1.01.2005

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» Q&A: The rates of return on the assets in a portfolio are normally distributed...
» Q&A: What are some of the ways to duplicate a bond index?
» Q&A: What are the cashflows of the following interest rate swap?
» Q&A: What are the lower bounds for European and American calls and puts with...
» Q&A: What are the net payment at the end of 1st year...
» Q&A: What are the payoffs for a 4-year Cap with a 6% Cap rate...
» Q&A: What is the appropriate arbitrage strategy given the following involving...
» Q&A: What is the forward price given the following...
» Q&A: What is the market value of the European receiver swaption....
» Q&A: What is the price of a call option that expires in 2 years...

Q&A: The rates of return on the assets in a portfolio are normally distributed...

...with a mean of 20% and a standard deviation of 12%. What's the probability that the return on an asset falls between -3.52% and 50.96%?

Answer :

  • the z-value of -3.52%: z = (-3.52% - 20%)/12% = -1.96
  • the z-value of 50.96%: z = (50.96% - 20%)/12% = 2.58
  • The 95% confidence interval is -1.96 to 1.96. 99% is -2.58 to 2.58
  • Since a normal distribution is symmetrical, P(-1.96<=Z<=2.58) = 95%/2 + 99%/2 = 97%
    • Or with Excel: NORMSDIST(2.58) - NORMSDIST(-1.96) = 97%.

Bonus Points

  • z-score / z-value / z-statistic: 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 the formula: z = (X - m)/s, where x, m, s are a score, mean and standard deviation from the original distribution.
    • z distribution will only be a normal distribution if the original distribution (X) is normal.
  • The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

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

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Q&A: What are some of the ways to duplicate a bond index?

Answer :
  • Pure bond indexing: try own all the bonds in the index in proportion to their market value weights.
    • it's difficult and costly to implement because a bond index typically consists of thousands of issues.
  • Simple sampling: the sample selected may not accurately reflect the risk factors of the index.
  • Stratified random sampling: divide the population of index bonds into groups with similar risk factors (e.g. issuer, duration/maturity, coupon rate, credit rating, call exposure, etc). Each group is called a stratum or cell.
    • Select a sample from each cell proportional to the relative market weighting of the cell in the index.
    • A stratified sample will ensure that at least 1 issue in each cell is included in the sample.

Bonus Points

  • In investment analysis, it is often impossible to study every member of the population. Sampling is the process of obtaining a sample
    • A simple random sample is a sample obtained in such a way that each element of the population has an equal probability of being selected.
    • A biased sample is one in which the method used to create the sample results in samples that are systematically different from the population.
    • it is the method used to create the sample not the actual make up of the sample itself that defines the bias. A random sample that is very different from the population is not biased: it is by definition not systematically different from the population. It is randomly different.
  • In stratified random sampling, the population is subdivided into subpopulations (strata) based on one or more classification criteria. Simple random samples are then drawn from each stratum (The sizes of the samples are proportional to the relative size of each stratum in the population). These samples are then pooled.
    • Stratified random sampling guarantees that population subdivisions of interest are represented in the sample. The estimates of parameters produced from startified sampling have greater precision -- that is, smaller variance or dispersion -- than estimates obtained from simple random sampling.

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

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Q&A: What are the cashflows of the following interest rate swap?

...On December 15 an investor enters into $50 million NP swap with a dealer. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. The investor will pay 7.5% fixed and the dealer will pay LIBOR. Interest based on exact day count and 360 days (30 per month).

Answer : For each period, the net payment = 50,000,000 x (LIBOR - 0.075) x (days/360)

time 12/15 03/15 06/15 09/15 12/15
LIBOR 7.68 7.50 7.06 6.06  
Days in Period   90 92 92 91
The dealer Owes   7.68%*50000000*(90/360)=960,000 7.50%*50000000*(92/360)=958,333 7.06%*50000000*(92/360)=902,111 6.06%*50000000*(91/360)=765,917
The investor Owes   937,500 958,833 958,833 947,917
Net to the investor   22,500 0 -56,222 -182,000

Bonus Points

  • The swap is determined in advance and paid in arrears: the interest amount is reset on each coupon reset date and paid one period after. For example, On day 0, the floating rate is set for the first period and the interest to be paid at that rate on day 90. Then on day 90, the rate is set for the second period and the interest is paid on day 180. This process continues so that on day 270 the rate is set for the last period, and the final interest payment and the principal are paid on day 360.

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

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Q&A: What are the lower bounds for European and American calls and puts with...

...exercise prices of $60, given that all options expire in 60 days, in which the current price of the underlying is $50 and the risk-free rate is 5%? What if the exercise price is $40 instead?

Answer

  • Time to expiration (T) = 60/365 = 0.1644.
  • American Call: C(0) >= Max [0, S0 - X] = MAX[0, 50 - 60] = 0.
  • European Call: C(0) >= Max [0, S(0) - X / (1 + r)^T] = MAX[0, 50 - 60/(1 + 5%)^0.1644] = MAX[0, -9.95] = 0.
  • American Put: P(0) >= Max [0, X - S0] = MAX[0, 60 - 50) = 10.
  • European Put P(0) >= Max[0, X / (1 + r)^T - S(0)] = Max[0, 60/(1 + 5%)^0.1644 - 50] = MAX[0, 9.95) = 9.95.
  • the higher the exercise price, the lower the price of a call and the higher the price of a put. (try with excise price of $40)

Bonus Points

  • A Call option Payoff
    • min (European/American) = 0
      • American: C0 >= Max (0, S0 - X),
      • European: c(0) >= Max [0, S(0) - X / (1 + r)^T] . Construct a portfolio consisting of a long call and risk-free bond and a short position in the underlying asset. First we need the ability to buy and sell a risk-free bond with a face value equal to the exercise price and current value equal to the present value of the exercise price. We buy the European call and the risk-free bond and sell short (borrow the asset and sell it) the underlying asset. At expiration we shall buy back the asset.
    • max (European/American) = the underlying price
  • A Put option Payoff Boundary
    • min (European/American) = 0
      • American: P0 >= Max (0, X - S0)
      • European: p(0) >= Max[0, X / (1 + r)^T - S(0)] . Construct a portfolio consisting of a long put, a long position in the underlying, and the issuance of a zero-coupon bond. This combination produces a non-negative value at expiration so its current value must be non-negative. For this situation to occur, the put price has to be at least as much as the present value of the exercise price minus the underlying price.
    • max
      • American = the exercise price
      • European = PV of the exercise price: X/(1 + r)^t

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

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Q&A: What are the net payment at the end of 1st year...

...for an interest rate swap agreement in which an investor has agreed to pay a dealer 8% fixed interest on a $1,000,000 notional amount ($80,000) for the next five years. In return, the dealer has agreed to pay the investor an interest rate tied to the 1 year LIBOR rate plus 1% on a notional amount of $1,000,000 for the next five years. The swap is determined in advance and paid in arrears. The 1 Year LIBOR rates for the following 2 years are shown below

Period Year 0 Year 1 Year 2
LIBOR 6.50% 6.75% 7.00%

Answer : the deal is obligated to pay (6.50% + 1%) * $1,000,000 = $75,000

Bonus Points

  • the LIBOR rate for year 0 is used as the swap is determined in advance.
  • The net payment is the difference between the obligations of the two counter parties.

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

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Q&A: What are the payoffs for a 4-year Cap with a 6% Cap rate...

...a $100 million notional, a quarterly settlement frequency, a 3-month LIBOR reference rate, and the LIBOR for the next 4 quarters as shown below?

Period 1 2 3 4
3-mon LIBOR 5.7% 6% 6.25% 5.7%

Answer : There is no payoff if the cap rate exceeds 3-month LIBOR. For period 3, $100 mil * (3-month LIBOR - cap rate)/4 = $62,500.

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

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Q&A: What is the appropriate arbitrage strategy given the following involving...

...call options with an exercise price of $100 expiring in 6 months. The risk-free rate is 10%. The call is priced at $7.5, and the put is priced at $4.25. The underlying price is $99.

Answer : According to the put-call parity, c(0) + X/(1 + r)^T = p(0) + S(0)

  • the fiduciary call (lhs) = 7.5 + 100/(1.10)^0.5 = 102.85.
  • the protective put (rhs) = 4.25 + 99 = 103.25
  • Since protective put > fiduciary call, that means the protective put is overpriced.
  • We could sell the protective put: sell the put and sell short the underlying. Doing so will generate a cash inflow of $103.25. The we buy fiduciary call, paying out $102,85, netting a cash inflow of $0.4.
    • At expiration, if the price of the underlying is above 100:
      • the bond matures, paying $100. use the $100 to exercise call, receiving the underlying.
      • deliver the underlying to cover the short sale.
      • the put expires with no value.
      • net effect: no money in or out.
    • Similarily If the price of the underlying is below 100 at expiration.

  • So we receive $0.4 up front and do not have to pay anything out.

Bonus Points

  • Put-call parity: c(0) + X/(1 + r)^T = p(0) + S(0) : the lhs is known as fiduciary call, rhs protective put
  • If the underlying assets make cash payments: c0 + X/(1 + r)T = P0 + [S0 - PV(CF, 0, T)] where PV(CF, 0, T) represents the present value of these cash flows.
  • The position is perfectly hedged and represents an arbitrage profit.

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

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Q&A: What is the forward price given the following...

...The options and a forward contract expire in 90 days. The continuously compounded risk-free rate is 5%, and the exercise price is 95. The call price is 10, and the put price is 3.9.

Answer : According to the put-call-forward parity, c0 + [X - F(0, T)]/(1 + r)^T = p0 (where [X - F(0, T)]/(1 + r)^T is the bond PV)

  • F(0, T) = X - (p0-c0)*(1 + r)^T = 95-(3.9-10)*1.05^(90/365) = $101.17
  • A synthetic forward contract is a combination of a long call, a short put and a zero-coupon bond with face value (X - F(0, T)).
  • we may either short or long this bond, depending on whether the exercise price of these options is lower (ie. short if X < F(0, T)) or higher than the forward price.

Bonus Points

  • To see the put-call-forward parity: Consider two portfolios
    • Portfolio A consists of a long call and a long position in a zero-coupon bond with face value of X - F(0, T)
    • Portfolio B consists of a long put and a long forward.
    • At initiation, the value of Porfolio A = c0 + [X-F(0, T)]/(1+r)^T, Porfolio B = p0.
    • At expiration the value of the portfolios are
      • if S(t) <= X: Porfolio A = X-F(0,T) = Porfolio B
      • if S(t) > X: Porfolio A = S(t)-F(0,T) = Porfolio B
    • As two portfolios have exactly the same payoff, their initial investments should be the same as well. That is: c0 + [X - F(0, T)]/(1 + r)^T = p0
  • For a synthetic forward contract: consider a portfolio consisting of a long call, short put and a long position in a zero-coupon bond with face value of X - F(0, T). At expiration the value of the portfolio is:
    • if ST <= X: 0 (value of long call) + [-(X - ST)] (value of short put) + [X - F(0, T)] (value of long bond) = ST - F(0, T)
    • if ST > X. : [ST - X] (value of long call) + 0 (value of short put) + [X - F(0, T)] (value of long bond) = ST - F(0, T)
    • As a forward contract's payoff at expiration is also ST - F(0, T), the portfolio's initial value must be equal to the initial value of the forward contract (which is 0): c0 - p0 + [X - F(0, T)]/(1 + r)^T = 0

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

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Q&A: What is the market value of the European receiver swaption....

... at expiration if it has an exercise rate of 10% on this two-year swap and expires in 1 years?

Answer : PV = Notional Principal * Max(0, R-X) * (days/360) * (DF(1) + DF(2) + ... + DF(N)). To value a receiver swaption at expiration, we take the difference between the exercise rate and the market swap rate, adjusted for its present value over the life of the underlying swap.

  • From this post, we know: R = 9.75%, DF(1) = 0.9569, DF(2) = 0.9112, DF(3) = 0.8673, DF(4) = 0.8264
  • Max {0, (10%-9.75%) * (180/360) * (0.9569 + 0.9112 + 0.8673 + 0.8264) } = 0.004
  • $20 million * 0.018 = $200,000,000 x 0.004 = $800,000

Bonus Points

  • Receiver swaptions are equivalent to calls on bonds, payer swaptions put on bonds. * At expiration, the market value of a bond with face (exercise price) of $1 and annual coupon of 10% is (10% * 180/360) x (0.9569 + 0.9112 + 0.8673 + 0.8264) + 1*(0.8264) = 1.004. The payoff on a call option on this bond with exercise price of $1 is Max [0, (1.004 - 1)] = 0.004
  • The market value of a swaption at expiration can be received in one of four ways:
    • By exercising the swaption to enter into the underlying swap.
    • By exercising the swaption and entering into an offsetting swap that keeps both swaps in force.
    • By exercising the swaption and entering into an offsetting swap that eliminates both swaps and pays a series of payments equal to the net difference in the fixed rates on the two swaps.
    • By exercising the swaption and receiving a lump sum cash payment.

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

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Q&A: What is the price of a call option that expires in 2 years...

Suppose the price of a stock is $100. It can either rise or fall by 10% in any 12 months. The risk-free rate is 6%. The exercise price of a call option on the stock is $100.

Answer

  • Risk-neutral "up" probability: p = ( (1 + r)^t - d) / (u - d) = (1.06 - 0.9) / (1.1 - 0.9) = 0.8
  • Stock in 2 years (at expiration) (forward traversing the tree )
    • S++ = 100 x 1.1 x 1.1 = $121
    • S+- = S-+ = 100 x 1.1 x 0.9 = $99.
    • S-- = 100 x 0.9 x 0.9 = $81. * Call Prices in 2 years (at expiration)
    • c++ = Max (0, S++ - 100) = $21
    • c+- = c-+ = Max (0, S+- - 100) = $0.
    • c-- = Max (0, S-- - 100) = $0.
  • Call Prices in 1 year (traverse the tree backwards)
    • c+ = ( c++*p + c+-*(1-p) ) / (1 + r) = (0.8 x 21 + 0.2 x 0) / 1.06 = $15.85.
    • c- = ( c+-*p + c--*(1-p) ) / (1 + r) = 0
  • c = ( c+*p + c-*(1-p) ) / (1 + r) = (0.8 x 15.85 + 0.2 x 0) / 1.06 = $11.96.

Bonus Points

  • Different hedge ratios at each time point:
    • n+ = (c++ - c+-) / (S++ - S+-)
    • n- = (c-+ - c--) / (S-+ - S--)
    • n = (c+ - c-) / (S+ - S-)

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

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