And yet it moves

The aim of this article is to explain the main idea behind the concept of volatility arbitrage by using Delta neutral portfolios composed by a combined position in a derivative instrument sensible to changes in (expected) volatility of the underlying, in our case an option contract, and in the underlying itself, in our case a common non dividend-paying stock.

The main concept here is to exploit situations where our view on actual volatility differs from the current market view, and to make a profit out of it. We’ll look in particular to cases where our expectations involve a future volatility of the underlying greater than the one currently priced in by the market.

Since the majority of what follows is based on the very theory behind the Black-Scholes-Merton model for option pricing, we give a brief recap of its main assumptions. But, before starting, let’s provide some definitions.

Although volatility can be generically thought as the attitude of market prices, rates and returns to change in an unpredictable and unjustified manner, and yet this is the meaning that is usually given to it in business and financial newspapers, especially concerning bearish market environments, in this case we simply refer to volatility as to the standard deviation of market log returns, denoted by σ.

Within this definition of volatility, and referring to market returns for the underlying of an option contract, we distinguish between actual volatility -that is the average size of deviations from the mean value that has actually occurred within a given time period, σ(a) – and implied volatility, which is the amount of noise, still expressed as a standard deviation of returns, that the market is currently pricing for the residual life of the option, and it will be denoted by σ(i).

The main assumption behind the Black-Scholes-Merton model (B&S) is that prices of stocks follow a log-normal distribution and their evolution through time can be described by the following geometric Brownian motion:

Where S is the stock price, μ is the drift rate for the stock returns, σ as said is the volatility, and W is a standard Brownian motion, which is the input for prices’ stochastic behaviour.

Following these assumptions, the price at time  for a European call option on a non dividend-paying stock S with strike K and maturity T is given by

 Where r is the risk free rate and where

Now that we’ve set the rules of the game, we give a more clarifying definition of implied volatility that will be fundamental in what follows: it is the value that the parameters  has to take in the B&S pricing formula such that the price given by the model equals the actual price observed in the market place.

Here comes the interesting part. An European call option is a contract who gives the buyer the right to buy the underlying asset, in our case a stock, at a given price K and at a given maturity T; imagine that today we bought an At the money call option on stock S as the one described above and we observed its price up to the expiration date T, and imagine that what we’ve observed is something like the process in Fig.1

Fig.1  A simulation for the Stock price (S) and the Option price (V) with μ=10%, σ=30%, K=100, r=5%

It is quite evident that, given the realized path for the underlying stock, our call option would have expired Out of the money and hence worthless. So where does the profit come from?

Let further imagine that we have a forecast for the volatility of the stock over the residual life of the option (σ(a)) of 30%, and that this forecast is constant through time. Assume that the price we observe today in the market for the option is 10.44 €; this is, given  and given the parameters for the risk free rate, the time to maturity and the strike price, fully consistent with an implied volatility of around 20% (note that since the option pricing formula is not invertible analytically with respect to the parameter σ this has been obtained through iterated substitutions). That is, the market expectation for the stock volatility over the residual life of the option is of around 20%, i.e. 0.1 less than our forecast (σ(a) > σ(i)).

In order to profit from this situation we then set up a Delta neutral portfolio, that is a portfolio insensitive to variations in the price of the underlying: such portfolio is made of a long position in the option that we think is under-priced and of a dynamically adjusted short position in the underlying stock.  That is we buy the option and we delta hedge until maturity.

Again, we provide some definitions: The delta (Δ) of an option is simply defined as the rate of change of the option price with respect to the change in the price of the underlying asset. It can be thought as the first derivative of the (call) option pricing formula with respect to S that is

Now, since the delta of a portfolio is equal to the sum of the deltas of each position we see that, given that the delta of S is one (the rate of change in the stock price with respect to changes in the stock price is clearly equal to one), and provided that we want to obtain a portfolio with a delta equal to zero (i.e. delta neutral), we can find the correct number of stocks to be short-sold, allowing for non integer quantities, by solving

with respect to q. But since Δ(S) is equal to one it is then clear that q=Δ(V)=N(d1). Since the payoff profile of long position in the call option is convex (i.e. non linear) with respect to S, with convexity time varying, one should compute at each time step different values for Δ(V) and should therefore adjust from time to time his short position in the underlying (this is where the expression dynamically comes from).

Given the definition we discussed above for the term N(d1) one could ask what number should be put in place of parameter σ to compute the delta term, i.e. one could ask if we should use the actual volatility (our forecast. i.e. 30%) or the implied volatility (20%) in order to perform delta hedging. We’re going now to show how different choices can lead to quiet different results (for those interested, full derivation of results, and much more, can be found in R. Ahmad & P. Willmott, 2005).

Imagine that our forecast for the stock’s volatility turns out to be correct, and that we decided to delta hedge by putting σ(a) (our forecast for the correct level of volatility) inside the equation for the option’s delta. By doing so we’ve replicated a short position, in terms of exposure to changes in the underlying, in a correctly priced option, so that the final profit from the strategy is given by

That is the difference at time t (i.e. today) between the price of a correctly priced option and the market price that we’re currently observing. Given that V(S,t; σ(a) ) = 14.22 € we’re guaranteeing ourselves a positive profit of around 3.78 € from the strategy.

The issue arises when we consider how such profit is obtained on a mark-to-market basis. To understand this point let’s imagine how our delta neutral portfolio is composed as of today; we see that it is made of

We notice that we can make a risk free return of r out of the cash component of the portfolio by simply leaving the money in the bank account. This leads to a mark-to-market profit of

By using Ito’s Lemma we can obtain

Where Γ(i) is called the gamma of the option and it is defined as the second derivative of the option’s price with respect to S (it can also be seen as the speed at which the delta changes with respect to changes in the stock’s price, which in turns determines the speed of the hedging adjustments). The expression above is nothing but the mark-to-market profit over one time step; the important point here is that we still spot the  term dW, which, as stated above when we discussed about dS was the source of randomness. This means that when we delta hedge with actual volatility σ(a) we get a certain positive final payoff (provided that our forecast is correct) but a random mark-to-market profit, i.e. on a mark-to-market basis we can lose before profiting. This can be seen in Fig. 2, where it is plotted the mark-to-market P&L deriving from such a strategy implemented in the example above (note that the final payoff is fairly close to 3.78 €).

Fig. 2 mark-to-market P&L from delta hedging with actual volatility, with σ(a)=0.3, σ(i)=0.2

More interesting is the case where we delta hedge by substituting σ(i) inside the N(d1) term, which is indeed the most popular and used approach among option traders. By doing so we get the nice property to have a deterministic and positive profit on a mark-to-market basis, provided that σ(a) > σ(i). In fact, by buying the option today, hedging with implied volatility and put any cash in the bank account earning r we make, over one time step,

with

Where the second result is derived from the B&S Partial Differential Equation.

As stated above, since in the expression for the mark-to-market profit from delta hedging with implied volatility there’s no dW term, we can say that ‘overnight’ we’re making a deterministic amount of money. The overall profit to be made in this case from today up to the expiration date is given by

This is always positive, but highly path-depend; meaning that, depending on the value taken by gamma, the final profit will in turn depend, in addition to S on the price path of the underlying stock. In particular it can be shown that the profit is maximized for a value of μ (remember that this is the drift rate for stock returns) that makes the final price being close to the strike price (i.e. the option ends up At the money), where gamma is maximized. An example of this path-dependence can be seen in Fig.3.

Fig. 3 mark-to-market P&L from delta hedging with implied volatility for different paths of the price of the underlying, with σ(a)=0.3, σ(i)=0.2. It is well evident the path dependence for the final payoff. Note that, when hedging with implied volatility, we don’t even need to know the exact value for realized volatility, as it is simply required to be greater than the implied volatility.

The intuition behind the fact that we’re gaining on gamma is again due to the convexity of the payoff function of a long option position with respect to the price of the underlying. This convexity, as measured by gamma, tends to be greater for At the money options, and it increases as the option becomes closer to maturity. The fact that we’re hedging a positive gamma position (a long call option) with a zero gamma position (a short position in the underlying stock) guarantees that for big changes in the price of underlying we’re making more/losing less money on the first position that on the second. Note that by ‘big changes’ we mean that we’re profiting from high volatility, which is indeed our starting idea. A profile for gamma with respect to time to maturity and the price of the underlying stock for a correctly priced call option as the one described before can be seen in Fig. 4.

Fig. 4 Graphical representation for the gamma of a call option on S with respect to the price of the underlying and time to maturity (chronological ordering). It can be seen how as time passed gamma tends to peak around the strike price (where at the limit, i.e. at maturity, tends ideally to infinite) and to decrease for ITM and OTM region. Parameters are σ=30%, K=100, r=5%

So far we’ve discussed on how to make a profit out of mispriced options (specifically under-priced call options) in the Black and Scholes world, that is a world of constant volatility. We’ve implicitly defined the change in the option price as

By looking at this expression we can spot a term that we’ve ignored so far; in particular the last term of the equation expresses the rate of change in the option price with respect to changes in its implied volatility and it’s called the vega of the option. This coefficient represents a deviation from the classic Black and Scholes hypothesis where, by assuming constant volatility through time (in our examples both σ(a) and σ(i) were treated as parameters and hence fixed for the whole residual life of the option), we simply considered it as being equal to zero. In reality things are more complicated, and implied volatility is ‘made’ free to vary across time, to the point that option market makers quote not prices but implied volatilities, and that for different options with the same maturity on the same underlying we actually observe different implied volatilities (such phenomenon is known as ‘volatility skew’ or, sometimes, as ‘volatility smile’).

The problem transforms now in the following way; we still want to make a profit out of under-priced options by dynamically delta hedging, but, since (according to our view, underestimated) implied volatility is not set to remain constant through the remaining life of the option, we’re not managing the position up to maturity. Rather we’re taking a directional position on implied volatility, as we want to profit from a later increase, i.e. from a re-alignment of markets expectations to our forecast.

The game works as before: whenever we observe an implied volatility materially lower than our forecast we set up a delta neutral portfolio made of a long position in the option and we delta hedge until this discrepancy has disappeared. Everything that has been said before still holds, with the only exception that positive changes in implied volatility will be our main contribution to the final profit (i.e. we’re vega positive), while the gamma coefficient will be useful to offset the negative contribution to the changes in the option price given by the time decay coefficient theta (i.e. what we have defined as the partial first derivative with respect to time). In Fig. 5 it is illustrated an example of such a strategy.

Fig. 5 A simulation of the payoff of a delta neutral strategy (lhs) with stochastic time varying implied volatility (rhs- simulated series), λ=0.98, n=1000

We simulated a time series for the implied volatility (right axis) by first defining stock returns as

where

That is we assumed stock log returns as extracted from a distribution that is a linear combination of a student’s t and a Gaussian distributions with random weights taken from a uniform distribution. This has been done in order to reproduce empirically observable features such as fat tails and volatility clustering (since we’re moving out the B&S world by letting implied volatility change over time we can forget about the normality hypothesis for stocks’ returns). The implied volatility has been simply computed as an exponential weighted moving average, that is

In particular we took as our forecast the average value for the returns’ volatility (computed as an EWMA) over 10000 observations, and traded according to the following rules: we opened our position as soon as we observed a level for implied volatility lower than two standard deviations below our (fixed) forecast value, and we closed it as soon as it crossed the one standard deviation level below the mean value, so that we traded in the range (v(a)-2σ, v(a)-σ), where the σ now indicates the standard deviation of the empirical distribution for the stock’s volatility (i.e. the ‘volatility’ of the volatility(!)). As an alternative way we could have increased the complexity of the model by introducing subjective expectations and correcting the forecasted value accordingly, in the same fashion as this is done in asset allocation models such as the Black&Litterman one.

The bottom line. So far we have briefly shown the theoretical framework behind delta hedging and volatility arbitrage, which is a way to profit from large movements in the underlying’s price without taking a directional exposure on them, in principle without even having a view on the direction of the market. This represents a nice feature of trading with derivative instruments such as vanilla options, and is made more interesting by the statistical properties of volatility that makes it in principle way easier to estimate than market expected returns. We should warn anyway that setting up such delta neutral strategies, and especially dynamically adjusting the position can be a very expensive task because of transaction costs. Delta hedging (and potentially even gamma and vega hedging) is usually performed by option market makers and large financial institution who actually write options (i.e. take a short position in the option), so that they can take advantage of large trading volumes and reduce their transaction costs. For individual investors it is more feasible to use other kind of strategies in order to bet on changes of implied volatility such as static option positions like (long/short) straddles or by taking direct exposure to volatility by trading derivatives instruments directly linked for example to the VIX index or again ETFs tracking volatility indexes. Finally, what we’ve seen so far was about taking a long position in the option and betting on underestimated implied volatility with respect to the actual future value, with all the benefits that this involved (i.e. non last a positive exposure on gamma). Empirically such situations are quite hard to find, as it usually holds the other way around, with an implied volatility that almost systematically overshoots the actual volatility rate.

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