15.437  Options and Futures Markets

Dear all,

Q2(b): use the risk-neutral method as follows:

• k(Z0(1)+z0(2)+z0(3)) = z0(1)*(exp(r0)-1) +z0(2)*(((exp(r1u)-1)+(exp(r1d)-1))/2)+z0(3)*(((exp(r2uu)-1)+(exp(r2dd)-1)+2*(exp(r2ud)-1))/4).

Q3: to estimte the volatility in Ho-Lee tree using historical data, my suggestion is:

• Using 3m Treasury rates with frequency very 3m months.
• Suppose you got the time series r_1, r_2, ..., r_T (annualized rates). Now, take the first-order difference: r_2 - r_1, r_3 - r_2, ..., r_T - r_{T-1}.
• Calculate the standard deviation of the difference sequence, denoted as v.
• \sigma = v*2.

Q4: In order to find the implied volatility of underlying asset using BS formula, you can use the maturity 8 years. But, for the purpose of evaluating the default probability, you should use maturity 1 year.

 

Best wishes,

Winston

Dear class,

Dear all,

I provide some hints and clarifications as follows:

Q1:

• In part (2), you can still achieve perfect dynmaic hedging by selling any derivative that purely negatively depends on the underlying stock.

Q2:

• This question is about pricing American option on a binomial tree. I asssue that you guys all are familiar with pricing European option on binomial tree model (you should!). The only difference that you need to be careful is that, at each non-terminal note on the tree, the value of the option is the maximum between the early  the "present value" of the value of the next two notes either by risk-neutral pricing or dynamic replication method. Note that in European option pricing, we only use the latter without considering the maximum at each non-terminal note.
• To handle the dividend at each note, it is exactly the same as before in the European case. The exercise decision in each period is always made right before/after the dividend payment, depending call or put.
  

Q3:

• The process for V which is a discretization of CIR process cannot guarantee that V is always positive. In your simulation, if V happens to be negative, you need to get ride of the path of V.
  
• Rho <0 is crucial to for Heston model to generate smirk in implied volatility.
  

Q4:

• In (a) and (b), since you need to hold the options up to their maturity dates. The returns are just
  

                                              max(S_t - K, 0) / P_{t-1} - 1, where P_{t-1} is the option price you paid.

• In (c) and (d), since you need to short the options, you need to put down margins at broker. According to CBOE, the Margin Account Initial Requirement for the position of short uncovered puts on broad-based index is 100% of option proceeds plus 15% of underlying index value less out-of-the-money amount, if any, to a minimum of option proceeds plus 10% of the put exercise price for puts. More precisely, the margin requirement is M_t = P_t + max {15%S_0 − (S_0 − K)+, 10%K} . Then, the return is R_{t+1}= −(K − S_{t+1})+ +M_t*e^{r/12}/(M_t − P_t).
  .
  

Q5:

 

• make sure that write out the final payoff function in terms of S_T and plot it.
• In (2), the percentage sensitivity is dC/dS*S/C.
• In (2), you need to calculate betas at different S for the contract with maturities 13 monts, 7 monts, and 1month.
  

 

Best wishes,

Winston

Dear class,

Dear class,