#delta hedging

[jkumari]

https://bigul.co/en/index.php/what-is-delta-hedging/

[deltahedgingdeskinsurat]

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[financeoholic-blog]

Delta Hedging

Hey all!

I apologize for being MIA for so long. I’ve been swamped with studying for the CFA exam during the first half of this year. The exam was on June 4th and so I’ve finally been able to catch up on life (TV, games, social, and drinking. Lots and lots of catching up on drinking).

With the recent “Black Swan”-like event of Brexit happening, the markets have been in turmoil. Today I’ve decided…

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[quantquestions-blog]

Black Scholes PDE

Question:

Derive the basic Black-Sholes equation for valuing options on a stock using a replicating portfolio.

Solution:

The stock process is assumed to follow Geometric Brownian Motion or a Log-normal random walk as represented by

\[ dS_{t} = \mu S_{t}dt + \sigma S_{t}dX_{t} \]

Let the option price at time \( t \) before expiry be represented by \( V(S_{t}, t) \).

Using two-dimensional Ito,

\( dV = \frac{\partial V}{\partial t}dt + \frac{\partial V}{\partial S_{t}}dS_{t} + \frac{1}{2}\frac{\partial^2V}{\partial S_{t}^2}dS_{t}^2 \)

Using Quadratic Variation \( dS_{t}^2 = \sigma^2S_{t}^2dt\)

\( dV = (\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S_{t}^2\frac{\partial^2V}{\partial S_{t}^2})dt + \frac{\partial V}{\partial S_{t}}dS_{t} \)

\( \)

Delta Hedging:

Consider a portfolio \( \Pi \) consisting long an option and short a \( \Delta \) of stock.

\( \Pi = V - \Delta S_{t} \)

Change in the portfolio value would be

\( d\Pi = dV - \Delta dS_{t} \)

      \( = (\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S_{t}^2\frac{\partial^2V}{\partial S_{t}^2})dt + (\frac{\partial V}{\partial S_{t}} - \Delta)dS_{t} \)

      \( = (\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S_{t}^2\frac{\partial^2V}{\partial S_{t}^2} + \mu S_{t}(\frac{\partial V}{\partial S_{t}} - \Delta))dt + \sigma S_{t}(\frac{\partial V}{\partial S_{t}} - \Delta)dX_{t}\)

The above SDE has a drift/deterministic(\( dt \)) term and a diffusion/randomness(\( dS_{t} \)) term. We can remove the diffusion/randomness term by making the coefficient of \( dS_{t} \) to \( 0 \).

\( \Delta = \frac{\partial V}{\partial S_{t}} \)

\( \implies d\Pi = (\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S_{t}^2\frac{\partial^2 V}{\partial S_{t}^2})dt \)

Once we remove randomness the portfolio should grow at risk-free rate.

\( d\Pi = r\Pi dt \)

\( r\Pi dt = (\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S_{t}^2\frac{\partial^2 V}{\partial S_{t}^2})dt \)

\( \implies r(V - \Delta S_{t}) = \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S_{t}^2\frac{\partial^2 V}{\partial S_{t}^2} \)

\( \implies \frac{\partial V}{\partial t} + rS_{t}\frac{\partial V}{\partial S_{t}} + \frac{1}{2}\sigma^2S_{t}^2\frac{\partial^2 V}{\partial S_{t}^2} - rV = 0 \)

The equation is free of the drift \( \mu \) of the stock. I will write in another post about why the stock drift doesn't appear in BSE.

[optionworkshop-blog]

Credit Spread Adjustments: Delta Hedging with Stock

One of the most effective ways to adjust a broken out-of-the-money vertical spread is with stock.  So many of us in the retail world—having been introduced to the flexibility and/ or leverage of options—seem hotly opposed to taking a position in an underlying stock, ETF or future.  Many of us would rather torment a simple vertical spread with layer upon layer of complicated adjustments, so that what started out as a hands-off strategy becomes a position that must be constantly tweaked—the original thesis for the trade reduced to a footnote.

The Decision to Adjust

Often the biggest risk to a credit spread is price:  a trader establishes a position and price woefully moves in the complete opposite direction than anticipated.  When trading out-of-the-money credit spreads, the general assumption is that implied volatility will decline (either overall or at a particular strike) and price will either move in a particular direction or remain in a perceived range.  Both scenarios result in a decay of premium and allow the trader to buy back the spread at a cheaper price than originally sold.  In the case of a bull put spread, the trader is neutral-bullish on price and bearish on volatility.  If that outlook proves incorrect, often the best adjustment is simply to admit being wrong and take losses.  But if the original thesis still holds water when the trade is under pressure, an adjustment can help manage risk while waiting for price and volatility to perform as anticipated.

Delta Hedging

The trick with making adjustments is to do so without completely sacrificing the original rationale for the trade.  The common adjustment of buying an out-of-the-money put is usually counter to the credit spread’s purpose because the put has negative theta (it decays) and positive vega (it profits from an increase in IV).  So if price does not continue south after the put purchase, the adjustment becomes a liability as it bleeds value (not to mention any complications from changes in vertical skew or if buying a different month, changing term structure).

Stock on the other hand, has no gamma; no vega; and no theta.  So the trader that shorts stock to protect a bull spread does not disrupt the spread’s rate of decay or volatility outlook.  And unlike a put, the deltas of a short stock position do not change with time, volatility or price.  An underlying stock position is “Delta” in the purest form.

For those new to this, allow me to explain.  Let’s say our credit spread is short ten 30 delta puts and long ten 20 delta puts.  We can add these together to get the delta of our overall position:  10 contracts  x (-30 + 20) = 100.  What this means is that the spread will lose $100 for every dollar the stock moves against it (ignoring gamma, theta and vega).  This is the primary definition of delta: the rate of change for every dollar movement in the underlying.  An underlying stock position is long one delta per share.  Our credit spread, being long 100 deltas is the rough equivalent of being long 100 shares of stock.

For the sake of argument, let’s say that our perception of risk, for whatever reason (be it emotional, strategy related or whatever) tells us that our position cannot withstand losing $100 per point of underlying movement.   We can reduce these deltas by shorting the stock.  How many shares we short will depend on our strategy and the level of risk.  For this illustration, let’s say our strategy demands we reduce our price risk to losses of $50 for every point in the underlying.  In this case, we would short 50 shares of stock (this can be done on margin in most accounts) and thereby reduce the overall position delta to 50.

Because this adjustment has no effect on gamma, theta or vega, the original credit spread’s potential to produce income from time decay and declining volatility is unimpeded.  The hedge allows the trader to return to a neutral-bullish bias, as he’d unwittingly become a hopeless bull when the position was threatened.  In fact, if the underlying remains range bound below the adjustment point, both the credit spread and the hedge can profit.

The Whipsaw

Some drawbacks are evident, the most obvious being the whipsaw.  Since trading stock is pure delta, it makes or loses money on a dollar for dollar basis.  Option traders are sometimes not used to that—it is akin to trading the deepest in-the-money option possible without the huge bid/ask spread.  However with stock, traders are not bound to hundred-lot contracts.  The trader can purchase or short the exact number of deltas required.

No Gamma

Another point that can’t be ignored is that the lack of gamma.  If price continues to move against the position or time ticks away, the vertical spread will pick up additional deltas that are not hedged by the long stock.  Since the underlying position is dynamic and the hedge is not, the trade may require additional hedging.

When to Exit the Hedge

One thing that should be very clear is that it that the desired outcome of the delta hedge by itself is a loss.  The trader expects gains from the larger underlying credit spread to outweigh the losses of the hedge.  Scenarios where the credit spread and the delta hedge both profit should be considered outliers rather than the goal of this particular adjustment.  With this in mind, it is wise to apply a budget to the hedge.  Commonly, this is 10-20% of the credit spread’s maximum profit.  In the case of the bull spread with the short stock hedge, one would remove the hedge when it incurred losses of that amount or when it is clear the hedge is no longer needed—whichever comes first.  This means a trader cannot wait until too deep in the hole to hedge the trade because a reversal in price will force him to remove the hedge while the risk is still high, or lure him into leaving the hedge on too long; diminishing the already limited returns of the credit spread.  The late hedger is vulnerable to whipsaws as price triggers him to remove and re-hedge with the danger of repeating the whole process over again.

Hedging Profitable Positions

While credit spreads in trouble certainly benefit from this type of adjustment, I’ve also found it useful for positions nearing a profit target with suspicious underlying price charts.  Stocks that are primed for a pullback or a bounce that may force the spread into an extended holding period—or worse, turn a winner into a loser—can be neutralized with stock to protect profits on the reversal.

Other Adjustments

Like any type of trade or adjustment, there is no single source solution for every situation.  What makes one adjustment better than another will always be dependent on the trader’s outlook, the perceived edges in the price chart and volatility surface and the trader’s own sense of comfort and expression.  The long put that I discounted at the beginning of this article is in fact, at times a wonderful hedge (see Vanna and the Put Explosion).  But for the trader looking to hedge a credit spread without disrupting the original outlooks of price, volatility and time, hedging with stock is a simple, easy to manage adjustment.

Trading Vertical Spreads with Edge

While I believe the most important element of position management begins with an intelligent entry, adjustments are often necessary and when done correctly, can significantly reduce risk while increasing the probability of profit.  In our upcoming course, tentatively titled “Trading Vertical Spreads with Edge,” we will discuss various adjustment strategies to fix broken trades.  The course will consist of roughly two months of live webinars and will be offered at a minimum cost.  I anticipate being able to offer this course in another month or so.  To be notified when this is available, send an email to [email protected].  Your email will never be shared elsewhere and you will never receive spam from me.  To those of you who have already notified me:  thank you.  I apologize for the wait.  I have a growing list and will be notifying you all as soon as everything is ready.