G12 Lecture 4

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G12 Lecture 4

• Introduction to Financial Engineering

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Financial Engineering

• FE is concerned with the design and valuation of
  derivative securities
• A derivative security is a contract whose payoff
  is tied to (derived from) the value of another
  variable, called the underlying
• Buy now a fixed amount of oil for a fixed price
  per barrel to be delivered in eight weeks
• Value depends on the oil price in eight weeks
• Option (i.e. right but not obligation) to sell
  100 shares of Oracle stock for 12 per share at
  any time over the next three months
• Value depends on the share price over next three
  months

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What are these financial instruments used for?

• Hedge against risk
• energy prices
• raw material prices
• stock prices (e.g. possibility of merger)
• exchange rates
• Speculation
• Very dangerous (e.g. Nick Leason of Berings Bank)

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Characteristics of FE Contracts

• Contract specifies
• an exchange of one set of assets (e.g. a fixed
  amount of money, cash flow from a project)
  against another set of assets (e.g. a fixed
  number of shares, a fixed amount of material,
  another cash flow stream)
• at a specific time or at some time during a
  specific time interval, to be determined by one
  of the contract parties
• Contract may specify, for one of the parties,
• a right but not an obligation to the exchange
  (option)
• In general the monetary values of the assets
  change randomly over time
• Pricing problem what is the value of such a
  contract?

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Dynamics of the value of money

• Time value of money receiving 1 today is worth
  more than receiving 1 in the future
• Compounding at period interest rate r
• Receiving 1 today is worth the same as receiving
  (1r) after one period or receiving (1r)n
  after n periods
• Investing 1 today costs the same as investing
  (1r) after one period or (1r)n after n
  periods
• Discounting at period interest rate r
• Receiving 1 in period n is worth the same as
  receiving 1/(1r)n today
• Investing 1 in periods costs the same as
  investing 1/(1r)n today

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Continuous compounding

• To specify the time value of money we need
• annual interest rate r
• and number n of compounding intervals in a year
• Convention
• add interest of r/n for each in the account at
  the end of each of n equal length periods over
  the year
• If there are n compounding intervals of equal
  length in a year then the interest rate at the
  end of the year is (1r/n)n which tends to exp(r
  ) as n tends to infinity
• (10.1/12)121.10506.., exp(0.1)1.10517...
• Continuous compounding at an annual rate r turns
  1 into exp(r ) after one year

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Why continuous compounding?

• Cont. comp. allows us to compute the value of
  money at any time t (not just at the end of
  periods)
• Value of 1 at some time tn/m is
  (1r/m)n(1tr/n)n
• (1tr/n)n tends to Exp(tr) for large n
• Can choose n as large as we wish if we choose
  number of compounding periods m sufficiently
  large
• X compounded continuously at rate r turn into
  exp(tr)X over the interval 0,t

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Net present value of cash flow

• What is the value of a cash flow x(x0,x1,xn)
  over the next n periods?
• Negative xi invest xi,, positive xi receive
  xi
• Net present value NPV(x)x0x1/(1r)xn/(1r)n
• Discount all payments/investments back to time
  t0 and add the discounted values up
• If cash flow is uncertain then NPV is often
  replaced by expected NPV (risk-neutral
  valuation)
• Benefits and limitations of NPV valuations and
  risk-neutral pricing can be found in finance
  textbook under the topic investment appraisal
• Lets now turn to asset dynamics

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A simple model of stock prices

• Stock price St at time t is a stochastic process
• Discrete time Look at stock price S at the end
  of periods of fixed length (e.g. every day),
  t0,1,2,
• Binomial model If StS then
• St1uSt with probability
• St1dSt with probability (1-p)
• Model parameters u,d,p
• Initial condition S0

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The binomial lattice model
u4S
u3S
State
u3dS
u2S
u2dS
uS
u2d2S
udS
ud2S
S
dS
ud3S
d2S
d3S
d4S
Time
t0
1 2 3 4
5
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Binomial distribution

• Stock price at time t St can achieve values
• utS,ut-1dS, ut-2d2S,, u2dt-2S,udt-1S, dtS
• P(Stukdt-kS)(nCk)pk(1-p)t-k
• Here (nCk)n!/((n-k)!k!)

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A more realistic model

• St1utSt, t0,1,2,
• where ut are random variables
• Assume ut, t0,1,2, to be independent
• Notice that utSt1/St is independent of the
  units of measurement of stock price
• Call ut the return of the stock
• What is a realistic distribution for returns?

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An additive model

• Passing to logarithms gives
• ln St1 ln St ln ut
• Let wt ln ut
• wt is the sum of many small random changes
  between t and t1
• Central limit theorem The sum of (many) random
  variables is (approximately) normally distributed
  (under typically satisfied technical conditions)
• Most important result in probability theory
• Explains the importance and prevalence of the
  normal distribution

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Log-normal random variables

• Assume that ln ut is normal
• Central limit theorem is theoretical argument for
  this assumption
• Empirical evidence shows that this is a
  reasonably realistic assumption for stock prices
• however, real return distributions have often
  fatter tails
• If the distribution of ln u is normal then u is
  called log-normal
• Notice that log-normal variables u are positive
  since uelnu and with normally distributed ln u

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Distribution of return

• Assume that the distribution of ut is independent
  of t
• Under log-normal assumption the distribution is
  defined by mean and standard deviation of the
  normal variable ln ut
• Growth rate ?E(ln ut), Volatility ?Std(ln ut)
• Typical values are
• ?12, ?15 if the length of the periods is one
  year ?1, ?1.25 if the length of the periods
  is one month
• Recall 95 rule 95 of the realisations of a
  normal variable are within 2 Stds of the mean
• Careful if ln u is normal with mean ? and
  variance ?2 then the mean of the log-normal
  variable u is NOT exp(?) but E(u)exp(??2/2) and
  Var(u)exp(2? ?2)(exp(?2)-1)

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Model of stock prices

• St1utSt, t0,1,2,
• uts are independent identically log-normal
  random variable with
• E(u) exp(??2/2)
• Var(u) exp(2? ?2)(exp(?2)-1)
• Model is determined by growth rate ? and
  volatility ?, which are the mean and std of ln ut
• Values for ? and ?2 can be found empirically by
  fitting a normal distribution to the logarithms
  of stock returns

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Simulation

• Find ? and ? for a basic time interval (e.g. ?
  14, ? 30 over a year)
• Divide the basic time interval (e.g. a year) into
  m intervals of length ?t1/m (e.g. m52 weeks)
• Time domain T0,1,,m
• Use model ln St 1 ln St wt
• Know ln Sm ln S0 w1wm
• w1wm is N(?,?2)
• Assume all wi are independent N(?,?2),
• ? E(w1wm)m?, hence ? ?/m
• ?2V(w1wm)m ?2, hence ?2 ?2/m

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Simulation

• Hence ln St?t ln St wt,
• wt is normal with mean ??t and variance ?2?t
• If Z is a standard normal variable (mean0,
  var1) then
• ln St?t ln St ??t ?Zsqrt(?t)
• Such a process is called a Random Walk
• Can use this to simulate process St

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Simulation

• Inputs
• current price S0,
• growth rate ? (over a base period, e.g. one year)
• volatility ? (over the same base period)
• Number of m time steps per base period (?t1/m is
  the length of a time step)
• Total number M of time steps
• Iteration
• St1 exp(??t ?Zsqrt(?t))St
• Z is standard normal (mean0, std 1)

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Options

• Call option Right but not the obligation to buy
  a particular stock at a particular price (strike
  price)
• European Call Option can be exercised only on a
  particular date (expiration date)
• American Call Option can be exercised on or
  before the expiration date
• Put option Right but not the obligation to sell
  a particular stock for the strike price
• European exercise on expiration date
• American exercise on or before expiration date
• Will focus on European call in the sequel

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Payoff

• Payoff of European call option at expiration time
  T
• MaxST-K,0
• If STgtK purchase stock for price K (exercise the
  option) and sell for market price ST, resulting
  in payoff ST-K
• If STltK dont exercise the option (if you want
  the stock, buy it on the market)

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Pricing an option

• Whats a fair price for an option today?
• Economics the fair price of an option is the
  expected NPV of its risk-neutral payoff
• Risk-neutral payoff is obtained by replacing
  stock price process St by so-called
  risk-neutral equivalent Rt
• St1 exp(??t ?Zsqrt(?t))St
• Rt1 exp((r- ?2/2)?t ?Zsqrt(?t))Rt
• Recall that the expected annual return of the
  stock is ???2/2 expected annual return of the
  risk-neutral equivalent is r
• Volatility of both processes is the same

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Option pricing by simulation

• Model
• Generate a sample RT of the risk-neutral
  equivalent using the formula
• RT exp((r- ?2/2)T ?Zsqrt(T))S0
• Compute discounted payoff
• exp(-rT)maxRT-K,0
• Replication
• Replicate the model and take the average over all
  discounted payoffs

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The Black-Scholes formula

• Risk-neutral pricing for a European option has a
  closed form solution
• The value of a European call option with strike
  price K, expiration time T and current stock
  price S is
• SN(d1)-Ke-rTN(d2),
• where

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Key learning points

• Stochastic dynamic programming is the discipline
  that studies sequential decision making under
  uncertainty
• Can compute optimal stationary decisions in
  Markov decision processes
• Have seen how stock price dynamics can be
  modelled by assuming log-normal returns
• Risk-neutral pricing is a way to assign a value
  to a stock price derivatives
• European options can be valued using simulation
  (also for more complicated underlying assets)