Chapter 7 Hedging
I reported this to Mr. J. P. Morgan, adding that I was willing to gamble half the sum from my own funds. “I never gamble,” replied Mr. Morgan. Hedging is a means to minimize risk. Hedging is a form of insurance.
§1. Delta Hedging Assume that you just wrote a call option on 1,000 shares of stock. You have the risk, if the price of the stock rises. Naïve way to protect the risk: Buy 1,000 shares of stock now. This is called a covered call: write an option at the same time buy a share of stock.
Let’s look at the value of the portfolio of a covered call at expiration: If S > X, the call option will be exercised, so The portfolio value = X If S < X, the call option will be cancelled, so The portfolio value = S The net profit is C – S0 + X if S > X C – S0 + S if S < X
Why is it positive? Why is it negative? Profit C + X – S0 X S C – S0 Thus you have the following problems: (1) If the price of stock drops, you lose money, (2) Buying 1,000 shares requires you to borrow lots of money.
This is the Delta Suppose we know that for every $1 drop of stock price, the call price goes down $0.50, then we can simply buy 0.5X1,000 = 500 shares of stock to protect the risk. Hedging Rule: To hedge the sale of one call option, buy Δshares of the stock.
Example: We just sold options on 1,000 shares of stock. The strike is X = 40 and expiration T = 1 year. The volatility of the stock is σ = 0.30. Currently the stock price is S = 50 and the interest rate is r = 0.05. Thus we buy 855 shares of stock to hedge the risk. But the delta is changing continuously!
What are drawbacks in real markets? • Costs is high • Trading in smaller lot • Buying high and selling low.
S – (p + S0) when S > X X – (p + S0) when S < X §2. Methods for Hedging a Stock or Portfolio Hedging with puts Buy a share of stock and put option at the same time. (Protective put) Net payoff
Hedging with spreads Vertical spread: Buy one call option and sell another call option with higher strike but same expiration date. Discuss the net payoff at the expiration date. Horizontal spread: Buy one call option and sell another call option with longer expiration date but with the same strike price. Discuss strategies with spreads.
Hedging with collars Buy a share of stock and a put option with strike price = S0 at the same time sell a call option with strike price X> S0 but with the same expiration date. Net payoff? What if the strike price of the call < S0?
Other Strategies Butterfly spread: Buy a call with strike X1 and another call with strike X3 at the same time sell two call options with strike X2 but with the same expiration date. Strip: Buy a call and sell two puts with the same strike and expiration date. Strap: Buy two calls and sell a put with the same strike and expiration date. Strangle: Buy a call and a put with the same expiration but different strike prices.
In the same industry Hedging with paired trades Good company A Bad company B The strategy is: Buy a share of stock of Company A at the same time short a share of stock of Company B. Discuss why this is a good idea?
Correlation-based hedges Suppose Company A and Company B are negatively correlated, that is, Good news for A is bad news for B Bad news for A is good news for B. Thus if A is up, then B will be down If A is down, then B will be up. Strategy: Buy stock of both A and B. Discuss this strategy.
§3. Implied Volatility In the Black-Scholes formula all the following parameters can be easily and accurately obtained S0 = current stock price X = strike price T = time to expiration r = risk-free interest rate But the volatility σhad to be estimated from past stock price.
On the other hand, the price of the option can be obtained from the market. We thus can solve for the volatility from the Black-Scholes formula. This is called the implied volatility. In theory, the implied volatilities calculated from different options on the same stock should be the same. But, in reality, we have what is called “volatility smile”.
We already have Thus Using Black-Scholes equation and Black-Scholes formula, we also have §4. Parameters: Greeks
Thus, by Black-Scholes equation, we have We already discussed the role of Δ in hedging. • This is consistent with the observation close to the expiration: • If S > X, then C and S moves in lockstep, • If S < X, then C is almost worthless.
The role of Γ We first observe that Γ > 0, and Since Γmeasures the change in Δ, it tells us that if Γis small we can re-adjust less frequently in Δhedging.
Another role: We have, from Taylor expansion, We have Suppose originally we have
After three weeks the stock price reaches 44, Thus dt = 3/52 and dS = 44 – 43 = 1. We can estimate Which is very close to 6.276 from Black-Scholes formula.
§5. Delta Hedging a Stock Purchase Strategy: hedge the purchase 1 share by selling 1/Δ call options. Advantage: We have various type of call options to choose with different strike prices. Net expense to set up hedging is H = C(1/Δ). But we already knowΔ = N(d1). C is given by the Black-Scholes formula.
S d1 1/Δ Δ C from BS Thus we usually choose large X to make the hedging expense low. When we re-adjust the hedging as stock price changes, we see