Gregory Berkolaiko/MATH 425 (Math of Contingent Claims)

Past Exams Hints and Answers

• Fall 2023 Midterm
  1. Profit from 90.95 to 103.10
  2. Call price 13.29 (deltas 0.79, 0.54, 095). Put price 2.18 (deltas -0.21, -0.46, -0.05). \(\Delta_{put} = \Delta_{call}-1\).
  3. Support \([0,\infty)\). PDF is \(\lambda e^{-\lambda x}\).
• Fall 2024 Final
  1. (1) Premium paid 3.74. Profit from 173.74 to 186.26. (2) Since premium must be greater than 0 (by No Arbitrage Principle), the price for the 180 call is less than 25.22.
  2. Call price 14.13. FInal balance close to 0.
  3. The distribution is \( N(0.18h, 0.17^2h) \), so the mean is \(0.18h\) and stdev is \(0.17 \sqrt{h}\).
  4. \(dY = 0.17 Y dt + 0.4 Y dW_t\)
• Fall 2021 Midterm
  1. Profit from 52.91 to 97.09
  2. Market maker profit: 0.05 and interest plus 0.05 (up to rounding errors). Speculator's return on investment is 77.5%
  3. Time-0 value equation is \(0 = S_0 - F e^{-rT}\)
• Fall 2021 Final
  1. Identical Payoffs \(\Longrightarrow\) Identical Price (No Arbitrage Principle)
  2. (1) Option price 12.41; (3) Writer earns \(11-10.54\) and interest (up to numerical errors).
  3. \( dS = 0.06 S dt + 0.2 s dW_t \).
  4. (3) cov = 5.
• Fall 2019 Midterm
  1. Profit is always positive — contradicts "No Arbitrage". One of the options costs too little (less than its intrinsic value).
  2. \(P = 11.92\).
  3. Integrate over positive \(x\) only! The probability is \(1-e^{-\lambda}\). \(Y \sim N(\mu, \sigma^2/n)\).
• Fall 2018 Final
  1. Profit for \(S > 184.85\). Call price is between \(22.65\) and \(27.80\).
  2. Option value \(10.67\). At every node, shares and cash replicate the option value (up to rounding errors) for the next two possible nodes in the tree.
  3. Approximate value is \(5.67\).
  4. The limit is 1 for \(S>E\) and 0 for \(S < E\).
• Fall 2018 Midterm
  1. Profit in \(191.49 \lt S \lt 218.51\) limited by \(\$8.51\); loss unlimited
  2. \(2018p\) and \(2018 p(1-p)\)
  3. Fair value \(2.83\); balance after hedging \(0.17\)
  4. See Notes; Verification example: \(2.83 \geq 20 - 18\times e^{-0.1\frac3{12}} = 2.44 \).
• Fall 2018 Final
  1. Profit in \(35.40 \lt S \lt 44.60\).
  2. \(0.01Ydt - 0.2YdW\).
  3. 1. Substitute \(S_u, S_d, q\) and simplify. 2. Early exercise means \(V_u = E-S_u, V_d=E-S_d\); substitute into the formula for \(\widetilde{V}\) and show that the result is smaller than \(EE = E-S\).
  4. Fair value \(16.28\); early exercise after 2 months, at price level \(S=99\).
• Spring 2018 Midterm
  1. Made money; linear function starts from 0 at \(S_1=S_d\), achieves its maximum value at \(S_1=E\), then falls to 0 at \(S_1=S_u\).
  2. 24.25
  3. Complete the square in the exponent, the sum is \(N(0,2)\).
• Spring 2018 Final
  1. If \(E_1 \leq E_2\) then payoff of \(E_1\) call is bigger than the payoff of \(E_2\) call. By PL, the current prices have to follow the same inequality.
  2. \(d(\ln S) = 0.03 dt + 0.3 dW\).
  3. 12.41; early exercise at \(t=1\); final balance close to 0.
  4. \(\frac{\partial C}{\partial E} = -e^{-r\tau} N(d_2)\). Call price decreases as \(E\) goes up.
  5. \(V = S_0 N(d_1)\).
• Spring 2017 Midterm
  1. 3. By 1.45 (the total premium), either direction. 4. Put-call parity takes the form \(C = P + S_0 - Ee^{-rT}\). But \(E=S_0\) and both \(r\) and \(T\) are very small so \(e^{-rT}\approx0\), so \(C\approx P\).
  2. 1. \(3.56\) 2. Lower precision balance \(-0.01\) (or thereabouts). 3. Higher precision balance \(0.00\). Bonus: our price is higher than markets, therefore we assume higher volatility.
  3. \(f_Y(y) = \frac1{\sqrt{2\pi x}} e^{-x/2}\).
• Spring 2017 Final
  1. 1. Search online for "long call condor". 2. Profit around the spot price: it is a bet that the price will stay flat. 3. long 70 put, short 65 put, short 55 put, long 50 put.
  2. \(C \to \max(0, S-Ee^{-r\tau})\)
  3. 1. 21.36 2. Early exercise at \(t=2\), balance -0.11 (numerical error happens to be particularly large in this example).
• Spring 2016 Midterm
  1. 3. No, the profit is negative in all market conditions
  2. \(f_Y(y) = \frac{1}{|a|} f_X\left(\frac{y-b}{a}\right) \)
  3. 1. Payoff = \(S_T - E = S_T - S_0 e^{rT}\). 2. Consider portfolio of one stock and \(-S_0\) cash. 4. It matches the properties of a forward contract: no upfront cost and the payoff equal to stock price minus the forward price \(S_0 e^{rT}\).
  4. Price is 5.88; final balance is close to 0 (9 cents if all numbers rounded to cents).
• Spring 2016 Final
  1. decreases; buy; doesn't matter; sell; increase.
  2. 2. Price of an option is the price of volatility. So option price increases with vol.
  3. Consider 3 portfolios: empty; cash \(E\) and 1 short stock; cash \(E\) and compare them with the put payoff at time \(T\)
  4. 2. Complete the square int the exponent.
  5. Option price = 12.11; early exercise at time t=2; final balance 0.05 if rounded to cents (closer to 0 with more precision).
• Spring 2015 Midterm
  1. \(N(x) = \frac12(1+\mathrm{erf}(x/\sqrt{2})\).
  2. 1. \(\$50\) (inclusive of mark-up). 2. Total profit is \(1000\) (which is the mark-up times 100) at either node. 3. If the index ends up at \(2320\) (outside the predicted range), you will suffer a loss of \(\$200\).
  3. 1. Differentiate the PCP and use the call delta. 2. Deep in the money \(\Delta \approx -1\), deep out of the money \(\Delta \approx 0\). Deep in the money the payoff fluctuates linearly with stock price (-dollar for dollar), so we should hold the stock with the same factor. Deep out of the money the option is likely to expire without payoff and we should remove our exposure to stock fluctuations.
  4. 1. \(E_2-E_1\). 2. \((E_2-E_1)e^{-r\tau}\) 3. The guaranteed fixed payoff \(E_2-E_1\) should cost its present value to achieve. 4. Box spread combines options to create a risk-free growth of investment and should be taxed at the higher (ordinary interest) rate.
• Spring 2015 Final
  1. Part 2: since payoff in any eventuality is nonnegative, the price must be nonnegative.
  2. Substitute \(V(t,S) = W(t, S e^{r(T-t)})\) into the PDE, use (multivariable) chain rule, simplify.
  3. At expiration, the price is precisely the payoff, i.e. \(\max(S-E, 0)\).
  4. Option price: 15.15. Early exercise at price nodes \(S=95\) and \(S=104.50\).

This file was last modified on Wednesday, 10-Jan-2024 16:51:37 CST.