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6 votes

Transform of payoff function $w_c=(\sqrt{y}-K)^+$

The fourier transform is \begin{equation} \hat{w}_c= \int_{-\infty}^\infty (\sqrt{y}-K)^+ e^{-i\phi y}dy = \int_{K^2}^\infty (\sqrt{y}-K) e^{-i\phi y}dy \end{equation} Now do a change of variable ...
Freelunch's user avatar
  • 1,086
4 votes

Architecture of a global pricing library with immutable payoffs

That's the best question that nearly no one asks. I'm with you on Quantlib and Strata, haven't really seen a very good design around but I've seen quite a few bad ones. It is definitely doable and has ...
user47501's user avatar
4 votes

What does "first-order effect" mean?

Assume you have a consumption $c$ and an asset with the payoff $x$. Cochrane states that if you add "a little bit of this asset" in your portfolio first you care about the correlation between the ...
Koval  Boris's user avatar
3 votes
Accepted

Replication of the payoff of a chooser option

Consider a European chooser option which allows you to choose at time $\tau$ if you want to receive a put option and call option with maturity $T>\tau$. At time $\tau$, using the put-call parity, ...
Kevin's user avatar
  • 16k
3 votes
Accepted

What does it mean by "A one period bond is a claim to a unit payoff." from Cochrane?

A bond repays its notional face value (plus interest sometimes), not the original purchase price. Do not assume the the price you pay for a bond is its face value. Sometimes a law or a regulation (...
Dimitri Vulis's user avatar
3 votes

What does it mean by "A one period bond is a claim to a unit payoff." from Cochrane?

That simply means that a bond pays one unit of the currency in any state (regardless what happens in the future, i.e. there is no default risk about the payoff of a bond). So you will receive 1 in ...
Kevin's user avatar
  • 16k
3 votes

Discontinuous derivative payoff approximation

We should be able to replicate the payoff exactly in each of the two regions $S_{T}\leq k_{1}$ and $S_{T}\geq k_{2}$. From the first, $$a_{0}+a_{1}S_{T}+a_{3}(k_{2}-S_{T}) =S_{T}$$ so, matching ...
Ali's user avatar
  • 131
2 votes

Explaining an Option product: SIX Discount Certificates

From @noob2's link, it looks like the product behaves like a basket of a long position in the underlying and short an out-of-the-money call option; thus the discount vs the price of the underlying is ...
Phil H's user avatar
  • 3,679
2 votes

Construct a portfolio of European call options with a certain payoff function

You can check my answer to this question for general details on how to solve this kind of problem. Let $C_X(S_T)$ and $P_Y(S_T)$ be a call and a put option with strikes $X$ and $Y$ respectively, ...
Daneel Olivaw's user avatar
2 votes
Accepted

How do we calculate option payoff before expiration?

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 ...
Kevin's user avatar
  • 16k
2 votes
Accepted

Value of the logcontract $Q^T(t,S)$ with payoff $Q(T,S)=-2lnS_T$

In the Black-Scholes model, risk-neutral the dynamics of $S_t$ are given by $$ dS_t = (r-q)S_t dt + \sigma S_t dW_t. $$ Using Itô's lemma, we can find the dynamics of the log-process $$ d\log S_t = \...
Achrbot's user avatar
  • 373
1 vote
Accepted

Understanding American option payoff at T+0

In general American options behave similar to European options but face another arbitrage boundary because of early exercise. For example in the case where the underlying price rises to $S = \\\$120$ ...
MrLCh's user avatar
  • 175
1 vote

Constructing payoff with options

This is essentially some form of commodity-linked structured note (this case being oil). In this case, such a structured is a Leveraged Participation note with Cap in industry lingo. In terms of how ...
Kai's user avatar
  • 123
1 vote
Accepted

How can I price this option?

A butterfly (option) is an option strategy with the payoff structure like below (disregard the axis labels, just take note of the structure): There are 4 ranges you mentioned in your question: $0 \...
KaiSqDist's user avatar
  • 1,454
1 vote
Accepted

Adequate model to payoff

I would not let the chosen model depend on the payoff function. For instance, consider a financial derivative where the underlying asset is a perfectly deterministic function of time. Then, your ...
rrnl's user avatar
  • 56
1 vote
Accepted

What's the difference between a normal Autocall and a Phoenix Autocall?

For "Classic Autocall" or "Athena", the coupons are indeed accumulated and paid on the event of autocall, either pre-maturity or at maturity (but for the later it will not be ...
Pierre_G's user avatar
1 vote

What is the probability of a lookback option ending in the money (CRR-model)

In a CRR binomial model, it would seem that the path-wise minimum is a function of the total number of down moves along that singular path. For example, let us fix $N=3$, resulting in $2^N=8$ ...
Kermittfrog's user avatar
  • 6,737
1 vote

Construct a portfolio of European call options with a certain payoff function

For question (i), you simply buy one EU call with strike A and sell one EU call with strike B - this is called a bull call spread. Try using the put-call-parity to construct the corresponding bull ...
AdB's user avatar
  • 714
1 vote

Architecture of a global pricing library with immutable payoffs

Payoffs can be broken down into actual cashflows driven by underlying variables (calculated from simulated assets/rates etc). On top of that there would be a list of features e.g. barriers, ...
user47501's user avatar
1 vote
Accepted

Get expected joint-payoff price of digital options from individual payoffs

Suppose you would like to compute \begin{align} Q_1(x_1,x_2;B) &= \Bbb{E}[X_1\max(B-X_2,0)]\\ Q_2(x_1,x_2;B) &= \Bbb{E}[X_2\max(X_1-B,0)] \end{align} where you know the marginal probability ...
Quantuple's user avatar
  • 14.7k
1 vote

Finding optimal drift, importance sampling, least square monte carlo

$h(\theta)=\nabla H(\theta)=\mathbb{E}\left[(\theta-Z)G^2(Z)e^{-\theta Z+\frac{1}{2}\theta^2}\right]$ so just take a bunch of paths and evaluate $$ (\theta-Z)G^2(Z)e^{-\theta Z+\frac{1}{2}\theta^2} $$...
Mark Joshi's user avatar
  • 6,983

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