The Greek Letters University of Macedonia E Executive i MBA Financial Engineering Textbook: Hull, J.C. “Options Futures and Other Derivatives”, 6e, Prentice Hall, 2006 Achilleas Zapranis Hedging the Risk of the Issuer z If an issuer can sell an option for more than its worth and then hedge away all the risk for the rest of the option’s option s life, life he has locked a guaranteed, risk-free profit Financial Engineering The Greek Letters 2 Hedging g g the Risk of the Issuer continued z z z z A bank has sold for $300,000 a European call option p on 100,000 , shares of a non dividendpaying stock S0 = 49,, K = 50,, r = 5%,, σ = 20%,, T = 20 weeks, μ = 13% The Black Black-Scholes Scholes value of the option is $240,000 How does the bank hedge its risk to lock in a $60,000 profit? Financial Engineering The Greek Letters 3 Naked & Covered Positions • Naked position Take no action • Covered C d position iti Buyy 100,000 , shares todayy Both strategies leave the bank exposed to significant risk Financial Engineering The Greek Letters 4 Naked & Covered Positions Payoff Diagrams Profit/Loss K Profit/Loss ST Naked position K ST Covered position Financial Engineering The Greek Letters 5 Stop-Loss Strategy z z This involves: Buying 100 100,000 000 shares as soon as price reaches $50 S lli 100 Selling 100,000 000 shares h as soon as price falls below $50 This deceptively simple hedging strategy does not work well Financial Engineering The Greek Letters 6 Stop-Loss p Strategy gy continued Caveats: •The cash flows to the h h hedger d occur at different times and must be discounted •Purchases and sales cannot be made at exactly th same price the i Financial Engineering The Greek Letters 7 Delta z Delta (Δ) is the rate of change of the option price with respect to the underlying Option price Slope = Δ B A Financial Engineering The Greek Letters Stock price 8 Delta Hedging z z z z z Suppose a stock call option with Δ = 0.6 Every time that the stock price changes the option price changes by 60% Imagine an investor who has sold 20 call option contracts – that is options to buy 2,000 shares If the th price i off the th option ti is i c = $10 h he can h hedge d his position by buying 0.6 x 2,000 = 1,200 shares The gain/loss of the position would then tend to be offset by the loss/gain on the stock position Financial Engineering The Greek Letters 9 Delta Hedging z If the stock price goes up by 1 (producing a gain of $1,200 on the shares purchased), the option price will tend to go up by 0.6 x $1 = $0.60 (producing a loss of $1,200 on the options written) z If the stock p price g goes down by y 1 (p (producing g a loss of $1,200 on the shares purchased), the option price will tend to down by 0.6 x $1 = $0.60 (producing a gain of $1,200 on the options written) Financial Engineering The Greek Letters 10 Delta Hedging z z z This involves maintaining a delta neutral portfolio The delta of a European call on a stock paying dividends at rate q is N (d 1)e– qT The delta of a European put is e– qT [N (d 1) – 1] Financial Engineering The Greek Letters 11 Delta Hedging continued z The hedge g p position must be frequently q y rebalanced z Delta hedging a written option involves a “buy “b hi high, h sellll llow”” trading di rule l Financial Engineering The Greek Letters 12 Simulation of Delta hedging g g Option Closes ITM Financial Engineering The Greek Letters 13 Using Futures for Delta Hedging z The delta of a futures contract is e(r-q)T times the delta of a spot contract z The position required in futures for delta hedging is therefore e-(r-q)T times the position required in the corresponding spot contract Financial Engineering The Greek Letters 14 Delta of a Portfolio z It is the weighted average of the Δ of the individual positions in the portfolio n Δ = ∑ wi Δ i i =1 z Suppose a financial institution in the US has the following 3 positions in GBP: z z z A long gp position in 100,000 call options p with K = 0.55 and expiration date in 3 months. Δ = 0.533 A short position in 200.000 call options with K = 0.56 and p date in 5 months. Δ = 0.468 expiration A short position in 50.000 put options with K = 0.56 and expiration date in 2 months. Δ = -0.508 Financial Engineering The Greek Letters 15 Making the Portfolio Delta Neutral z Portfolio Δ: 100,000x0.533 – 200,000x0.468 – 50,000x(-0.508) = -14,900 z This means that Thi th t the th portfolio tf li can be b made d delta d lt neutral with a long position of 14,900 GBP z Another alternative is to take a long position on a 6month futures for buying14,650 y g GBP ((suppose pp that the risk-free rate in US is 8% and in UK is 4% 14,900e-(0.08-0.04)x0.5=14,605 Financial Engineering The Greek Letters 16 Theta z z Theta (Θ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time The theta of a call or put is usually negative negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of the option declines Financial Engineering The Greek Letters 17 Gamma z z z Gamma (Γ) is the rate of change of delta (Δ) with respect to the price of the underlying asset For a European call or put option on a non dividend paying stock: N ′(d1 ) Γ= Sσ T For a European p index call or p put option p with a dividend yield q: Γ= N ′(d1 )e − qT Sσ T Financial Engineering The Greek Letters 18 Gamma Financial Engineering The Greek Letters 19 Gamma Addresses Delta Hedging Errors Caused By Curvature Call price C'' C' C C Stock price S Financial Engineering The Greek Letters S' 20 Interpretation of Gamma z F a delta For d lt neutral t l portfolio, tf li ΔΠ ≈ Θ Δt + ½ Γ ΔS 2 ΔΠ ΔΠ ΔS ΔS Positive Gamma Financial Engineering The Greek Letters Negative Gamma 21 Making a Portfolio Gamma Neutral z A position on the underlying or on a future has zero Γ z The only way to change the portfolio Γ is by taking a position on wT options with ΓΤ z The portfolio Γ then becomes: wT ΓΤ + Γ Financial Engineering The Greek Letters 22 Making a Portfolio Gamma Neutral (continued) z z z Hence, the required position on the option in order for the portfolio to become gamma neutral is -Γ/ ΓΤ However, including the option in the portfolio changes the portfolio’s Δ, so the position on the underlying d l i h has to b be changed h d to maintain i i d delta l neutrality A time As ti passes, the th position iti on the th option ti iis being adjusted so that the portfolio remains gamma neutral Financial Engineering The Greek Letters 23 Making a Portfolio Gamma Neutral (Example) z z z z z Suppose that portfolio is delta neutral with Γ = 3,000 Suppose a call option with Δ = 0.62 and Γ = 1.50 The portfolio will become gamma neutral if we include a long position on 3,000/1.5 = 2000 call options However, the portfolio Δ will change from 0 to 2,000 x 0.62 = 1,240 A quantity 1,240 of the underlying asset must be sold Financial Engineering The Greek Letters 24 Relationship Between Delta, Gamma, and Theta For a portfolio of derivatives on a stock paying p y g a continuous dividend yyield at rate q 1 2 2 Θ + (r − q ) SΔ + σ S Γ = rΠ 2 Financial Engineering The Greek Letters 25 Vega z z z Vega (ν) is the rate of change of the value of a derivatives portfolio with respect to volatility Vega tends to be greatest for options that are close to the money A position on the underlying asset or on a future written on the underlying has zero vega Financial Engineering The Greek Letters 26 Vega (continued) •If V is the vega of the portfolio and VT is the vega of a traded option option, a position of of–V/V V/VT in the option makes the portfolio vega neutral •Unfortunately a portfolio that is gamma neutral will not in general be vega neutral, and vise versa •If a hedger requires a portfolio to be both gamma and vega neutral, at least two traded derivatives dependent on the underlying must be used Financial Engineering The Greek Letters 27 Vega (continued) z z z z z z Consider a portfolio that is delta neutral, with Γ = 5,000 and V=-8,000 Suppose a traded option with Γ=0.5, V=2, Δ=0.6 The portfolio can be made vega neutral by including a long position on 8,000/2 = 4,000 options This would increase Δ from 0 to 4,000 x 0.6 = 2 400 2,400 It is required 2,400 units of the asset to be sold for the portfolio to remain delta neutral Γ will change from –5,000 to -5,000 + 0.5 x 4,000 = -3,000 , Financial Engineering The Greek Letters 28 Vega (continued) z z z To make the portfolio vega and gamma neutral two options are needed Suppose that there is a 2nd option with Γ=0.8, V=1 2 Δ=0 V=1.2, Δ=0.5 5 If w1 and w2 are the quantities of the two options included in the portfolio, portfolio then it is required that -5,000 5 000 + 0 0.5 5 x w1 +0.8 +0 8 x w2 = 0 -8,000 + 2.0 x w1 +1.2 x w2 = 0 Financial Engineering The Greek Letters 29 Vega (continued) z The solution to these equation is w1 = 400 w2 = 6,000 z The Δ of the portfolio will change to 400 x 0.6 + 6,000 x 0.5 = 3,240 z Hence, 3,240 H 3 240 units it off the th underlying d l i assett mustt b be sold to maintain delta neutrality Financial Engineering The Greek Letters 30 Managing Delta, Gamma, & Vega • z Δ can be changed g by y taking gap position in the underlying To adjust Γ & ν it is necessary to take a position in an option or other derivative Financial Engineering The Greek Letters 31 Rho z z z z Rho is the rate of change of the value of a derivative with respect p to the interest rate For currency options there are 2 rhos For example, suppose a European put index with Rho = -52.67 52 67 That means that for 1% change in the risk-free i kf rate t (e.g., ( f from 8% to t 9%) the th value of the option increases by 0.5267 Financial Engineering The Greek Letters 32 Hedging in Practice z z z Traders usually ensure that their portfolios are delta-neutral delta neutral at least once a day Whenever the opportunity arises, they improve gamma and vega As portfolio becomes larger hedging becomes less expensive Financial Engineering The Greek Letters 33 Scenario Analysis A scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities Financial Engineering The Greek Letters 34 Hedging vs. Creation of an Option Synthetically z z When we are hedging we take positions that offset Δ, Γ, ν, etc. When we create an option synthetically we take positions that match t h Δ, Δ Γ Γ, & ν Financial Engineering The Greek Letters 35 Portfolio Insurance z z In October of 1987 many portfolio managers g attempted p to create a p put option p on a portfolio synthetically This involves initially selling enough of the portfolio (or of index futures) to match the Δ of the put option Financial Engineering The Greek Letters 36 Portfolio Insurance continued z z As the value of the portfolio increases, the Δ of the put becomes less negative and some of the original portfolio is repurchased As the value of the portfolio decreases decreases, the Δ of the put becomes more negative and more off the th portfolio tf li mustt be b sold ld Financial Engineering The Greek Letters 37 Portfolio Insurance continued The strategy Th t t did nott workk wellll on O October t b 19 19, 1987... Financial Engineering The Greek Letters 38 Assignment Questions Hull, Chapter 15, p. 371 Try to answer assignment questions: 15.25 Optional This presentation is an augmented version of the presentation accompanying Chapter 15 of J. Hull’s textbook “Option’s Futures and other Derivatives”, 6e, Prentice Hall, 2006, downloaded from http://www.rotman.utoronto.ca/~hull Financial Engineering The Greek Letters 39

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