The Tangency Portfolio: 1

⇒

≝ 0

⇒

⇔

∙

The tangent line is thus given by: ∙ The special case with two risky assets and one risk‐free asset: 2

∙

∙ ∙

∙

Optimization Relative to a Benchmark: ∑

Objective function:

∑

Side conditions:

∑

1 ,

∑

→

Optimization in a Shortfall Context: ,

⇒

In the case of an efficient frontier with both a risky and a risk‐free asset: ≝

∗

⇒

∗

In the case of an efficient frontier with only a risky asset: 1

∗

⇒

With a multi‐period model we have: ⁄

⋯

⇒

√

Mean‐LPM Optimization (M/LMP): U

,…,

Investment vector: Individual portfolio return:

∑

Additionally:

∑

Additionally:

∑

∑ ∑ ∑ ⇒

∙

,…,

∙

,…,

∑

∙

subject to:

,…,

⟶ ∑

∑ ∑

,…, ∑

⇒ ⇒∑

Objective function:

∑

,…,

Defining the objective function:

,…,

and the correspondent portfolio return:

, ∑

! 0

International Asset Allocation

∙

, ,

∙

1≔

1, … ,

∙ 1

, ,

2 2

∆

Due to the IRP (interest rate parity) the forward premium is given by: 1

1 1

0, 0,

∈ 0,1 ≔

∙

Optimization methods: Naïve hedge: the hedge ratio is given exogenously and the weights Optimal currency hedge: the hedge ratio and the weights

are determined thereafter

are determined simultaneously

Currency overlay: the weights are determined first by setting determined given the already computed weights

0 and then the optimal hedge ratio is

Estimation Risk and Asset Allocation Shrinkage towards the MVP (Jorion 1985/‐86): 1, … ,

The Bayes/Stein method: 1, … ,1 is a 1 ̂

1 vector.

vector. , , ,

≔ and

are

matrices.

∙ ̂ 1

is a 1

1, … ,

posterior covariance matrix The Bayes/Stein tangency portfolio (BST) is given for:

2 ̂

This parameter of Jorion, based on the statistical decision theory, manages to minimize the estimation risk. The denominator shows how much the sample average returns ̂ scatter around the expected return of the MVP. For 0 the traditional tangency portfolio of the CAPM is delivered whereas for 1 the MVP is delivered. For ⟶ ∞ the sample average returns (which are actually the unbiased population returns) are delivered, which means that the Bayes/Stein estimator includes these returns as a special case. Shrinkage towards the Market Portfolio (Pastor 1999/2000): This approach combines the sample information and the implications of an asset‐pricing model. In this case it is the CAPM, but the approach can also accommodate multi‐factor models like the APT. Assumption: the prior expected excess returns are set equal to the expected excess return implied by the CAPM.

⁄

⁄ ⁄ 1

is the risk premium of the market portfolio. ̂ denotes the sample means of the excess returns (over the risk‐ free rate). is the Sharpe ratio of the market portfolio. is the average variance of the residual terms in the multivariate regression of the asset’s excess returns on the excess returns of the market portfolio. measures the dispersion of the asset’s alphas (i.e. the deviation of the asset’s expected returns from the values implied by the CAPM). and are

1 vectors.

, and

are

matrices. 33

The Capital Asset Pricing Model (CAPM) (ex‐ante) Check the Derivations chapter for further details and definitions. Capital Market Line (CML): U

∙

Model is valid only for efficient portfolios in a Tobin (Two‐Fund‐Theorem) context. Security Market Line (SML): U

∙ ,

∑

∙

Model is valid for efficient and inefficient portfolios as well as individual securities.

The Market Model (ex‐post) ∙

∙

If the CAPM holds and the markets are efficient, then 0.

should not be statistically different from zero and

Arbitrage Pricing Theory (APT) Traditional Form: Assumption: the market portfolio does not exhibit any non‐systematic (idiosyncratic) risk.

1, … ,

≡

Modern Form: Assumption: the market portfolio exhibits non‐systematic (idiosyncratic) risk.

1, … ,

≡

Generally:

Specifically:

∙

1 1

∙

3.3 Fixed Income

Values 1

∙

⁄

Zero Bond:

⁄

Consol:

Fixed Coupon: 1

∙

Floating Rate: ,…,

∙

Returns

∙

1 Yield to Call

∙

Yield to Average Life

∙

∑

Realistic Return : ∙ 1

⇒

∙

Term Structure of Interest Rates ,…,

,…,

is the interest rate curve and 1

1⇔

the discount curve at time .

Forward Rates 1

∏

⇔

No arbitrage constraint: ,…,

∙

Single‐factor Duration Models under a Flat Term Structure Duration:

1 ∆

∆

1 ∆

∑

∙∆

∑

∙ ∆

∆ 1

∙

1 1

⇔

For a portfolio we have: Convexity: 1

∙ ∆

≔

∑

1 ∙ ∆

∆

∙ ∆

∙∆

1 2

∙ ∆

1 2

∙ ∆ 1

1 1

1 2

∙∆

exp 2

1 1

∑

≔

∆

1 1

In continuous time: ∑

∑

The Babcock Relation: ∙ 1

∙

⇔

∙ 1

In continuous time: ∙

∙

through the Taylor approximation on the right side we obtain ∙

Cash Flow Matching :

, :

Objective function:

∑

Constraints:

∑

⟶

, 0

1, … ,

∈

Observations: no reinvestment risk; no transaction costs from reallocation of assets; no need for the specification of expected interest rates

Duration Matching Immunization conditions for a single liability:

Immunization conditions for multiple liabilities:

Single‐factor Duration Models under a Non‐flat Term Structure Fisher/Weil Approach: Note: immunization against additive shifts (of the interest accumulation factor, not the interest) due to the continuous time approach. The Fisher/Weil and Macaulay durations are equivalent in nature as both present the sensitivity and immunizing features. ,…, 0

⋯ ≔

∗

, ∈

∗

: 1

, ∀

∙

Fisher/Weil‐duration: 1

,…,

∙ ∆

,…,

⇒

,…,

∙

∆

Fisher/Weil Convexity: 1

∆

M‐squared Sensitivity Measure (Fong/Vasicek): Note: immunization against twists. 0

∑

Generally:

2 ∙

∑ ,

⇒

Key‐Rate Duration Assumption: the key‐rate periods correspond to the maturity periods of the coupons. 1

,…,

1 ∙ 1

∆ ,…,

∆

1 1

∙

∆

∙∆

,…, 1

,…,

∙

,…,

∙

⋯

∙∆

In continuous time: ,…,

,…,

∙

,…,

∆

,…,

∆

,…,

1 2

∙∆

∙ ∆

⋯

If the term structure is flat with ,…,

, we then have:

If the assumption is dropped and the key‐rate periods do not correspond to the maturity periods of the coupons, the key‐rate periods): the following methodology is then applied (for the coupon maturity period and , ⇒

∆

For a portfolio we have:

∙

∑

⁄

, ,

Methods for the Identification of Term Structures The Recursive Approach (bootstrapping):

⋮ ⋯

∑

⇒

⋮

Note: is the discount factor for period . The Svensson Parametric Approach: 1

Note: ,

, , are parameters to be estimated.

The Multi‐factor (statistical‐econometric, non‐parametric) Approach: ~

0,1

1, … ,

Using the multi‐factor model in the key rate‐duration:

∆

∙

∑

∆

∙

∆

∙

∑

∆

∑

∆

∙

Dirty Price of a Coupon Bond ∙ 1

⇒

∙

1 1

⇒

Practical approximation : ⇒

∙ ∙

∙

3.4 Derivatives

Forwards and Futures Forward/Futures Price:

Basis and Cost of Carry (CoC): 1

Futures P&L position (Marking to Market): ,

⋯

Forward Cost‐of‐Carry Value: ,

Short Hedge Position

Underlying with Income:

backwardation area

Underlying with Dividend (Stock):

contango area

;

super contango area

Practical approximation: ,

∙ 1

∙

Currency as an Underlying: ,

Money Market Instrument as an Underlying:

∙ 1

⇒

Practical approximation: 30 365

365

∙ 1

30 365

30 ⇒ 365

1 ∙

365

365 30

Coupon‐bearing Instrument as an Underlying:

⇒

Generally:

∙ 1

∑

⋯

or simplified:

∙ 1 1

∙ 1

1 1

≝

1 ∙ 1

Cheapest to Deliver (CTD) approach: ∙ 1

with accrued interest (simplified): ∙ 1

Practical approximation:

∙ 1

365

∙ 1

365

1 44

∙ ∙ 1

365 © Andrei Costea, 2008

Hedging with Futures The Minimum Variance Hedge (MVH):

≝0⇒

∙

⇒

∙

∙ ,

∙ ∙ 1

∙ 1

∙

⇒

Hedging with Stock Futures:

∙

2∙

∙

∙ 1

∙

≝0⇒

∙

Analysis of the beta factor: ⇒

⇒

, 2

⇒

0⇒ 45

Hedging with Interest Rate Futures: ∆

Assumption: flat term structure with additive shift

∆

≝ 0

Approximation 1: ∙∆

∆

∙∆

∙∆ ≝0⇒

and ∆

and

∙ ∆ and ∆

∆ ≝0⇒ ∗

and

∙ ∆ with

≝ 0.

are the Macaulay durations. ∗

∙ ∙

⇒

are the absolute durations, while

∙ ∆ and ∆

0⇒ ∗

∗

∙

∗

∙

∗

∙

Approximation 2: ∆

1 2

∙∆

1 2 1 2

∙∆

⇒

∙ ∆

∙∆

∙ ∆

∙∆

1 2

∙ ∆

≝ 0

∙ ∆

and

are the Macaulay durations. ∙ ∆ ∙ ∆

⇒

1 2

Tailing the Hedge: Assumption: Marking to Market with

⇒

Without considering the interest rate effects, we have that ⇒

Options Binomial Delta Valuation: ∆∙

∆∙

∆

∆∙

Binomial Risk‐neutral Valuation: 1

∙ 1

∙

1 ∙

∙ 1

∙

1 1

∙ 1

∙

1 ∙

∙

1 ∙

∙ 1

∙

1 ∗

for multiplicative binomial trees:

∗

⁄

∗

⁄

Generally:

1 ∙

∙ 1

∙

,0 ∙

∙

max ∙

∙

∙ 1

∙

; ;

⇔

∙

; ;

∙ 1

∙

min

; ;

Put‐call Parity: ∙

Put‐call Inequality (American options): ∙

European Call Option: ∙

∙

European Put Option: ∙

∙

∙ 1 2

∙

∙ √

√

Delta: ,

∆

∙

∆

1 ∙

∙

Gamma: ,

∙

∙ √

Theta:

2√

2√ Rho: U

Vega: √

The Risk‐free Portfolio using Itô’s Lemma: ,

⇒

∙

⇒ 1 ∙ 2

∙

1 ∙ 2

fundamental PDE: ∆ ∙ ∙

∙

For a delta neutral portfolio (in PDE form): ∙

∙

Lévy Processes (càdlàg processes, fr. càdlàg: “continue à droit, limitée à gauche”): U

The Wiener process: 0 is almost surely continuous ~ 0, . . . 0 ∙ √ ∙ ~ 0,1

1) 2) 3)

, ∙

min t, s 0

The Poisson process: is a Poisson process which follows a Poisson distribution with associated parameter , where is called the intensity. It is characterized by a number of events in the time interval , . The process can be homogeneous, with constant over time, or non‐homogeneous, with changing over time. ∙ ! ,

∙

0,1, …

Itô’s Lemma: , ,

∙

1 ∙ 2

∙

Fundamental PDE (Partial Differential Equation): U

1 ∙ 2

∙ ∙

∙

Geometric Brownian Motion (GBM): U

∙

ln ln

∙ ~

∙ 1 2

⇒ ln

∙

Ornstein/Uhlenbeck Process: ∙

∙

Generalized ABM (using the Ornstein/Uhlenbeck process): U

∙

∙ ∙

Stochastic Volatility Continuous Jump Model (SVCJ): U

∙

∙ ∙

∙

Option Pricing with other Underlyings Options on Currencies: ≔

∙

∙ 1 2

≔

∙

√

∙

1 2

max

∙ ∙

Options on Futures: ≔

max

Case 1: ∙

∙

1 2 √

√

Case 2: ∙

∙

It should be that . If we also have that and the European call options are in both cases equal. The American call options would not be equal as they depend on the path of the basis. For neither options are equal in the two cases. Options on Interest‐bearing Instruments: Option on a zero bond: ≔

max

Option on a coupon bond: ≔

∑

∙

∙ ,

max ∑

max

∙ ,

∙

⋯

√

Option on a cap rate: ∙

1, … ,

is determined by the market interest rate , with tenor , for the period of to The caplet due in and by the cap rate . The valuation of the caplet can thus be risk‐free. We consider 1. max ,

∙ max 1

⇒

∙ ,0

∙

∙ max

∙ ,0

∙

Due to the similarity of the payoff to a put option on a zero bond

with a strike price of 1

∙

value of the cap can be interpreted as a portfolio of a number of 1

∙

, the

caplets.

Option on a floor rate: The specifications coincide with those for the option on the cap rate. max 1

∙ ,0

∙

∙ max

∙

Due to the similarity of the payoff to a call option on a zero bond

with a strike price of 1

∙

the value of the cap can be interpreted as a portfolio of a number of 1

∙

floorlets.

Cap‐floor parity with 1

;

: ∙

; ,

∙

⇒

;

1 ,

;

∙

;

∙

; ,

∙

∙ 1

∙

, ,

;

∙

;

Collar with

;

max

, ,

;

∙ ,0

max

∙ ,0

Options Trading Strategies Call‐Put Combinations: Straddle: max

min

Strangle: max

min

Spreads:

Vertical Call‐Spread: max

min

⇒

Vertical Put‐Spread: max

min

⇒

Vertical Butterfly‐Call‐Spread: ⇒

max

2 ∙ min

min

2 ∙ max

max

2∙

min

2∙

max

2∙

min

2∙

Vertical Butterfly‐Put‐Spread: ⇒

max

2 ∙ min

min

2 ∙ max

,0 ,0

Hedging: n:m Put Hedge: ∙

∙ max

Covered Short Call: min

Collar: min

max

Synthetic Put Hedge: ∙

∙

⇒

∙ ∙

∙

0 Synthetic Put Hedge using Futures: ∙

Assumptions: ≔

∙

⇒

⇔

∙ 1 ∙

∙

Constant Proportion Portfolio Insurance (CPPI) ≔

≝

≔

≔ ≝

≔

≝

∙

≔

∙

min

⇒

∙

≝ ∙

≔

≝ min

∙

≔

∙

≝

Portfolio‐shifting Algorithm: ≝

∙

0⇒

min

For

∙

∙ 1

0. The higher the cushion, that is the higher the portfolio value relative to the floor, the more

Note that

should the stock quota increase. ≝

The critical point is given by ⇔

∙ 1

and as such we have the relationship:

∙

≔

or alternatively ∗

0 ∗

or alternatively

⇔

∙

⇔

Proof of the Relationship: ∙

∙

∙ ≝

∙

≝ ∙ 1

∙

from

∙

⇒

∙

we have that: ∙

∙ 1 ∙ ∙

∙

∙ 1

∙

1 ∙

∙

Exotic Options Asian Option: max

Barrier Option: max

,0 ∙

max

,0 ∙

max

,0 ∙

max

,0 ∙

Option with Multiple Underlyings: max

max max

Exchange Option: max

Option with Discontinuous Payoffs: ∙ ∙

Forward At The Money Option: ⇔

European call option with ⇒ ⇒

1 √ 2

⇔

Swaps Interest Rate Swaps: Types of interest rate swaps: fixed/floating interest rate swap; floating/floating interest rate swap Fixed for floating payer asset swap: fixed

fixed

Debtor

capital

Payer

Receiver

capital

Debtor

floating

fixed

Fixed for floating payer liability swap: capital

Creditor

Payer

Receiver

floating

Creditor

floating

capital

fixed

Fixed for floating payer asset/liability swap: fixed

Debtor

Payer

Receiver

Creditor

capital

floating

capital

fixed

capital

Creditor

Payer

Receiver

floating

Debtor

floating

Valuation of Interest rate Swaps: 0

⋯

⇒

∙

, ,

The value

∙

1, … ,

∙

of the payer swap in is zero:

⇒

∙

∙∑

Generally:

∙ 1

Considering that 1 ∙∑

∙

∙∑

0 we can determine the fair swap rate

as:

Currency Swaps: Types of currency swaps: fixed/fixed currency swap; floating/floating currency swap (basis swap); combined swap Step 1: capital exchanges at the prevailing exchange rate capital 1 in currency 2

Party 1

Party 2 capital 2 in currency 1

Step 2: interest exchanges at the prevailing local market rate interest for capital 2 at rate 1 in currency 2

Party 1

Party 2 interest for capital 1 at rate 2 in currency 1

Step 3: capital is exchanged back at the initial exchange rate capital 2 in currency 2

Party 1

Party 2 capital 1 in currency 1

Equity Swaps: loss from equity

Payer

profit from equity

Receiver

market rate ± margin

Types of synthetic replications:

Swap Duration: ∙

∙

Swap Derivatives T+S Forward Swap:

; , ∙

∙

Under the measure the forward swap rate is a martingale and is given by: ,

∙∑

;

Swaptions: Receiver Swaption: present value

present value

Long receiver swaption (long call)

Short receiver swaption (short call)

swaption premium

swap strike rate

swap rate

swap strike rate

swap rate

swaption premium

∙

;

1⇒

max

,0 ∙

max

∙

Payer Swaption: present value

present value

Long payer swaption (long put)

Short payer swaption (short put)

swaption premium

swap strike rate

swap rate

swap rate swaption premium

;

∙

⇒

max

∙

max

,0 ∙

∙

, ,

Credit Derivatives Credit Default Swap (CDS): CDS spread

Protection Buyer

Protection Seller face value + accrued interest at credit event

The CDS spread should be equal to the difference between the risky bond and the risk‐free rate. CDS Forwards:

present value

Long CDS forward

Short CDS forward

spread forward rate

CDS spread

If a credit event occurs the CDS forward ceases to exist. CDS Options:

present vale

present value

Long CDS option (long call)

Short CDS option (short call)

CDS option premium

spread strike rate

CDS spread

spread strike rate

CDS spread

CDS option premium

If a credit event occurs the CDS option ceases to exist. Total Return Swap:

bond coupons and interest

Gain in bond value

Payer

Receiver Loss in bond value

LIBOR + spread

Interest Rate Derivatives (additional) Diskontierungsfunktion (discount function):

1 ∀

Kassazins (spot rate): ∙

⇔

1 ln 1

Kassazinsstruktur (spot rate curve):

Terminzins (forward rate): ⋯

, ∙

∙

⇔

, ∙

∙

ln ,

∙

, ∙

∙ , ∙

⋯

The spot rate is thus a weighted average of the forward rates. 1

∙ 1 ,

ln 1

⇔

, ,

Terminzinsstruktur (forward rate curve):

Kurzfristiger Terminzins (short rate): ,

The short rate is given by:

exp

lim

∆ ↘

∆

exp

∆

lim

∆ ↘

≡

⋯

∆ ∆

∆

exp

∆

Certainty and Uncertainty:

For interest rate uncertainty:

For interest rate certainty:

Arbitrage‐free conditions for uncertainty: 60

Interest Rate Models ,

Generalized Single‐Factor Model: ,

Black/Derman/Toy Model (BDT): Ho/Lee model:

ln ∞ ∀

The Vasicek Model: ,

∗

The Extended Vasicek Model (Hull/White model):

Cox/Ingersoll/Ross Model (CIR): ,

√

∗

√

∗ ∗

∗

The Heath/Jarrow/Morton Model (HJM): ,

In reduced form:

The forward drift‐restriction:

The forward intensity under P:

, ,

The forward intensity:

, ,

The forward intensity under Q:

4. Derivations

The Black/Scholes Formula European Call Option: max

∙

Numéraire‐based pricing using Martingales; normalizing the MMA with the stock: ∙

⇔

∙

Itô’s Lemma:

∙

1 ∙ 2

∙

1 ∙2 2

∙ 1 ∙2 2

∙

Martingale transformation: ∙ ∙

⇒ The Process of ln

∙

≝0⇔

under the terminal measure :

∙

Computing the probability under the terminal measure:

ln ln

ln Φ

2 √

Φ d

Numéraire‐based pricing using Martingales; normalizing the stock with the MMA: 64

∙

⇔

∙

Itô’s Lemma:

∙

1 ∙ 2

∙

∙ ∙

∙

Martingale transformation: ∙ ⇒

∙

≝0⇔

under the risk‐free measure :

The Process of ln

∙

Computing the probability under the risk‐free measure:

ln Φ

Φ d

√

∙Φ

⟹

∙

∙Φ

∙

European Put Option: ∙ ⇔

∙

∙Φ

⇔ ⇔

⇔

∙ 1 ∙

Φ ∙Φ

∙Φ

∙Φ ∙

∙ ∙ 1

∙Φ

∙ ∙ Φ

∙

∙ Φ

Portfolio Selection (MPT) 65

The Minimum Variance Portfolio (MVP) and the Efficient Frontier without the Risk‐free Asset: ,…,

,…,

Portfolio return:

,…,

∑

Expected portfolio return:

≔

Portfolio variance:

≔

∑ ∑

⇒

⟶ ∈

;

1 ,

≝0⇔

1≝0⇔

1⇔

Definitions:

∗

For the global MVP we have that

≔ ≔

For all the MVPs we thus have: The MVP curve is given by:

∗

≔

∗

≔

∗

For all the MVPs we have that:

⇒

subject to:

⇒

Lagrange function:

Optimization without the short‐sell constraint Objective function:

The efficient frontier is given by:

The Minimum Variance Portfolio (MVP) and the Efficient Frontier with the Risk‐free Asset: 66

The risk‐free weight is given by:

∑

1 ,…,

The risk premium or excess rates are given by: Portfolio return:

Expected portfolio return:

≔

Excess portfolio return:

≔

Portfolio variance:

≔

Objective function:

⟶

∑

subject to:

(the short‐sell constraint is not needed anymore as it is already implicit)

⇒

Definitions:

⇒

∗

The tangency portfolio

∗

0 is given by:

In an economical context only the case

√

we have that:

∗

√

with

∗

∗ ∗

∗

The tangent line is given by: For all efficient portfolios

∗

√

can be valid.

Portfolio Optimization in a Two Risky Assets and One Risk‐free Asset Case: 67

∙

1 ∙

∙ 1

2 ∙

∗

∙

2 1

∙

2 1

∙

≝0⇔

∙

∙ ∙

2 ∙

∙

∗

∙

2 ∙

2 2 2

4 2

∗

2 2

∙

2 ∙

∙

The capital market line and the tangency portfolio: ∙ ∗

∙

≝

∗

⇔ ⇔

4 ⇒4

∙

∙2 ∙

≝ 0 ∙

⇔

∙

∙ ∙

∙

0⇔

2 ∙ 2

∙

The shortfall line and the shortfall portfolio: ∶

⇔Φ

≝

⇔

∙

⇔

∙

∶

The Capital Asset Pricing Model (CAPM) The Capital Market Line Approach: The market demand portfolio is given by: ∙

is the proportion of the portfolio of investor that is invested in the tangency portfolio . There are investors rationally acting in the market with homogenous expectations. ,…,

is the investment vector.

The market supply portfolio is given by: ∙

∑

is the market capitalization with securities in composition. ,…,

,…,

is the investment vector.

Given that per definition the market is in equilibrium, such that demand is equal to supply, and that the investment ), we then arrive to the conclusion that the vectors of the demand and supply sides must also be equal ( . available values (monetary resources) are also be equal ∑ Thus, the market portfolio is equivalent to the tangency portfolio with: ∙ Note that

∙

is equal to the Sharpe ratio

, which is the Market Price of Risk.

The Security Market Line Approach: The Sharpe ratio ,…,

of the market portfolio exhibits the maximum value possible. ∑

⁄

,…,

∑

⁄

This is the Sharpe ratio of a random risky portfolio consisting of the initial risky assets, which has to be maximized: 1 2

,…,

≝ 0

∑

⇒

In the optimum we also have, due to the equilibrium condition, that , ∑ Consequently:

∑

, ,

, ⇔

and therefore:

, ∑

The Gordon‐Growth Formula ∙

1 1

∙

∙ 1

1 1

⇔

∙

⇔

∙ 1

⇔

∙ 1

1 1

⇔

∙

∙ 1

1 1

⇔

∙ 1

lim

→

1 1

⋯

∙ 1

1 1

∙

∙ 1

1 1