Background to the Black-Scholes Formula

The Black-Scholes formula is used for expressing the present value of a European call option that fails to pay any dividend before the option expires. The call value is expressed by the formula in terms of the current stock price multiplied by a probability factor N(d1), minus the discounted exercise payment multiplied by another probability factor N(d2).

The stock price multiplied by the probability factor N(d1) represents the current value of receiving the stock if and only if the option finishes in the money. Meanwhile, the discounted exercise payment multiplied by N(d2) represents the current value of paying the exercise price in that event.

To briefly state it, N(d2) represents the risk-adjusted probability that the option will be exercised.

On the other hand, N(d1) multiplied by the current stock price and the riskless compounding factor equals the expected value, which is calculated using risk-adjusted probabilities, of getting the shares at option expiration in the event of the option finishing in the money. Consequently, the factor by which the present value of the stock's contingent receipt is greater than the current stock price is N(d1).

Straddles & Win Rates

The volatility of an asset is just its standard deviation of returns, which is often an annualized numerical figure. This means that if the S&P 500 Index has a volatility of 15%, it can be expected to return to ±15% about 68% of the time.

In such cases, using the average expected move, also known as straddle, is more preferred.

But what is the difference between straddle and standard deviation?

‍Turning Volatility into Straddle and Vice Versa

The at-the-money straddle approximation is a useful formula that every trader knows. Technically, this approximation is for the at-the-forward strike, not the ATM strike. Since the ATF strike is the carry-adjusted spot price, it is the strike where the call and put are equal.

Straddle =0.8*S*σ*√T

Where S = Stock Price

σ = Implied Volatility, and

T = Time to Expiry

Now, supposing that the Nifty is at ₹100, the time to expiry is 1 year, and the implied volatility is 15%, the straddle value obtained by using the formula comes to ₹12 or 12%.

Now, Straddle/S= 0.8*σ*√T

Speaking simply in annualized terms, the value of T can be assumed to be 1, thus simplifying the equation to:

Straddle as % of Spot = 0.8*σ

This, of course, means that when the annualized straddle price as a percent of spot is known, volatility can be calculated by going in reverse as follows:

σ = Straddle as % of Spot x 1.25

How is this useful? For example, if, based on a stock’s previous earnings, it is noticed that it generally moves by 5% each day, it can be said that the earning day straddle of that stock is 5%. In that case, the standard deviation can be calculated as:

5% x 1.25= 6.25%

The thus obtained standard deviation represents the volatility that can be annualized to be plugged into an options model, which, in turn, would give out a straddle price of 5%. Annualizing the standard deviation and plugging it, in this case, would give out:

6.25% x√252 = 99.2%

(Note: Instead of using 365 for a year’s time in days, 252 has been used in order to align it with the average working days of the market in a year.)

Before trying to estimate the term volatility for a much longer period, including the earning day, it is useful to figure the 1-day implied volatility out.

But how is volatility practically different from straddles?

Volatility is a figure that you enter into a model to determine the price of an actual trading instrument, which, in this case, is a straddle. For example, plugging 15% volatility in the abovementioned example, it is found that a 1-year straddle costs 12% of spot. Buying this straddle would yield a return that will be equal to the absolute value of the Nifty Index return –12%.

The worst-case scenario is that the Nifty remains unchanged and the 12% premium is, thus, lost. This brings us to expectancy and win rates.

Expectancy

Expectancy takes into account the mean size of gains and losses.The main point of the model is to produce a price that is reasonable for the specified volatility. In the abovementioned example, 12% was the fair theoretical value for an asset with 15% volatility. This means that if you pay 12% straddle on an asset that has 15% volatility, your expectancy becomes zero.

Win Rates

Expectancy and win rates are different from each other. The most one can lose is 12%; yet, as the stock has no upper limit, the potential gain is virtually limitless. Therefore, the probability of the straddle paying off balances out of its expectation.

Since wins are theoretically greater than losses, one should anticipate losing more often than winning in order for the expectancy to be zero.

A fairly priced straddle quoted as a percentage of spot incurs volatility of 80%. Since it is known that roughly 68% of a distribution falls into a 1-standard deviation range, what percent of the distribution would be encompassed in the 0.8 standard deviation range?

The cumulative distribution function obtained by applying Excel's NORM.DIST(0.8,0,1,True) formula yields 78.8%, which means that 21.2% of the time the Nifty rises by more than 0.8 standard deviations. On doubling it, since there are two tails, a win rate of 42% is obtained.

In the context of the Black-Scholes world, buying a straddle for correctly priced volatility, the expectancy becomes zero, while one can expect to lose 58% of the time.

Outside of Black-Scholes World

In the Black Scholes model, asset prices are assumed to follow a lognormal distribution. This results in compounded or log returns that are normally distributed. In this scenario, the straddle represents 80% of the annualized volatility as a percentage of spot.

In this given scenario, buying a fairly priced straddle results in a loss of 58% of the time. Being fairly priced means that the expectancy is zero, of course. But what would happen when the distribution is changed? Looking at the example of SPX, it is observed that:

• SPX goes up by 5.55% 90 percent of the time.
• On the other hand, the remaining 10 percent of the time, the SPX falls by 50%. (Source: Investopedia)
• This gives us the expected move size as: 90% x5.55% + 10% x 50% = 10%

The expected move size is the same as the straddle, which is 10% worth of spot. The expectancy from owning it is 0.

Had it been the Black-Scholes scenario, the volatility would have been 10% x 1.25 = 12.5%, which is again not annualized. But since it’s not the Black-Scholes scenario, the distribution is binary instead of a lognormal one. Now, what is the standard deviation of this binary asset?

The standard deviation can be computed as simply as we do it for coin tosses or dice throwing:

σ= √ (0.9x 0.05552 + 0.1 x 0.502) (Following Basic Standard Deviation Formula)

σ = 16.7% (again, not annualized for us to compare)

Now, the straddle is 10%, but the volatility is at 16.7%. The ratio now does not remain at 80% that was seen in the lognormal distribution; instead, it is now at 60%.

It must be noted that the steps mentioned earlier to compute the win rate cannot be repeated in this scenario as the distribution is not normal. However, in the case of binary distribution, the win rate is much easier calculated. In this example, when paying 10% of the bridge, a loss can be incurred 90% of the time.

Even after paying 6% for the straddle, a loss is still incurred 90% of the time. However, purchasing the straddle for such a low price would lead your expectancy to become greatly positive!

Conclusion

• Straddles as a percent of the spot is 80% of the volatility.
• Straddles give information about the average move.

• Fair straddles have zero expectancy.
• The win sizes are larger than the losses.
• Skewed distributions alter the relationship between win rates and expectancy while also changing the relationship between straddle prices and standard deviation.

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