Active questions tagged delta-hedging - Quantitative Finance Stack Exchange most recent 30 from quant.stackexchange.com 2019-08-23T12:17:51Z https://quant.stackexchange.com/feeds/tag/delta-hedging http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://quant.stackexchange.com/q/41842 3 Expected value of delta-hedged portfolio Paul https://quant.stackexchange.com/users/35882 2018-09-21T20:27:26Z 2019-08-21T17:13:49Z <p>Consider portfolio in black-scholes world </p> <p><span class="math-container">$\Pi = \Delta S - V$</span>, where <span class="math-container">$S$</span> is the stock price and V is the price of the option.</p> <p>I have read that if we set <span class="math-container">$\Delta = \frac{\partial V}{\partial S}$</span> then we obtain <span class="math-container">$d\Pi = (...)dt + 0 * dW$</span>, where <span class="math-container">$W$</span> is brownian motion. And by no-arbitrage we have <span class="math-container">$d\Pi = r \Pi dt$</span>, where is risk-free interest rate, so that <span class="math-container">$\Pi_T = (\Delta_0S_0 - V)\exp(rT)$</span>.</p> <p>I came across some lecture notes, that claim that if <span class="math-container">$\Pi = \Delta S - V$</span> is <span class="math-container">$\Delta$</span>-hedged then value of such portfolio is <span class="math-container">$0$</span> at time of expiration of the option <span class="math-container">$T$</span>.</p> <p>But I would be expecting such a portfolio to have a value of <span class="math-container">$\Pi_T = (\Delta_0S_0 - V)\exp(rT)$</span>, could someone help to figure out what is going on?</p> <p>Thank you </p> https://quant.stackexchange.com/q/47148 1 FX Spot Delta market standard calculation (Trader View) NewNY1990 https://quant.stackexchange.com/users/31477 2019-08-16T16:39:29Z 2019-08-16T16:39:29Z <p>I am just writing my thesis about FX instrument and hedging and one question popped up which I can't solve. Maybe it is silly but cant find anything about it how the delta of a fx spot is defined and I want to hedge it with an option in USD Deltas. The delta of an option is easy just the first derivative of the Garman-Kohlhagen option pricing formula. </p> <p>I have a GBP/USD FX-SPot trade with T+2 settlement period and the deal is made today on the 8/13/2019. The spot date would be 8/15/2019 (physical exchange). I have the folling parameters:</p> <p><span class="math-container">$$\Delta_{USD_{T+2}} \approx Notional_{GBP} * pips$$</span></p> <p>The question is how can I discount the delta to be the value of today. </p> <p>How would I now discount the delta to today in terms of T+2 to T? </p> <p>I would use the instantaneous fx spot rate: <span class="math-container">$$FX_{instantaneous_{GBP/USD}} = FX_{Spot}-(ON+TN)$$</span> but how can I use it in the approximation above? </p> <p>If I would look into a USD/CHF FX-Spot trade the delta would look like:</p> <p><span class="math-container">$$\Delta_{USD_{T}} \approx Notional_{USD} * pips = \Delta_{CHF}/FX_{instantaneous_{USD/CHF}}$$</span></p> <p>So my two questions:</p> <ol> <li>How can I discount the USD <span class="math-container">$$\Delta$$</span> for GBP/USD FX-Spot?</li> <li>Does the approximation makes sense for USD/CHF?</li> </ol> <p>If not what approach should I use? </p> https://quant.stackexchange.com/q/47023 0 In literature, is IV constantly adjusted during option delta hedging? confused https://quant.stackexchange.com/users/31372 2019-08-08T19:58:47Z 2019-08-09T05:21:36Z <p>In a lot of literature, they like to compare the performance of buying an option, and then delta hedging either at that options implied volatility (IV) or the true future volatility. This is under the BSM framework.</p> <p>My question is for the former case. Let's assume the option IV is 20% and the true future volatility is 30%. We buy the option at 20% IV and delta hedge using 20% IV. As we move forward in time, if the option that we bought has an IV that changes, let's say now it is at 25% IV. Do we now delta hedge at 25% IV or do we continue to hedge at 20% IV? The literature never makes this part clear.</p> <p>Thanks!</p> https://quant.stackexchange.com/q/40430 1 Heging against stochastic interest rate benSlash https://quant.stackexchange.com/users/34401 2018-06-22T09:56:22Z 2019-08-05T18:53:58Z <p>I am working on an Index and I am trying to price Call options on it. I work with the 3 Months LIBOR as Cash.</p> <p>I use the following Black-Scholes formula $$C_{t} = S_{t}e^{-q_{t}(T-t)}\mbox{N}[d_{1}(t)] - K e^{-r_{t}(T-t)}\mbox{N}[d_{0}(t)]$$ with the usual notations. $r_{t}$ is the LIBOR rate and $q_{t}$ is the dividend of my Index. I cannot use the classical formula $R(t,T) = \frac{1}{T-t} \int_{t}^{T} r_{s} ds$ since the rate are not deterministic. </p> <p>I implemented a classical delta-hedging. My delta-hedging works well, except for too deep In the money call options, where the P&amp;L behaves exactly like the interest rate.</p> <p><a href="https://i.stack.imgur.com/pPlKE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pPlKE.png" alt="enter image description here"></a> This happens only for Call with maturities of several years and strikes far in the money. The conclusion of my manager is that I have to hedge against the stochasticity of the interest rate, using a zero-coupon bond.</p> <p>I looked at some documentations on zero-coupon, and saw that we can have a closed formula under the Hull-White model. However, I am not so sure how to calibrate it as my only inputs are the LIBOR rates.</p> <p>Besides, I do not know which amount I should invest in my zero-coupon bond : should I invest the $\rho$ ( $\frac{\partial C_{t}}{\partial r_{t}}$) ?</p> <p>I do not really know where to start so any help would be appreciated :)</p> <p>Thank you!</p> https://quant.stackexchange.com/q/23267 11 Delta of binary option user11128 https://quant.stackexchange.com/users/19439 2016-02-12T22:54:08Z 2019-07-29T20:39:35Z <p>What is the Delta of an at-the-money binary option with a payo out $0$ at $&lt;100$ dollars, and payout $1$ at $&gt;100$ dollars, as it approaches expiry?</p> <p>This is from a sample interview exam. I understand that Delta essentially measures the change in the derivative price relative to the change in the asset price, as trading on the open market. </p> <p>How do I actually go about computing Delta for a particular situation like the one above? I've been unable to find a formula for it on Google which is a bit weird? My naive guess is that the answer should be 0.5 but I'm not sure why?</p> https://quant.stackexchange.com/q/46722 0 Market Maker ETF Hedging Strategy Leopardl https://quant.stackexchange.com/users/41843 2019-07-21T07:18:55Z 2019-07-21T14:09:44Z <p>Some thoughts about ETF hedging; feel free to leave comments!</p> <p>Scenario 1:</p> <p>An investor sells 1M ETF shares to a Market Maker(MM) at bid price. MM has a long position and will need to offload the shares bit by bit. How does MM hedge its position prior to the long position? My guess will be using option - MM has a positive delta and therefore needs a put option to bring down the +ve delta.</p> <p>Scenario 2:</p> <p>An investor wants to buy 1M ETF shares from MM. Is there any hedging strategy involved in this case? If so how? </p> https://quant.stackexchange.com/q/46432 1 Why does it make sense to delta hedge a deep OTM option given the very low delta exposure? [closed] charm93 https://quant.stackexchange.com/users/41678 2019-07-03T15:41:22Z 2019-07-04T07:05:46Z <p>I am not sure if this is actually done in practice as I'm not a derivatives trader, but I can only think of reducing the cost of the OTM option as a reason for delta hedging a deep OTM option, which is likely to be pricey/expensive. </p> <p>Appreciate the help on this </p> https://quant.stackexchange.com/q/44747 1 Options Delta Meaning of Term [closed] Shyam https://quant.stackexchange.com/users/39316 2019-03-23T12:13:33Z 2019-06-27T22:23:34Z <p>not able to understand delta in options. Whilst I understand, it is how much the option moves when the underlying moves by 1 unit, I fail to understand, when someone books a currency option, why does the delta need to be hedged.</p> https://quant.stackexchange.com/q/46329 1 Books and techniques to hedge options that expire tomorrow? confused https://quant.stackexchange.com/users/31372 2019-06-27T16:53:08Z 2019-06-27T16:53:08Z <p>Can anyone point me to books or resources that talk about best techniques to hedge ATM or close to ATM options that expire tomorrow. I am particularly interested on how to hedge if you are short the option.</p> <p>Thanks!</p> https://quant.stackexchange.com/q/37406 8 What really is Gamma scalping? Hans https://quant.stackexchange.com/users/6686 2017-12-17T20:22:18Z 2019-06-26T22:45:01Z <p>How does Gamma scalping really work? It seems there is no true profit scalped. If we look at the simplest scenario, Black-Scholes option price $V(t,S)$ at time $t$ and the underlying stock price at $S$ with no interest, the infinitesimal change of the overall portfolio p&amp;l under delta hedging, assuming we have the model, volatility, etc., correct, is $$0=dV-\frac{\partial V}{\partial S}dS=\big(\Theta+\frac12\sigma^2S^2\Gamma\big)dt.$$ So the Gamma effect is cancelled by the Theta effect. Where does so called Gamma scalping profit come from?</p> <p>Note: My condition implies that $$P\&amp;L_{[0,T]} = \int_0^T \frac{1}{2} \Gamma(t,S_t,\sigma^2_{t,\text{impl.}})S_t^2( \sigma^2_{t,\text{real.}} - \sigma^2_{t,\text{impl.}})\,dt$$ coming from the misspecification of volatility is $0$.</p> https://quant.stackexchange.com/q/46090 1 transactions costs and leland modified volatility user40929 https://quant.stackexchange.com/users/40929 2019-06-14T09:44:06Z 2019-06-17T13:43:42Z <p>When there are transactions costs, we are in a situation of incomplete market. What does the modified volatility of Leland (<em>Option Pricing and Replication with Transactions Costs</em>, 1985) bring us? can we replicate the option when computing/hedging the risks with this volatility?</p> https://quant.stackexchange.com/q/45802 0 Delta hedging theta pnl InnocentR https://quant.stackexchange.com/users/7117 2019-05-26T13:17:51Z 2019-05-26T20:30:18Z <p>Say I sell a swaption and delta hedge it, and the breakeven daily move in the underlying is <span class="math-container">$x$</span> bps. Then if on any given day the actual move in the underlying is <span class="math-container">$y$</span> bps <span class="math-container">$( y &lt;x)$</span>. Then I, as option seller, get to keep some theta decay as my pnl for that day. </p> <p>Question is what fraction of theta pnl do I get to keep as a function of <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. My guess is </p> <p><span class="math-container">$$1 - \frac{y^2}{x^2}$$</span> It looks correct in the limiting case of <span class="math-container">$x$</span> = <span class="math-container">$y$</span>. But could someone please correct or confirm</p> https://quant.stackexchange.com/q/45761 1 How should one hedge option positions on the date of expiry? user1559897 https://quant.stackexchange.com/users/23846 2019-05-23T13:24:44Z 2019-05-23T15:41:24Z <p>Let's say we are looking at a non-liquid equity ticker and a slightly OOM option on it. The problem is that if we buy delta to hedge it, it could move the underlying market and push the option to be ITM. </p> <p>How do we delta hedge this position on the date of expiry?</p> https://quant.stackexchange.com/q/45627 1 Delta hedging pnl to recover option price InnocentR https://quant.stackexchange.com/users/7117 2019-05-15T20:17:10Z 2019-05-17T08:10:26Z <p>In Black Scholes framework, assuming zero interest rates and realized volatility to be same as implied volatility, gamma pnl is exactly same and opposite of theta pnl. So if I buy an option and delta hedge then I make money on gamma but lose on theta and these two offset each other. </p> <p>Then how do I recover option price from delta hedging i.e. shouldn't my pnl be equal to the option price paid?</p> <p>Note: I realize if you hedge discretely rather than continuously there will be a hedging error, but please ignore this error for the purpose of this question.</p> https://quant.stackexchange.com/q/45576 0 Understanding delta based strike selection in an Iron Condor CL40 https://quant.stackexchange.com/users/36062 2019-05-12T20:39:18Z 2019-05-12T20:46:34Z <p>I am reading a small book on the proper use of Iron Condors (<a href="https://books.google.com/books?id=kkuOjyWMrhAC" rel="nofollow noreferrer">link</a>). I do not use these strategies as I have had a very hard time being profitable on them. This book mentions some strategies to creating an Iron Condor I didn't consider.</p> <p>I am trying to understand the following statement:</p> <blockquote> <p>Selecting short strikes at a particular level of delta exposure allows the width of the iron condor to change automatically with changes to Implied Volatility. </p> </blockquote> <p>I understand how IV effects the price of options - however I am confused at the use of the term "automatically adjust". Isn't the width of an Iron Condor fixed at the difference between the two short strikes? How does selecting say, the 40 delta strikes, help the Iron Condor deal with Implied Volatility? The author implies this is some sort of automatic risk factoring being done. It seems too good to be true, which means I am missing something here.</p> <p>I'm interested not only in an explanation but perhaps some mathematical treatment to this as well. I find this very interesting.</p> https://quant.stackexchange.com/q/45552 1 Implied volatility as break-even delta hedge volatility ilovevolatility https://quant.stackexchange.com/users/34971 2019-05-11T19:06:25Z 2019-05-12T12:59:24Z <p>There have been some posts on this topic, but not what I am looking for, so a new post on an old topic..</p> <p>I think some/most of us here are familiar with the following formula expressing implied volatility as the break-even <em>constant</em> BlackScholes hedge volatility to make the expected final P/L equal to zero. After some re-arranging we get the familiar formula, and restricting now to a pure stochastic volatility model as the true dynamics:</p> <p><span class="math-container">$$\Sigma^2(S_t,t,K,T) = \frac{E_t^Q \int_t^T \sigma_u^2 S_u^2 \Gamma^{BS}(S_u,K,\Sigma(S_t,t,K,T)) du}{E_t^Q \int_t^T S_u^2 \Gamma^{BS}(S_u,K,\Sigma(S_t,t,K,T)) du}$$</span></p> <p>where <span class="math-container">$\sigma_u$</span> is the stochastic volatility, and <span class="math-container">$Q$</span> is the risk-neutral measure for this SV model. Note that the implied volatility in the Black-Scholes gamma terms is always the initial implied volatility (this formula is afterall obtained under the assumption of hedging at constant initial implied volatility).</p> <p>Now this is where I think I am either making an error, or where things get interesting:</p> <p>Assume that the correlation between the spot and vol is zero always. Then we can apply conditioning. So, for a given path of volatility we can take the expectation of the dollar gamma terms, and then take expectation over all volatility paths. </p> <p>But for a given path of realized variance, we know that the dollar gamma is a martingale, and so the resulting initial gamma appears in both denominator and numerator and can therefore be cancelled out. What is left is then the following formula (when <span class="math-container">$dSd\sigma = 0$</span>)</p> <p><span class="math-container">$$\Sigma^2(S_t,t,K,T) = \frac{1}{T-t} E_t^Q \int_t^T \sigma_u^2 du$$</span></p> <p>This cannot be true of course since that would mean for all strikes the break even implied vol is the variance strike and the smile would be flat. Not what we observe in a SV model with zero correlation.</p> <p>The formula for implied volatility as break even constant delta hedge volatility is used by many (e.g. Bergomi in his book, Reghai in his book, Gatheral, and others) as a starting point for expansions and the like. These are very knowledgeable guys so I am sure they won't use something that doesn't make sense logically.</p> <p>My question is, in which step did I make the brain-fart?</p> <p>Thanks.</p> https://quant.stackexchange.com/q/45367 3 When should we delta hedge? Victor https://quant.stackexchange.com/users/27045 2019-04-29T18:54:19Z 2019-04-30T15:29:47Z <p>Let's say I'm the seller of a European call option on a non-dividend paying stock. I pocket the premium <span class="math-container">$c_0$</span> of the call at <span class="math-container">$t=0$</span>.</p> <p>If I start to delta-hedge right away, this is equivalent to replicating the call and the cost of the strategy will converge to the price of the option <span class="math-container">$c_0$</span> if the option is priced fairly (according to the Black-Scholes formula) and if the delta-hedging is done sufficiently frequently. </p> <p>So, it seems unless at <span class="math-container">$t=0$</span> the option is priced higher than its theoretical price, delta-hedging right away leads to an average null profit. Am I correct? So what are the precise scenarios when we should use delta-hedging?</p> <p>For example, if at a time <span class="math-container">$t$</span>, the call price <span class="math-container">$c_t$</span> is below the price at <span class="math-container">$t=0$</span> (<span class="math-container">$c_t &lt; c_0$</span>), the delta-hedging strategy will cost in average <span class="math-container">$c_t$</span>. Does it mean I will lock an average positive profit?</p> https://quant.stackexchange.com/q/41979 0 How is FX cross rates options are priced? Gleb Yarnykh https://quant.stackexchange.com/users/27775 2018-09-29T17:19:39Z 2019-04-27T23:01:10Z <p>Say I have market for EUR/USD and also USD/CAD, how would EUR/CAD would be priced and hedged in practice? What are good papers/book chapters to read on that? (Assuming basic knowledge already on option pricing/hedging)</p> https://quant.stackexchange.com/q/44670 2 Profit and Loss on delta-hedged portfolio user101998 https://quant.stackexchange.com/users/39275 2019-03-19T14:44:45Z 2019-03-19T15:54:06Z <p>The overnight profit formula from a textbook (possibly Derivative Markets by McDonald) is the following:</p> <p><span class="math-container">$$\Delta _{t}(S_{t+h}-S_{t})-(V_{t+h}-V_{t})-(e^{rh}-1)(\Delta_{t}S_{t}-V_{t}),$$</span></p> <p>where Delta is the delta of the option, S is the stock price, t is a specific point in time, h is a small movement in time, V is the value of the option, r is the continuous risk-free rate.</p> <p>Assuming the market maker sold a put and delta hedged by shorting the stock.</p> <p>My question is that if this is applied in a multi-period fashion, are we assuming that we are losing interest on the option we sold?</p> <p>For example: at time <code>t</code>, we sold a put for <code>7</code> dollars and shorted <code>50</code> worth of stock to delta hedge. At time <code>t+h</code>, the put is worth <code>15</code> dollars and we sold additionally 40, making it <code>90</code> to delta hedge. Our profit from the last term of the equation is (a positive number since delta of a put is negative).</p> <p><span class="math-container">$$-(e^{rh}-1)(-50-7)$$</span></p> <p>Then in the next period, the interest would be</p> <p><span class="math-container">$$-(e^{rh}-1)(-90-15)$$</span></p> <p>To me, this is counter-intuitive, because if we bought a call for 5 dollars and delta hedged at time t, and if the call price shot up 10 dollars at t+h, it would mean we're earning interest on the 15 dollars from time t+h to t+h+j, for some j in the future, when we only paid for the call at time t for 5 dollars.</p> <p>A somewhat related question that may be too simple to start a new question is that is there a way to arrive at the profit/loss number just by looking at the portfolio value alone? For example, say MM sold a put and shorted stocks to delta hedge--the portfolio consists of a shorted put, shorted stock, and cash that is presumably invested in risk-free bonds. From time to time, the value of the stock and put changes, and so does the portfolio. Will it be possible to construct a portfolio that consists of the shorted put, shorted stock, risk-free bonds, and the interest earned, so that the profit/loss can be ascertained from the closing value of the portfolio value? </p> <p>Thanks!</p> https://quant.stackexchange.com/q/44578 3 Uniqueness of the Hedging strategy quallenjäger https://quant.stackexchange.com/users/25867 2019-03-15T11:25:46Z 2019-03-15T12:31:39Z <p>I am currently reading the book "Nonlinear Option Pricing" by Julien Guyon. In the book they defined an attainable payoff <span class="math-container">$F_T$</span> as a <span class="math-container">$\mathcal{F}_T$</span> measurable random variable for which there exists an admissible portfolio and a real number <span class="math-container">$z$</span> such that <span class="math-container">$$z+\int_0^T \Delta_s\mathrm{d}\tilde{X}_s+D_{0,T}F_T=0$$</span> and <span class="math-container">$\int_0^T\Delta_s\mathrm{d}\tilde{X}_t$</span> should be a true <span class="math-container">$Q$</span>-martingale.( <span class="math-container">$D_{0,T}$</span> is the discount factor and <span class="math-container">$\tilde{X}$</span> is the discounted stock price.)</p> <p>Next, they claim that the pair <span class="math-container">$z,\Delta_s$</span> is unique because suppose there is a <span class="math-container">$z',\Delta'_s$</span>, then <span class="math-container">$$\int_0^T(\Delta_s-\Delta_s')\mathrm{d}\tilde{X}_s=z'-z$$</span>.</p> <p>Now since <span class="math-container">$\tilde{X}$</span> is a <span class="math-container">$Q$</span>-Martingale, <span class="math-container">$\Delta_s=\Delta'_s$</span> and thus <span class="math-container">$z=z'$</span>. </p> <p><strong>Question:</strong> I don't understand the last argument. Why is <span class="math-container">$\Delta_s=\Delta_s'$</span> necessarily? <a href="https://math.stackexchange.com/questions/115276/stochastic-integral-which-is-almost-surely-zero-at-fixed-time">This</a> question on Math.stachexchange shows one can find a non-trivial previsible process, such that the stochastic integral is almost surely equal to zero. Or do I miss something?</p> https://quant.stackexchange.com/q/44562 1 Break-even volatility for delta hedge portfolio DasBoot https://quant.stackexchange.com/users/39217 2019-03-14T06:49:27Z 2019-03-14T06:49:27Z <p>After simulating practical and theoretical PnL of a delta hedged portfolio on some data from the SPX500 under 0.15 management Vol I want to find the Vol which gives me an accumulated PnL of 0.</p> <p>Initially I tried using the formula squared-vol = (-2 * theta) / (sigma * S^2) but that just returns whatever management Vol I calculate the portfolio under.</p> <p>Can someone tell me what I'm missing?</p> https://quant.stackexchange.com/q/41818 4 Option order imbalance Bougias A. https://quant.stackexchange.com/users/29745 2018-09-19T15:40:38Z 2019-03-05T02:44:08Z <p>Currently studying the paper:</p> <p><strong><em>HU, Jianfeng. Does Option Trading Convey Stock Price Information?. (2014). Journal of Financial Economics. 111, (3), 625-645. Research Collection Lee Kong Chian School Of Business.</em></strong></p> <p>To test the impact of option order flow affects stock order flow. Author defines the measure of option order imbalance:</p> <p><span class="math-container">$\text{OOI}_{it}=\frac{\sum_{j=1}^{N} 100\text{ Dir}_{itj}\text{Delta}_{itj}\text{Size}_{itj}}{\text{Num_Shares_Outstanding}}$</span></p> <p>Where, The option order imbalance,<span class="math-container">$\text{OOI}_{it}$</span>, is measured for stock i on day t. <span class="math-container">$\text{Dir}_{itj}$</span> is a dummy variable equal to 1 if the jth option trade on stock i is initiated by the buyer, and -1 if the trade is initiated by the seller, according to certain trade signing algorithms.<span class="math-container">$\text{Delta}_{itj}$</span> is the option price sensitivity to the underlying stock price, and <span class="math-container">$\text{Size}_{itj}$</span> denotes the trade size in option lots (100 shares of the underlying stock).</p> <p>My question is:</p> <p>If option trade denotes an executed trade, why do we need the dummy variable? Shouldn't buy trades be cancelled out by sell trades? Since whenever one buys someone else sale the respective quantity( price differs due to bid-ask spread).</p> https://quant.stackexchange.com/q/31567 14 How to, from various hypotheses on the P&L, get known models (BS, Heston etc ...) ujsgeyrr1f0d0d0r0h1h0j0j_juj https://quant.stackexchange.com/users/13470 2016-12-18T17:11:07Z 2019-02-28T07:02:27Z <p>Usually models in quantitative finance are taught by giving, let's say, stochastic differential equations, initial conditions, and then pricing, under the model, various derivatives written on the underlying. Qualifying a model as good or bad wrt the pricing of a given derivative is often done by saying that the model captures, in a rather pertinent way, "quantities" the derivative is sensitive to. (For instance the Black-Scholes model would be bad to price forward starting options as it fails, least to say, to capture movements of the forward implied volatility.) Then, we can compute the discounted P&amp;L of the porfolio consisting in the sold derivative and underlyings (corresponding to a delta-hedge of the derivative).</p> <p>Now consider a european option of pay-out function <span class="math-container">$\varphi$</span> and maturity <span class="math-container">$T$</span> written on a (tradable) underlying <span class="math-container">$S$</span>. Consider the aforementioned portfolio associated to this option and note <span class="math-container">$\Delta_t$</span> the number of shares that we have in the portfolio at time <span class="math-container">$t$</span>. We can write (and it's quite classic) the discounted P&amp;L of the portfolio over the options's lifetime as <span class="math-container">$$P\&amp;L_0 = -e^{-\int_0^T r_S ds} \varphi (S_T) + \pi_0 + \int_0^T e^{-\int_0^t r_s ds} \Delta_t \left( dS_t - S_t r_t dt\right)$$</span> where <span class="math-container">$\pi_0$</span> is the price we make a time <span class="math-container">$t=0$</span> on the option. Note that I only stress the dependance of <span class="math-container">$\Delta$</span> in time, but it can, of course, depend on anything else. Note also that <span class="math-container">$P\&amp;L_0$</span> does not depend on any model specification for <span class="math-container">$S$</span>.</p> <p>My question is the following : without making any model hypothesis, that is, without specifying <span class="math-container">$dS_t$</span> or <span class="math-container">$r_t$</span>, is there a correspondance</p> <p><span class="math-container">$$\{\textrm{set of hypotheses on P\&amp;L_0}\}\to \{\textrm{models on S under a certain measure}\}$$</span></p> <p>such that for any given "known" model (BS, Heston, SABR, 3/2, 4/2, Bergomi's P1 etc) there exist a set of hypotheses on <span class="math-container">$P\&amp;L_0$</span> (for instance on its expectation, its variance that one would like for instance to minimize, on its others moments etc under some measure or under another, or something else) that <em>lead</em> to the given model and the fondamental theorem of pricing associated to it ?</p> <p>By <em>lead</em> I mean : wanting to prescrible/minimize some quantities associated to <span class="math-container">$P\&amp;L_0$</span> will lead to functional equations (variational, in fact) on the function <span class="math-container">$\Delta$</span> that will lead to a PDE that will ultimately lead (through Feynman-Kac) to a model on <span class="math-container">$S$</span>.</p> <p>First of all : how to introduce dependance of <span class="math-container">$\Delta$</span> in a parameter (in <span class="math-container">$S$</span> for instance, to begin with, or later in realized variance) through assements/hypotheses on the <span class="math-container">$P\&amp;L_0$</span> ? How to retrieve the Black-Scholes model ? Other models ?</p> https://quant.stackexchange.com/q/44303 1 Correct beta weighted delta options formula? 4thSpace https://quant.stackexchange.com/users/2666 2019-02-26T15:30:22Z 2019-02-26T15:30:22Z <p>Is this the correct formula for beta weighted delta: <a href="http://www.nishatrades.com/blog/beta-weighted-delta" rel="nofollow noreferrer">http://www.nishatrades.com/blog/beta-weighted-delta</a></p> <p><a href="https://i.stack.imgur.com/b4L9b.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/b4L9b.jpg" alt="enter image description here"></a></p> <p>I've seen this <a href="https://quant.stackexchange.com/questions/15131/what-is-the-formula-for-beta-weighted-delta-and-gamma/44045#44045">What is the formula for beta weighted delta and gamma?</a> but they seem to be doing something completely different. Beta weighted delta doesn't involve Black Scholes.</p> <p>For the above, they have a delta of 30, which means .30. Why is it necessary to multiply the delta by 100?</p> https://quant.stackexchange.com/q/43770 0 replicate option by dynamic hedging bramvs https://quant.stackexchange.com/users/20006 2019-01-29T14:16:18Z 2019-01-29T14:16:18Z <p>I've just started working for a company with a decent commodity exposure. They manage this by as they call it dynamically hedging it. Basically when they start the hedging they identify a market price and a budget price x% above the market. This budget price is the maximum they are willing to pay on average for the commodity. To achieve this they hedge a percentage of the portfolio and increase this if prices go up and decrease this when prices go down. I'm pretty convinced by this strategy they replicate buying an option at the strike of the budget price but I'm struggling to mathematically prove this. Any help?</p> https://quant.stackexchange.com/q/43435 1 why gamma decreases when option is deep in the money? [closed] kave ka https://quant.stackexchange.com/users/38305 2019-01-11T17:17:58Z 2019-01-11T19:18:39Z <p>Gamma decreases when a call option goes either deeper in, or deeper out of the money. That is due the demand for the call option. I can imagine the demand for the option would decrease as it goes deeper out of the money, but I would expect the demand for the option should increase as it goes deeper in the money because it would make more profit for the holder of the option. Why is this not true? In other words, why does the demand for the option decrease despite the fact that a deep in the money option is more profitable?</p> https://quant.stackexchange.com/q/43404 1 Delta hedging/Gamma PnL babaji https://quant.stackexchange.com/users/31829 2019-01-10T01:50:45Z 2019-01-10T06:29:30Z <p>Suppose I am long USDIDR straddle with my start of the day delta being USD10m long IDR and USDIDR gamma being $5m. </p> <p>There is a 1% intra-day IDR strengthening, so my delta becomes roughly long IDR 15m. I execute a 15m long USDIDR 1Y NDF to re-balance my delta (bid/ask spread 20 fwd points, 1Y USDIDR NDF mid 14850).</p> <p>How do I calculate total <strong>risk sensitivities based</strong> PnL from delta balancing including transaction costs from trading new NDF? Thanks. </p> https://quant.stackexchange.com/q/43150 2 Confused by Solution to the Expected Profit when Hedging an option using Implied Volatility (from Wilmott 2006) Nel https://quant.stackexchange.com/users/25882 2018-12-20T15:01:53Z 2018-12-20T15:01:53Z <p><em>Paul Wilmott on Quantitative Finance 2nd Ed</em> (section 12.5.1) gives a solution to the initial expected profit when hedging using delta based on implied volatility as</p> <p><span class="math-container">$$\frac{1}{2}(σ^2 - σ̃^2) \,\int_{t_0}^{T}e^{-r(s-t_0)}S^2\Gamma\,ds$$</span></p> <p>from which he then derives the single integral <span class="math-container">$$\frac{Ee^{-r(T-t_0)}(σ^2 - σ̃^2)}{2\sqrt{2\pi}} \, \int_{t_0}^{T} \frac{1}{\sqrt{σ^2(s-t_0) + σ̃^2(T-s)}} \\ \times \exp\left( - \frac{(log(S/E) + (\mu - 0.5σ^2)(s-t_0) + (r - D - 0.5σ̃^2)(T-s))^2}{2(σ^2(s-t_0) + σ̃^2(T-s))} \right)\\$$</span></p> <p>However, and very confusingly, in order to get similar results to those shown by him when comparing expected profit versus various growth rates (as per figures 12.4, 12.5, 12.6) I have to set the initial term to </p> <p><span class="math-container">$$\frac{Ee^{-r(T-t_0)}(σ^2 - σ̃^2)}{\frac{2}{\sqrt{2\pi}}}$$</span> instead of <span class="math-container">$$\frac{Ee^{-r(T-t_0)}(σ^2 - σ̃^2)}{2\sqrt{2\pi}}$$</span> as set out in the text.</p> <p>My implementation uses numerical integration from the Python SciPy package.<br> When implementing it as shown, the magnitude of the expected profit is clearly excessive but with the change in denominator it's nearly spot on.</p> <p>In the VBA code accompanying <em>Paul Wilmott Introduces Quantitative Finance 2nd Ed</em> he has an approximate solution which also uses <span class="math-container">${\frac{2}{\sqrt{2\pi}}}$</span> instead of <span class="math-container">${2\sqrt{2\pi}}$</span></p> <p>Have I misread his equation?<br> I can't find errata for either book and am really confused as to why his implementation differs from the derivation shown.<br> Am I missing something (obvious)?</p> <p>Thanks</p> https://quant.stackexchange.com/q/42183 0 Hedging with machine learning John Doe https://quant.stackexchange.com/users/32314 2018-10-14T20:17:53Z 2018-12-14T13:36:45Z <p>I’ve been thinking about an interesting problem lately: Suppose I have a position in an exotic derivative. How can I automate the hedging process?</p> <p>Traditionally, one build a pricing model and calculate sensitivities to the risk factors. Then one uses various products like stocks, bonds, futures, swaps etc. to hedge each risk factor. The algorithm would need to determine how to hedge at each discrete time point. I think this has been covered in many papers so far. </p> <p>I’ve also heard of people using AI to hedge. So suppose we have a preferred share ETF for example. Then the question becomes: given a discrete set of products with their price history, how can one optimally hedge? We need to calculate the weights of the portfolio and minimize the tracking error of the hedging portfolio. This would result in the optimal hedge. </p> <p>What else could I try? Can someone point me towards some papers in this area? I’ve been thinking about this and might want to pursue the idea for my thesis. Thanks!</p> https://quant.stackexchange.com/q/34806 2 Tracking error Black Scholes Methamortix https://quant.stackexchange.com/users/25997 2017-06-22T07:36:33Z 2018-12-14T01:58:22Z <p>Suppose an asset follows the SDE</p> <p>$$d S_{t}^{1} = \mu S_{t}^{1} dt + \sigma_{t} S_{t}^{1} d W_{t}$$ Furthermore assume that$r = 0$and a trader who uses Black-Scholes for pricing and hedging with volatility$\sigma^*$for a terminal value claim with payoff$h(S_T)$. Then the price of the claim under BS is given as the solution of the PDE</p> <p>$$h_t^BS(t,S) + \frac{1}{2} (\sigma^*)^2 S^2 h_{SS}^{BS}(t,S_t^1) = 0$$</p> <p>with$h_t^BS(t,S) = h(S)$.<br> The tracking error of his hedge is then given by</p> <p>$$e_T = h(S_t) - V_T$$</p> <p>It can be shown that in this case it is equal to</p> <p>$$e_T = \frac{1}{2} \int_0^T ( S_{t}^{1})^2 (\sigma_t^2 - \sigma^*) h_{SS}^{BS}(t,S_t^1) dt$$</p> <p>Now suppose that the trader sells a plain vanilla call option and replicates this option with a stock postion equal to the delta of the call. Hence, at this time point$t$he is delta neutral. </p> <p>Now suppose the true volatility$\sigma_t^2$is bigger than the volatility$\sigma^*$he used for replication. According to the formula$(\sigma_t^2 - \sigma^*) &gt; 0 $and the gamma of the position is $$h_{SS}^{BS}(t,S_t^1) &gt; 0$$ Hence, $$e_T &gt; 0$$ and the trader makes a loss.</p> <p>From an intuitive stand point this is clear for me - the trader is short a call option which has a convex payoff - for a large move of$S_t$the option will gain more value than this hedging portfolio and since he is short the option he will have a loss. However, it is not clear for me why $$h_{SS}^{BS}(t,S_t^1) &gt; 0$$ - the call option has a positive gamma since again it has a convex payoff but he is$\textbf{short}\$ the option hence the gamma would be negative as far as I see.</p>