Adjusting Option Pricing Models
Derivatives-based Portfolio Solutions


The Black-Scholes model assumes that an investor knows the future volatility of a stock, in addition to being able to continuously delta hedge it. In order to discover what the likely profit (or loss) will be in reality, we stretch these assumptions. If the future volatility is unknown, the amount of profit (or loss) will vary depending on the path, but buying cheap volatility will always show some profit. However, if the position is delta-hedged discretely, the purchase of cheap volatility may reveal a loss. The variance of discretely delta-hedged profits can be halved by hedging four times as frequently. We also show why traders should hedge with a delta calculated from expected – not implied – volatility, especially when long volatility.

While there are a number of assumptions behind Black-Scholes, the two which are the least realistic are the assumption of continuity and ex-ante knownledge of future realized volatility; and the ability to continuously delta hedge one's position.
We therefore investigate the four . We assume that options are European (can only be exercised at maturity), although most single-stock options are American (can be exercised at any time).

  • Continuous delta hedging with known volatility. In this scenario, the profit (or loss) from volatility trading is fixed. If the known volatility is constant, then the assumptions are identical to Black-Scholes. Interestingly, the results are the same if volatility is allowed not to be constant (while still being known).
  • Continuous delta hedging with unknown volatility. If volatility is unknown, then typically traders hedge with the delta calculated using implied volatility. However, as implied volatility is not a perfect predictor of future realised volatility, this causes some variation in the profit (or loss) of the position. However, with these assumptions, if realised volatility is above the implied volatility price paid, it is impossible to suffer a loss.
  • Discrete delta hedging with known volatility. As markets are not open 24/7, continuous delta hedging is arguably an unreasonable assumption. The path dependency of discrete delta hedging adds a certain amount of variation in profits (or losses), which can cause the purchase of cheap volatility (implied less than realised) to suffer a loss. The variance of the payout is inversely proportional to the frequency of the delta hedging. For example, the payout from hedging four times a day has a variance that is a quarter of the variance that results if the position is hedged only once a day. The standard deviation is therefore halved if the frequency of hedging is quadrupled (as standard deviation squared = variance).
  • Discrete delta hedging with unknown volatility. The most realistic assumption is to hedge discretely with unknown volatility. In this case, the payout of volatility trading is equal to the sum of the variance due to hedging with unknown volatility plus the variance due to discretely delta hedging.

In a Black-Scholes world, the volatility of a stock is constant and known, while the trader is also able to continuously delta hedge. In each unit of time, the stock can either go up or down. As the position is initially delta-neutral (ie, delta is zero), the gamma (or convexity) of the position gives it a profit for both downward and upward movements. While this effect is always profitable, the position does lose time value (due to theta). If an option is priced using the actual fixed constant volatility of the stock, the two effects cancel each other and the position does not earn an abnormal profit or loss as the return is equal to the risk-free rate. There is a very strong relationship between gamma and theta (theta pays for gamma)

Profit from delta hedging is equal to the difference between price and theoretical price
The theoretical price of an option, using the known volatility, can be extracted by delta hedging. Should an option be bought at an implied volatility less than realized volatility, the difference between the theoretical price and the actual price will equal the profit of the trade. As there is no path dependency, the profit (or loss) of the trade is fixed and cannot vary.

As theta and gamma are either both high or both low, profits are not path dependent
If a position is continuously delta hedged with the correct delta (calculated from the known future volatility over the life of the option), then the payout is not path dependent. The cause of this relationship is the fact that, while an ATM option would earn more compared to an OTM option due to delta hedging, the total theta cost is also higher (and exactly cancels the delta hedging profit).

Profits are path independent, even if volatility is not constant (but still known)
While Black-Scholes assumes a constant known volatility, there are similar results for non-constant known volatility. This result is due to the fact that a European option payout depends only on the stock price at expiry. Therefore, the volatility over the life of the option is the only input to pricing. The timing of this volatility is irrelevant.

As it is impossible to know in advance what the future volatility of a security will be, the implied volatility is often used to calculate deltas. Delta hedging using this estimate causes the position to have equity market risk and, hence, it becomes path dependent (although the average or expected profit remains unchanged). Figure 91 below shows that the profits from delta hedging are no longer independent of the direction in which the underlying moves. The fact that there is a difference between the correct delta (calculated using the remaining volatility to be realised over the life of the option) and the delta calculated using the implied volatility means returns are dependent on the direction of equity markets.

If implied volatility = realised volatility, profits are path independent
If the implied volatility is equal to the realised volatility, then the estimated delta calculated from the implied will be equal to the actual delta (calculated from the realised). In this case, profits from hedging will exactly match the theta cost for all paths, so it is path independent.

With continuous hedging, buying a cheap option is always profitable
If there is a difference between the actual delta and estimated delta, there is market risk but not enough to make a cheap option unprofitable (or an expensive option profitable). This is because in each infinitesimally small amount of time a echeap option will always reveal a profit from delta hedging (net of theta), although the magnitude of this profit is uncertain. The greater the difference between implied and realised, the greater the market risk and the larger the potential variation in profit.

While assuming continuous delta hedging is mathematically convenient, it is impossible in practice. Issues such as the cost of trading and minimum trading size make continuous trading impossible, as do fundamental reasons, such as trading hours (if you cannot trade 24 hours then it is impossible to trade overnight and prices can jump between the close of one day and start of another) and weekends.

Discrete hedging errors can be reduced by increasing the frequency of hedging
The more frequent the discrete hedging, the less variation in the returns. If 24-hour trading were possible, then with an infinite frequency of hedging with known volatility the returns converge to the same case as continuous hedging with known volatility (ie, Black-Scholes). In order to show how the frequency of hedging can affect the payout of delta hedging, we shall examine hedging for every 5% and 10% move in spot:
  • Hedging every 5% move in spot: If an investor delta hedges every 5% move in spot, then an identical profit is earned if the underlying rises 10% as if the underlying rises 5% and then returns to its starting point. This shows that the hedging frequency should ideally be frequent enough to capture the major turning points of an underlying.
  • Hedging every 10% move in spot: If the investor is hedged for every 10% move in the underlying, then no profit will be earned if the underlying rises 5% and then returns to its starting point. However, if the underlying rises 10%, a far larger profit will be earned than if the position was hedged every 5%. This shows that in trending markets it is more profitable to let positions run than to re-hedge them frequently.

Hedging error is independent of average profitability of trade

As the volatility of the underlying is known, there is no error due to the calculation of delta. As the only variation introduced is essentially ‘noise’, the size of this noise, or variation, is independent from the average profitability (or difference between realized vol and implied vol) of the trade.

With discrete hedging, cheap options can lose money
With continuous delta hedging (with known or unknown volatility) it is impossible to lose money on a cheap option (an option whose implied volatility is less than the realized volatility over its life). However, as the error from discrete hedging is independent from the profitability of the trade, it is possible to lose money on a cheap option (and make money on an expensive option).

Hedging error is halved if frequency of hedging increased by factor of four
The size of the hedging error can be reduced by increasing the frequency of hedging. An approximation (shown below) is that if the frequency of hedging is increased by a factor of four, the hedging error term halves. This rule of thumb breaks down for very high-frequency hedging, as no frequency of hedging can eliminate the noise from non-24x7 trading (it will always have noise, due to the movement in share prices from one day’s close to the next day’s open)

The most realistic assumption for profitability comes from the combination of discrete delta hedging and unknown volatility. Trading hours and trading costs are likely to limit the frequency at which a trader can delta hedge. Equally, the volatility of a stock is unknown, so implied volatility is likely to be used to calculate the delta. The variation in the profit (or loss) is caused by the variation due to discrete hedging and the inaccuracy of the delta (as volatility is unknown).

As an example, let us assume a Dec08 SX5E ATM straddle was purchased in April 2008. In theory, it should be very profitable as the realised volatility of 39% was more than 50% above the 25% implied. However, most of the volatility came after the Lehman bankruptcy, which occurred towards the end of the option’s life. If implied volatility was used to calculate the delta, then the time value would be assumed to be near zero. As equity markets had declined since April, the strike of the straddle would be above spot, hence we would have a delta -100% (the call would be OTM with a delta 0, while the put would be ITM with a delta ≈ 100%). To be delta-hedged, the investor would then buy 100% of the underlying per straddle. If the delta was calculated using the actual volatility (which was much higher), then the time value would be higher and the delta greater than -100% (eg, -85%). As the delta-hedged investor would have bought less than 100% of the underlying per straddle, this position outperformed hedging with implied volatility when the market fell after Lehman collapsed (as delta was lower, so less of the underlying was bought).

Hedging with delta using implied volatility is bad for long volatility strategies

Typically, when volatility rises there is often a decline in the markets. The strikes of the option are therefore likely to be above spot when actual volatility is above implied. This reduces the profits of the delta-hedged position as the position is actually long delta when it appears to be delta flat. Alternatively, the fact that the position hedged with the realized volatility over the life of the option is profitable can be thought of as due to the fact it is properly gamma hedged, as it has more time value than is being priced into the market. Hence, if a trader buys an option when the implied looks 5pts too cheap, then the hedge using delta should be calculated from a volatility 5pts above current implied volatility. Using the proper volatility means the profit is approximately the difference between the theoretical value of the option at inception (i.e., using actual realized volatility in pricing) and the price of the option (i.e., using implied volatility in pricing).
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