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5

For a sufficiently smooth function $f$, positive constant $a$, and $x>0$, Note that, \begin{align*} f(x) -f(a) &= \int_a^{x} f'(v) dv \\ &= \int_a^{x} \big[f'(v) -f'(a) + f'(a) \big] dv \\ &= f'(a) (x-a) + \int_a^{x}\!\! \int_a^v f''(u)du dv\\ &= f'(a) (x-a) + \int_a^{x}\!\! \int_u^{x} f''(u)dv du\\ &= f'(a) (x-a) + \int_a^{x}f''(u)(x-...


5

Yes, these are rules that the exchange uses to start trading new options as the old ones expire (if no new options are introduced trading will come to a halt...). These rules guarantee that a "reasonable number" (which is subjective, of course) of future expiration dates are in existence at all times. But these rules are not that important (not worth ...


5

The vega of an option is very dependent on the spot price. The vega of a variance or volatility swap is not.


5

[Short answer] IMHO there is a fundamental problem with wanting to extract a sound implied volatility figure out of a deep ITM option's price. You should use out-of-the-money forward options (OTMF) instead: put options for strikes smaller than the forward price (left wing of the volatility surface) and call options otherwise (right wing of the volatility ...


5

stochastic vol and Levy process models are popular. Jump diffusion less so. FT techniques are definitely used. These days most of the focus is on valuation adjustments for vanilla products rather than how to price structured products. It tends to use both MC and lattice methods. If you want to be topical, I'd advise something related to valuation ...


4

1: Follow the calculations in The Complete Guide to Option Pricing Formulas. The book has many formulas, sample values and outputs. Highly recommended for validating your results. Apparently, this is one of most popular books used by real-world quants (simple and fast). 2: You can still use QuantLib to price with year fractions. I have an example: ...


4

Clearly, from a theoretical point of view, a varswap is a better way of capturing volatility change, since as mentioned by Mark Joshi a varswap has, by construction, a Vega that does not vary with the stock price. For a single option on the other hand the Vega is at maximum at a stock price $S^*$ roughly comparable to the strike price X and decays in a "bell ...


3

Note that \begin{align*} \frac{S_T-S_t}{S_t} &= \frac{S_T-K +K-S_t}{S_t}\\ &=\frac{(S_T-K)^+-(K-S_T)^+ +K-S_t}{S_t}. \end{align*} Then, \begin{align*} E\left(\frac{S_T-S_t}{S_t} \mid \mathcal{F}_t \right) &= \frac{e^{rT}}{S_t}(C_t-P_t)+ \frac{K-S_t}{S_t}. \end{align*} where \begin{align*} C_t &= e^{-rT} E\left((S_T-K)^+ \mid \mathcal{F}_t \...


3

For $0 < T_0\le T$, consider the option with payoff, at the option maturity $T_0$, of the form \begin{align*} \max(F_{T_0, T}-K, \, 0).\tag{1} \end{align*} Note that \begin{align*} F_{T_0, T} &= F_{0, T}\exp\left(-\frac{\sigma^2}{2}\int_0^{T_0} e^{-2\lambda (T-t)} dt+\sigma \int_0^{T_0}e^{-\lambda (T-t)} dB_t\right). \end{align*} Let \begin{align*} \...


3

Although the answer of @SRKX is right on spot, I was already writing a solution along the lines of how you had specifically approached the problem. I think it might still be useful to you, so here it goes The price of the chooser option, as seen of today $t=0$ is by definition \begin{align} V_0 &= \underbrace{e^{-r T_2}}_{\text{Payoff dicount factor}}...


3

You can refer to one of my previous answers here for a detailed development. There are actually two ways you can price this: - the price of a call plus a put with adjusted strike (like above) - a put plus the price of a call with an adjusted strike (like in my answer). The only difference is whether you do $\max( a, b ) = b + ( a - b )^+$, or $\max( a, b ...


2

To elaborate on the explanation provided by @Alex, the reasoning is because when we look at the PDE we notice that the $S$ terms appear in pairs with the $\dfrac{\partial}{\partial S}$, i.e. $S\dfrac{\partial}{\partial S}$ and $S^2\dfrac{\partial^2}{\partial S^2}$. What this says it that if we were to try a polynomial function of $S$ then after applying ...


2

It's only true if the claim can be replicated by dynamically hedging with the tradeable assets. So any proof should certainly refer to that property. My proof would be: There is a dynamic portfolio that replicates the claim, i.e. which is self-financing, pre-visible, and has terminal value equal to the value of the call option The value of any portfolio, ...


2

Generally we consider this issue for every $T$-claim contingent $\Pi(t,X)$. However, there are two main approach in this context. As you mentioned, for first approach we should demand that the extended market $\Pi(.,X)\,,\,S_0\,,S_1,...,S_N$ is free of arbitrage possibilities. Indeed we demand that there should exist a martingale measure $Q$ for the ...


2

$d$ is a vector that collapses the $n$-dimensional vector into a real number. In the BS case $d=1$. There is nothing to be estimated. Also not that in practice affine pricing is done through FFT (and variants) rather than the direct transform you quote.


2

Under GBM $$ \frac {dS_t}{S_t} = \mu dt + \sigma dW_t $$ we get $$ S_T = S_0 e^{(\mu - \frac{1}{2}\sigma^2)T + \sigma W_T} $$ suggesting that $$ S_T \sim \text{ln}\mathcal {N} ( \tilde {\mu}, \tilde {\sigma}) $$ where \begin{align} \tilde {\mu} &= \ln S_0 + (\mu - \frac{1}{2}\sigma^2)T \\ \tilde {\sigma} &= \sigma \sqrt {T} \end{align} Now if $X \...


2

I would definitely recommend Volopta as a reliable source of self-contained and commented financial engineering source codes (useful for prototyping/understanding but clearly not production code). I have for instance copy-pasted, the explicit PDE solver you are looking for (centred in space, backward in time) below (+ edited for clarity + improved ...


1

If you compute the cumulative sum of the 'Shares Purchased' column you will find that in Week 9 the company owns a total of 78,700 shares. Each share is worth 53.00 (see 'Stock Price' column), so the value of the shares held in Week 9 is 78700*53 = 4,171,100. The increase in share value is 4,171,100-2,557,800 = 1,613,300 The loss in the option position is -...


1

I would tend to do the following: If, under your working modelling assumptions, there exist closed form formulas, then compare your results to them. "The Complete Guide to Option Pricing Formulas" in @Student T is indeed a nice reference for that. Beware of true formulas vs. approximations though. Now if it's not the case: Compare different pricers' ...


1

two things I would try...and this is really off the top of my head... is 1). to use put-call parity to check that your work makes financial sense. Call = Spot + Put - (strike price)/(1+risk_free_rate)^Time 2). see if you can recreate anything close to present/past market (Yahoo finance?) data prices, i.e. testing your model against reality. good luck


1

IMHO the simplest way would be to: (1) fit a probability distribution to the $T$-period returns you've historically observed. This can be done by moment-matching the sample variance/skewness/kurtosis statistics you've already computed, or using kernel density estimation (2) compute European option prices by numerically integrating the $T$-period returns pdf (...


1

For historical volatility I actually like this article: http://www.todaysgroep.nl/media/236846/measuring_historic_volatility.pdf it provides several of the better known methods for calculating historical vol, which of course could be done manually. Just being aware of the upsides and downsides of each method. As for implied vol, yes as onlyvix has said it'...


1

CSI: Expensive, but the data is not bad (quality wise) SIX Financial (former Telekurs): Middle tier price-wise, OK data CRB: Terrible customer service, but reasonable pricing CQG (don't know about their pricing)



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