Questions and Answers of Finance With Monte Carlo

In the following problems, create a calculator for the given exotic option and use it to calculate a table of prices for various option parameters.A barrier option.
The following technique exploits the Central Limit Theorem to create approximate samples Z from the standard normal distribution. (An exact method is given in Section A.9.) The mean of a uniformly
Let Xt describe a Brownian particle with parameter σ = 4 starting at X = 0 when t = 0 and moving for a time t = T /4. Now let Ys, for s = 0 to s = 3T /4, describe the motion of the same particle
(a) Suppose Brownian motion is used to model stock prices (instead of geometric Brownian motion). If S0 = 10, μ = 0 per year, volatility = 1 per square root year, and T = 1/12 years (about 30 days),
Same question as in Problem 3 but assume prices follow GBM with the same parameters. Compare the less than 10 values.Data from in problem 3(a) Suppose Brownian motion is used to model stock prices
Starting from S0 = 100, run 10,000 trials of a GRW with the following parameter sets (take Δt = 1/365 in every case). What fraction of outcomes are greater than S0? less than S0? equal to S0? (a)
Run 10,000 trials of a GRW with the following parameters: Δt = 1/365, T = 60 days, σ = 40 %, μ = 3 %. What fraction of outcomes: (a) End between 105 and 115? (b) End between 95 and 100? (c)
Answer the questions in Problem 5 by constructing a 6-step binomial tree. As in the text, Δt must equal T/n where n is the number of steps. Then all the parameters must be converted to use the same
Answer the questions in Problem 6 by constructing a 6-step binomial tree. Again, Δt must equal T/n where n is the number of steps, so Δt = 10 days here. All the parameters must be converted to use
If Z is distributed as N(0, 1), how is X = 3+6Z distributed? How is S = ex distributed? What is the mean and variance of S?
Let St, 0 ≤ t ≤ T be a Geometric Brownian Motion (GBM) random variable with drift μ and volatility parameter σ. Suppose S0 = 1 and σ2/2 = μ. What is the mean of log ST ? What is the mean of
(a) Write a program to test the hypothesis that stock prices are up just as likely as down from one trading day to the next. Test that hypothesis on a database of stock prices of your choice. (b)
Research another technical analysis indicator and test it. See [Ach00] for an extensive list. [Ach00] Achelis, S.B.: Technical Analysis from A to Z. McGraw-Hill, New York (2000)
MACD is the difference between a short term moving average and a long term moving average, MACD = maShort − maLong. Typically the short term average is 12 days and the long term is 26 days. When
What should be the price of a 2 year $100 zero coupon bond in order that the investment earns 6 % per year?
Algorithm 6 assumes that the equity is purchased at the beginning of the dividend period. If the stock is held for an exact multiple of the dividend period, then the annualized return should be
An annuity starts with $501,692 and pays out $10,000 per month. If the remaining principle earns 4 % annual interest compounded continuously, for how many months will the annuity pay?
Write a program to display a piecewise linear approximation of a price path as follows. Let S0, S1, S2, ..., S365 be a 1 year sequence of prices. Select a subset of these, for example monthly S0,
Use Algorithm 8 to construct several figures such as Fig. 2.4 and observe the extent to which correlated prices trend together (recall that it is the increments that are correlated, not the prices
Run Algorithm 9 (pp. 54) with various correlations between the two stocks and the market. What is the risk of loss when: (a) ρ1 = 1, ρ2 = 1?, (b) ρ1 = 1, ρ2 = −1?, (c) ρ1 = 0, ρ2 =
Investigate the probability of losing money for the investment of Example 2.8, if the stock is correlated with the market and, variously, ρ = 0.8, ρ = 0, ρ = −0.5. Do this for various market
Calculate the VaR at the 99 % level over, variously, 1 month, 3 months, and 6 months, for a stock whose initial price is $45, whose drift is 2 %, and whose volatility is 23 %.
Find the VaR at the 99 % level over 2 months by simulation for a portfolio of two stocks with parameters: for the first: S0 = 20, μ = 3%, volatility = 26 %, for the second: S0 = 40, μ = 1 %,
Investigate how the probability of loss in Example 2.8 varies as a function of volatility. Make a graph of loss vs. volatility. Example 2.8. Using the parameters as in Fig. 2.2: So = 100, drift p =
Find the VaR for the stocks in Problem 9 by the historical method. For their price histories, use the GBM model to generate 2 months worth of prices for the equities. Only treat the ρ = 0.2
An investor has a choice between two ventures A and B. As the investor sees it, the future holds three possibilities: (bull) A returns 12 %, B returns 3 %, (bear) A return −4 %, B returns 4 %, or
Use the Monotonicity Theorem to show that all risk-free assets must have the same return rate. Theorem (Monotonicity) If portfolios A and B are such that at every possible state of the market at time
If the risk-free rate is 3 % in Problem 12, what is the market point?Data Given in Problem 12 An investor has a choice between two ventures A and B. As the investor sees it, the future holds three
Obtain recent price data for some security from among the list: AAPL, MON, KO, F, MCD, FDX. Along with data for the S&P-500, use it to calculate daily returns over the last month and to calculate
Use the results of Problem 14 to calculate the risk premium for that stock.Data given in Problem14Obtain recent price data for some security from among the list: AAPL, MON, KO, F, MCD, FDX. Along
In Example 3.1 what is the value of the forward contract at 5 months if the stock price at that time is $48? Example 3.1. What should be the price of an 8 month forward contract for 100 shares of a
A portfolio consists of a long 55 put and a short 50 put. Show how to replicate this portfolio using stocks and calls. Comparing the payoff graphs of the two puts, which put has greater value no
(a) Price the option in Problem 7 by Monte Carlo. Use the Numerical Integration algorithm.(b) Same question using Algorithm 13. How many trials are required to get 3 correct digits in both? What are
A company whose current stock price is $43.44 has been paying a quarterly dividend of $0.77 per quarter. What is the annual yield? What should be the price of an 8 month forward contract on this
Let S0 = 20, σ = 0.3, r = 0.06 and Δt = 1 week. Construct a 2-week binomial tree (2 steps), use the p = 1/2 method, and calculate the no-arbitrage call price C for K = 21. If the market price CM
Simulate the tree in Problem 5. (Start with S = S0 and randomly choose “up” with probability p or “down” with probability 1 − p, do this twice. Note the ending price ST and note the payoff
(a) Use a 4-step binomial tree to price a call option with these particulars: S0 = 36, K = 34, r = 0.02, σ = 0.3, T = 4 weeks (28 days). (b) What is the Black-Scholes price? What is the probability
(a) Price the call option of Problem 7 under the assumption that the volatility increases by 10 % each week.(b) Same question under the assumption the volatility decreases by 10 % each week.Data in
This problem is for gaining experience with simulated annealing. Write an annealer to find the minimum value of the following function defined for 0 < x < 10, 1 1 6 f(x) = 20- 3(x-3)² +0.18 (x-7)²
(a) Price the option in Problem 7 if the company has announced it will give a $0.66 per share dividend in 7 days. (b) Same question except the option is an index fund whose dividend yield is 7.3
A 90 day call option with strike price $100 is valued at $8.23. The stock price is $102, and the risk-free rate is r = 3 %. Write a bisection or other numerical solver to find the implied volatility.
A 120 day put option with strike price $80 is selling for $5.33. The risk-free rate is 3 % and the current stock price is $82. Presently the VIX shows volatility at 22.6 %. Is the option over priced?
Use a 4-step binomial tree to price an American put option with these financial parameters: S0 = 60, K = 60, r = 0.10, σ = 0.3, T = 90 days. Compare with the Black-Scholes price. What is the
Use an annealer or a genetic algorithm or other stochastic optimizer to solve the American put problem stated in the caption of Fig. 3.12. (A genetic algorithm is available at the web page for this
Write a program to calculate Asian options. Try it out for a 60 day ATM call option with S0 = 100, and r = 3 %. Let the averaging take place over the last 30 days. Plot the option price as a function
Write a program to calculate correlated basket options. Extend the results of Table 4.3 to T = 90 days.Table 4.3 Call, S = 53 = $8 = 100, K 01 02 03 P12 P13 0.2 0.2 0.2 1 0 0.2 0.2 0.2 -1 0 0.2 0.2
Repeat Problem 1 for a floating strike Asian option.Data given in Problem 1Write a program to calculate Asian options. Try it out for a 60 day ATM call option with S0 = 100, and r = 3 %. Let the
Price a 90 day 100 strike Bermudian option with 15 day early exercise periods. Assume r = 1 % and σ = 20 %. Use the binomial tree solution method. Plot the price of the option versus originating
Same question as Problem 4 but use the exercise boundary method.Data given in Problem 4Price a 90 day 100 strike Bermudian option with 15 day early exercise periods. Assume r = 1 % and σ = 20 %. Use
Find the price of a 365 day exchange option between equities A and B. Assume r = 6 %, σB = 20 % and the current price of B is $60. Plot the price as a function of the current price of A for σA =
In the following problems, create a calculator for the given exotic option and use it to calculate a table of prices for various option parameters.A barrier option.
In the following problems, create a calculator for the given exotic option and use it to calculate a table of prices for various option parameters.A chooser option.
In the following problems, create a calculator for the given exotic option and use it to calculate a table of prices for various option parameters.A lookback option.
In the following problems, create a calculator for the given exotic option and use it to calculate a table of prices for various option parameters.A spread option.
In addition to the Greeks as in Problem 1, for the 46 day 7.50 puts delta is −0.0528, gamma is 0.0738, and vega is 0.0036. Can you set up a delta-gamma-vega neutral position? Do so if possible.Data
Conduct a delta hedge similar to that in Section 5.4.3. In one case assume the stock price increases modestly over the 42 days until expiration. In another, generate a GRW 42 day price sequence.Data
At the present time BAC is selling for 9.52. Its 18 day 10 dollar calls have these Greeks: Δ = 0.3366, Γ = 0.3509, ν = 0.0077 (directly from a brokers web site). (a) What are the corresponding
Answer the second “dilemma” in the section on maximum variables, Section 5.3. If the volatility is 20 % and the time to expiration is 2 months, what is the probability that the stock price
Analyze the strategy of selling covered calls 5 or 6 days before expiration. Experiment with different volatilities, drifts, and stock/strike relationships (i.e. ITM, ATM, etc.).
Using the parameters in the header of Table 5.1, except for the drift, and analyze the OTM call trade (85 strike) for various values of drift; for example μ = 0.02, μ = 0.04, and μ = 0.06.Data
Using the parameters in Table 5.2 for the 45 day 82.50 covered call, analyze the following stop outs: exit the trade if the stock price falls to 79, to 78, to 77. What is the gain rate in each
Using the parameters in the header of Table 5.5, investigate the effect on the gain rate of the 80/85 spread if the volatility: becomes 30 % just after the trade is established, becomes 10 % under
Analyze the dual trade to the third butterfly trade in Table 5.8. Thus go long a put at 80, short 2 puts at 75, and long a put at 70. Let the starting price be S0 = 70 and assume the drift is −2
Show that if X ∼ Po(λ1) and Y ∼ Po(λ2), then X + Y ∼ Po(λ1 + λ2). Pr(X + Y ≤ z) = Pr(X ≤z-yY= y)Pr(Y = y) y=0 11 y=0 dĩ e-site-re -1₂ (z - y)! y!
The skew of a random variable X is defined as Given data x1, x2,...,xn an estimator for skew iswhere x̄ and s̅ are empirical mean and standard deviation. Being a symmetric distribution the normal
The kurtosis of a random variable X is defined as Given data x1, x2,...,xn an estimator for kurtosis is where x̄ and s̅ are empirical mean and standard deviation. The kurtosis of the normal
The Gamma distribution, G(α, λ) has density given byHere Γ(α) is the gamma function of Section 6.8 and equals (α−1)! if α is a positive integer. Show that the Gamma is infinitely divisible
(a) Make a chart similar to Fig. 6.9 showing the price of a put option using the jump diffusion model with lognormal jumps for stock prices versus the GBM model. In order to compare the results with
(a) From market price data make a graph of implied volatility σ versus strike price K for call options on the S&P-500 for expiration maturities of T on the order of 30 days (near as possible). Do
(a) Make a chart similar to Fig. 6.9 showing the price of a put option using a difference IG model for stock prices versus the GBM model. Use a− = a+ = 41 and b− = b+ = 8. What are the mean and
Work the Bermuda option Problem 5 of Chapter 4 assuming prices follow a symmetric differential IG model, use equation (6.39). Be sure to report your model’s parameters.Data given in problem 5Price
Recalculate Table 4.2 for barrier options assuming prices follow a jump diffusion model with normal sized jumps. Recall that the simulation must go event by-event.Data given in table 4.2 Table 4.2 So
(a) Work the Bermuda option Problem 5 of Chapter 4 assuming prices follow a jump diffusion with normal sized jumps. Be sure to report your jump parameters. (b) Repeat (a) using lognormal sized
Recalculate Table 4.1 for Asian options assuming prices follow a differential IG process.Data given in table 4.1 Table 4.1. Asian versus European option prices So = 100, f = 3%, o = 20%, At(days) =
Analyze covered calls as in Table 5.2.Data given in table 5.2 In Table 5.2 we show simulation results for some covered call trades. Table 5.2 Gain expectation for covered call trades So = $80.00, r
Analyze creditspreads as in Table 5.5.Data given in table 5.5 Table 5.5 Gain expectation for credit spreads So = $80, r= 1%, T = 20(days), o= 20% Trade sell strike/ buy strike 80/85 calls 82.50/85
A portfolio consists of 100 shares each of stock A: S0 = 60, μ = 8 %, σ = 40 %; and B: S0 = 40, μ = 3 %, σ = 20 %. Their correlation is ρ = 0.3. After 6 months what is the probability of losing
Analyze iron condors as in Table 5.9.Data given in table 5.9 Table 5.9 Gain expectation for iron condors T = 20, r = 0.01, costs and losses as noted in the text Trade: strikes at 55:60:70:75 Stock
Work the VaR Problem 9 of Chapter 2 assuming prices follow jump diffusion with normal sized jumps.Data given in problem 9Find the VaR at the 99 % level over 2 months by simulation for a portfolio of
Implement the 60-40 game over 300 iterations and show the fortune of the game for several runs. Assume the return upon success is 88 %. Experiment with several betting fractions above and below the
Analyze the straddle strategy as in Table 5.7.Data given in table 5.7 Table 5.7 Gain expectation for straddles So $60, buy 60 call, buy 60 put Trade Price = Drift IV SV Total Gain amt at (%) (%) (%)
In Table 5.1 the second line gives the gain expectation for buying ATM calls as 0.161 with the probability of a gain as 0.37. Assuming complete loss of investment is the complementary probability,
In Table 5.2 the fifth line gives the gain expectation for $5 OTM 30 day covered calls as 0.242 with probability of a gain as 54 %. Since the gain could be anywhere between 0 and 5.36, assume it is
Repeat Problem 7 but now use a dynamic allocation strategy such as Algorithm 31.Data given in Problem 7Implement Algorithm 30 on the investment of Problem 2. Show the fortune after 30 iterations of
Implement Algorithm 30 on the investment of Problem 2. Show the fortune after 30 iterations of the strategy, after 60. Estimate the mean and variance of these ending fortunes. Use at least 10,000
What fractions of a portfolio should be allocated to the following independent investments: A has 70 % chance of succeeding and payoff multiplier α = 1.5; B has 40 % chance of succeeding and payoff
Same question as Problem 4 if A and B are correlated with coefficient ρ = 0.2.Data given in Problem 4What fractions of a portfolio should be allocated to the following independent investments: A
Throughout this chapter (and in Problem 1 above) it has been assumed that the return upon success was fixed, e.g. 88 %. But what if the return is only an average? Rework Problem 1 as follows: upon
Using the parameters in the header of Table 5.4 except for the starting stock price S0, analyze the 80/75 put spread for various starting prices ITM; for example S0 = 79, S0 = 77, S0 = 75.Data given