1.01.2005

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» Q&A: What is the price of a call option that expires in 6 months...
» Q&A: What is the price of a put option that expires in 6 months...
» Q&A: What is the probability that the penalty paid is...
» Q&A: What should be the price of a European call...
» Q&A: What should the swap fixed rate?
» Q&A: What will be the value of his investment on the exercise date if...
» Q&A: What's sampling error in mean if...
» Q&A: What's the % price change to...
» Q&A: What's the actual futures price if a T-bill futures...
» Q&A: What's the arbitrage strategy given...

Q&A: What is the price of a call option that expires in 6 months...

...assuming a stock price is $80 and in the next year it will either 90 or 75. The risk free interest rate is 6%. A call option on this stock has an exercise price of $85.

Answer : price via the binomial options model

  • Diagram Stock Price Dynamics and Option Values on "Trees"
    • Stock Price
      • Up state: $90
      • Down state: $75
    • Call Price
      • Up state: c+ = Max (0, 90 - 85) = $5.
      • Down state: c- = Max (0, 75 - 85) = $0.
  • Compute Risk Neutral Probabilities of Up and Down States
    • Up state price relative: µ = (S+ / S0) = 90/80 = 1.12
    • Down-state price relative: d = (S- / S0) = 75/80 = 0.94.
    • Risk-neutral "up" probability: p = ( (1 + r)^t - d) / (u - d) = ( (1.06)^0.5 - 0.9375) / (1.12 - 0.94) = 0.49
  • Compute Expected Value of Call Option: c = ( c+*p + c-*(1-p) ) / (1 + r)^t = (0.49* 5 + (1-0.49)* $0) / (1.06)^0.5= $2.39

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

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Q&A: What is the price of a put option that expires in 6 months...

...assuming a stock price is $80 and in the next year it will either 90 or 75. The risk free interest rate is 6%. A put option on this stock has an exercise price of $85.

Answer : price via the binomial options model

  • Diagram Stock Price Dynamics and Option Values on "Trees"
    • Stock Price
      • Up state: $90
      • Down state: $75
    • Put Price
      • Up state: p+ = Max (0, 85-90) = 0.
      • Down state: p- = Max (0, 85-75) = $10.
  • Compute Risk Neutral Probabilities of Up and Down States
    • Up state price relative: µ = (S+ / S0) = 90/80 = 1.12
    • Down-state price relative: d = (S- / S0) = 75/80 = 0.94.
    • Risk-neutral "up" probability: p = ( (1 + r)^t - d) / (u - d) = ( (1.06)^0.5 - 0.9375) / (1.12 - 0.94) = 0.49
  • Compute Expected Value of Put Option: p = ( p+*p + p-*(1-p) ) / (1 + r)^t = (0.49* 0 + (1-0.49)* $10) / (1.06)^0.5 = 4.9

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

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Q&A: What is the probability that the penalty paid is...

...greater than 7% and less than 12%, assuming that the penalty for withdrawing funds early from a certain account follows a uniform distribution on the interval from (5%, 10%).

Answer

  • Because the density ends at 10, there is no probability associated with X values greater than 10. So P(7 < X < 12) is the same as P(7 < X < 10)
  • F(10)-F(7) = 1-F(7) = 1 - (x - a)/(b - a) = 1 - (7-5)/(10-5) = 0.6

Bonus Points

  • aka a rectangular distribution. can be discrete or continuous.
  • For a continuous uniform random variable,
    • The probability density function is: f(x) = 1 / (b - a) for a <= x <= b; or 0 otherwise.
    • The cumulative density function is: F(x) = 0 for x <= a; (x - a)/(b - a) for a <= x <= b; 1 for x >= b.
    • the distribution is symmetric: the mean = median.
    • the mean = (a + b)/2, and the variance = (b - a)^2/12.

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

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Q&A: What should be the price of a European call...

...with an exercise price of $30 expires in 90 days, given that a European put with the same exercise price, expiration date and underlying is selling for $6. The underlying is selling for $40, and the risk free rate is 10%.

Answer : compute the value of the synthetic call.

  • c(0) = p(0) + S(0) - X/(1 + r)^T = 6 + 40 - 30/(1 + 10%)^(90/365) = $16.7
    • time to expiration T = 90/365 = 0.2466
  • Since the synthetic call and the actual call have the same payoff, they must have the same price ($16.70) as well.

Bonus Points

  • Synthetic positions enable us to price options, because they produce the same results as options and have known prices (assuming the underlying assets make no cash payments)
  • c(0) = p(0) + S(0) - X/(1 + r)^T : the rhs is known as a synthetic call, consisting of a long put, a long position in the underlying, and a short position in the risk-free bond.
  • p(0) = c(0) + X/(1 + r)^T - S(0) : the rhs is known as a synthetic put, consisting of a long call, a short position in the underlying, and a long position in the risk-free bond.
  • p(0) + S(0) - c(0) = X/(1 + r)^T : the lhs is known as a synthetic bond.

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

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Q&A: What should the swap fixed rate?

An investor enters into a two-year $20 million notional principal interest rate swap in which it promises to pay a fixed rate and receive payments at LIBOR. The payments are made every six months based on the assumption of 30 days per month and 360 days in a year. The term structure of LIBOR interest rates is given as follows:

Term (days) Rate
180 9%
360 9.75%
540 10.2%
720 10.5%

What should the fixed rate be?

Answer :

  • Compute the bond discount factor: the PV of a 6-mon LIBOR zero coupon bond for maturity of T days is DF(T) = 1 / [1 + L(0, T) * (T/360)]. think of this discount factor as the value of spot LIBOR deposit that pays $1 T days later.
Term (days) Spot LIBOR Rate Discount Factor
180 6m LIBOR = L(0, 180) = 9% B(0, 180) = 1 / (1+L(0, 180)*(180/360)) = 1 / (1+9%*(180/360)) = 0.9569
360 12m LIBOR = L(0, 360) = 9.75% B(0, 360) = 1 / (1+L(0, 360)*(180/360)) = 1 / (1+9.75%*(360/360)) = 0.9112
540 18m LIBOR = L(0, 540) = 10.2% B(0, 540) = 1 / (1+L(0, 540)*(540/360)) = 1 / (1+10.2%*(540/360)) = 0.8673
720 24m LIBOR = L(0, 720) = 10.5% B(0, 720) = 1 / (1+L(0, 720)*(720/360)) = 1 / (1+10.5%*(360/360)) = 0.8264
  • R = (360/T) * (1 - DF(T)) / (DF(1) + DF(2) + ... +DF(N)) = (360/180) * [(1 - 0.8264) / (0.9569 + 0.9112 + 0.8673 + 0.8264)] = 0.0975
  • Swap fixed payments would be $20,000,000 x 0.0975 x 180/360 = $975,000

Bonus Points

  • The interest amount is reset on each coupon reset date and paid one period after, but its value on reset date is par.
Suppose today is day 360, and the LIBOR on that day is l. Looking ahead to day 540, we anticipate receiving 1.0, the final principal payment, plus l*180/540. What is the value of this amount on day 360?

We would discount it by the appropriate 6-mon LIBOR: Value on day 360 = (Payment on day 540) * (One-period discount factor) = [1.0 + l * (180/540)] * {1 / [1 + (l * (180/540))]} = 1.0

  • R = (360/T) * (1 - DF(T)) / (DF(1) + DF(2) + ... +DF(N))
    • The PV of a $1 fixed-rate bond = R * ( DF(1) + DF(2) + ... +DF(N) ) + DF(N)
    • Initial swap PV ($1 principal) = PV(fixed) - PV(float) = V(fixed) - $1 = 0

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

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Q&A: What will be the value of his investment on the exercise date if...

...an investor buys one share of stock, a put option on the stock and simultaneously sells a call option on the stock with the same exercise price?

Answer : stock_price + Max{0, excercise_price - stock_price} - Max{0, stock_price - exercise_price}, which results in two scenarios

  • S + 0 - S + X = X, Or
  • S + X - S + 0 = X

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

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Q&A: What's sampling error in mean if...

...a sample of 10 observations is drawn from a population with mean 20.1. The mean of the observations equals 18.3 and the sample standard deviation equals 4.8.

Answer : Sampling error refers to the difference between a sample statistic and the corresponding population parameter: 18.3 - 20.1 = -1.8

Bonus Points

  • The sampling distribution: A sample statistic itself is a random variable and therefore has a probability distribution.
    • The sampling distribution of a statistic is the distribution of all the distinct possible values that the statistic can assume when computed from samples of the same size randomly drawn from the same population. For example, imagine sampling 10 numbers and computing the mean over and over again, say about 1,000 times, and then constructing a relative frequency distribution of those 1,000 means. This distribution of means is a very good approximation to the sampling distribution of the mean. As the number of samples approaches infinity, the relative frequency distribution approaches the sampling distribution.

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

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Q&A: What's the % price change to...

...a 12%, 12-year bond selling for $125 for +/-75 bp shocks?

Answer : approximate percentage price change for +75 bps = - duration x change in yield x 100 + convexity adjustment = -5.41

  • From this earlier post, we know the duration is 7.35 and Convexity 17.93.
  • Duration adjustment = - 7.35 * .0075 * 100
  • Convexity adjustment = 17.93* .0075 * .0075 * 100

Do the same for -75bps.

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

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Q&A: What's the actual futures price if a T-bill futures...

...has a discount rate of 5%?

Answer : T-bill Futures Price = (1 - r/100 * (90/360)) * $1,000,000 = (1 - 0.05 x (90/360)) * 1,000,000 = $987,500.

  • The quoted price is 100 - 5 = 95

Bonus Points

  • Treasure bill futures are contracts in which the underlying is a 90-day $1,000,000 of a US Treasury bill
  • Treasury bills are sold at a discount from par value. The price of a Treasury bill futures is quoted as 100 minus the discount rate used by the futures market to derive the contract price, or simply: price = 100 - rate. The value is called the IMM Index (IMM stands for International Monetary Market).

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

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Q&A: What's the arbitrage strategy given...

...the following information on put and call options on a stock: Call price: 3.1, Put price: 9, Exercise price: 60, Forward price: 55, Days to option expiration: 180 days, The continuously compounded risk-free rate: 4%.

Answer : As the actual put is more expensive, we should sell the put and buy the synthetic put (long call, short forward and long bond).

  • bond PV = [X - F(0, T)] / (1 + r)^T = (60 - 55) / 1.04180/365 = 4.9
  • The price of a synthetic put: p0 =short forward + c0 + [X - F(0, T)]/(1 + r)^T = 0 + 3.1 + 4.9 = 8.
  • The initial up-front cash: -3.1 (long call) - 4.9 (long bond) + 0 (short forward) + 9 = 1.
  • At expiration
    • short forward = -(ST - 55)
    • long bond = 60 - 55 = 5
    • If ST < 60: the portfolio would generate 0 (long call) - (ST - 55) (short forward) - (60 - ST) (short put) + (60 - 55) (long bond) = 0.
    • If ST >= 60: the portfolio would generate (ST - 60) (long call) - (ST - 55) (short forward) + 0 (short put) + (60 - 55) (long bond) = 0.
    • The strategy would generate 1 up-front without any investment or any amount to pay back later.

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

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