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So I'm enrolled in a Mathematics for Finance course, and we received this question on the last Problem Set. I'm completely stuck on how to solve this problem. I tried applying the formula xS(t) + yA(t) = C(t), but I get stuck with figuring out A(t) - which represents the price of bonds.

Any help on this would be appreciated. I'm baffled on how to continue with this.

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Maria Sanchez
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    Typically, there is an assumption regarding the second moment of the distribution of the underlying asset that is absent here. This is often given as $u=e^{\sigma \Delta t}$. Without it, this problem hangs. – Mark Viola Sep 14 '15 at 22:26

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Hints: In a one-period market the discount factor is $1/(1+i)$ where $i$ is the risk-free interest rate. The price of the put option is the discounted expected value of the payoff, $\max(0,K-S(T))$, where the expectation is taken with respect to risk-neutral probabilities $q_u$ and $q_d$. In the risk-neutral world the expected future stock price is the forward price $S_0(1+i)$. Solve for the risk-neutral probabilities by equating the forward price with this expectation. Then go back and price the option.

RRL
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  • There is one degree of freedom here. Typically, there is assumption regarding the second moment of the distribution that is absent here. This is often given as $u=e^{\sigma \Delta t}$. Without it, this problem hangs. – Mark Viola Sep 14 '15 at 22:25
  • Dr. MV: You can use the the spot forward relationship to circumvent that and $q_d = 1 - q_u$ I would simply write the equation but OP should make an attempt. Or use the hedging argument but that takes more work. – RRL Sep 14 '15 at 22:33
  • That is still missing an equation. You have $u=1/d$, $q_d=1-q_u$, and the arbitrage free condition. That is three equations with 4 unknowns. We need one more equation. As you probably know, with these models, we need to assume a distribution. Without the second moment, the distribution is not prescribed. – Mark Viola Sep 14 '15 at 22:35
  • $S_0(1+i) = q_uuS_0 + (1-q_u)dS_0$. That means $q_u = [1+i - d]/(u-d)$. Now the probability is determined in terms of known quantities. Now option price is just the discounted expected value using this probability. – RRL Sep 14 '15 at 22:40
  • We are not given $u$ or $d$. So, we are short one equation. If you are certain that you are correct, please feel free to write the solution. But I believe that we cannot do this. – Mark Viola Sep 14 '15 at 22:43
  • If you are given $u$ -- so you are correct. – RRL Sep 14 '15 at 22:43
  • Yes, we need $u$. And often $u=e^{\sigma \Delta t}$, where $\sigma$ is given. – Mark Viola Sep 14 '15 at 22:44
  • But this is the solution in terms of $u$. – RRL Sep 14 '15 at 22:44
  • Yes. If the solution is in terms of $u$, then we you are correct! – Mark Viola Sep 14 '15 at 22:45
  • Thanks -- good catch. I assumed that $u$ was given. Option can't be priced without knowing the volatility. Without it the problem is ill-posed and it might have been an error in the problem statement. – RRL Sep 14 '15 at 22:47