Exercise :

Let $K>0$. A European Cash-or-Nothing put option $P$ has the following pay-out profile : $$P=K^2\cdot \mathbf{1}_{\{S_T < K\}}$$ Let $P_0^n$ be the no-arbitrage value at time $0$ and the $n$-th binomial model. Calculate the limit $\lim_{n \to \infty} P_0^n$. You can use fact that the random variable $S_T^n$ follows the log-normal distribution when $n \to \infty$ and that $\lim_{n \to \infty} \mathbb{E}_\mathbb{Q}[\mathbf{1}_{\{S_T < K\}}] = \Phi(-d_2)$.

Note : For the binomial models, it is : $$S_T^n = \prod_{i=1}^n(1+R_i^n)$$

Question :

Now, I know how one shows what the limit of $P_0^n$ is when simply $P=K$, but I am really struggling to manipulate the case of this special given $P$. Any help will be greatly appreciated.

  • $\begingroup$ I think a good starting point would be to use the Monotone Convergence Theorem en.wikipedia.org/wiki/Monotone_convergence_theorem $\endgroup$ – FunnyBuzer Feb 19 '19 at 12:54
  • $\begingroup$ @FunnyBuzer Hi. I don't know how to use the $P$ given. It's not standard as when we say it pays off $K$. Can you help me with the initial expression of $P_0^n$ ? $\endgroup$ – Rebellos Feb 19 '19 at 12:58
  • $\begingroup$ Not sure I understand you comment. You do know how to solve it when the payoff is $K \mathrm{1} \left\{ S_T < K \right\}$ and struggle with it being $K^2$? $\endgroup$ – LocalVolatility Feb 19 '19 at 17:09
  • $\begingroup$ @LocalVolatility Yes. We proved the limit of $P_0^n$ at class when the payoff is simply $K$ by using the expected martingale value of $(K-S_T)^+$ and expanding using the Triangular Central Limit Theorem, followed by the definition integrals and leading to the expressions involving $d_1$ and $d_2$. But, how would $K^2$ change that ? Would it just be the same but with $(K^2-S_T)^+$ ? $\endgroup$ – Rebellos Feb 19 '19 at 17:11
  • 1
    $\begingroup$ I think you are overcomplicating things. An option that pays $K^2 \mathrm{1} \left\{ S_T < K \right\}$ has expected payoff which is just $K$ times that of an option that pays $K \mathrm{1} \left\{ S_T < K \right\}$. It is just a constant that you can take out of the expectation. Also, I don't see why your starting point is the call payoff as opposed to just the payoff the product you are interested in. $\endgroup$ – LocalVolatility Feb 19 '19 at 17:14

I’m not sure I understand your question. In the limit as n goes to infinity, the binomial price approaches the continuous price. Your answer is simply k * N(-d2) where N(.) is the CDF of a standard normal with parameters given by a standard put option in BS framework


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.