Research Experience for Undergraduates

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Research Experience for Undergraduates

• 2005 North Carolina State University
• Financial Mathematics REU
• Jennifer Geis

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What is an REU?

• REU is provides a research experience for
  undergraduates.
• You work under professors in a very specific
  field of a subject such as mathematics.
• There is compensation and provisions made for
  your time and effort.
• You meet other students like you from across the
  U.S. and sometimes the world.

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Who should go to a REU?

• If youre planning on going to graduate school,
  its a great application booster.
• If youre not sure if graduate school is for you,
  it will provide a similar grad-school-setting
  experience.
• If you want to learn more about a specific area,
  it provides experience in a specialized area.

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Research in Financial Mathematics Option
Pricing Made Cents
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Definition of an Option

• An option gives one the right, but not the
  obligation, to buy or sell an asset, such as a
  stock share, under specified terms.
• A call option gives the right to buy.
• A put option gives one the right to sell

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• A call option, priced at 5, for a share of
  Google stock currently trading at 192 may be
  bought.
• It has a 60 day duration at the end of which the
  stock may be bought for the strike price of 200.
• If at the end of the 60 days the stock price is
  less than the strike price, the option is
  worthless and isnt exercised.
• Else, if the stock price is
• greater than the strike price,
• the option is exercised and
• a payoff is received of the
• current stock price less than
• the strike price.

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• Say you buy 1000 options for 5000 with a strike
  price of 200.
• The price of a share of stock in 60 days is 274.
• The final payoff is (274-200)1000-500069,000

14.8
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Types of Options European

• European Options have an explicit duration with
  the ability to be exercised only at the end of
  the duration and a set strike price.

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Types of Options Barrier

• Barrier Options are European Options. However, if
  the stock price ever exceeds the barrier price,
  the option cannot be exercised and is worthless.

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Type of Options American

• American Options have a set strike price, but may
  be exercised at any point from the time of
  purchase to the end of the duration when seen fit
  by the owner.

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Why is option pricing important?

• If an option is not priced correctly, someone is
  guaranteed to make money! This is called
  arbitrage.
• If someone is guaranteed to make money, someone
  else is guaranteed to lose money.
• Which one are you? You could win big or lose big.

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Unfair Pricing Arbitrage Situation with Put
Option

• Consider stock XYZ currently trading at 12.
• The stock price of XYZ will increase to 24 with
  probability ½.
• The stock price of XYZ will decrease to 6 with
  probability ½.
• No interest. Duration of one discreet period.
  Strike Price of 16.

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Naive Put Option Pricing

• The naïve approach to option pricing is taking
  the simple expected value of the payoff.
• Market Assumptions
• No Interest or Fees
• Stocks are Perfectly Divisible

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Unfair Pricing Portfolio Value

• You invest in three put options and two shares of
  stock.
• Either way, you make a profit! This is arbitrage!

Dont exercise Stock 224 48 Less initial
costs -39 Portfolio Value 9
Stocks 2-12 -24 Options 3-5
-15 -39
Do exercise Stock 26 12 Options 316
48 less spent -39
Portfolio Value 21
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Fair Option Pricing Risk Neutral Probability

• Can we find a different probability such that
• where St is the stock price at time t and r is
  the current interest rate.

• Yes! Let p be the risk neutral probability, u be
  the probability that the stock increases, and d
  be the probability that the stock decreases.
  Solve for p

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Finding fair option prices

• Using the risk neutral probability, there are
  various methods for finding fair option prices.
• Very useful are precise formulas yielding precise
  prices Binomial Formula and Black-Scholes.

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Explicit Formula

• A binomial formula finds an exact fair price.

N units of time. k increase per unit of time
the discount factor
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Explicit Formula

• Black-Scholes Formula

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Formula verses Simulation

• The option price is the expected value of the
  payoff under a risk-neutral random walk.
• An analytical or numerical or simulation should
  give the same answer.
• Monte Carlo Simulations are easily programmed

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Monte Carlo Simulation

• We can use computers to simulate a random walk
  from a starting stock price. The random walk
  simulates random changes to the stock price
  determined by the risk neutral probability p.

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An Example of Monte Carlo Simulation

• flip a fair coin 7 times.
• Generate 7 random numbers
• p.5
• h,h,t,h,h,t,h

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Moving to a continuous model Euler Scheme

• Euler Scheme allows us to create a continuous
  simulation.
• Instead of taking discrete intervals, we use very
  small partitions of the time intervals.
• The small the interval, the more continuous our
  model is.

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Things to Consider

• Is it expensive? If there are too many formulas
  to calculate, the overall run-time is affected.
• What is the run-time? The run-time needs to be
  reasonable.
• How accurate are our results?
• How much fluctuation is present between the Euler
  Scheme paths?

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Solving These Issues Control Variates

• Currently, we have a variance to the order of
• To reduce this, we use the following formula that
  contains a control variate
• where f(X) is the simulated option to be priced,
  g(X) is the simulated control variate option-
  another option picked to be simulated based upon
  the same data generated for the option to be
  priced, and Eg(X) is the expected value of the
  control variate option based upon a deterministic
  formula.

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Example Barrier Option with European Control
Variate

• f(X) is the Barrier Call Option
• g(X) is the European Call Option
• If the stock path does NOT drop below the Barrier
  Price, we have the Barrier Call Option Price
  equal to the European Call Option Price.
• Otherwise, we have the difference between the
  expected and simulated European Call Option
  Prices.

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Results from Example

• Comparing Standard Monte Carlo Method with
  Control Variate Monte Carlo Method
• The following are the expected Barrier Option
  Prices found with these methods for all
  simulation runs, combined
• Determined Option Price
  
• By Standard Monte Carlo Method
  
• 13.2834228
  
• By Control Variate Monte Carlo Method
  
• 13.2770358
  
• Variances For Each
  
• Standard Monte Carlo Method
  
• 0.0056024
  
• Control Variate Monte Carlo Method
  
• 0.0000907
  

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Least Squares Monte Carlo Approach for Pricing
American Put Options

• American Put Options are difficult to price to
  due the many possible exercise points.
• Least Squares Monte Carlo approach by Longstaff
  and Schwartz considers many stopping points
  simultaneously verses only one in standard Monte
  Carlo.
• At each potential stopping point for many
  different simulated stock paths, we consider what
  happens if we exercise as well as if we continue
  without exercising the option.
• We build a function that predicts the payoff at
  the next stopping time in order to create a
  stopping point matrix.
• After determining where the stock point is for
  each simulated stock path, we discount the payoff
  to time 0 and average all of the payoffs to
  determine the appropriate option price.

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Improving the Least Squares Approach

• Our research was improving the least squares
  Monte Carlo approach by implementing control
  variate variance reduction technique.
• The following provides an example of our
  improvements for various initial inputs for
  pricing an American put option with a European
  put control variate.
• Note that the options prices are very similar
  without and with the control variate. However,
  the variance is reduced greatly with the control
  variate.

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American Put with European Put as Control
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So is this the end?

• Of course not.
• We can test of different types of control options
  in hopes of finding a more correlated option.
• Other research consists of testing also different
  based options and using more variance reduction
  techniques.
• We can also see how this simulation holds up as a
  model and make comparisons with actual stock
  market data.

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Thank You!

• I would like to thank my mentors at North
  Carolina State University, Professor Fouque and
  Professor Pang, for their time and guidance and
  graduate student, Stephen Zhou. I would also like
  to acknowledge the other REU students I worked
  with TJ Deems and Troy Tingey.