5

Let's take a step back to look at what implied volatility (IV) really is. If we know the price of a call option, the interest rate (we can use the spot rate corresponding the option maturity) then Implied volatility is that level of volatility that will result in the option price when putting into the Black-Scholes formula for a call option value. If we ...


5

There are several derivations and interpretations for $N(d_1)$ and $N(d_2)$. As you know, \begin{align*} C(t,S_t)=S_te^{-q(T-t)}N(d_1) -Ke^{-r(T-t)}N(d_2). \end{align*} We can also show that \begin{align*} \mathbb{Q}_S[\{S_T\geq K\}]&=e^{-q(T-t)}N(d_1), \\ \mathbb{Q}[\{S_T\geq K\}] &=e^{-r(T-t)}N(d_2). \end{align*} Thus, $N(d_i)$ may be seen as ...


3

There are two ways to look at it, a mathematical way or an alternative, intuitive way. The alternative way can be to look at F as an alternative S with 0 interest rate discounting because we still have the cash (minus a small posted margin, and ignoring this) which earns the interest rate. So for the F’s value itself every day’s time value of money effect ...


3

To make this clear. The article considers a bond that is paying a coupon infinitely, so there is no expiration. In this case, the value of the bond is the sum of the discounted coupons $\frac{100}{(1+r)}+\frac{100}{(1+r)^2}+\frac{100}{(1+r)^3}+\dots$ which eqals $\frac{100}{r}$. So in case of $r=0.10$ the bond price is $\frac{100}{0.1}=1000$ and in case of $...


2

Your question is answered by the no-arbitrage principle. The payoff of your put option is $\max\{K-S_T,0\}\leq K$. Thus, their time $t$ prices need to have the same relationship (everthing else creates arbitrage opportunities), i.e. $$ P_t(K,T)\leq KZ_t(T).$$ This statement is kind of related to the law of one price which is implied by assuming an arbitrage-...


2

I can confirm there is no error in @Sanjay graph. I obtain the same plot with Obloj correction for the SABR formula. In fact, the popular SABR approximation formulas (Hagan or the further corrections) use as hypothesis a small vol of vol. In your case, the vol of vol $\nu$ is very large ($\nu=7$) and it is not too surprising that the approximations break ...


2

Typically structures like this are traded as notes. They will be sold at a face value of 100%, where that is normally the combination of a zcb (ie 1y usd, say 97.5%), expected coupon (say +10%), short Knock In put (also knocked out by the autocall feature, say -8%), and some profit for the issuer (in this case, 100%-97.5%-10%+8%=0.5%). Sometimes these are ...


1

These papers study delta-hedging of equity options with different models. @Article{, author = {Gurdip Bakshi and Charles Cao and Zhiwu Chen}, title = {Empirical Performance of Alternative Option Pricing Models}, journal = {Journal of Finance}, year = 1997, volume = 52, number = 5, pages = {2003--2049}, } @Article{, author = {...


1

Answer was provided to me in the comments so I may as well close the question non. My computation is right, and the Investopedia article is not saying what I thought it was. (What the article really says is still unclear.)


1

Keeping it simple, your payoff at time t is: $S_t-100$ The present value of the stock is $S_0$, it’s current price; and the present value of 100 is its discounted value as you correctly explained in your question.


1

Somewhere must be a little error, here I used $r=0.02$ and $\sigma=0.25$. In black you have the payoff and in red the current price of the portfolio. Note that as the time to maturity decreases, the red line converges towards the black line. In grey and and yellow (on the secondary axis), you can see the individual call option prices which form your ...


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