CAS 3 Fall 2007 Notes

Contents

1 Statistics and Stochastic Processes 3

1.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Point Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Hypothesis Tests for Normal Means . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Testing Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.3 Chi Square Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.4 UMP Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.5 Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Regression: Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Regression: Goodness of Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7 Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.8 Random walk and Martingale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.9 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Life Contingency 12

2.1 Survival Distribution: Survival Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Survival Distribution: Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Survival Distribution: Recursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Survival Distribution: Mortality Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7 Premiums: Equivalence Principle and Loss at Issue . . . . . . . . . . . . . . . . . . . . . . 22

2.8 Premiums: Percentile and Variance of Loss at Issue . . . . . . . . . . . . . . . . . . . . . . 23

2.9 Reserves: Prospective and Retrospective Formulas . . . . . . . . . . . . . . . . . . . . . . 23

2.10 Reserves: Other Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.11 Reserves: Variance and Recursive Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1

CONTENTS CONTENTS

2.12 Multiple Lives: Probabilities and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.13 Multiple Lives: Premiums and Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.14 Multiple Lives: Reversionary Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.15 Multiple Decrements Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Financial Economics 28

3.1 Forwards and Prepaid Forwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Put-Call Parity and European and American Options . . . . . . . . . . . . . . . . . . . . 28

3.3 Binomial Option Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Black-Scholes Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Delta Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6 Exotic Options: Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 Exotic Options: Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.8 Exotic Options: Compound, Gap and Exchange Options . . . . . . . . . . . . . . . . . . . 35

3.9 Lognormal Model for Stock Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.10 Bonds and Interest Rate Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.10.1 Pricing Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.10.2 Pricing Interest Rate Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.10.3 Pricing Forward Rate Agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.11 Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2

1 STATISTICS AND STOCHASTIC PROCESSES

1 Statistics and Stochastic Processes

1.1 Probability

Definitions and Formulas

Moments RelatedWhat? Equation What? Equation What? Equation

n E(X )n Kurtosis 44

Skewness33

MGF, MX(t) E(etX) Coefficient of variation /

Iterated Conditional IdentitiesIterated Expectation EX(X) = EY EX(X|Y )Iterated Variance Var(X) = VarY (EX(X|Y )) + EY (VarX(X|Y ))

Expected Value of Various Distribution and Nonnegative RV

For X N(, 2), what is E(eX)?

e+0.52

Mixture distribution: fZ(z) =

i wifZi(zi) where sum of wi is 1. What is the kth moment?

E(Zk) =i

wiE(Zki )

For nonnegative N , discrete or continuous, E(N) is the sum of its survival function from 0 to .

E(N) =n=0

nP(N = n) =n=0

nk=1

P(N = n)

=k=1

n=k

P(N = n) =k=1

P(N k) =k=0

P(N > k)

0

sN (n)n = 0

n

fN (y)yn

= 0

y0

nfN (y)y = 0

yfN (y)y = E(N)

Random sumGiven random N and random sample X1, . . . , XN , find the expected value and variance of S defined by

S =Nk=1

Xk

E(S) =n

E(S|N = n)P(N = n) =n

nE(Xk)P(N = n) = E(Xk)E(N) (1)

Var(S) = E(N)Var(Xk) + Var(N)E(Xk)2 by iterated variance formula (2)

3

1.2 Point Estimation 1 STATISTICS AND STOCHASTIC PROCESSES

Order StatisticsX(k) is the kth largest value of the sample X1, X2, . . . , Xn. What is the PDF of X(k)?

fX(k)(x) =n!

(k 1)!(n k)! [F (x)]k1f(x)[1 F (x)]nk

(k 1) of the values smaller than, (n k) values larger than, and 1 value equal to X(k). Constant is thenumber of ways of ordering the sample.

Chi-square, t Distribution, and F distribution

If Zi are standard normal, what is distribution ofni=1

Z2i ?

2n

If Xi are normal with mean and variance 2, what is the distribution ofni=1

(Xi

)2?

2n

IfXi are normal with unknown mean and variance 2, what is the distribution ofni=1

(Xi X

)2?

2n1

If Z is standard normal and X is 2r, then

T =ZX/r

tr

If X is 2r and Y is 2s, thenX/r

Y/s F (r, s)

1.2 Point Estimation

Type of Estimators

1. MoM (Method of Moments). Set E(Xk) = n1ni=1

Xki and solve for .

2. MLE (Maximum Likelihood). Find that maximizes the likelihood L() or log-likelihood `().Invariance property: If is MLE of , so is T () of T ().

3. Equivalence of the two when the underlying distribution is: binomial with fixed n, Poisson, neg-ative binomial with fixed r, exponential, gamma with fixed , and normal when estimating bothparameters or just with fixed .

Evaluating Estimators

1. Bias: E() . Unbiased if equals to 0. Asymptotically unbiased if bias converges to 0.2. Consistency: lim

nP(| | > ) = 0.

3. MSE: Bias2 + Var(). If MSE converges to 0, consistency follows.

4. UMVUE: No other unbiased estimators have a lower variance. To find one, find T () that is anunbiased estimator of and is also sufficient and complete (find in exponential families).

4

1.3 Hypothesis Testing 1 STATISTICS AND STOCHASTIC PROCESSES

1.3 Hypothesis Testing

Definition and Notation

1. 0 and 1 are the sets of parameters specified in null and alternative hypothesis respectively.

2. Significance level: Probability of rejecting H0 (size of the critical region). Denoted .

3. P-value: Probability of data observations given that the null hypothesis is true. Reject H0 if p-value< .

4. Power: Probability of rejecting H0 when H0 is false. Denoted () for 1.5. Type of errors: Type I, rejecting H0 when it is true and Type II, rejecting H1 when it is true.

Following relations hold:

P(Type I Error) = and P(Type II Error | 1) = 1 ()

1.3.1 Hypothesis Tests for Normal Means

Given one sample (X1, . . . , Xn) normally distributed with known variance testingH0 : X = 0,H1 :X > 0, use Z test.

Given one sample (X1, . . . , Xn) normally distributed with unknown variance testing H0 : X =0,H1 : X > 0, use T test and estimate variance with sample variance.

Given two samples, (X1, . . . , Xm) and (Y1, . . . , Yn), they are assumed to be normally distributedwith the same variance. To test the hypothesis: H0 : X = Y , calculate t statistic

T =X Y

s

1m +

1n

where s is the sample pooled standard deviation, square root of s2 = (m1)s2x+(n1)s2y

m+n2 . Have toassume same variances for it to work. Test it against a t-distribution has n + m 2 degrees offreedom.

Given two samples and normally distributed with different variance and large samples, we canapproximate to normal.

Z =X Ys2xm

+s2yn

1.3.2 Testing Variances

Given unbiased sample variance S2, what is its distribution?ni=1

(Xi X

)2=

(n 1)2

S2 2n1

This can be used to test hypotheses involving S2 and form confidence intervals for S2. If is known,the chi square distribution has n degrees of freedom.

Describe a test that compares two variances.

Look at ratio of sample variances and perform hypothesis test (or confidence intervals) using Fdistribution. If testing H1 : 2x >

2y, reject for large F .

5

1.4 Regression: Basics 1 STATISTICS AND STOCHASTIC PROCESSES

1.3.3 Chi Square Tests

One dimensional Chi-sqare test: testing cell probabilities are equal. For n observations, Oi in k categoriesand probability of each category pk, compute

Q =ki=1

(Oi npi)2npi

and test for 2k1.

Two dimensional chi-square test: test cell probabilities distributed among classes (2 sets of categories).Let R be the number of categories in the row class and C be the number of categories in the column classand pr and pc be respective categories probabilities. Calculate

Q =Rr=1

Cc=1

(Orc nprpc)2npcpr

and tests for 2(R1)(C1).

1.3.4 UMP Test

Definition of UMP test?A hypothesis test in class of hypothesis tests, C, and with power (), is a uniformly most powerfultest of size if () () for every 1 and every power function () in C.

State the Neyman-Pearson Lemma (NP Lemma).Testing simple hypothesis H0 : = 0,H1 : = 1, a test with critical region (rejection region) Rthat satisfies

R ={all the X such that LR =

L(0)L(1)

k for any 0 k 1}

and P(X R | H0) =

is a UMP level test. If H1 is a composite hypothesis, such a region may not exist.

How to apply NP Lemma?Want LR to be as small as possible since it would mean H1 is more likely. If the ratio is amonotonically decreasing function of T (X), some statistic defined on X, R can be of the form{x : T (x) > k}. The inequality reverses otherwise.

1.3.5 Likelihood Ratio Test

Likelihood ratio is

=max0 L()max L()

Form of the likelihood ratio test is < k for 0 k 1. If H0 and H1 are simple hypotheses, LRT reducesto NP Lemma.

1.4 Regression: Basics

Assumptions of simple linear regression:

In the model Yi = 0 + 1Xi + i, Xi are fixed.

i IID N(0, 2)

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1.5 Regression: Goodness of Fit Tests 1 STATISTICS AND STOCHASTIC PROCESSES

0 and 1 are fixed.

The problem: min0,1

ni=1

[Yi 0 1Xi]2 max0,1

ni=1

exp{ [Yi 0 1Xi]

2

22

}What is the least squares solution (maximum likelihood estimator)?

0 = Y 1X, 1 =

i(Xi X)(Yi Y )i(Xi X)2

=

iXiYi nXYiX

2i nX2

What are the variances of the least squares estimates?

Var(1) =2

i(Xi X)2Var(0) = 2

(1n+

X2i(Xi X)2

)Other relevant facts:

Sum of residuals is zero. So i(Yi Yi) = 0 i(Xi X) = 0

1.5 Regression: Goodness of Fit Tests

Notations and Formulas

SSE, SSR, SST represent sum of squares error, sum of squares regression, and sum of squares total.In formula

SSE =i

(Yi Yi)2 SSR =i

(Yi Y )2 SST = SSE + SSR =i

(Yi Y )2

Shortcut for SSR:

SSR = 21i

(Xi X)2

Variance of the regression:

s2 =SSE

n qq is degree of freedoms lost. For two-variable model, q = 2.

Coefficient of determination: R2 = SSR/SST . It is the percentage of variability in Yi explained bythe variability in Xi.

Goodness of Fit Tests for Regression

1. F-test to test the significance of the entire regression.Calculate

F1,Nq =SSR/1

SSE/(n q)2. T-test to test the significance of only one particular coefficient.

To test: H0 : = b, calculate

T = bSEb

If 2 is known, then we use Z instead of T .

7

1.6 Markov Chains 1 STATISTICS AND STOCHASTIC PROCESSES

1.6 Markov Chains

Notation

Qn: transition matrix at time n Mn: the state of chain at time n. kQ: transition matrix after k periods, kQn: transition matrix from time n to time n+ k. kP (i): probability of staying in i for next k periods given currently in i, kP (i)n : is the same butcurrent time is n.

What? Homogeneous Nonhomogeneous

Transition probabilities depend on time? No Yes

Transition matrix after k periods kQ = Qk kQn = QnQn+1 Qn+k1

Staying probability for k periods (in state i) kP (i) =[Q(i,i)

]kkP

(i)n = Q

(i,i)n Q

(i,i)n+1 Q(i,i)n+k1

Table 1: Markov Chain Comparison

Absorbing ChainIf Q has an absorbing state S and starts in non-absorbing state i, the probability kQ(i,S) is the probabilityof starting from i and going to S within k periods and not during period k.

Sum Triple ProductActuarial present value can be calculated as summation of a product of three values: probability ofcashflow occurring, values of cashflow and discount factor.

1.7 Poisson Processes

Definition and Notation

What is a counting process and what properties must a process Nt satisfy o be a counting process?Counting process represents the total number of arrivals by time t. Nt 0, Nt is integer-valued, ifs < t. then Ns Nt, and for s < t,N(t) N(s) represents the number of arrivals in the interval(s, t].

What does independent increments and stationary increments mean?A process has independent increments if number of arrivals in two disjoint periods of time areindependent of each other. A process has stationary increments if the distribution of Ns+t Ns isindependent of s or t.

Let Xk be the time the interarrival time between (k 1) occurrence and kth occurrence. DefineWn = X1 +X2 + . . .+Xn for n > 0

This is the waiting time until the occurrence of the nth arrival

What is a Poisson Process and its properties?Poisson process is a counting process with exponential interarrival times (i.e. Xk are exponential),independent increments and stationary increments. Other properties include

8

1.7 Poisson Processes 1 STATISTICS AND STOCHASTIC PROCESSES

1. N0 = 0

2. P(Nh = 1) = h+ o(h)

3. P(Nh 2) = o(h)Homogeneity means the process has constant intensity .

The intensity of a nonhomogeneous Poisson Process is a function of time, denoted (t). What isthe mean number of arrivals between time a and b? b

a

(t)t

Interarrival timesWhat is the distribution of Wn for Poisson Process? Since Xk is exponential for a Poisson Process withintensity , Wn is Gamma with parameters n and 1/.

Fewer arrivals means waiting time is longer. More arrivals means waiting time is shorter. Therefore,

Wn t Nt n

Combining, Mixing and Thinning a ProcessWhat do we get from combining two independent Poisson Processes and from selecting a fraction ofarrivals, p, from a Poisson process? The latter is called thinning a process.Combining two independent Poisson Processes with intensities 1 and 2 gives a Poisson Process withintensity 1 + 2. Thinning a Poisson Process gives a new Poisson Process with intensity p.

Whats the difference between mixing and thinning?Thinning a process is decomposing arrivals into subsets of arrivals from a main process with intensity .Mixing a process is having different sets of arrivals but they are not obtained from decomposing down alarger process. For example, if we think of arrivals as claims, mixing involves different types of insured:high risk, average, low risk and thinning involves different types of claims: claims over 50k and claimsunder 50k.

Most important difference is that the number of arrivals in a mixed process follows a mixture of twoPoisson distributions and is not a single Poisson. In mixing, differs by risk and thus it is random. Forproblems like this, they may require Bayes Theorem to solve.

Number of Different Arrivals

Given n independent Poisson Processes with intensities 1, , n, what is the probability the firstarrival comes from process k?

kni=1

i

Given 2 independent Poisson Processes with intensities 1 and 2, what is the probability of seeingn events from process 1 before k events from process 2?It is equal to the probability of seeing AT LEAST n events from process 1 before the (n + k)th

arrival. Sum up the binomial probabilities from n to n+k1 using success probability, 1/(1+2). Given n independent Poisson Processes with intensities 1, , n, what is the probability of seeing

9

1.8 Random walk and Martingale 1 STATISTICS AND STOCHASTIC PROCESSES

at least one of each type before time t?

P(every process with at least one arrival by t) =ni=1

P(at least 1 arrival from process i by t)

=ni=1

(1 eit)

Compound Poisson ProcessSuppose now for each arrival, we get a process pair (Ns, Ys). Ys can be anything, for example, for arrivalof claims, Ns, Ys can be claim amount. This is a compound poisson process. What is the expected valueand variance of the aggregate value of Y s, S, over a period of t units?

This is a random sum problem. Using formulas (1) and (2), we have

E(S) = tE(Y ) Var(S) = tE(Y 2)

1.8 Random walk and Martingale

What is a random walk? What is a symmetric random walk?Let Xt be 1 with probability p and 1 with probability 1 p for non-negative integers t. ThenSt =

ti=1Xt is a random walk. A symmetric random walk is a random walk with p = 0.5.

What is a martingale?A random process is a martingale if the conditional expected value of a future value given the cur-rent is the current. E(Xn+1 | Xn) = Xn.

Example: Symmetric random walk.

E(Sn+1 | Sn) = E(Sn +Xn1 | Sn) = Sn + E(Xn+1 | Sn) = Sn

1.9 Brownian Motion

Standard Brownian Motion

Standard Brownian Motion Z(t) satisfies what basic properties?Z(t) is continuous, Z(0) = 0, Z(t + s) Z(t) N(0, s), and Z(t + s1) Z(t) is independent ofZ(t) Z(t s2).

What is the distribution of Z(t)?Z(t) N(0,t)

What is the relationship between Standard Brownian Motion and a symmetric random walk?Take a symmetric random walk, take m steps of size 1/

m and each step takes time 1/m. As

m, it approaches a Standard Brownian Motion.Let t represents the time in between steps so t = 1/m. Thus, each step is size 1/

m =

t.

Then, for a symmetric random walk with each step requiring t and sizedt, we have a standard

Brownian Motion on (0, 1). So,Z(t) = St

t

and is also a martingale.

Arithmetic and Geometric Brownian Motion

10

1.9 Brownian Motion 1 STATISTICS AND STOCHASTIC PROCESSES

How to write Arithmetic Brownian Motion X(t) in terms of Z(t)?X(t) = t+ Z(t), is called the drift and is the volatility.

Arithmetic Brownian Motion X(t) satisfies what properties?It is continuous, has independent increments and X(t+ s)X(t) N(s, 2s).

X(t) is a Geometric Brownian Motion if logX(t) is a .....? What is the distribution of logX(t)logX(0)?

X(t) is a Geometric Brownian Motion if logX(t) is an Arithmetic Brownian Motion.

log(X(t)X(0)

) N(( 0.52)t, 2t)

Geometric Brownian Motion X(t) satisfies what properties?X(t) is continuous, X(t+ s)/X(t) is lognormal with parameters s and

s, and ratios of disjoint

time intervals are independent.

11

2 LIFE CONTINGENCY

2 Life Contingency

2.1 Survival Distribution: Survival Probabilities

Notations:

1. X is the age at death. T (x) is the number of years until death for (x) (also known as completesurvival time).

2. tpx = P(X > x+ t X x) is the probability that (x) survives an additional t years.3. tqx = 1t px is the probability that (x) dies within t years.4. t|uqx = P(x+ t < X x+ t+ u X x)5. `x is the expected number of lives in year x and dx is the expected number of deaths in year x.

6. x is the mortality rate given survival to age x. Valid if 0

xx =.

Formulas:

What? Relations to Equals to

tpx = upx tpx+u `x `x+t`x

qx `xdx`x

=`x+1 `x

`x

tqx s(x)s(x) s(x+ t)

s(x)

t|uqx `x`x+t `x+t+u

`x= tpx t+upx

x s(x)f(x)s(x)

= x

log s(x)

tpx x exp{ x+tx

x x

}

t|uqx x, tpx x+t+ux+t

fT (t)t = x+t+ux+t

tpx x+t t

t|uqx qx+t, tpx tpx uqx+t

t

tpx x, tpx tpxx+t

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2.2 Survival Distribution: Moments 2 LIFE CONTINGENCY

2.2 Survival Distribution: Moments

Notations

1. ex = E[T (x)]: mean number of years until death for (x). Also known as complete life expectancy.

2. ex:n = E[min{T (x), n}]: mean number of years lived within the next n years. Also known as n-yeartemporary complete life expectancy.

3. ex = E[K(x)]: mean future lifetime not counting last fraction of year (number of complete yearslived).

FormulasThe first and second moments can be derived using integration by parts.

4 ex = 0

t tpx x+tt = ttpx]0 + 0

tpxt t

tpx = tpxx+t

know this= 0

tpxt limt t tpx = 0

4 E [T (x)2] know this= 2 0

t tpxt

4 ex:n = n0

t tpx x+tt+ n npx

know this= n0

tpxt

4 E[min{T (x), n}2] know this= 2 n0

t tpxt

4 ex know this= x2 if the remaining lifetime for (x) is uniform

4 ex know this= x+ 1

If the remaining lifetime for (x) is beta

4 ex:n know this=n1k=0

k k|qx + n npx =n

k=1

kpx

4 E[min{K(x), n}2] know this=n1k=0

k2 k|qx + n2 npx =n

k=1

(2k 1) kpx

2.3 Survival Distribution: Recursions

Notations

1. Tx is the total future lifetime of a group of `x individuals. Tx = 0

`x+tt

2. nLx is the total future lifetime of a group of `x individuals over the next n years.

nLx = n0

`x+tt n1k=0

`x+k + `x+k+12

13

2.4 Survival Distribution: Mortality Laws 2 LIFE CONTINGENCY

3. mx, also known as central death rate, is the number of deaths divided by the average number aliveduring the period. Also, nmx is the average death rate for n years. For n > 1, it is smaller thannqx since the latter is the total death rate. In general,

nmx =ndx

nLx

4. a(x) is the fraction of the year lived by those dying during the year.

a(x) =Lx `x+1

dx

5. Percentile pi of survival time is time t that satisfies tpx = 1 pi.

Life Expectancy Identities

What? Using Equals to

ex quantities from notations aboveTx`x

ex:n quantites from notations abovenLx`x

Recursion IdentitiesFormulas for recursive computations of life expectancies. Use when mortality changes.

ex = ex:n| + npx ex+n (3)

ex =

ex:n| + npx ex+n for n 6= 1

px(1 + ex+1) for n = 1

(4)

ex:n| =

ex:m| + mpx ex+m:nm| for 1 < m < n (can be replaced by ex)

px

(1 + ex+1:n1|

)for n = 1

(5)

Think of life expectancy being broken into two pieces: (1) life expectancy for first n years, in addition,(2) life expectancy at (x+ n) weighted by the probability of surviving n years.

2.4 Survival Distribution: Mortality Laws

Mortality Laws: Parameterized models for X

1. Exponential, F (x) = 1 ex (has memoryless property: probability of survival from age x to x+ tis the same as from 0 to t). As a result, x = is constant.

2. Uniform(0, ), F (x) = x x = 1x . This is the basic De Moivre Law.

14

2.4 Survival Distribution: Mortality Laws 2 LIFE CONTINGENCY

3. Generalized De Moivre Law. x =

x . Beta distribution with parameters = , a = 1, b = .

4. Gompertz: Death rate is the sum of age dependent factors: x = BCx. Makeham states that it isthe sum of independent and dependent factors: x = A+BCx.

5. Weibull: F (x) = 1 exp{(x )}

Uniform Distribution of Deaths (UDD)Assumes a uniform distribution of deaths within each year of age.

For 1 s 0, `x+s = `x s dx

In words, the meaning: Number of remaining lives after s years for (x) decreases linearly with time on aunit time interval. Using this, it can be shown that sqx = sqx and spx = 1 sqx for 0 s 1.

x+t = ddt(tpx)

tpx=

qx1 tqx

Other formula:

sqx+t =sqx

1 tqx for 0 s+ t 1

DeMoivres LawDifferent from UDD. Uniform not on unit intervals [x, x+1] but on an entire lifetime [0, ]. The force ofmortality for the uniform case and in general is

x =1

x and x =

x

Constant Force of MortalityForce of mortality is constant, x = and the survival probability is exponential, i.e. tpx = exp{t}

What? DeMoivres Law Modified DeMoivres Law Exponential

s(x) x

( x

)ex

x1

x

x

tpx 1 t x

(1 t

x)

et

ex x2

x+ 1

1

ex:n n npx + n2 nqx

ex x 1

2

Table 2: Mortality Formulas for DML and Constant Mortality

15

2.5 Insurance 2 LIFE CONTINGENCY

2.5 Insurance

NotationsZ is the actuarial present value of the benefit paid out by a life insurance policy. bt and vt are the benefitand discount rate at time t respectively. K(x) is the curtate future lifetime of (x). This is the discreteversion of T (x).

For payment at time of death: E(Zn) = 0

bnt vnt pxx+tt

For payment at end of year: E(Zn) =k=0

bnkvn(k+1)P(K = k) =

k=0

bnkvn(k+1)

k|qx

Life Insurance Symbol for E(Z) Z (at death) Z (end of year)

Whole Ax vT vK+1

n-year term A1x:n| vT if T n, 0 otherwise vK+1 if K < n, 0 otherwise

n-year deferred n|Ax vT if T > n, 0 otherwise vK+1 if K n, 0 otherwise

n-year deferred m-year term n|mAx vT if n < T n+m, 0 otherwise vK+1 if n < K n+m, 0 otherwise

n-year pure endowment A 1x:n| or nEx vn if T n, 0 otherwise same

n-year endowment Ax:n| vmin{T,n} vK+1 if K < n, vn otherwise

Table 3: Table assumes bt = b = 1. For end of year payment insurance, there are no bar on top of A.

What? DeMoivres Law Exponential

Axax x

+

Axax x

ax1

+

Table 4: Insurance and Annuity Formulas for DML and Constant Mortality

Useful Facts and Identities

What is variance of the present value variable Z?

Actuarial present value at double the force of interest, 2, minus the square of actuarial presentvalue at .

A whole life end-of-year payment insurance is the sum of n-year term and (x + n) whole life end-of-year insurance discounted back n years.

Ax = A1x:n + vnpxAx+n = A1x:n +A

1x:n| Ax+n = A1x:n + (Ax:n A1x:n) Ax+n

16

2.5 Insurance 2 LIFE CONTINGENCY

Rules for actuarial PV notations

The bar over A indicates payment at the moment of death. No bar means payment at end ofyear. Subscript to the left of A indicates deferral period. The superscript 1 right on top of n paysa benefit only if T > n. To the left of n means benefit is paid only if T n.

How to find the variance of n-year endowment?

PV of endowment (Z) is the sum of the PV of n-year term insurance (Z1) and PV of n-yearpure endowment (Z2). Then apply definition for a variance of a sum. So,

Z = Z1 + Z2 Var(Z) = Var(Z1) + Var(Z2) 2E(Z1)E(Z2) since Z1 and Z2 are disjoint

What is the distribution of Z for a large group of insured?

Find E(Z) and Var(Z) for one person and apply Central Limit Theorem.

How to find percentile of Z?

Assume benefit is level.

P(Z < z) = p P(vT < z) = p P(T > z) = p

So 100p percentile of Z is 100(1 p) percentile of T . For Z, a monotonic function of T , t thatsatisfies tpx = p is 100(1 p) percentile of T and 100p percentile of Z. To calculate 100p percentilefor n-year deferred insurance, find x > n such that P(n T x) = 1 p. In figure 1, if we want tofind 80th percentile of Z, let vt equal to the value of Z that is exceeded by only 20% of the valuesof Z. Then find t such that 5px tpx = 0.20.

Another way to derive FZ(z) for Figure 1 is to realize its equal to 5qx + tpx.

Recursive Formulas

Ax = vqx + vpxAx+1 ends until px = 0

Ax:n = vqx + vpxAx+1:n1 ends until x+ n 1n|Ax = vpx n1|Ax+1 ends when deferral period ends

The term vqx is the term insurance for 1 year. For continuous payment insurance, replace A withA and vqx with Ax:1|.

Increasing and Decreasing Life Insurancen-year increasing insurance pays k if death occurs in year k until year n. n-year decreasing insurancepays n if death occurs in year 1, (n 1) in year 2, etc. The symbols are (IA)1x:n and (DA)1x:n forn-year increasing and decreasing term insurance. Some recursive identities:

(IA)1x:n = A1x:n + vpx(IA)

1x+1:n1

(DA)1x:n = nA1x:1 + vpx(DA)

1x+1:n1

17

2.5 Insurance 2 LIFE CONTINGENCY

Z

T5 t

(t,z)

Figure 1: Pain graph of 5-year deferred life insurance.

nA1x:1

(DA)1x+1:n1

(DA)1x:n = nA1x:1 + vpx(DA)

1x+1:n1

Figure 2: Rectangle shortcut to derive recursive relation for increasing/decreasing insurance

18

2.6 Annuities 2 LIFE CONTINGENCY

2.6 Annuities

Formulas and Notes for Continuous Annuities

Denote actuarial present value for annuities by E(Y ) n-year certain and life annuity is the sum of n-year annuity certain and n-year deferred life. For ax,

ax = 0

at| tpxx+tt (Expectation formula)

= 0

vt tpxt (Current payment technique)

ax = 1 Ax

and ax:n| =1 Ax:n|

. So,

n|ax = ax ax:n =Ax:n| Ax

Life Annuity Symbol Description Payment logic

Whole ax Pays until death T 1, t T

n-year temporary ax:n pays until earliest between T and n 1, t min(T, n), 0 otherwise

n-year deferred n|ax starts after n years until death 1, n < t T, 0 otherwise

n-year deferred n|max starts after n years 1, n < t n+m and t Tm-year temporary and ends n+m in years 0 otherwise

n-year certain and life ax:n| Until latest date between n and death 1, t max(T, n)

Table 5: Summary of life annuities with continuous payments.

Formulas and Notes for Discrete Annuities

Type Annuity-Due Annuity-Immediate

Whole ax =k=1

vk1 k1px ax =t=1

vt tpx

Temporary ax:n| =n

k=1

vk1 k1px ax:n| =nt=1

vt tpx

Table 6: Discrete Annuities: Current Payment Technique

19

2.6 Annuities 2 LIFE CONTINGENCY

Useful Identities

ax:n = ax:n+1| 1 sx:n = Ex+nsx:n+1| 1

nEx

ax:n = ax:n| 1 + nEx sx:n = sx:n| + 1 1nEx

ax:n = ax nExax+n

Shortcuts for Whole Life Annuity-due and immediate:

For K, curtate lifetime, aK+1| =1 vK+1

d. Take expected value to get actuarial present value.

ax =1Ax

d

For whole life annuity immediate, denoted aK =1 vK

i. So,

vax =v Ax

i iax = 1 (1 + i)Ax

Temprary Life Annuity-due and immediate:For temporary life annuity-due,

ax:n| =1Ax:n|

d

Temporary life annuity-due by definition of expectation:

ax:n| =n

k=1

ak| k1pxqx+k1 + an| npx

k1pxqx+k1 is the probability of living k 1 full years and fewer than k full years.

For temporary life annuity-immediate (dont memorize),

ax:n + 1n Ex = ax:n| =1Ax:n|

d ax:n = 1 (1 + i)Ax:n + inEx

i=

1Ax:n + iA1x:ni

Annuity-due, in general: E(Y ) =t=1

btvt1

t1px, bt is payment in beginning of year t.

Accumulated Values

sx:n| =ax:n|nEx

Variance of AnnuitiesUse the shortcut definition for calculating life annuities (involving insurance).

Var(aT ) = Var(1 vT

)=

2Ax A2x2

The same applies to n-year temporary life annuity (use shortcut definition with endowment). For annuity-due, the denominator is d2.

20

2.6 Annuities 2 LIFE CONTINGENCY

Percentile of AnnuitiesY , the present value of annuities, increases with T . So percentile p of Y is percentile p of T . Thereforeto find the percentile of Y , first find the same percentile of T , call it t. Then plug in t into the formulafor Y , i.e.

Y =1 vT

to get the percentile of Y .

Recursive Formulas

Whole Life Annuities

ax = E(vK | K 1)P(K 1) + E(vK | K < 1)P(K < 1) = (v + vax+1)px + 0 qx = vpx + vpxax+1

ax = E(vK+1 | K 1)P(K 1)+E(vK+1 | K < 1)P(K < 1) = (1+vax+1)px+1 qx = 1+vpxax+1ax = ax:1 + vpxax+1

Temporary Life Annuitiesax:n| = vpx + vpxax+1:n1|

ax:n| = 1 + vpxax+1:n1|

ax:n| = ax:1 + vpxax+1:n1|

n-year certain and life annuities: If death occurs, annuity-certain payments. If no death, recursefurther.

ax:n| = v + vqxan1 + vpxax+1:n1|ax:n| = 1 + vqxan1 + vpxax+1:n1|

ax:n| = a1 + vqxan1 + vpxax+1:n1|

Best to work out these formulas from graphs than to memorize them. Take following as example.

ax+1:n1ax:n

1 1 1 1 1

Figure 3: Annuity recursion picture shortcut. Suppose n = 5

We know 1 is the first payment of ax:n. So ax:n = 1 + something. From the picture, the future expectedpayment (consists of all payments in the future) is ax+1:n1, if (x) survives the current year. So wediscount ax+1:n1 by the probability of surviving the current year and interest rate.

21

2.7 Premiums: Equivalence Principle and Loss at Issue 2 LIFE CONTINGENCY

2.7 Premiums: Equivalence Principle and Loss at Issue

Definition

Equivalence Principle: Actuarial present value of benefit premiums must equal to the actuarialpresent value of death benefits.

Fully continuous insurance: Death benefit is paid at the moment of death and the premiums arepayable continuously.

Full discrete insurance: Death benefit is paid at the end of year and the premiums are payable atbeginning of the year

Loss at issue: Mathematically it is

0L = APV(Benefits) APV(Premiums)

So the random variable is the excess of benefits over premiums or how much the insurer is expectedto lose. Equivalence principle determines the premium to ensure E(0L) = 0.

Premium Formulas by Equivalence Principle

Name Fully Continuous Fully Discrete

Whole Life P (Ax) =Axax

Px =Axax

n-payment Whole Life nP (Ax) =Axax:n

nPx =Axax:n

n-year Endowment P (Ax:n) =Ax:nax:n

Px:n =Ax:nax:n

n-year Term P (A1x:n) =A1x:nax:n

P 1x:n =A1x:nax:n

Other Premiums Formulas by Equivalence PrincipleDeferred Insurance: n-year deferred insurance assuming payments are made during the deferral periodonly,

P (n|Ax) =n|Axax:n

Deferred Annuity : n-year deferred annuity assuming payments are made during the deferral period only,

P (n|ax) =n|axax:n

Premium RefundsFor an insurance that refunds all premiums without interest, then the refund of premiums forms anincreasing insurance. If pi is the premium and lasts forever,

APV(Death Benefits) = Ax + pi(IA)x

22

2.8 Premiums: Percentile and Variance of Loss at Issue 2 LIFE CONTINGENCY

If there are only n payments of pi, then

APV(Death Benefits) = Ax + pi[(IA)x:n + n n|Ax]

2.8 Premiums: Percentile and Variance of Loss at Issue

Percentile of Loss at IssueBy definition, a 100p percentile of 0L, L satsifies

P(0L < L) = 1 p (6)0L decreases as T increases. For level non-deferred insurance, the probability (6) corresponds to findingt of (t, L) such that

1 p = P(T t) = 1 P(T > t) = 1 tpxVariance of Loss at IssueThe variance of loss at issue for continuous case is

Var(0L) =(2Ax A2x

) (1 +

pi

)2For the discrete case,

Var(0L) =(2Ax A2x

) (1 +

pi

d

)2These work for only endowment and whole life insurance.

Derivation:

0L = vT piaT = vT(1 +

pi

) pi

Var(0L) = Var(vT )(1 +

pi

)2=

(2Ax A2x

) (1 +

pi

)22.9 Reserves: Prospective and Retrospective Formulas

Definition

tL represents the prospective loss after t. It is the excess of the APV of death benefits after t overthe APV of premiums that will be received after t.

E(tL | T > t) = tV is the reserve after t. Its how much to save for years after t. In other words,the reserve and the APV of the premiums should be enough to cover the APV of death benefits.

APVt(Premiums) + tV = APVt(death benefits)

Prospective loss is always correct in calculating reserves. Retrospective loss at t is the accumulated value of benefit premiums minus the accumulated costof insurance to time t. If premiums determined through equivalence principle, reserves may becalculated retrospectively.

tkx is the accumulated cost of the past insurance. In most cases,

tkx =A1x:t|tEx

23

2.10 Reserves: Other Formulas 2 LIFE CONTINGENCY

Type of Reserves Prospective Retrospecitve

Whole Life kVx = Ax+k Pxax+k kVx = Pxsx:k kkx

Endowment kVx:n = Ax+k:nk Px:nax+k:nk kVx:n = Px:nsx:k kkx

Term kVx:n = A1x+k:nk P 1x:nax+k:nk kVx:n = P 1x:nsx:k kkx

h-pay whole life hkVx:n =

Ax+k hPxax+k:hk k < h

Ax+k k hhkVx:n =

hPxsx:k kkx k < h

hPxsx:h

khEx+h kkx k h

Table 7: Reserves Prospective and Retrospective Formula

2.10 Reserves: Other Formulas

Premium Difference Formula:Factor out the annuity from the prospective formula. For example,

tV (Ax:n|) =[P (Ax+t:nt|) P (Ax:n|)

]ax+t:nt|

Paid Out Insurance Formula:Factor out the insurance from the prospective formula. For example,

tV (Ax:n|) =

[1 P (Ax:n|)

P (Ax+t:nt|)

]Ax+t:nt|

Three Premium PrincipleGiven two insurances with identical death benefits through time n, difference of benefit reserves attime n is the actuarial AV at time n of their benefit premiums. Note that 1/sx:n = P 1x:n.

Between FormulaEndowment and term (Px:n P 1x:n)sx:n = 1

Endowment and whole life (Px:n Px)sx:n = (1 nVx)Endowment and n-pay life (Px:n nPx)sx:n = (1Ax+n)n-pay life and whole life (nPx Px)sx:n = (Ax+n nVx)n-pay life and term (nPx P 1x:n)sx:n = Ax+nwhole life and term (Px P 1x:n)sx:n = nVx

Table 8: Three Premium Principle

Annuity ratio and Insurance RatioWorks for whole life insurance and endowment with premiums rated under equivalence principle.

tVx:n = 1ax+t:ntax:n

annuity ratio

tVx:n =Ax+t:nt Ax:n

1Ax:n insurance ratio

24

2.11 Reserves: Variance and Recursive Definition 2 LIFE CONTINGENCY

2.11 Reserves: Variance and Recursive Definition

Variance of prospective lossUnder equivalence principle,

Var(tL | T (x) t) = Var(Z)(1 +

P

)2general

=2Ax+t A2x+t(1Ax+t)2 whole life

=2Ax+t:nt A2x+t:nt

(1Ax+t:nt)2endowment

RecursionWorks for fully discrete reserves only: Reserve up to t accumulated to t+ 1 must be enough to cover fordeaths during t+ 1 and those who die after (t+ 1).

(tV + pit)(1 + i) reserve accumulated from t to t+ 1

= qx+t bt+1 pay bt+1 if death

+ px+t t+1V keep rest for survivors

Rearrange and gett+1V = (tV + pit)(1 + i) qx+t(bt+1 t+1V )

(bt+1 t+1V ) is net amount at risk and tV + pit is the initial reserve during [t, t+ 1].

2.12 Multiple Lives: Probabilities and Moments

Notation

Joint life probability pxy or px:y Last survivor probability pxy or px:y

What Equals totpxy under independence tpx tpytqxy under independence 1 tpx tpy

tpxy tpx + tpy tpxytqxy tqx tqy

t|uqxy tpxy t+upxy or tpxy uqx+t:y+tt|uqxy tpxy t+upxyexy exy + exy = ex + ey

T (xy) + T (xy) T (x) + T (y)T (xy) T (xy) T (x) T (y)

Table 9: Multiple Lives Probabilities and Moments Properties

Other Formulas

exy = 0

tpxy dt exy = 0

tpxy dt

exy:n = n0

tpxy dt Var[T (xy)] = 2 0

t tpxy dt e2xy

25

2.13 Multiple Lives: Premiums and Insurance 2 LIFE CONTINGENCY

Cov(T (xy), T (xy)) = Cov(T (x), T (y)) + (ex exy)(ey exy)

Derivation of covariance formula:

Cov(T (xy), T (xy)) = E[T (xy) T (xy)] ET (xy)ET (xy)= E[T (x) T (y)] ET (xy) E(T (x) + T (y) T (xy))= E[T (x) T (y)] (exyex + exyey e2xy) + exey exey= Cov(T (x), T (y)) + (ex exy)(ey exy)

Caution! Wrong Formulas for Last Survivor Probabilities

1. t+upxy = tpxy upx+t:y+t

2. t|uqxy = tpxy uqx+t:y+t

2.13 Multiple Lives: Premiums and Insurance

Multiple Lives Identities for

1. Discount factor vvT (x) + vT (y) = vT (xy) + vT (xy)

2. Annuityax + ay = axy + axy

3. InsuranceAx + Ay = Axy + Axy

For non whole-life insurance, put cap symbol on top of xy in Axy to represent that x and y aretreated as a joint status. Endowment as an example:

Ax:n +Ay:n = A xy :n +Axy:n

Note: Above works for all kind of insurance and shortcut formulas for annuity and insurance stillwork in multiple lives.

2.14 Multiple Lives: Reversionary Annuity

Reversionary annuity pays an annuity to one status after another status has failed. Annuity that pays(y) after (x) has died is ax|y

ax|y = ay axyOther kind of annuities:Paying while either one of them is alive: ax + ayPaying while one of (x) and (y) is alive and the other is dead: ax|y + ay|x

26

2.15 Multiple Decrements Models 2 LIFE CONTINGENCY

2.15 Multiple Decrements Models

Notation

J (or j) represents type of failure or decrement. tq(j)x is the probability of (x) failing within t years due to decrement j. If t =, its te probabilityof ever failing due to decrement j.

p() is the probability of surviving all the decrements. Distribution function FT,J(t, j) = P(T t and J j)

b(j)t is benefit paid at time t for decrement j

Formulas

What Equals to

t1|q(j)x t1p

()x q

(j)x+t1

tq(j)x

t0

sp()x

(j)x+ss

single benefit premiumt=1

vt t1p()x

j

q(j)x+t1b

(j)t

tp()x exp

[ t0

()x+ss

]Table 10: Multiple Decrement Model Formulas

Caution!

tp()x = exp

[ t0

()x+ss

]BUT NOT: tp(j)x = exp

[ t0

(j)x+ss

]Formula Derivation

t1|q(j)x = t1p()x q

(j)x+t1

Comments: Must survive all decrements first before dying due to (j). First probability represents theprobability of surviving all decrements for t 1 years.

FT,J(t, j)j

= P(T t and J = j) = tq(j)0

(j)x+t =

f(t, j)

tp()x

= tq

(j)x

t

1

tp()x

tq(j)x = t0

sp()x

(j)x+ss

Comments: f(t, j) is the joint pdf of T and J and is also the derivative of F (t, j). Second equationderived from definition of force of mortality.

27

3 FINANCIAL ECONOMICS

3 Financial Economics

3.1 Forwards and Prepaid Forwards

Forward (F0,T ) Prepaid Forward (FP0,T )No dividend S0erT S0

Discrete Dividend S0erT FV(Dividends) S0 PV(Dividends)Continuous Dividend S0e(r)T S0eT

3.2 Put-Call Parity and European and American Options

Put-Call ParityNo strike asset:

C(S,K, T ) P (S,K, T ) = FP0,T KerT

Exchange options (want to purchase S1 with S2 as strike):

C(S1, S2, T ) P (S1, S2, T ) = C(S1, S2, T ) C(S2, S1, T ) = FP0,T (S1) FP0,T (S2)

Currency options: Similar to exchange options in that theres a currency rate you want to purchase andone that serves as strike rate. For S0 in the prepaid forward pricing, use the spot exchange rate x0. For aC1-denominated option on C2, strike price and premiums are in the currency C1 and C2 is the rate gettingpurchased.

Early exercise or sell American OptionsFor non-dividend paying stocks, it is not better to exercise an American call early (as opposed to sellingit). Put-call partiy shows that the American call premium is strictly greater than the early exercise payoffat time t, St K. For dividend paying stocks, it is possible that it is more optimal to exercise early.Again, put-call parity shows this. For American put options, there is a possibility that it is optimalto exercise early regardless of whether dividends are paid. For calls, dividends are the reason toreceive stock early and for puts, interest on the strike is the reason to receive the strikeprice early.

Different Strike PriceAssume strike prices K1 < K2 < K3. Here are three inequalities that apply to option premiums (fornon-arbitrage to hold).

1.

C(S,K1, T ) C(S,K2, T )P (S,K1, T ) P (S,K2, T )

Arbitrage opportunity: If first not true, buy low-strike call and sell high-strike call. If second nottrue, buy high-strike put and sell low-strike put.

2.

C(S,K1, T ) C(S,K2, T ) K2 K1P (S,K2, T ) P (S,K1, T ) K2 K1

Arbitrage opportunity: If first not true, sell low-strike call and buy high strike call. If second nottrue, buy low-strike put and sell high strike put.

28

3.3 Binomial Option Pricing Models 3 FINANCIAL ECONOMICS

3. Interpretation: Call premiums decrease at a decreasing rate as strike increases and put premiumsincrease at an increasing rate as strike increases.

C(S,K1, T ) C(S,K2, T )K2 K1

C(S,K2, T ) C(S,K3, T )K3 K2

P (S,K2, T ) P (S,K1, T )K2 K1

P (S,K3, T ) P (S,K2, T )K3 K2

Arbitrage opportunity: If either condition were not true, then there is an asymmetric butterflyspread with positive profits at all prices.

3.3 Binomial Option Pricing Models

Notation

1. S0 is price of stock at time 0. S is stock price after a sequence of movements 2{u,d}. B:amount that we lend at period 0. : number of shares invested inside the replicating portfolio. ris the risk-free rate and is the continuous dividend rate. is the volatility (standard deviation) ofstock movements annualized. p is risk neutral probability. C is the option premium or also knownas arbitrage-free price.

2. u and d are up and down factors that satisfy d < e(r)h < u. C after movements . If binomialmodel is built using forward rates, then for a h-year binomial model,

u = e(r)h+h and d = e(r)h

h

It ensures no arbitrage will occur.

3. is the true discount rate for the option when risk-neutrality is not used (p the true probability isused). is the discount rate of the stock (or the expected return on a stock).

Formulas:

What? Single Period N -Periods

,

(Cu CdS0(u d)

)eh

(C,u C,dS(u d)

)eh

B, B

(uCd dCu

u d)erh

(uC,d dC,u

u d)erh

pe(r)h d

u de(r)h d

u d

C (option premium), C S0+B = erh [pCu + (1 p)Cd] erh [pC,u + (1 p)C,d]

Table 11: For an European option with h-year periods

Derivation for Formulas of and BSolve the equations

uS0eh +Berh = Cu and dS0eh +Berh = Cd

29

3.3 Binomial Option Pricing Models 3 FINANCIAL ECONOMICS

Options on Futures Contracts and CurrencyFutures contract does not pay dividends. A futures contract of T years has price F = S0e(r)T . Also,u, d = exp(h) and p = 1dud .

For currency, S0 is the spot exchange rate and dividend yield is the foreign (currency you are buying)risk-free rate.

Discount Rate of Options under pWhat is the formula for true probability of going up?

S0eh = (uS0)ehp+ (1 p)(dS0)eh p = e

()h du d

What is the formula for the true discount rate of a replicating portfolio?

The discount rate for replicating portfolio is weighted average of the discounting rates of the stock andbond. So

eh =S0

S0 +Beh +

B

S0 +Berh Ceh = S0eh +Berh (7)

Difference from using p or p and different discount rates?

The value of the option is the same whether one uses risk neutral probability with r as discount rate oruses true probability p with .

Schroder MethodGiven an asset with fixed dividends, the stock may end up negative in a binomial model. To avoidthat, Schroder method suggests using the pre-paid forward price as the new starting value inbinomial tree.What is the starting price and the formula for F , volatility of pre-paid forward? Also how about u, d, p?

F = S PV(Dividends) and FS

=S

F

Calculations for u, d, p remain the same.

How to find the call premium Cw?

For an American call option, the real stock price at is S = S + PV(dividends) and

C = max{(S K)+, erE(C+1)}

For an European call option, the stock price remains the same and

C = erE(C+1)

whereE(Cw+1) = pC,U + qC,D

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3.4 Black-Scholes Formula 3 FINANCIAL ECONOMICS

Other Topics

1. Cox-Ross-Rubinstein Tree: u, d = exp{h}. Arbitrage wont happen for small h.2. Lognormal Tree: u, d = exp{(r 0.52)h h}. Arbitrage happens only for very large values

of and h.

3. To estimate volatility , find the unbiased sample variance of log(St/St1). To annualize, find 2h,where h is the number of observations that would be gathered in a year. Square root that to getthe estimated volatility.

3.4 Black-Scholes Formula

Black-Scholes FormulaThe call and put premiums are given by

C(S,K, , r, t) = FP (S)(d1) FP (K)(d2) and P (S,K, , r, t) = FP (K)(d2) FP (S)(d1)

where

d1, d2 =1

hlog[FP (S)FP (K)

]

h

2

For an option on a t-year future contract and forward price F , FP (S) = Fert.

Greeks (Definition, Formula, Graph)

1. Def: Increase in option price as stock price increases. Formula: CSCan be interpreted as number of shares necessary to replicate the option. A call can be replicatedthrough purchasing = et(d1) shares of stock and buying Kert(d2). The signs of d1 and d2reverse for put options.

2. Def: Increase in as stock price increases. Formula:

2CS2

Always positive and hump shaped. So call and put options are convex.

3. VegaDef: Increase in option price for every percentage point increase in volatility. Formula: 0.01CHump usually to the left of strike price. Longer the period to expiration, usually larger the Vega.

4. Def: Increase in option price as time to expiry t decreases. Formula: 1365

Ct

Mostly negative for calls with short expiration (especially at-the-money). Usually negative for putsunless in-the-money. For longer expirations, as function of S is flatter than shorter expirations:gradual decrease unless is large.

5. Def: Increase in option price as r increases. Formula: 0.01Cr is positive for call and negative for put. As function of stock price, it is increasing (the morein-the-money, its worth more).

6. Def: Increase in option price as increases. Formula: 0.01CAs a function of stock price, it is decreasing. Negative for calls and positive for puts.

31

3.5 Delta Hedging 3 FINANCIAL ECONOMICS

Change in Portfolio ValueTo find the change of a position due to change in r, , or , multiply the respective greek to the percentagechange in r, or . So change in option value due to 2% change in r would be 2.Put-Call Parity and GreeksPut-call parity can be used to derive formulas involving greeks between calls and puts. For :

C P = Set Kert C P = et after taking partial derivative with respect to S

As a result, C = P .

Useful Measures

Greek of a portfolio: Greeks are linear in nature: Greek of portfolio is sum of Greeks of individualderivatives.

Elasticity: The ratio of percent change in option price to the percent change in stock price. Let be change in stock price.

E =/C/S

=SC

Option Volatility: C = S |E| Risk Premiums and Sharpe Ratio: Risk premium is the excess of return rate of an asset over therisk free rate. For stock, its r and for option it is r. From (7), ( r) = E( r)

Sharpe ratio is the risk premium over the volatility. rC

=E( r)ES

= rS

Miscellaneous Topics

Profit on options before maturityA N -day European option that is h days old is the same as an N h day European option. Theprofit of the option after h days is

CNh FV(CN ) compounded over h365 years

Calendar SpreadsCalendar spread is an option spread where the options you sell and buy have the same strike priceon the same underlying stock but with different expiration dates. The one that is purchased hasa longer expiration date. Usually a desirable option strategy is the market is neutral, that is thestock price is not very volatile. If the market is neutral, value of the calendar spread increases withtime; the shorter option expires without value and the long option gains profit.

3.5 Delta Hedging

Basic Delta-HedgingIf a market maker sells a call, he can hedge his position by purchasing shares because of fears stockrises in price. This is delta-hedging. It is also possible to sell a put and that position is hedgedby shorting shares of underlying stock.

Steps of delta-hedging?Delta hedging follows 3 steps:

1. Sell an option (in some cases buy an option)

32

3.5 Delta Hedging 3 FINANCIAL ECONOMICS

2. Purchase shares of underlying stock

3. Borrow money for the first two transactions

Overnight profit of delta-hedging?The overnight profit of delta-hedging is

[C(Sh) C(S0)] + (Sh S0)(erh/365 1

)(S0 C(S0)) (8)

How to approximate change in option value using Delta-gamma-theta approximation?Use Taylor series with f(x) = C(S) around x0 = S0. Then include the effects of time on optionpremium, h. has to be given on per-year basis. The Delta-gamma-theta approximation states

C(Sh) C(S0) = (Sh S0) + 12(Sh S0)2 + h (9)

What is market maker profit from delta-hedging using Delta-gamma-theta approximation?For small h,

(erh/365 1) = rh. Then, the profit is

rh(S0 C(S0)) 12(Sh S0)2 h (10)

This can also be used to approximate profit in general. Suppose we are not delta hedging, butbought and sold a call at time 0. Then rh(S0 C(S0)) in equation (10) can be replaced by theaccumulated interest on the difference in the call premiums paid and received at time 0.

For what stock price does the market-maker break even?One standard deviation up or down in stock price for a delta-hedged position. S Sh

What is the Black-Scholes equation?Set (10) to 0 and let (Sh S0) = S0

h.

rC(S0) = rS0 +122S20 + (11)

Why cant we gamma-hedge or theta-hedge with stocks? What can we use to hedge with other Greeksand how?Greeks are defined as partial derivatives with respect to stock price and time so Greeks for thestock are all zero except for which is 1. To hedge other Greeks, buy options as well as stocks.Suppose we purchased option P and want to delta-gamma hedge it with option Q and stock. Weneed to equate the Greek of the hedged position to the Greek of our portfolio (which contains P ).To equate ,

number of stocks S + number of options Q = Pand to equate ,

number of stocks S + number of options Q = Pwith S = 1 and S = 0.

RehedgingAssume market-maker has written a call, delta-hedged by buying shares of stock and rehedged atregular time intervals h. If stock price moves Z standard deviations over a time period of length h, inother words, Sh = S0 ZS0

h, then the market maker profit is

122S20h(1 Z2)

33

3.6 Exotic Options: Asian Options 3 FINANCIAL ECONOMICS

So the variance of the profit is 21(2S20h)2. In a total time period of s during which market-maker

rehedges s/h times, the variance of the total profit is the sum of the variance in each individual periodsof length h. So the variance would be

s

hh2

(2S20h)2

2= sh

(2S20h)2

2

So more frequent market maker rehedges, the variance of total profit decreases.

3.6 Exotic Options: Asian Options

Two ways to average stock price

Arithmetic average: n1n

k=1

Sk Geometric average:

(n

k=1

Sk

)1/n

Remember not to include S0 in the average.

Differences between average price option and average strike option

In payoff: Payoff of average price option is based on the average price of the asset over a periodof time and the strike is set. Average strike option has no set strike. The payoff is the differencebetween the final price of the asset and the average price. So the strike is the average price for anaverage strike option.

Payoff Call PutAverage Price (A(S)K)+ (K A(S))+Average Strike (S A(S))+ (A(S) S)+

Table 12: Payoff of Asian option

Value compared to European option: Average price options worth less than European options sinceaverage prices are less volatile than stock price. Geometric average is less than arithmetic average.So geometric average price calls worth less than arithmetic average price calls. The opposite is truefor average price puts. Geometric average strike calls worth more than arithmetic average strikecalls. The opposite is true for average strike puts.

Binomial Model for Asian OptionsTrees do not recombine because of the averaging. For a N period binomial model, consider all 2N endnodes. Calculate the probability of reaching each node using the risk neutral probability

e(r)h du d

Then find the expected payoff through weighting (using risk neutral probabilities as weights). Thenpresent value the expected payoff back N periods.

3.7 Exotic Options: Barrier Options

Knock-out options:Up-and-out means if price rises to the barrier, the option doesnt pay. Down-and-out means if pricefalls to the barrier the option doesnt pay.

34

3.8 Exotic Options: Compound, Gap and Exchange Options 3 FINANCIAL ECONOMICS

Knock-in options:Unlike before when the option wont pay if the asset hits the barrier, this type of options pays onlyif the asset hits the barrier.

Rebate options:Pays a fixed amount if the barrier is hit. Payment can be made end of year or immediately whenits hit.

Most important: Down-and-in and down-and-out call/put options having the same barrierand strike K correspond to having a normal call/put option. Same for up-and-in and up-and-out options.

For which type of knock-in/knock-out options is worth more, consider which makes more money for thebuyer. The more it can make, its worth more. For example, a knock-out call gets more expensive asbarrier increases.

3.8 Exotic Options: Compound, Gap and Exchange Options

Definition and Notation

Compound option is an option to buy/sell another option which expires at a later time. 4 types of compound options?PutOnCall (sell call option), PutOnPut (sell put option), CallOnCall (buy call option), CallOnPut(buy a put option)

K strike price on the stock, x strike price of compound option, volatility of underlying stock, t1expiration date of compound option, T expiration date of underlying option

K2 is trigger of gap option

Compound Option Parity and Pricing American OptionsCompound options can be priced by a binomial option pricing model but also can be priced by put-callparity.

CallOnCall PutOnCall = C(S,K, , r, t, ) xert1CallOnPut PutOnPut = P (S,K, , r, t, ) xert1

Compound options can be used to price American call option with one discrete dividend. Assume a stockpays a single dividend of D at time t1 prior to expiration T of the American call option. At the time ofdividend payment, we can either exercise the option at strike K and receive St1 +D or not exercise theoption. If we dont exercise the option after dividend payment, we dont exercise it until T .

At the time of dividend payment t1, what is the value of the American call?

PV(max[St1 +D K,C(St1 , T t1)])

By put-call parity, C(St1 , T t1) = St1 Ker(Tt1) + P (St1 , T t1). So

max[St1 +D K,C(St1 , T t1)] = St1 +D K +max[0, P (St1 , T t1) +K

(1 er(Tt1)

)D

]Present value back to time 0, the call premium is

S0 Kert1 +CallOnPut[S,K,D K

(1 er(Tt1)

)]35

3.9 Lognormal Model for Stock Prices 3 FINANCIAL ECONOMICS

Gap OptionsGap options pay when the stock price is either higher or lower than the trigger K2. Can have a negativepayoff.

Gap Options Call PutPayoff S K if S > K2 K S if S < K2

Option Premium SeT(d1)K1erT(d2) K1erT(d2) SeT(d1)d1, d2 same except use the trigger as strike in the formula

Table 13: payoff and premiums of gap options

Exchange OptionsInstead of a strike price, exchange options use a strike asset and compares that to the underlying asset.The option pays off if an underlying asset outperforms the strike asset. If S and Q are the underlyingand strike assets, then the payoff of a exchange call is (ST QT )+ at time T . Black-Scholes formula:

SeST(d1)QeQT(d2) for callVolatility of exchange option is

Var(S Q) =2S +

2Q 2SQ

3.9 Lognormal Model for Stock Prices

Let R be the continuously compounded return from 0 to t. So

St/S0 = eR log(St/S0) = RR is expected to be ( )t. Suppose its variance scales with time, i.e. 2t. Black-Scholes formula isderived under the assumption stock price is modeled by Geometric Brownian Motion. So R follows anArithmetic Brownian Motion. In other words,

R = log(StS0

) N(( 0.52)t, 2t)

Questions

Why subtract 0.52?So that the expected value of R comes out to be ( )t.

StS0

= exp{( 0.52)t+ tZ}

E(St) = S0e(0.52)tE exp{tZ}

= S0e(0.52)te0.5

2t = S0e()t

What differential equation does an Arithmetic and Geometric Brownian Motion satisfy?

For St = S0 + ( )t+ Zt, differential is dSt = ( )dt+ dZt.

For log(St) = log(S0)+(0.52)t+Zt, the differential is d(log(St)) = (0.52)dt+dZtor dSt = ( )Stdt+ StZt

36

3.10 Bonds and Interest Rate Caps 3 FINANCIAL ECONOMICS

Whats dt dt, dZ dt, dZ dZ?

dZ dZ = dt, rest are 0. What is Itos Lemma?

If C(St, t) is a twice-differentiable function of S and t then

dC =C

SdS + 0.5

2C

S2(dS)2 +

C

tdt

What is the Sharpe ratio for two processes based on the same dZ?

Equal

3.10 Bonds and Interest Rate Caps

Notation

Pt(T, T + s): price to be paid at time t to purchase a zero-coupon bond for 1 issues at time Tmaturing at time T + s. If t = T , subscript t can be left out, e.g. P (T, T + S).

Ft,T [P (T, T + s)]: forward price at time t for agreement to purchase a bond at time T maturing atT + s.

Let RT be interest rate during year T + 1 and K be the interest rate cap

3.10.1 Pricing Bond Options

What is the formula for Ft,T [P (T, T + s)]?

P (t, T )Ft,T [P (T, T + s)] = P (t, T + s)

Right-hand side is how much to pay at time t to receive a bond that matures for 1 at T + s. Another wayto get 1 at T + s is to enter a forward agreement at time T for a bond that matures at T + s - by payingFt,T [P (T, T+s)] at time T . To accumulate that much at T , we pay the quantity on left-hand side at time t.

What is the premium of an option at time 0 to buy/sell a bond at time T which matures at time T + s?Prepaid forward prices are P (0, T )F , where F = F0,T [P (T, T+s)] and similarly prepaid strike is P (0, T )K.So,

Call premium = P (0, T )[F(d1)K(d2)] (12)Put premium = P (0, T )[K(d2) F(d1)] (13)

where d1, d2 =log(F/K)h

2

h and h is number of years until maturity.

3.10.2 Pricing Interest Rate Caps

What is an interest rate cap?Provides someone with a variable rate loan protection against increases in interest rates.

37

3.10 Bonds and Interest Rate Caps 3 FINANCIAL ECONOMICS

How to price it using Binomial Tree?The payoff at the beginning of year T + 1 (end of year T ) is max

(0,RT K1 +RT

).

To value the cap using binomial tree, calculate payoff at each node (not just the leaf nodes), multiplyeach payoff by their path probabilities, and sum the products.

What is a caplet?Caplets are individual payments of an interest rate cap. For a cap bought over n years, its paymentis the sum of the caplet payments: payment at time 0, after year 1, . . . , after year n 1.

How to price a cap using Black Formula?Price individual caplets using Black Formula and sum up the values.

Caplet is a put on a 1-year bond; caplets pay interest for 1 year which is like selling a bond withmaturity 1, equivalently, is a put on a 1-year bond. Mathematically, payoff of a caplet is(

RT K1 +Rt

)+=(1 +RT (1 +K)

1 +Rt

)+= (1 +K)

(1

1 +K 11 +RT

)+So it is equivalent to (1 + K) puts with strike 1/(1 + K). To calculate its premium, use BlackFormula for bond puts and then multiply by (1 +K).

3.10.3 Pricing Forward Rate Agreements

Standard Forward Rate AgreementA standard forward rate agreement is an agreement to pay for the difference a prevailing rate and aforward rate R, R(t, T ) R, priced so that the expected value of the agreement is 0. A standard FRAsettles at maturity and end of the year. So the forward interest rate is the implicit forward rate. So

R =P (0, t)P (0, T )

1

Eurodollar-style forward agreementPays R(t, T )R at time t, the time load is made, instead of at time T . If rates are modeled by a binomialtree,

R =Discounted[R(t, T )]

P (0, t)

38

3.11 Interest Rate Models 3 FINANCIAL ECONOMICS

3.11 Interest Rate Models

Binomial TreeA binomial tree model for interest rates can be used to price zero coupon bonds or options on zerocoupon bonds. Assume N period binomial tree with period of length h. Denote Rt be the interestat period t.

What is value of zero coupon bond paying 1 that matures at Nh?

E(exp

{h

N1t=0

Rt

})(14)

In words, sum the rates to every end node, derive the bond price for every end node, and find theexpected bond price (multiplying each bond price by path probabilities).

How to find the value of a n-period option on a zero coupon bond that matures in N n periods?

Start pricing zero coupon bonds using (14) from period n. Then calculate the option premiumunder the usual binomial tree procedure.

Discrete Time Model: Black-Derman-ToyBuilds Binomial Tree model for interest rates based on observed yields to maturity and volatility, soit is a model calibrated to real world data. A tree of effective annual rates rather than continuouslycompounded rates. Upper node larger than lower node by a factor of exp{2h}. Path probabilitiesassumed to be 1/2.

R0Rh

Rhe21

h

R2he22

h

R2he42

h

R2h

Year 0 Year 1 Year 2

Figure 4: Black-Derman-Toy. Rates are effective rates.

Equilibrium Interest Rate Models

Notation

r is the short term interest rate and P (r, t, T ) is the price of a zero-coupon bond purchased attime t maturing at time T with short term interest rate r.

Pr = and Prr = are derivatives of P (r, t, T ) with respect to r.

Hedge ratio, N , is the amount of the second bond needed to be purchased to hedge the firstbond.

39

3.11 Interest Rate Models 3 FINANCIAL ECONOMICS

Rendelman-Bartter Vasicek CIRNonnegative interest rates Yes No Yes

Volatility varies by interest rates Yes No YesMean reversion property No Yes Yes

Differential dr = ardt+ rdZ dr = a(b r)dt+ dZ dr = a(b r)dt+ rdZSolution to equation not derived in text P (r, t, T ) = A(t, T ) exp (B(t, T )r(t))

B(t, T ) aTt:aGreeks: and not derived in text = BP (r, t, T ) and = B2P (r, t, T )

Table 14: Equilibrium Interest Models

What is the Sharpe ratio of a Vasicek bond pricing model?

It is constant, independent of r or t. If we write the model as

dP (r, t, T )P (r, t, T )

= (r, t, T )dt+ q(r, t, Y )dZ

then Sharpe ratio equals to

(r, t) =(r, t, T ) rq(r, t, Y )

=(r, t, T ) rB(t, T )(r)

What is the hedge ratio for duration hedging and delta hedging?

For duration hedging: N = T1P (r, 0, T1)T2P (r, 0, T2)

For delta hedging: N = Pr(r, 0, T1)Pr(r, 0, T2)

T2 is the maturity of the bond you are using to hedge.

40