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Valuing an option in a risk-neutral world is essentially saying that the risk preferences of investors do not impact option prices. That seems strange at first: given that options are risky investments, shouldn't they be affected by investor's risk preferences? The answer is no, and the reason is clear: we are valuing the option in terms of the underlying ...


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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 ...


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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 = {...


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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 ...


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It is simpler than the other Greeks, and the reason you don't hear a lot about $\rho$ is because it has smaller impact in the scheme of things. Let's say we are in the BS world, then the rho formulae for a call or put are rather simple: $\rho_{\mathrm{Call}} = K { e^{- r_{d} \tau} }\tau { N\left (d_{2} \right ) }$ $\rho_{\mathrm{Put}} = - K { e^{- r_{d}...


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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 $...


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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.)


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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 ...


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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 ...


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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-...


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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.


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Please see below: Copied from Quantitative Methods in Derivative Pricing by Tavella.


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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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Note that the tree is recombining. You have $u=1.2$ and $d=0.8$ with $ud=0.96$. Your tree for the asset price reads as At time zero: 100 At time one: 80 or 120 At time two: 64 or 96 or 144 The transition probabilities are $q_u=0.55$ and $q_d=0.45$. For your put option with strike price $K=104$, you thus obtain by backward induction At time two: 40 or 8 ...


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The PDE, in its original form, has got variable coefficients -depend on S and t - e.g., co-efficient of $\frac{\partial^2 V}{\partial S^2}$ has got $\sigma(t)$ and S. They are hard to solve and analyse, and if one can find some transformations of variables, that reduce it to constant coefficient, then it gets a lot easier. P Wilmott has chosen the three ...


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The time value of an option, in a sense, captures all the stochastic influence on your option. Think about a call option with payoff $\max\{S_T-K,0\}$. The intrinsic value, which is just the payoff if you exercised the option immediately, depends on $S_0$ and $K$. In the Black Scholes model there are further variables such as the interest rate, dividend ...


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