# Tag Info

4

It is not the fact that volatility is time varying that creates the skew per se, but the fact that volatility is negatively correlated with the spot. That is to say, as the stock/index price declines volatility will tend on average to increase, and vice versa. Time varying volatility itself would create a more symmetric 'smile'. Edit: Suppose that you ...

3

Let $t=1$ and $T=2$. The value at time $t$ is given by \begin{align*} &\ e^{-r(T-t)}\max\left(E\left((S_T-K)^+\mid \mathcal{F}_{t}\right), \, E\left((K-S_T)^+\mid \mathcal{F}_{t}\right)\right) \\ =&\ e^{-r(T-t)}E\left((K-S_T)^+\mid \mathcal{F}_{t}\right) +e^{-r(T-t)}\max\left(E\left((S_T-K)\mid \mathcal{F}_{t}\right), \, 0\right)\\ =&\ ...

2

I solved it the following way, just want make sure I'm not missing something obvious. Set up a portfolio $PF$ consisting of long $S$ and short $P$ at time $t = 0$. Choose arbitrary time $0 < t < T$. If $S_t > P_t$ then $PF_t = S_t - P_t$ which coincides with the value of the option. If $S_t$ hits $P_t$ from above, then dissolve the portfolio by ...

2

Let's define $t=0$, $T_1 = 1$ and $T_2 = 2$. I believe the interviewer is looking for the price of the "global" option $V_t$ for $t \leq T_1 \leq T_2$. Let's define the payoff at time $T_1$: it is the maximum between the value of a call or a put on the same underlying with maturity at $T_2$. $$\text{Payoff}_{T_1} = \max( c_{T_1}, p_{T_1} )$$ where ...

1

The value of an option is the premium that is paid to own this option. For this paylater option, since nothing is paid upfront, the value of the option is zero. That is, \begin{align*} e^{-rT}E\big((S_{T}-K)^{+}-P1_{S_{T}>K}\big)=0, \end{align*} or \begin{align*} E\big((S_{T}-K)^{+}-P1_{S_{T}>K}\big)=0. \end{align*}

1

The option payoff is equivalent to $Z_{\tau \wedge T}-1$ where $\tau=\inf\{t | Z_t = 1\}$ provided that $Z_t$ is assumed to be continuous. Since $Z_t=S_t/P_t$ is a martingale under $Q_P$, we have $E_P[Z_{\tau \wedge T}]=Z_0$ and the option value is $P_0 (Z_0 - 1)=S_0-P_0$ regardless of the model.

1

The option payoff at maturity $T$ is defined by \begin{align*} (S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1}. \end{align*} Let $Q$ be the risk-neutral probability measure and $E$ be the corresponding expectation operator. Let $Q_p$ be a probability measure defined by \begin{align*} \frac{dQ_p}{dQ}\big|_t = \frac{P_t}{e^{rt} P_0}. ...

1

Certainly, you must agree that $$C_{T}-P_{T}=\left(S_{T}-K\right)^{+}-\left(K-S_{T}\right)^{+}=S_{T}-K.$$ Therefore, since $$C_{t}=e^{-r\left(T-t\right)}E_{Q}\left[C_{T}\right]\text{ and }P_{t}=e^{-r\left(T-t\right)}E_{Q}\left[P_{T}\right]$$ it follows by the linearity of $E$ that  C_{t}-P_{t}=e^{-r\left(T-t\right)}E_{Q}\left[C_{T}-P_{T}\mid ...

1

Yes, you are right. It appears to be a trivial typographical error in the book. I checked the formulas on Wikipedia https://en.wikipedia.org/wiki/Greeks_%28finance%29 and they agree with yours. The signs are obvious also since N(.) is between 0 and 1, i.e. non-negative. Now, about the reasoning starting with "from a logical point of view". Are you familiar ...

1

If you have many strikes of european-exercise options for two dates $T_1$ and $T_2$, then the option skew $\sigma_{1,2}(x)$ implies model-free risk-neutral probability distributions $p_1, p_2$ for each of these dates, $$p_i(x) = {\left. \frac{\partial^2 }{\partial x^2}\right|} BS_{\text{Call}}(S_0, x, \sigma_i(x), r, T_i, q)$$ ...

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