1.01.2005

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» Q&A: What's the min & max possible futures price movement...
» Q&A: What's the no-arbitrage value of the call option...
» Q&A: What's the no-arbitrage value of the put option...
» Q&A: What's the payoff to the holder of an option...
» Q&A: What's the porfolio's duration...
» Q&A: What's the price of the options using Black-Scholes-Merton model
» Q&A: What's the probability of a normal random variable X...
» Q&A: What's the probability that the return on an asset in a portfolio...
» Q&A: What's the probability that the student passes the interview test?
» Q&A: What's the PVBP of a 10%, 15-year bond...

Q&A: What's the min & max possible futures price movement...

...during any day's futures trading?

Tick size and daily price limit respectively.

For example, the Chicago Board of Trade (CBOT) establishes that the minimum price fluctuation , or tick size for the US Treasury bond futures contract is 1/32. Suppose that the price limit is $5, and the previous settlement price is $110. Today, all trades must take place between $110-/+5.

Category: C++ Quant > Finance > Derivatives

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Q&A: What's the no-arbitrage value of the call option...

in this Q&A?

Answer

  • construct a portfolio of stocks and calls in which final payoff (at expiration) is up/down state-independent: S- - c- * n = S+ - c+ * n
    • Buying stocks and selling calls ((ie. covered call) so that regardless of which way the underlying moves, the portfolio value should be the same (perfectly hedged).
    • Calculate the hedge ratio (shares per call) : n = ($5 - $0) / ($90 - $75) = 0.3333.
  • To form a perfectly hedged portfolio, an investor needs to buy 1/3 of a share of stock for each call that written (sold), or buy 1 share of stock and sell (write) 3 calls. i.e. consider the position of the portfolio at the expiration of the call if the investor writes 1 call:
Value S+ = $90 S- = $75
1/3 share Stock $30 $25
Call -$5 $0
Expiration $25 $25
    • Guaranteed outcome is $25 for this portfolio: PV = $25 / 1.06^0.5 = $24.28
  • Investor would have to buy 1/3 share of the stock and sell 1 call option: $24.28 = S/3 - c
    • c = $80/3 - $24.28 = $2.38

Category: C++ Quant > Finance > Derivatives > Valuation

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Q&A: What's the no-arbitrage value of the put option...

in this Q&A?

Answer

  • construct a portfolio of stocks and calls in which final payoff (at expiration) is up/down state-independent: S- + p- * n = S+ + p+ * n
    • Buying stocks and calls (ie. protected put) so that regardless of which way the underlying moves, the portfolio value should be the same (perfectly hedged).
    • Calculate the hedge ratio (shares per call) : n = ($10 - $0) / ($90 - $75) = 0.66.
  • To form a perfectly hedged portfolio, an investor needs to buy 2/3 of a share of stock for each put, or buy 1 share of stock and 3/2 puts. i.e. consider the position of the portfolio at the expiration of the call if the investor writes 1 call:
Value S+ = $90 S- = $75
2/3 share Stock $60 $50
Put $0 $10
Expiration $60 $60
    • Guaranteed outcome is $60 for this portfolio: PV = $60 / 1.06^0.5 = $58.28
  • Investor would have to buy 2/3 share of the stock and 1 put option: $58.28 = 2*S/3 + p
    • p = $58.28 - $2*80/3 = $4.9

Category: C++ Quant > Finance > Derivatives > Valuation

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Q&A: What's the payoff to the holder of an option...

...expiring in 90 days on 180-day LIBOR, with an exercise rate of 5.5 percent and a notional principal of $10 million, assuming that the 180-day LIBOR is 6 percent on the expiration day?

Answer : $10,000,000 * (0.06 - 0.055) * (180/360) = $25,000.

Bonus Points

  • The call is in-the-money.
  • By convention this money is not paid at expiration but 180 days later.

Category: C++ Quant > Finance > Derivatives > Options

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Q&A: What's the porfolio's duration...

...given the following 3 types of option free bonds?

Bond Price # of Bonds
10%, 5-year $100 40,000
8%, 15-year $84.6275 50,000
14%, 30-year $137.8586 10,000

Answer : Portfolio Duration = w(1)D(1) + w(2)D(2) + ... + w(k)D(k), where w(i) is the market value of bond i / market value of the portfolio (each yield must change by 100 basis points for the duration measure to be useful.)

  • Calculate yields for all bonds with Excel's Rate function: turns out yield1 = yield2 = yield3 = 10%
  • Calculate durations for all bonds with MDuration: D1 =MDURATION("jan 01, 2005","jan 01, 2010",10%,10%, 2) = 3.861. Repeat the same for D2=8.047, D3 = 9.168
  • Market Values: m1 = 40,000 * 100 = $4,000,000, m2 = 50,000* 84.6275 = $4,231,375, m3 = 10,000*137.85 = $1,378,586
  • w1 = $4,000,000/$9,609,961 = 41.6%; w2 = $4,231,375/$9,609,961 = 44%; w3 = $1,378,586/$9,609,961 = 14.4%
  • duration = 0.416*3.861 + 0.440*8.047 + 0.144*9.168 = 6.47.

For a 100 bp change in the yield of all three bonds, the porfolio value will change by about 6.47%. For a 50 bp change in the yield of all three bonds, the porfolio value will change by about 3.24% (= 6.47% / 2)

Category: C++ Quant > Finance > Debt > Valuation

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Q&A: What's the price of the options using Black-Scholes-Merton model

Consider call and put options with a common exercise price of $100 and 150 days until expiration. The underlying stock trades for $102, and that you expect the stock to pay a $3 dividend in 90 days. The continuously compounded interest rate is 9%, and the standard deviation for the stock is 0.30.

Answer : Anything that affects the underlying price at expiration will affect the price of the option. use S0 - PV(CF, 0, T) in the Black-Scholes-Merton model instead of S0 to adjust for cash flows of the underlying.

  • The $3 dividend PV = $3*e^( -9%*(90/365)) = $2.93. In Excel, this can be done with EXP() function.
  • The adjusted stock price = $102 - $2.93 = $99.07
  • Apply the Black-Scholes-Merton model
BSM Call Put
Adjusted for dividends 8.91 6.21
Unadjusted 10.74 5.11
  • The difference in prices is substantial, amounting to almost 20%.

Bonus Points

  • Adjusting the stock price reduces the value of the call and increases the value of the put.

Category: C++ Quant > Finance > Derivatives > Valuation

Your Turn!

 

Q&A: What's the probability of a normal random variable X...

...equal to a particular value, say 15? what about P(X<mean)?

Answer :

  • P(X=15) = 0. Can only calculate the probability of x when it falls in a range, such as 14 < x < 16.56 > x > 54.
  • P(X<=mean) = P(X>=mean) = 50%. therefore P(X<mean) < 50%. The total area under the bell-shaped curve is 1.

Bonus Points

  • The normal distribution is completely described by two parameters: the mean (m) and the standard deviation (s).
  • For normal distributions with the same mean but difference variances: The higher the standard deviation the more speard out the distribution.
  • For normal distributions with the same variance but difference means: the mean locates the axis of symmetry.
  • The normal distribution is symmetric: mean = median = mode. it has a skewness of 0, a kurtosis (it measures the peakedness of a distribution) of 3, an excess kurtosis (which equals kurtosis less 3) of 0.
  • A linear combination of two or more normal random variables is also normally distributed.

Category: C++ Quant > Finance > Probability > Distribution

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Q&A: What's the probability that the return on an asset in a portfolio...

... falls between -3.52% and 50.96% if the rates of return on the assets in the portfolio are normally distributed, with a mean of 20% and a standard deviation of 12%?

Answer

  • If the return on the asset is -3.52%: z = (-3.52% - 20%)/12% = -1.96.
  • If 50.96%: z = (50.96% - 20%)/12% = 2.58.
  • For a standard normal random variable X, the 95% confidence interval is -1.96 to 1.96, the 99% confidence interval is -2.58 to 2.58
  • A normal distribution is symmetrical, so the probability of z-value falling between -1.96 to 2.58 is 95%/2 + 99%/2 = 97%.

Bonus Points

  • The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
  • There is unlimited number of normal distributions, each with a different mean or standard deviation. Therefore, it's impractical to provide a table of probabilities for each combination of mean and standard deviation. However, we can standardize the actual distribution for a normal random variable to a standard normal distribution with z = (X - m)/s, where X, M, s are a score, the mean, and the standard deviation from the original normal distribution respectively. aka z distribution.
  • For a normal random variable X, the exact confidence intervals are
    • 90% confidence interval for X is : x-bar - 1.645s to x-bar + 1.645s, with 10% of the observations fall outside the 90% confidence interval, with 5% on each side.
    • 95%: x-bar - 1.96s to x-bar + 1.96s: 5% fall outside the 95% confidence interval, with 2.5% on each side.
    • 99%: x-bar - 2.58s to x-bar + 2.58s, 1% fall outside the 99% confidence interval, with 0.5% on each side.

Category: C++ Quant > Finance > Probability > Distributions

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Q&A: What's the probability that the student passes the interview test?

There are 5 multiple choice questions on an interview test, each having 4 answers. Each question is worth 5 points and only one answers per question is correct. To pass the test one needs at least 20 points. Suppose the student guesses the answer to each question, and his or her guesses from question to question are independent.

Answer : This is similar to the coin flipping problem (i.e what is the probability of obtaining exactly 4 heads if a fair coin is flipped 5 times) except in this case the probably is not 1/2.

  • to get at least 20 points, one must get at least 4 questions right. so we are interested in P(X>=4)
    • The random variable X is binomial with n = 5 and p = 0.25 (4 choices)
  • P(X>=4) = P(4) + P(5) = 0.0155.
    • P(4) = C(4, 5) * p^4 * (1-p)^(5-4) = {5!/[(5 - 4)! x 4!]} * 0.25^4 * (1-0.25)^(5-4) = 0.0146. Or with Excel,
      • P(4) = ( fact(5)/(fact(5-4)*fact(4)) ) * 0.25^4 * (1-0.25)^(5-4)
      • P(4) = BINOMDIST(4,5,0.25,FALSE)
    • P(5) = BINOMDIST(5,5,0.25,FALSE) = 0.0009

Bonus Points

  • For a binomial random variable, the probability, p, of success must be constant for all trials, and the trials are independent.

Category: C++ Quant > Finance > Quantitative Analysis > Probability

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Q&A: What's the PVBP of a 10%, 15-year bond...

...selling for $125.50?

Answer

  • Yield: Nper = 15*2, Pmt=10%*100, PV=-125.50, FV=100, leads to Rate=3.6%*2 = 7.2%
  • Modified Duration: With Excel's MDuration, Settlement = "January 1, 2005", Maturity = "January 1, 2020" (any dates would do as long as maturity is 15 years), Coupon = 10%, Yield = 7.2%, Frequency = 2, leads to 8.50.
  • PVBP = Duration x Bond Price / 100^2 = $0.11

Another approach is to raise the yield by 1 basis point, then solve using PVBP = abs( initial price - PV(7.21%) ).

Followup Question: assume in the above example that the market interest rate is expected to shoot up by 200 basis points, what is the expected change in bond price?

-8.50 * 2% = -17%

Category: C++ Quant > Finance > Debt > Valuation

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