In the final post of this technical series on derivatives trading we discuss using options as insurance, why the Black-Scholes model is wrong and trader psychology.

In the previous three posts we introduced options and discussed various issues involving them - from both a trading perspective and how they are priced and behave. In this final post we will discuss the use of options as an insurance product and how the volatility (vol) adds additional complexity to their behaviour.

# Volatility Effects - Options as Insurance

Until now we ignored the effect of volatility on option prices, assuming volatility does not change. We discussed in the previous article how option price and volatility are synonymous: option prices are largely quoted by vol levels as this is a more stable measure. We also discussed how the vega of an option is always positive: higher volatility means higher option prices.

When using options, we need to pay attention to two price levels: the price of the underlying and the volatility level. Both impact the profitability of the option so understanding this behaviour is important: the stock price change is directional, and the volatility change is probabilistic.

When the stock price moves, the option loses or gains value due to the delta position - calls gain from a price rise, and lose from a price drop; puts are the other way around. Highly volatile stocks have a lot of uncertainty around where the price will be at expiration - insuring movements on this stock will be more expensive.

Volatility is representation of the future uncertainty of how the underlying will move - think of it as the cost of insuring against price moves.

Furthermore, the underlying price and the volatility level are not independent: vol levels react to changes in the price of the underlying (and the market as a whole). Options are forward-looking instruments: events that impact expectations of future price changes will impact the price of the option and hence the volatility level.

To give a simple example, imagine we trade an option on SPY, an ETF that tracks the S&P 500 Index of large public companies in the US. If we treat options as insurance against price movements, then puts are insurance against price falls, and calls are insurance against price rises.

Suppose the markets have a bad day, and fall by 2.5% - a reasonably large move by the standards of the last five years or so1. What will happen to the price of options on the SPY?

In theory, it depends. In practice, most participants in the markets are long stocks2, so this fall in price is bad for them. This increases the demand for insurance, and option prices go up, bringing up volatility. When market levels drop, volatility tends to go up.

Conversely, on a good day where the share price rises, demand for insurance drops, and vol levels fall.

Thus, there is a negative correlation between stock prices and volatility3.

Thinking of an option contract as an insurance policy helps explain why it is difficult to make money from owning options: insurance is often priced above fair value so that a profit is made from selling it. This means the vol level implied in the option price is higher than the price that the underlying is moving.

If you own the option, the option expires in the money but yields less than the premium paid to buy the option - losing money overall.

On the plus side, you will not go out of business overnight, but that may be scant comfort when looking at your account balance. Almost all successful option traders make money by selling options, accepting the risk that adverse market events may prove ruinous.

Diligent, informed and prudent risk management is essential for staying in business, so understanding and anticipating what will happen is hugely important.

# Volatility Smile and Skew

In most option pricing models, the volatility of the underlying is independent of the strike price of the option. A plot of volatility against strike price should be a flat, horizontal line. I have SPY option data from May 25, 2015 for the June 26, 2015 expiration.

##      symbol       date     expiry strike callput impliedvol underlying
## 1: SPY.AMEX 2015-05-26 2015-06-26    175       C   0.369487      210.7
## 2: SPY.AMEX 2015-05-26 2015-06-26    175       P   0.270437      210.7
## 3: SPY.AMEX 2015-05-26 2015-06-26    180       C   0.326222      210.7
## 4: SPY.AMEX 2015-05-26 2015-06-26    180       P   0.246886      210.7
## 5: SPY.AMEX 2015-05-26 2015-06-26    185       C   0.283991      210.7
## 6: SPY.AMEX 2015-05-26 2015-06-26    185       P   0.226441      210.7
##    use.implied atm.implied skew.factor
## 1:    0.270437    0.122421     2.20908
## 2:    0.270437    0.122421     2.20908
## 3:    0.246886    0.122421     2.01670
## 4:    0.246886    0.122421     2.01670
## 5:    0.226441    0.122421     1.84970
## 6:    0.226441    0.122421     1.84970


Now lets plot the strike against the implied volatility and see what we get (with calls and puts in different colors). Note that the closing price for SPY on that date was 210.70 USD

We see that for both calls and puts there is 'volatility smile' rather than a flat line. We looked at all strikes for that particular expiration, but there will be almost no liquidity very far away from the money - so lets zoom in closer to view strikes from 190 to 230. These options will be more liquid and hence have more trustworthy prices:

Implied volatility is higher both below and above the stock price - rounding out when the strikes are close to and at the money. Why might this be?

Also, it is interesting that the implied vols for the calls below the strike price end up cheaper than the puts, and the reverse above the strike price. Why might this happen?

Some important issues are at play here, so I'll list them first and discuss in the following sections:

1. The Black-Scholes Model is Wrong
2. Options are Insurance

## 1. The Black-Scholes Model is Wrong

The Black-Scholes pricing model is a model for pricing options.4 It makes a number of assumptions, the most important of which is that the underlying price follows a lognormal distribution.

This assumption is known to be wrong, and has been so from the very early days of option pricing theory. As a simplifying assumption it allows us to make progress and is undeniably useful. Nevertheless, it is wrong. Stock price movements are not lognormal, they have much heavier tails.

Consequently, the Black Scholes model tends to underestimate the probability of larger price movements.

As an exercise, what are the implications for pricing options? Are there any?

We may guess that it has implications - and we would be right. If option pricing models systemically underestimate the probability of very large movements it means that option prices far away from the money are systemically priced too cheaply. What happens to the implied vol for those strikes if we put our prices up? Implied vol has an increasing relationship with price, so the implied vols at those strikes are raised.

This makes sense - as strikes move away from the money, the prices rise above the price given by the model to account for the model error. In turn, this pulls up the implied vols as we move away from the at the money strikes creating the 'volatility smile'.

## 2. Options are Insurance

Options are insurance policies against the movement of the underlying stock, so when we buy an option we buy an insurance policy.

Another way to look at this is that someone else is selling us an insurance policy - taking on the uncertain risk of the stock moving adversely for them. As the option holder, we know how much we are putting at risk: the most we can possibly lose is the premium we paid for the option. For the seller, the opposite is true, they receive money upfront but may have to pay out much, much more in the future.

As a seller of risk, a trader wants enough premium upfront to make it worthwhile. This is more straightforward when the strike price of the sold option is around the money, but if it is far away there is more uncertainty - as we discussed, real-life price movements only approximately match the assumptions of the Black-Scholes model.

In the previous post we looked at the leverage effect of options: worthless options can become valuable fast when the markets move violently - and that turmoil often arrives without much warning. Even worse, when warnings do come they are often only obvious in retrospect5.

We try to think like an option seller. Suppose the volatility of a stock is around 20% and the stock is at 100. We are approached by a broker to sell 90 strike puts that expire in a month - put options deep out of the money (often called 'deep puts' for short).

AmericanOption(type          = 'put'
,underlying    = 100
,strike        = 90
,dividendYield = 0
,riskFreeRate  = 0.01
,maturity      = 20/252
,volatility    = 0.2)

## Concise summary of valuation for AmericanOption
##  value  delta  gamma   vega  theta    rho divRho
## 0.0617     NA     NA     NA     NA     NA     NA


According to model, this option is worth about 6c. The stock is 10 USD above the strike, so the chances of this option expiring in the money are very slim, but if something bad does happen in the next 20 days, we may lose much more than the 6c we earned by selling it.

There is a price for every risk though, so suppose we decide to quote 15c. What is the implied vol at this price?

AmericanOptionImpliedVolatility(type          = 'put'
,value         = 0.15
,underlying    = 100
,strike        = 90
,dividendYield = 0
,riskFreeRate  = 0.01
,maturity      = 20/252
,volatility    = 0.2)

## [1] 0.236759


By putting the price up to 15c we are now quoting an implied vol of 23.6%. This behaviour is wholly rational. Prior to the October 1987 crash, vol curves were flat, and many institutions and traders selling options or 'portfolio insurance' (a product that functioned like an option) lost huge. They were not being compensated appropriately for the risk undertaken.

One final contributor to this behaviour is human nature and natural responses to asymmetric risk. In our previous example we priced at option at a strike of 90. What if the strike were 80?

print(AmericanOption(type          = 'put'
,underlying    = 100
,strike        = 80
,dividendYield = 0
,riskFreeRate  = 0.01
,maturity      = 20/252
,volatility    = 0.2)
,digits = 6)

## Concise summary of valuation for AmericanOption
##   value   delta   gamma    vega   theta     rho  divRho
## 4.8e-05      NA      NA      NA      NA      NA      NA


According to the model, this option is worth less than 1% of 1c. No person will ever sell that risk at that price - the amount is too small and the asymmetric nature of the risk means it is not a sensible thing to do. At a very minimum, a trader may be willing to sell it for 1c, but will probably want to do it for more. We can see what vols are implied by these prices:

AmericanOptionImpliedVolatility(type          = 'put'
,value         = 0.01
,underlying    = 100
,strike        = 80
,dividendYield = 0
,riskFreeRate  = 0.01
,maturity      = 20/252
,volatility    = 0.2)

## [1] 0.301862

AmericanOptionImpliedVolatility(type          = 'put'
,value         = 0.03
,underlying    = 100
,strike        = 80
,dividendYield = 0
,riskFreeRate  = 0.01
,maturity      = 20/252
,volatility    = 0.2)

## [1] 0.342957


To get these prices of 1c and 3c we need to raise the vol to about 30% and 34% respectively.

# Volatility Skew

We discussed the volatility smile, but we often also observe a 'volatility skew' - vol is higher on the downside strikes than at equivalent upside strikes. The curve is not symmetric around the current stock price.

Again, we stop for a minute to think why? We observe the implied vols across the strikes and see a skew toward the downside. Can we infer anything from this?

In short yes, and this is discussed further in the next section, but for now we can state that stock price movements are skewed: the tendency is to have a large number of smaller positive price moves and a smaller number of large negative moves. When the market falls, it tends to be in larger drops that happen quickly.

Lets view the distribution of returns for a stock:

spy_data_xts <- getYahooData("SPY"
,start = 19900101
,end   = 20151231
,type  = 'price')

spy_returns <- (spy_data_xts$Close / lag(spy_data_xts$Close))[-1] - 1

ggplot() +
geom_density(aes(x = as.numeric(spy_returns))) +
xlab("SPY Daily Return") +
ylab("Probability Density")


The simple density distribution is shown above, and cumulative density shown below:

ggplot() +
geom_line(aes(x = seq_along(spy_returns) / length(spy_returns)
,y = sort(as.numeric(spy_returns)))) +
xlab("Cumulative Probability") +
ylab("SPY Daily Return")


The two curves show the distribution of returns is skewed to the right slightly. The mean of the distribution is not zero - the S&P 500 tends to grow over the long run. Also, there is a concentration of density in the smaller numbers above zero, and then a fatter density below zero.

Those plots are a little surprising, the distributions are not as skewed as expected. Then again, we have been in a cyclic bull market since the early 1980s due to falling interest rates so that affects the return distribution. The very large positive days in the market occurred around the Credit Crisis of 2008, and are anomalous.

The skewed distribution of returns induces the skew in the strike/implied vol curve. The skew is how the market 'corrects' for the mismatch between the basic assumptions of option pricing models and empirical reality.

# Miscellanous Issues

Before we start to wrap everything up, there are a few phenomena to discuss.

## Volatility and Stock Price Direction

Capital markets have a natural bias towards the bull side. A large majority of investors are long equities, and herding behaviour is common and well-known in market behaviour. Fear and greed are significant drivers of price changes.

Furthermore, capital markets institutions like the investment banks and the asset market divisions of large banks have huge operations around the issuing of new equity and debt, and facilitating IPOs (initial purchase offerings) - when private corporations go public and their shares become available for public trading on listed exchanges.

This creates a natural bias for the whole financial industry to be optimistic and bullish.

Setting aside options for a moment, there's a number of general observations we can make about the movement of markets.

In a falling market, panic sets in. Most investors have long positions, and the psychological effects of loss aversion prompts selling. Trading volumes rise. This exacerbates negative moves, so negative days are larger. Unless there is a total crash, the strong selling resolves the issue faster, running its course quicker.

While larger in magnitude, the count of those negative days is smaller.

Conversely, in a rising market, complacency sets in as everyone profits and is happy. Greed sets in as investors ride the wave of rising prices and do not take profits for fear of missing out of further gains, so trade volumes fall.

From a volatility perspective, things are a little complex. It is important to draw a clear distinction between realised volatility (the volatility of the underlying price movements), and implied volatility. Recall that implied vol is the vol input to the pricing model required to obtain the price of the options on the market.

Implied and realised vols are not the same, though they are coupled. Implied vol is best thought of as a measure of the price of insurance - higher implied vols means the option market is charging more to insure market moves.

Implied vols are forward-looking in nature, often moving ahead of the realised vol. For example, suppose a company expects bad earnings, but releases results that are better than expected. Realised volatility in the period of time after the announcement (say a day or two) will be high as the news is digested, but implied volatilities likely drop. From an insurance perspective, we have less uncertainty about the company, so the cost of insuring it is lower.

Another observed phenomenon is that stock price and implied volatility are often inversely correlated - when stock prices drop, implieds go up, and vice versa. This makes sense from both a mathematical and psychological point of view.

From the mathematical point of view, when markets fall, the falls are likely to be large, and get bigger, increasing the risk. The cost of insurance goes up and implied volatilities rise.

In rising markets, we have less risk: smaller, positive rises reduce the risk of adverse events (in the short term) and so the cost of insuring it go down. Implied vols fall.

Psychologically, in falling markets people panic and want to buy insurance, caring less than they probably should about the cost - they just want the protection. Supply does not increase to match the greater demand6, so the price of insurance and implied vols rise.

In rising markets, investors get complacent and do not want insurance - why spend premium on protection you do not need? Demand drops and options sellers are willing to sell more risk as it is lower, so prices and implied vols fall.

## A Recent Real-World Example

To illustrate this phenomenon, let's look at options on SPY around the time of the mini-Flash Crash of Aug 24, 2015. The market was quiet until about a week before that, when the market started to fall. So let's look at the Sep16 options around then (options that expire on Sep 18, 2015 - the options have about a month of lifetime before expiration):

      symbol       date     expiry strike callput price impliedvol
1: SPY.AMEX 2015-08-18 2015-09-18    210       C 2.860   0.120893
2: SPY.AMEX 2015-08-18 2015-09-18    210       P 3.485   0.137099


The closing price for SPY on August 18 was 209.93, so the 210 strike is at-the-money. We see the 210 call for the 2015-09-18 expiration is 2.86 and the put is 3.49. Implied vol is around 12%.

The market fell for the rest of that week, so vol drifted up, and then on Aug 24, we had the mini-flash crash and turmoil broke loose in the markets. SPY closed at 189.55, almost 20 USD below where it was a week before. What are the prices for those options now?

      symbol       date     expiry strike callput price impliedvol
1: SPY.AMEX 2015-08-24 2015-09-18    210       C  0.16   0.212698
2: SPY.AMEX 2015-08-24 2015-09-18    210       P 22.50   0.399226


The put gained about 20 dollars in value (which is reasonable), and the call is now worth 0.16 so most of its value. Had we bought a straddle, we paid 5.35 USD for a spread that is now worth 22.66 USD.

A lot of that profit is due to the stock move though. So what do the at-the-money (ATM) options look like on Aug 24?

      symbol       date     expiry strike callput price impliedvol
1: SPY.AMEX 2015-08-24 2015-09-18    190       C 5.925   0.305885
2: SPY.AMEX 2015-08-24 2015-09-18    190       P 7.910   0.374319


The ATM strike is now 190, the call for the same expiration is 5.93 and the put is 7.91, so despite a week of time decay the ATM options are worth more than ATM options from a week ago. The price of an ATM straddle has gone from 5.35 USD to 13.84 USD. This is because vol rocketed up.

On Aug 18, ATM vol was about 12%, by the end of the day on Aug 24, it had tripled to about 36%.

The relationship between stock price and implied vol is loose, and stronger for index-linked products like ETFs. Single stocks bring more individual risk so the pattern is not as strong.

For a long volatility position, periods of a slow price grind upwards will kill your PnL - regardless of whether you own calls or puts.

## Option Portfolios

When trading options seriously, it is rare to own individual contracts. In fact, it is rare to have a trade involving an option alone. Option trades are usually tied to the underlying stock trade to hedge the deltas. It is also common to trade option spreads - combinations of contracts mentioned in the second post in this series.

Call and put spreads are especially common: we buy and sell calls or puts at two different strikes but with the same expiration for example. This allows people to take bearish and bullish positions while reducing the risk from trading a lone contract.

This begs the question: how do you manage a portfolio of options? Is it possible to aggregate positions in some way so we can look at the portfolio as a whole, rather than have to think about it every time we need to make a decision?

Thankfully, we can, and can do so easily - at least for options on the same underlying. The greeks are derivatives, and are additive: if we are long 1500 deltas as a result of one option contract and short 800 from another, our net delta position is long 700 deltas.

Similarly, gamma, theta, and vega are also additive so a portfolio of options can be analysed by adding all the greeks of all our option positions. At a glance we can see what our net positions are in the greeks, making it easier for traders and portfolio managers to make decisions on managing risks.7

That said, it is important to pay attention to the individual contracts in the portfolio - the composition of the portfolio will have second order effects.

To illustrate, suppose we buy and sell a number of calls and puts on Monday, when the underlying stock was around 100 USD, ending up with a collection of positions with strikes ranging from 90 to 110. A few days pass, we do no trades and the stock is now around 80. Regardless of what our net greek positions are, it makes sense that the stock moving back towards 100 will be very different to the stock falling further. If the stock goes up, our overall gamma will pick up, as the stock is moving back near strikes where we have positions. If it falls, our gamma will get smaller. This will not be reflected in our aggregate Greek positions as it stands.

Reading some of the work on looking at further derivatives of $V$, such approaches may capture this additional knowledge: second derivatives like Vega-delta, $\frac{d^2V}{dS d\sigma}$, but I am not yet sure of their utility. Estimating them numerically is problematic due to numerical rounding issues, etc.

I could be wrong but my intuition tells me it may be best to stick with the standard greeks and keep the composition of the portfolio in mind when doing risk assessments. This is not a position I hold strongly, and could easily change.

# Final Thoughts and Conclusions

This series covered a lot of ground! We started with an explanation of what options are, how they are traded and the infrastructure that has built up around them. We then discussed some basics of option pricing and option spreads, and described the behaviour of option prices as the values of inputs change.

There is much, much more to the topic. Options seem straightforward when explained, but the practical issues of their usage is deceptively complex. The nonlinearity of their behaviour often surprises, and it is not well understood that their value is determined along two axes: the stock price and the volatility level. It is very possible to buy a call, have the stock rise in value, and still lose money on trade!8

Options, like insurance, are a fascinating topic for anyone interested in probability and statistics - and they are not well understood. It was my aim in this series to discuss some intricacies and issues I have not seen elsewhere in quantitative finance books, but there is a whole lot more in the topic.

I did not need to use too much code for this blog post series, but it is available to anyone on request - as always, get in touch with us if you have further questions or comments or would like access to the code.

Quantitative finance is a huge topic, and one closely tied to actuarial studies so understanding the basics is very useful in insurance.

1. Historically, seeing 1-2% daily returns was common as volatility was much higher. At the time of writing - May 2016 - there is an ongoing debate about potential reasons for the low volatility of the past few years. Central bank intervention, structural changes due to the reduce role of human trading and rise of automated trading are all plausible reasons. It is possibly temporary: a return to a much higher volatility regime is highly plausible.

2. North American equity markets are naturally bullish as most investors own stocks via mutual funds and pensions funds. It is possible to be short, but that is much trickier to do and is risky, especially in single stocks.

3. This is true regardless of whether the option is a call or a put. As we discussed, calls and puts are the same apart from their delta.

4. Regular readers of this blog are likely thinking of Patsy and Monty Python right now...

5. It is very easy to fool yourself into seeing a pattern in the markets and acting accordingly. This is a particularly efficient way to lose money. It is also much easier to identify the signal from the noise after the fact.

6. Supply will likely fall as the rise in uncertainty means less traders are willing to sell the risk.

7. While usually a big fan of thinking about what you are doing, from a trading perspective you do not want the trader to have to do arithmetic when making decisions - it is too fraught with error. It is much better to give the trader as much information as he or she needs to make decisions and keep their mind focused on the larger picture.

8. As a quick spot test for the reader, can you think of a scenario where this happens? If you can, I will be pleased. It means I have managed to successfully convey the core concepts to at least one other person!