Problem Set 1

Stochastic Foundations for Finance

Instructions: This problem set is due on 10/31 at 11:59 pm CST and is an individual assignment. All problems must be handwritten. Write full sentences to answer questions that require an explanation. Scan your work and submit a PDF file.

Problem 1 Consider the biased random walk defined by S_{n} = S_{0} + \sum_{i = 1}^{n} X_{i}, where \operatorname{P}(X_{n} = 1) = p and \operatorname{P}(X_{n} = -1) = 1 - p = q where p \neq q. If we denote \tau = \min \{n \geq 0: S_{n} = A \text{ or } S_{n} = - B\}, we would like to compute \operatorname{P}(S_{\tau} = A \mid S_{0} = 0). The proof that \tau is finite (almost surely) is similar to what we did in class, and therefore we will omit it and assume that \operatorname{P}(\tau = \infty \mid S_{0} = 0) = 0.

  1. Let f(K) = \operatorname{P}(S_{\tau} = A \mid S_{0} = K). Argue that f(K) = p f(K + 1) + q f(K - 1). \tag{1}

  2. Verify that f(K) = c_{1} + c_{2} \left(\frac{q}{p}\right)^{K} \tag{2} is a solution of equation (1).

  3. Find c_{1} and c_{2} in equation (2) so that f(-B) = 0 and f(A) = 1. (Hint: You should find that c_{1} = \frac{- \left(\frac{q}{p}\right)^{-B}}{\left(\frac{q}{p}\right)^{A} - \left(\frac{q}{p}\right)^{-B}} \quad \text{and} \quad c_{2} = \frac{1}{\left(\frac{q}{p}\right)^{A} - \left(\frac{q}{p}\right)^{-B}} if everything went right).

  4. Determine \operatorname{P}(S_{\tau} = A \mid S_{0} = 0).

  5. We want to compute the probability of winning $100 before losing $100 for different odds of winning. Complete the table below assuming a constant bet size of $1 on all rounds of the game.

    Chance of winning one round 0.500 0.495 0.490 0.480 0.470
    Chance to win $100

  6. Argue that it’s much better to bet $100 at once rather than $1 at a time if your chance of winning each round is less than 0.5.

Problem 2 Suppose that X is a normally distributed random variable with mean \mu=12 and standard deviation \sigma=20.

  1. What is the probability that X \leq 0?
  2. What is the probability that X \leq -4?
  3. What is the probability that X > 8?
  4. What is the probability that 4 < X \leq 10?

Problem 3 Suppose that X is a normally distributed random variable with mean \mu = 10 and standard deviation \sigma = 20. Compute the 90%, 95%, and 99% confidence interval for X.

Problem 4 Suppose that X=\ln(Y) is a normally distributed random variable with mean \mu=3.9 and standard deviation \sigma=15.

  1. What is the probability that Y \leq 6?
  2. What is the probability that Y > 4?
  3. What is the probability that 3 < Y \leq 12?
  4. What is the probability that Y \leq 0?

Normal Distribution Table

The table below computes \Phi(z) = \operatorname{P}(Z \leq z) if Z \sim \mathcal{N}(0, 1) and z \geq 0. The rows denote the first decimal whereas the columns denote the second decimal. If you need \Phi(-z), you can always use \Phi(-z) = 1 - \Phi(z).

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990