Hedging: Setting the stage
Example
Consider a financial institution that has sold for $300,000 a European call option on 100,000 shares on a non-dividend paying stock.
- Spot stock price (): 49
- Strike price (): 50
- Interest rate (): 5%
- Volatility (): 20%
- Time-to-maturity (): 20 weeks
Using the BS model, we can get the theoretical value is $240,000 while the call was sold for $60,000 more than its throretical value.
Then the institution is faced with the problem of hedging the option risk: there are three alternative strategies: Naked position, covered position, stop-loss stategy
Naked (or uncovered) position
- Do nothing (don’t hold any stock position)
If the stock price at maturity is low (below than ), the European call will not be exerciesed and the profit is $300,000.
If the stock price at maturity is above the call will be exercised. The financial institution needs to buy 100,000 shares at maturity.
If , the financial institution will suffer losses.
Covered positions
- Buy 100,000 shares immediately
If the stock price is high (above ), the European call will be exercised but there’s no loss.
If the stock price is low (below ), the European call will not be exercised but there’s is loss: For example, , then the financial institution will suffer losses of
Stop-loss strategy
Strategy:
- Buying 100,000 stocks if rises above
- Selling 100,000 stocks if falls below
This strategy ensures that the institution holds the stock, if the option matures ITM and does not hold the stock if the opetion matrues OTM
The ideal of this strategy is which is fixed at the beginning of the trading. But there are transaction costs which make the strategy not work perfectly. The stop-loss strategy does not work well in practice because if the stock prices crosses many times, the strategy is expensive.
Greeks & Hedging
Greeks
Greeks are sensitivity factors: Partial derivative of asset/portfolio price w.r.t. a variable that it depends on.
Partial derivatives useful for hedging purposes:
- Delta: ,
- Gamma: ,
- Vega: ,
- Theta: ,
- Rho: ,
Delta
Dleta () the rate of change of the derivative price, with respect to the price of the underlying asset
Delta & Black-Scholes world
When deriving the Black-Scholes p.d.e we set up a riskless portfolio: buy units of the stock & sell one option → delta-neutral portfolio.
From the call-put parity relation:
Delta Hedging
Construct a delta-neutral portfolio whose delta is equal to zero
Let a portfolio consist of assets whose price depends on a single asset with a price .
where is the quantity (not the proportion) & is the delta of the i-th asset. For delta hedging purposes, we choose appropriately to make .
Consider a portfolio consisting of one option and stocks:
The portfolio delta is
Since we have one option in the portfolio, i.e. . For stock we have that: . Thus, . Therefore, delta heding involves selling stocks for each option held.
Note:
- Delta is a dynamic measure and changes with
- Delta-neutrality is only maintained for a short time (instant), thus the hedge has to be adjusted periodically
- Delta heding of a written option involves a ‘buy high, sell low’ trading strategy which is very risky
Example 1
An investor sells 20 calls on a non-dividend paying stock. Each call is written on 100 stocks, call price c=10. Call delta (for long position)
To delta-hedge this position, we buy/sell stocks:
that is, to delta-hedge we buy 1200 stocks. And the gain (loss) from the option is equal to the loss (gain) from the stocks
Example 2
A trader owns:
- A long position in 100,000 calls with a Delta of 0.533
- A short position in 200,000 calls with a Delta of 0.468
- A short positoin in 50,000 puts with a Delta of -0.508
The Delta of this portfolio is:
To delta hedge we need to buy/sell stocks
Example 3
A financial institution sold for 300000 European call options on 100000 shares on a non-dividend paying stock: week
Assume that stock prices follow a GBM, then
- Buy 0.522 stocks per option sold, i.e. 0.522*100,000=52200 stocks
- Total cost for buying the stocks: 52200*49=2557800
- Weekly interest:
After a week, the stock price drops to 48.12:
- Now
- The financial institution sells (0.522-0.458)·100000=6400 stocks
- The borrowing is reduced to
Gamma
Gamma () is the rate of change of , with respect to the price of the underlying asset
Gamma is rate of change of delta as changes:
- Large Gamma: Delta is sensitive to price changes ⇒ frequent re-balancing of the portfolio is necessary
- Small Gamma: Delta is not very sensitive to price chagnes ⇒ infrquent re-balancing of the portfolio is necessary
- Gamma is maximised near-the-money
- Gamma takes the lowest value as away from the money
Delta heding error
Stock price changes from to
Delta hedging assumes that the option price changes from to . However, in reality the option price moves from to
- The hedging error
- HE depends on the curvature of the relationship between option & stock price
- The curvature is measured by gamma
Example: B-S model
Assume call option value follows Black-Scholes model: . Assume that
From the B-S model we have:
Delta-neutral portfolio: Buy one option & sell 0.672 stocks
Assume stock price increases by 0.1:
- The call price now i.e. higher by 0.067
- The short stock position is worth less 0.672·0.1= 0.0672 ⇒ still good hedge
Assume stock price increases by 10:
- The call price now i.e. higher by 7.875
- The short sotck position is worth less 0.672·10 = 6.72 ⇒ Bad hedge
The problem is that the position is not Gamma-neutral
Gamma-neutral portfolio
Let a portfolio consist of assets whose price depends on a single asset with a price :
Heding Gamma risk ⇒
Since the underlying asset (e.g. the stock) has a Gamma of zero, it cannot be used to hedge Gamma risk. We can use another derivative whose Gamma is to hedge.
Buy/sell options that have a gamma of , then the combined position has a Gamma of
Adjusting Gamma changes the Delta of the portfolio → Delta corrections
The Gamma-hedge needs to be re-balanced
Note:
- Delta-neutrality: Protection against small changes in
- Gamma-neutrality: Protection against large changes in
Vega
Vega () is the rate of change of the value of the derivative w.r.t. a change in the underlying (asset) volatility
Large vega → the derivative is very sensitive to changes in the volatility
Vega is biggest for at-the-money options
In B-S model: where
Vega-neutral portfolio
Let a portfolio consist of assets whose price depends on a single asset with a price
Heding vega risk →
Since underlying asset (e.g. the stock) has a vega of zero, it cannot be used to hedge vega risk.
Assume we hold a portfolio that has a vega of . We can buy/sell options that have a vega of . The combined position has a vega of:
Note:
- A gamma-neutral portfolio is not necessarily vega-neutral
- Vega-neutrality: Protection against changes in
simultaneous neutrality
Consider a delta neutral portfolio (i.e. ), with and . There are two options that can be traded:
- Option 1:
- Option 2:
Aim: construct a portfolio having zero gamma and vega: add units of option1 and units of option2 to initial portfolio.
Hence: and . We then get
The delta of the portfolio now is
To make the portfolio delta-neutral again, we buy/sell units the underlying asset:
that is, we sell 3240 units of the underlying asset.
Theta
Theta () is the rate of change of the value of the derivative w.r.t time
Theta is also referred to as time decay
In a Black-Scholes world:
where
Hedging in practice
- Traders ensure that their portfolio is delta-neutral at least once a day
- Whenever the opportunity arises, they improve gamma and vega, but it’s difficult to find options that can be traded in the volume required at competitive prices
- As portfolio becomes larger, hedging becomes less expensive because of the economies of scale in hedging
Implied Volatility
The Black-Scholes option prices: , we can observe but not
Implied volatility (IV): The volatility for which the Black-Scholes price is equal to the market price:
There is no closed-form solution, we can use numerical methods or trail & error.
The Numerical search
Methods:
- Obtain data on: option price (c or p), underlying asset price (), strike (), interest rate (), time-to-maturity ()
- Start with an arbitrary (but reasonable) volatility guess (e.g. the standard deviation of the underlying asset)
- Calculate the model option price and compare it with the market price
- Because c and p are both positive proportional to
- Thus, if , then we have to increase the guess and vice verse
Put-call parity
- for BS prices:
- for Market prices:
Subtracting the two equations from one another:
Considering the same strike (K) and time-to-maturity (T):
- Puts and calls should have identical dollar pricing errors
- The IV derived from must be the same as the from
- The volatility smile derived from must be the same as that from
- The volatility term structure derived from must be the same as that from
Volatility smile & Volatility smirk (skew)
Volatility smile is the relation between volatility and strike. (Volatility term structure: relation between time-to-maturity and strike)
In the Black-Scholes model, we assumes that is constant, both across
But in practice, the IV has a “smile”
It indicates that the Black-Scholes model cannot be true. Besides, IV is relatively low for at-the-money options and it increases as we move in- or out-of-the-money.
Several hypotheses explain the existence of volatility smiles. The simplest and most obvious explanation is that demand is greater for options that are in-the-money or out-of-the-money as opposed to at-the-money options.
However, for equity options (especially after 1987), the smile changes to volatility smirk
One straight explanation to this is that people (especially in stock market) prefer to hold long position and they do not hope the price goes down, thus the put option whose strike price is smaller than its spot price is much demanded. The demand increases the price and since we know IV is positively proportional to the option price, the IV is also higher. Similarly, for some assets people do not want their price go up, the option’s volatility smile will change to the “reverse smirk”.
Implied distribution
Using the risk-neutral valuation, in discrete situation:
- Binomial tree: is the risk-neutral probability of an up move
- is the risk-neutral expected payoff
- is the value of the option
In continuous time, the risk-neutral expected payoff is
First derivative of the option value w.r.t. is
Second derivative of the option value w.r.t. is
Hence:
Assume are the prices of T-year European call options with strike prices of , an estimation of is
Proof:
construct the portfolio by longing and short , the value of the position is
The value of the position can also be calculated by integrating the payoff over the risk-neutral probability distribution and discount it at the risk-free rate.
Since is small, we can assume that in the whole range where the payoff is nonzero. The area is thus the integration is also and the payoff is .
Therefore,
And then we can use numerical methods to simulate the distribution. [ Reference Method ]
With we can get : for discrete situation, it’s just the cumsum. And with more samples, we can get the continuous cases which shows (same methods, different results for smile type and smirk type)
For volatility smiles:
- Both tails are “fatter” than the log-normal distribution (derivated by B-S model)
- Implied distribution is “more peaked” than the log-normal one
For volatility smirks:
- The left tail is heavier & the right tail is less heavy than in the log-normal case
Implied volatility term structure
In addition to calculating a volatilit smile, traders also calculate a term structure for implied volatility.
The term structure shows the variation of implied volatility with time-to-maturity.
The volatility term structure tends to be downward-sloping when volatility is high and upward sloping when it is low
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