Highest voted questions tagged payoff - Quantitative Finance Stack Exchange most recent 30 from quant.stackexchange.com 2019-10-18T05:57:54Z https://quant.stackexchange.com/feeds/tag?tagnames=payoff&sort=votes https://creativecommons.org/licenses/by-sa/4.0/rdf https://quant.stackexchange.com/q/45664 2 Discontinuous derivative payoff approximation harve https://quant.stackexchange.com/users/41082 2019-05-17T15:27:09Z 2019-05-18T08:52:18Z <p>Consider a derivative of digital type which pays this kind of payoff at time <span class="math-container">$T$</span>: <span class="math-container">\begin{align*} g(S_T,k) &amp;= \begin{cases} P_0,~S_T&gt;k \\ S_T, ~S_T\leq k \end{cases} \end{align*}</span></p> <p>with <span class="math-container">$S_T$</span> being the current price of the underlying at maturity time <span class="math-container">$T$</span>, <span class="math-container">$P_0$</span> the price of the underlying at the issue time 0 and <span class="math-container">$k$</span> - kind of the strike price with barrier feature. </p> <p>Apparently, function <span class="math-container">$g$</span> is discontinuous at <span class="math-container">$S_T=k$</span> and has a jump there. The idea is to approximate it with a set options, call <span class="math-container">$c(S_T,k_1)$</span> and put <span class="math-container">$p(S_T,k_2)$</span> that have strikes: <span class="math-container">$k_1 &lt; k &lt; k_2$</span>. Then, to construct a linear piece-wise function that will look as following: <span class="math-container">$$\hat g(S_T,k_1,k_2)=a_0+a_1 S_T+a_2 c(S_T,k_1) + a_3 p(S_T,k_2).$$</span></p> <p>The question is how to get the coefficients. Which complementary equations may be used?</p> https://quant.stackexchange.com/q/39853 2 Explaining an Option product: SIX Discount Certificates chocolatekeyboard https://quant.stackexchange.com/users/33988 2018-05-16T11:23:23Z 2018-05-20T23:54:34Z <p><a href="https://i.stack.imgur.com/k8kAG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/k8kAG.png" alt="Option "></a></p> <p>So I have the option with the important info above. I am trying to generate a portfolio that represents the option. </p> <p>However I am stuck on the first hurdle as I believe it is a call option as the product makes a profit only if the underlying share price rises above 121.35 to 145.50 and it is capped at the 145.50 price.</p> <p>I believe the exercise price to be the 145.50</p> <p>The maturity also seems confusing as I believe it to be the 28th November, starting on the 3rd, which is 211 working days which seems a bit random.</p> <p>Is the issue price of 115.78 the cost of the option? as this seems quite high.</p> <p>I think this means that if the share price falls or doesn't hit the capped level then at expiry it is worthless and will simply provide the share price at the time.</p> <p>Basically is what I have said correct? Thanks.</p> https://quant.stackexchange.com/q/17622 2 "For any random variable $X$, someone will be willing to buy and someone to sell a financial instrument, whose final payoff is $X$." Evan Aad https://quant.stackexchange.com/users/15619 2015-05-02T16:46:00Z 2016-11-04T14:10:02Z <blockquote> <p>we will assume that for any random variable $X:\Omega\rightarrow\mathbb{R}$, some investor will be willing to buy and some investor will be willing to sell a 'financial instrument' whose final payoff is $X$. (Actually, this is one of the few assumptions about the market that we have made that is actually plausible.)</p> </blockquote> <p>This quote is taken from Steven Roman's "Introduction to the Mathematics of Finance Arbitrage and Option Pricing", 2nd edition, Springer 2012.</p> <p>Why is this a plausible assumption? (The market model under discussion in this part of Roman's book is a finite model (finite time, finite probability space, finite number of assets) with no arbitrage opportunity.)</p> https://quant.stackexchange.com/q/44450 2 Architecture of a global pricing library with immutable payoffs ujsgeyrr1f0d0d0r0h1h0j0j_juj https://quant.stackexchange.com/users/13470 2019-03-06T08:54:41Z 2019-03-06T16:03:03Z <p>By global pricing library I mean a library</p> <ul> <li>handling equity, rate etc, hybrid products</li> <li>having several models (BS, LV, SV, LSV)</li> <li>having several numerical methods (analytic formula, MC, PDE FD/FE)</li> </ul> <p>I never had to design a global pricing library, only had to write isolated MC or PDE FD pricing libraries, with BS, LV and SV mainly, in a purely front office setting, so I was quite free for the modelling and designing. In these cases I always used the following architecture (in the case of a toy MC) :</p> <ul> <li>a <code>Product</code> has a (reference to) a <code>PayOff</code></li> <li>a <code>PayOff</code> has a <code>Model</code> and a <code>ComputePayOff</code>method that computes the payoff on a path generated by the model</li> <li>a <code>Model</code> has a <code>RandomNumberGenerator</code> and a <code>GenerateMCPath</code> method that generates an MC path given dates with the given random generator</li> </ul> <p><code>PayOff</code> is abstract, as well as <code>Model</code> and <code>RandomNumberGenerator</code>, even if I always had issues with avoiding exponential increase of subclasses due to transverse functionality, as I am not a design pattern (bridge ?) expert.</p> <p>So that <code>PayOff</code> has a lot of "non-immutable" information. For instance if my <code>RandomNumberGenerator</code> is a Sobol, it may have a member that changes after generating random number, so that after a pricing, PayOff has a information that has changed. I never cared about that.</p> <p>Now, I have the task of laying out a poc for a global pricing library, with the constraint that <code>Product</code> and <code>PayOff</code> must not change (they going to be (de)serialized). I could of course, with a lot of contorsions, continue to do as in the previous toy-example, but it would be wrong.</p> <p>Still, after thinking, some things do not change : I indeed want to have three categories of "objects" :</p> <ul> <li>products (or payoffs to make it simple)</li> <li>models</li> <li>numerical methods</li> </ul> <p>and these categories may intersect, for instance :</p> <ul> <li>the intersection of european payoffs, BS model and closed formulae (a special kind of numerical method) yields the BS formula</li> <li>the intersection of european payoffs and Heston model yields as numerical methods either closed formulas, PDE FD2D or MC</li> </ul> <p>etc. In fact, the library needs to process a given payoff under a given model, using a given numerical method, keeping track of the fact that it cannot price everything in any model and with any numerical method ...</p> <p>Is there a classic way to design this ? As I do not intend to necessarily reinvent the wheel, I looked at QuantLib and Strata so far, but they both have "non-immutable" "payoffs".</p> https://quant.stackexchange.com/q/43937 1 Transform of payoff function $w_c=(\sqrt{y}-K)^+$ [closed] Michael https://quant.stackexchange.com/users/38693 2019-02-06T22:53:20Z 2019-02-07T13:02:19Z <p>I am working on a project where I price EU call options written on the VIX index.</p> <p>The payoff function of interest looks like </p> <p><span class="math-container">$w_c=(\sqrt{y}-K)^+$</span></p> <p>where K is the strike price and y is the value of <span class="math-container">$VIX^2$</span></p> <p>The Fourier transform of this function takes the form:</p> <p><span class="math-container">$\hat{w_c}=\frac{\sqrt{\pi}}{2}\frac{1-\text{erf}(K\sqrt{-\phi})}{(\sqrt{-\phi})^3}$</span></p> <p>Here <span class="math-container">$\phi$</span> is the transform variable.</p> <p>I would like to know more about how to get from <span class="math-container">$w_c$</span> to <span class="math-container">$\hat{w_c}$</span>.</p> <p>I have tried to look around in different transform tables, but with no luck.</p> <p>Any help is much appreciated. Thanks. </p> https://quant.stackexchange.com/q/27828 1 Finding optimal drift, importance sampling, least square monte carlo Elekko https://quant.stackexchange.com/users/18211 2016-06-28T00:06:26Z 2016-09-26T11:12:20Z <p>I am working with Importance sampling for Least Squared monte carlo and have now problems understanding the implementation of the Robbins-Monro algorithm for finding the <em>optimal drift</em> for finding minimum variance of my estimate. The original problem formulation that is now answered is given <a href="https://quant.stackexchange.com/questions/27600/importance-sampling-for-pricing-options-with-longstaff-and-schwartz/">here</a>. </p> <p>The article I am following for Robbins-Monro algorithm is this <a href="https://quanto.inria.fr/pdf_html/Arouna_doc/" rel="nofollow noreferrer">link</a></p> <p>The problem i want to solve is to find a optimal drift $\theta^*$ by solving:</p> <p>$H(\theta^*)=\min_{\theta}H(\theta)$</p> <p>Where $H(\theta)=\mathbb{E}\left[ G^2(Z)e^{-\theta Z+\frac{1}{2}\theta^2}\right]$, the second moment of the payoff function $G(Z)=\max(K-S(t),0)$. Indeed, we have: $\nabla H(\theta)=0$</p> <p>Now following the Morris monro algorithm in the link, the general formulation of the stochastic algorithm is given in equation (10) and is given by:</p> <p>$X_{n+1}=X_n-\gamma_{n+1}F(X_n,Z_{n+1})$</p> <p>and going further to equation (15) we have the second moment (the gradient of $H(\theta)$) given by:</p> <p>$h(\theta)=\nabla H(\theta)=\mathbb{E}\left[(\theta-Z)G^2(Z)e^{-\theta Z+\frac{1}{2}\theta^2}\right]$.</p> <p>Now I wonder, since I don't know the second moment, how should I approximate it numerically in order to evaluate the algorithm? Given in the article, they don't really explain how the second moment is found?</p> <p>Appreciate for help. Thank you!</p> https://quant.stackexchange.com/q/45783 1 Wheres is this method/notation of option portfolio payoff design from? jthg https://quant.stackexchange.com/users/32202 2019-05-24T22:06:01Z 2019-05-24T23:01:07Z <p>The "desired position" in the image is a set of slopes <span class="math-container">$(0,1,-1,0)$</span>, and a set of strike prices between these slopes <span class="math-container">$\mathbf{K}=(98,100,102)$</span>.</p> <p>The payoff is then designed by finding the positions <span class="math-container">$n_1,n_2,n_3$</span> in three call options </p> <p><span class="math-container">$$c_1=(0,1,1,1)\mathbf{K}$$</span> <span class="math-container">$$c_2=(0,0,1,1)\mathbf{K}$$</span> <span class="math-container">$$c_3=(0,0,0,1)\mathbf{K}$$</span></p> <p>So that they total the desired payoff <span class="math-container">$(0,1,-1,0)$</span>. </p> <p>In this case of a butterfly spread, the required postions are <span class="math-container">$n_1=1$</span> <span class="math-container">$n_2=-2$</span> and <span class="math-container">$n_3=1$</span>.</p> <p>Any points to litterature where this method is used is appreciated. <a href="https://i.stack.imgur.com/ZaKgV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZaKgV.png" alt="Designing a butterfly spread"></a></p> https://quant.stackexchange.com/q/25939 1 Differentiating a Payoff ThePlowKing https://quant.stackexchange.com/users/20543 2016-05-11T02:08:44Z 2016-05-11T04:15:42Z <p>Okay this is probably going to be an extremely easy/straightforward question but I thought I should post it here just to double check. Suppose I have a payoff $\Phi = (S_{T}-K)^{+}$. Now let's say I now have an equation: $u = s\partial_{s}\Phi - \Phi$, this means that given a payoff $\Phi$ as given above then, substituting this payoff into the equation and assuming $S_T = S_{0}\exp((r-1/2)T+\sigma\sqrt{T}Z_i))$ then I should get: </p> <p>$u = max(S_{T},0) - max(S_{T}-K,0)$, right?</p> <p>And from this equation, the possible solutions should be:</p> <p>If $S_{T} &gt; K$, $u = K$, if $S_{T} &lt; K$ and $S_{T} &gt; 0$, $u = S_{T}$, and if $S_{T} &lt; K$ and $S_{T} &lt; 0$ then $u = 0$.</p> <p>Is all of this correct? I know this is really trivial but I just thought I should check...</p> https://quant.stackexchange.com/q/9758 0 How is holding an European call option equivalent to holding an asset-or-nothing call option and writing a cash-or-nothing call option? user35777 https://quant.stackexchange.com/users/6792 2013-12-19T13:26:27Z 2013-12-30T09:14:48Z <p>The cash-or-nothing call option has a payoff that is equal to the strike price. All three options have the same expiry date.</p> https://quant.stackexchange.com/q/45002 0 Construct a portfolio of European call options with a certain payoff function ʎpoqou https://quant.stackexchange.com/users/38424 2019-04-09T08:40:20Z 2019-04-09T09:44:39Z <p>My question is similar to <a href="https://quant.stackexchange.com/questions/37419/replicate-a-portfolio-with-given-payoff">Replicate a Portfolio with Given Payoff</a> but I am not quite sure how to apply this to my problem. </p> <p>A portfolio of European call options on an asset <span class="math-container">$S_T$</span> has a payoff function given by <span class="math-container">$V_T$</span> where: <span class="math-container">$$V_T = 0,\ S_T &lt; A$$</span> <span class="math-container">$$V_T = S_T - A , \ A \leq S_T \leq B$$</span> <span class="math-container">$$V_T = B - A, \ S_T &gt; B$$</span></p> <blockquote> <p>(i) Construct a portfolio <span class="math-container">$H_1$</span> of European call options with this payoff function.</p> <p>(ii) Use Put-Call parity to construct a portfolio <span class="math-container">$H_2$</span> of European put options with this payoff function.</p> </blockquote> https://quant.stackexchange.com/q/46783 0 How do we calculate option payoff before expiration? Eka https://quant.stackexchange.com/users/16754 2019-07-24T10:55:10Z 2019-07-24T13:59:07Z <p>I am trying to simulate a bull spread option <a href="https://i.stack.imgur.com/xzMcw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xzMcw.png" alt="enter image description here"></a></p> <p>and I have used an online tutorial to calculate <code>payoff at expiry</code> but I am having difficulty simulating the payoff before expiration.</p> <p>What I have done so far,</p> <pre><code># payoff for long call long call premium = bs_model() long call payoff = max(spot-strike,0)-long call premium # payoff for short call short call premium = bs_model() short call payoff = -1*(max(spot-strike,0)- short call premium) # Theoretical P&amp;L theoretical p&amp;l= long call payoff + short call payoff * bs_model = Black Scholes Model </code></pre> <p>This theoretical P&amp;L I plotted to a graph but instead of getting the smooth sigmoidal curve like the image above I getting a weird graph? </p> <p>Edit:</p> <p>The above calculations are my own guess work of calculating theoretical P&amp;L. Can any one share a good link which explains the calculation of theoretical payoff before expiry? I searched all the web and cant find any?</p> https://quant.stackexchange.com/q/30395 0 Get expected joint-payoff price of digital options from individual payoffs stochastic_zeitgeist https://quant.stackexchange.com/users/24661 2016-10-01T11:10:39Z 2016-10-03T12:43:22Z <p>I am trying to model a joint distribution $f(X_1,X_2)$ </p> <p>(where $X_1$ and $X_2$ are market prices of the options) and then find from it the value of joint payoff price:</p> <p>$F(X_1, X_2; B_1, B_2) = E[ max(X_1-B_1,0) * max(B_2 -X_2,0)]$ </p> <p>where $B_1$ and $B_2$ are corresponding strike prices.</p> <p>I have limited samples of individual payoffs $Q_1$ and $Q_2$ for different values of $B_1$ and $B_2$ </p> <p>$Q_1(X_1,X_2;B2) = E[X_1 * max(B_2-X_2,0)]$ </p> <p>$Q_2(X_1,X_2;B1) = E[X_2 * max(X_1-B_1,0)]$</p> <p>$C(X;k) = E[max(0,X_T - k)]$ </p> <p>Is there a way to solve this with or without assuming that $X_1$ and $X_2$ are independent.</p> <p>I am new to modelling option-prices.</p> https://quant.stackexchange.com/q/9931 0 Expected payoff and weighted average price Lisa Ann https://quant.stackexchange.com/users/3058 2014-01-12T09:27:07Z 2014-01-13T20:54:59Z <p><strong>Settings</strong></p> <p>Let you're trading a security whose probability to be equal to $S_{T}$ at time $T$ follows a p.d.f. like the ones in the picture below.</p> <p><img src="https://i.stack.imgur.com/vEvRD.gif" alt="P.d.f. examples"></p> <p>(That is just an example found with Google images, assume you're considering just one of the expiry dates shown above, e.g. $T=60$).</p> <p>You are trading with a capital equal to $M$ dollars; your main goal is to maximise your expected payoff at time $T$, and you have some constraints to achieve this result:</p> <ul> <li>you cannot go short, this is long only;</li> <li>your capital, $M$, must be split in a number of parts which is equal or less than $k$, that is, you cannot average your position more than $k-1$ times in addition to the first entry. This means that your position average price will be the weighted average of a maximum of $k$ security prices;</li> <li>you cannot borrow money, then no leverage involved;</li> <li>at least one entry must be made.</li> </ul> <p>Below an instance in which using the whole capital, $M$, at the first trade was the best choice possible:</p> <p><img src="https://i.stack.imgur.com/jj9TP.gif" alt="An example of what the price could do in the future"></p> <p>From such a description my guess is that what a trader should try to do is to take advantage of the security's volatility: if my average price at $t=T$ is below $S_{T}$ I am earning moneys, then I am involved in a trade off between a greater but less likely profit and a smaller but more likely profit.</p> <p><strong>Possible objection</strong></p> <blockquote> <p>Hey, mate, if $E[S_{T}]$ (expected value) is above $S_{0}$ you've just to buy it now and keep it until $T$, isn'it?</p> </blockquote> <p>I don't think so. Consider the following security's path:</p> <p><img src="https://i.stack.imgur.com/qpOLw.jpg" alt="Google"></p> <p>It's easy to see that if the trader does not use the whole capital, $M$, buying the stock at $t=0$, but instead he uses the remainder to buy the valley (about June), his performance will be better. Yes, he's also running the risk to buy surging prices, as well...</p> <p><strong>Question</strong></p> <ol> <li>What the analytical formula of the expected payoff would be? I guess it <strong>cannot</strong> be the sum of probability-weighted returns due to overlapping events;</li> <li>once the aforementioned payoff formula has been discovered, how to maximise your expected payoff varying the $k$ entry prices along the time which goes from $t=0$ to $t=T$?</li> <li>If an analytical solution did not exist, what a possible simulative approach would be? Could some Monte Carlo simulations help?</li> </ol> <p><strong>Hint</strong></p> <p>Do not let a "classic" p.d.f. trick you: what if the security p.d.f. was something like this...</p> <p><img src="https://i.stack.imgur.com/ynyd9.png" alt="enter image description here"></p> https://quant.stackexchange.com/q/39147 0 Transform the payoff to be non-zero Sam Palmer https://quant.stackexchange.com/users/11762 2018-04-05T16:22:06Z 2018-04-05T17:55:41Z <p>Is there any way to transform the basic call option payoff $V(s,0) = \max(s-K,0)$ such that $g(V(s,0))\neq 0$ $\forall s$, where $g()$ is the transform function of the payoff. This is to use in a numerical method where the form $V(s,t) = f(s,t)g(V(s,0))$ is assumed, where $f(s,t)$ is to be approximated, hence $g(V(s,0)) \neq 0$ otherwise it kills off $f(s,t)$ for all OTM options. </p> https://quant.stackexchange.com/q/43490 0 Seagull Spread payoffs user403033 https://quant.stackexchange.com/users/38348 2019-01-14T17:45:07Z 2019-01-14T18:12:26Z <p>I'm looking at different option strategies and the ways that their payoffs differ (and therefore how they can differently be used).</p> <p>I'm looking at the long seagull (buy a call spread and sell a put), and wondering if taking the opposite positions in these would provide an unlimited payoff with decreasing strike and a limited loss with increasing strike?</p> <p>As an example: </p> <ul> <li>Buy a put with strike 1.2</li> <li>Sell a call with strike 1.3</li> <li>Buy a call with strike 1.4</li> </ul> <p>Should the two calls not cancel once the strike hits 1.4 and therefore this is your maximum loss? Whilst a strike 1.2 and below will result in a profit (ignoring premiums)?</p> https://quant.stackexchange.com/q/28203 -1 Swaption Corridor Payoff Diagram jake_r https://quant.stackexchange.com/users/20465 2016-07-21T16:23:50Z 2017-05-16T04:52:41Z <p>What does the payoff diagram look like for a long payer swaption corridor? </p> <p>For example, suppose that I am looking at a long-payer $1 \times 10$-year swaption with 10Y swaps as the underlying. If I am buying a 2.0% strike and selling a 2.5% strike, I'm trying to plot the payoffs at various future potential 10Y swap rates in one year (e.g. 1.5%, 2.0%, 2.5%, 3.0%, $...$). I haven't found a good example online (and am having trouble calculating in excel) and am concerned that the convexity of the bonds will make the payoff nonlinear as the 10Y market swap rate in one year increases above my high strike (unlike IR caps). If it is nonlinear, is there a quick, intuitive explanation?</p> <p>Any thoughts/guidance would be appreciated. Thanks.</p> https://quant.stackexchange.com/q/14671 -1 conservative approach payoff table user12008 https://quant.stackexchange.com/users/12008 2014-09-08T14:47:49Z 2015-01-07T17:19:08Z <p>With the conservative approach, we choose the decision which maximises minimum payoff. I was wondering which decision is chosen if 2 decisions have equal minimum payoff (which is the maximum)? </p> <p>Thanks</p>