Lecture 5

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Lecture 5

  1. 1. LECTURE FIVE a. Hedging Linear Risk b. Optimal hedging in linear risk 1
  2. 2. Part 1HEDGING LINEAR RISK a. Overview b. Basis Risk 2
  3. 3. 1. Overview • Risk that has been measured can be managed •Taking positions that lower the risk profile of the portfolio Our main goal will be: find the optimal position that minimize variance of the portfolio or limit the VaRPart 1. Hedging Linear Risk - Intro Then our portfolio consists of two positions: asset to be hedge & hedging instrument Initial consideration Short Hedge: Long Hedge A company that knows that it is due to •: A company that knows that it is due sell an asset at a particular time in the to buy an asset at a particular time inLecture 5 future the future Hedge by taking a short futures Hedge by taking a long futures position position
  4. 4. 1. Overview How hedge can be? Static hedging • Consists of setting and leaving a position until maturity of asset or contract.Part 1. Hedging Linear Risk - Intro • Appropriate if the hedge instrument is linearly related to the underlying asset price Dynamic hedging • Consists of continuously rebalancing the portfolio. • Associated with options which have non linear payoffs in the underlyingLecture 5 Hedging limits the losses, but also the potential profits. Only makes the outcome more certain – Risk management focus
  5. 5. 1. Overview Example US exporter who has been promised a payment of ¥125 millions in 7 monthsPart 1. Hedging Linear Risk - IntroLecture 5
  6. 6. 2. Basis Risk • Definition: Basis = Spot – Future • Occurs • when the hedge horizon does not match the time to futures expirationPart 1. Hedging Linear Risk - Intro • when the characteristics of the futures contract differ from those of the underlying. Some details • For investments assets ( stock indices, gold and silver, etc) the basis risk tends to be small., because there is a well-defined relationship between the future price and the spot price F0T=ertS0 • For commodities supply and demands effects can lead to large variation in the basisLecture 5 • Cross hedging, using a futures contract on a totally different asset or commodity than the cash position. Basis can be large
  7. 7. 2. Basis Risk • Some details (cont) Additionally of the assets, it is important to consider: • The choice of the delivery monthPart 1. Hedging Linear Risk - Intro • No the same date • Not the same volatility: Assets Volatility DateLecture 5
  8. 8. Part 2OPTIMAL HEDGING INLINEAR RISK a. Hedge Ratio: overview b. The model c. Regression analysis approach d. Applications of linear hedging 8
  9. 9. 1. Hedge Ratio - Overview Definition The “hedge ratio” is the ratio of the size of the position taken in futures contracts to the size of the exposure (up to now we havePart 2. Hedging Linear Risk – Hedge Ratio assumed a hedge ratio = 1). Or, how many future contracts to hedge a position A model of a single portfolio is considered with a known variance and size. To hedge the risk, (e.g. reduce the variance, of the portfolio only one assets is available). This assets is called the hedge instrument. The variance and correlation with the portfolio of hedge instrument is known.Lecture 5 Historic data could be used to compute the relevant variances and correlation or one could opt to use current market consensus
  10. 10. Part 2. Hedging Linear Risk – Hedge Ratio 1. Hedge Ratio - Overview Two scenarios Values of the portfolios: • Owns the product and sells the future • portfolio value is (S - hF) { h because hedges the position • change in value of the portfolio is ΔS - hΔF • Buys the future and is short the product • portfolio value is s hF – S • change in value of the portfolio is hΔF – ΔSLecture 5
  11. 11. Lecture 5Part 2. Hedging Linear Risk – Hedge Ratio 2. Hedge Ratio – The model
  12. 12. 2. Hedge Ratio – The model The model • Unique asset • To be hedged (reduce the variance) with one hedging instrumentPart 2. Hedging Linear Risk – Hedge Ratio • Variance and correlation of both instruments are known using historic data • Size of portfolio: w1 • Size of hedging instrument: w2 • Standard deviation of w1 is σ21 • Standard deviation of w2 is σ22 The variance of the un-hedged portfolio will be : The total variance (including asset and hedging instrument) will be: The hedge instrument isLecture 5 added to reduce variance or eliminate it al together
  13. 13. 2. Hedge Ratio – The model To find the optimum for the hedge instrument we just have to find the first derivative with respect to w2 (associated with the hedgingPart 2. Hedging Linear Risk – Hedge Ratio instrument) FOC to find the Vh minimum W2  2w2 2 2  2 w1 1 2  0 w2 =0 To find the optimal position in the hedge instrument, set equation equal to zero and solve for w2 1 w2   w1 Solve for W2 2 We can find that this is a minimum because  2Vh Minimum !!  2 2 2  0Lecture 5 w2 2 Second derivative is greater than zero
  14. 14. 2. Hedge Ratio – The model 1 w2   w1 2Part 2. Hedging Linear Risk – Hedge Ratio • The closer ρ is to one, and the larger is the variance of the product you are hedging, • the more you hedge • The larger is the variance of the product used to hedge the lower the hedge ratio. • It is even possible that h would be greater than 1.Lecture 5
  15. 15. 2. Hedge Ratio – The model 1 w2   w1 2Part 2. Hedging Linear Risk – Hedge Ratio Hull (2005) and Kocken (1997) proved this finding How? The variance using the hedge should be less than the variance without hedge Substitute the w2* in our initial equation Where w2 is the hedging instrument Obtain:Lecture 5 So, we have 1. Variance of portfolio with no hedge 2. Variance of portfolio with hedge Is this hedge efficient?
  16. 16. 2. Hedge Ratio – The model Hull (2005) and Kocken (1997) proved this finding To check that our finding is accurate, compare the Variance including the hedging instrument with the variance without hedging instrumentPart 2. Hedging Linear Risk – Hedge Ratio The mathematical reduction leads to: H  2 So, this hedging is indeed effective in reducing variance !!!Lecture 5
  17. 17. 2. Hedge Ratio – The model Hull (2005) and Kocken (1997) proved this finding The proposed hedging strategy using the optimal hedge ratio can be compared to a less optimal, to illustrate the case:Part 2. Hedging Linear Risk – Hedge Ratio Take the opposite position! Not very optimal Solve for w2 (that is the hedging instrument) w1 1 w2   2 And as done previously, use w2 in the formula for variance The variance of the portfolio including hedge is given by:Lecture 5
  18. 18. 2. Hedge Ratio – The model Hull (2005) and Kocken (1997) proved this finding Again, I can compare both variancesPart 2. Hedging Linear Risk – Hedge Ratio Rearranging the equation leads to: We have to models to compare: 1 w1 1 w2   w1 w2   2 2Lecture 5 Kay point is H  2 correlation!!
  19. 19. 2. Hedge Ratio – The model Hull (2005) and Kocken (1997) proved this finding The reduction in variance is given byPart 2. Hedging Linear Risk – Hedge Ratio H  2 •Both are equivalent when p =1 (when correlation between the portfolio to be hedged and the hedge instrument is prefect.) •BUT once correlation drops to 0.5 the linear strategy does not yield any variance reduction, while the optimal strategy still produce some reduction is variance. Variance goes to zeroLecture 5 Variance is far from zero
  20. 20. Part 2. Hedging Linear Risk – Hedge Ratio 2. Hedge Ratio – The model Conclusion • Using a simple model it was shown that various hedging strategies can influence the total variance reduction • In our case an optimal hedging ratio was found for a simple model. There is no reason why this same technique wouldnt work form more complicated models.Lecture 5
  21. 21. 2. Hedge Ratio – The model Example Airline company knows that it will buy 1million gallons of fuel in 3 months.Part 2. Hedging Linear Risk – Hedge Ratio S • St. dev. of the change in price of jet fuel is 0.032. h   • Hedger could be futures contracts on heating oil (St. dev is F 0.04 • ρ =0.8 S 0.032 h  0.08  0.64 This is the ratio! F 0.04 • One heating oil futures contract is on 42,000 gallons. 1000000 0.64  15.2 42000Lecture 5
  22. 22. 3.The Regression Analysis approach It is also possible to estimate the optimal hedge using regression analysis. The basic equation isPart 2. Hedging Linear Risk – Hedge Ratio S    hF Remember the role of r2 Using OLS theory, it is known that beta is  xy   xy  2   x  y  y So beta (the hedge instrument) will be What is this expression? This is the solution to the minimizing the original objective functionLecture 5
  23. 23. 3.The Regression Analysis approach The regression analysis approachPart 2. Hedging Linear Risk – Hedge Ratio It is useful to note that the regression analysis also provides us with some information as to how good a hedge we are creating. The r-square of the regression tells how much of the variance in the change in spot price is explained by the variance in the change of the futures price.Lecture 5
  24. 24. 3.The Regression Analysis approach An additional considerationPart 2. Hedging Linear Risk – Hedge Ratio Futures hedging can be successful in reducing market risk BUT They can create other risks • Costs and daily balance: Futures contracts are marked to market daily , they can involve large cash inflows or outflowsLecture 5 • Liquidity problems, especially when they are not offset by cash inflows from the underlying position
  25. 25. Part 2. Hedging Linear Risk – Hedge Ratio 4. Application of this linear hedging Duration Hedging Beta HedgingLecture 5
  26. 26. 4. Application of this linear hedging Duration Hedging The modified duration is given by: P  ( D * P)yPart 2. Hedging Linear Risk – Hedge Ratio Dollar duration Duration for the cash and future positions S  ( DS * S )y •Duration for each asset •Where S, F are quantities F  ( DF * F )y of S and F Variances and covariances are:  2 S  ( DS * S ) 2  2 (y )  2 F  ( DF * F ) 2  2 (y )Lecture 5  SF  ( DF F )( DS S ) 2 (y ) * *
  27. 27. 4. Application of this linear hedging Duration Hedging And using the expression that we already found: 1  1, 2Part 2. Hedging Linear Risk – Hedge Ratio w2   w1  2 Why? 2 2  SF * * ( DF F )( DS S ) * DS S h*   2    *  F * ( DF F ) 2 DF FLecture 5
  28. 28. 4. Application of this linear hedging Duration Hedging  SF * * ( DF F )( DS S ) * DS S h*   2    *  F *Part 2. Hedging Linear Risk – Hedge Ratio 2 ( DF F ) DF F Example: Portfolio 10M Duration 6.8 years Time to be hedged: 3 months Future price: 93-02 Notional: $100.000 Duration: 9.2 a. Notional of the future contract b. Number of contracts This is just convert 93-02  2 93  Lecture 5 6.8 * $10,000,000  h*    79.4  32  *100.000  93,062.5 9,2 * $93,062.05 100
  29. 29. 4. Application of this linear hedging Beta Hedging Beta, or systematic risk, can be viewed as a measure of the exposure of the rate of return on a portfolio i to movements in the “market”:Part 2. Hedging Linear Risk – Hedge Ratio Where • β represents the systematic risk • α - the intercept (not a source of risk) • ε - residual. It is easy to interpret the β as: The change of the spot is a function of the sensitivity to the market movement (beta and the And solving for ΔS and ΔF in the ΔV formula change of market)Lecture 5
  30. 30. 4. Application of this linear hedging Beta HedgingPart 2. Hedging Linear Risk – Hedge Ratio When N* = Δ(S / F), ΔV=0 So: The optimal hedge with a stock index futures is given by beta of the cash position times its value divided by the notional of the futures contract.Lecture 5
  31. 31. 4. Application of this linear hedging Beta HedgingPart 2. Hedging Linear Risk – Hedge Ratio Example Portfolio : $10,000,000 Beta : 1,5 (SPX V Stock) Current future prices 1400 Multiplier : 250 a. Notional of futures contract $250 x 1400 = $350.000 b. Number optimal of contractsLecture 5 1.5 * $10,000,000   42.9 1* $350,000
  32. 32. 4. Application of this linear hedging ConcludingPart 2. Hedging Linear Risk – Hedge Ratio REASONS FOR HEDGING AN EQUITY PORTFOLIO • Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.) • Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will outpeform the market.)Lecture 5
  33. 33. Part 3HEDGING NON LINEARRISK a. Initial considerations (pricing) b. From Black-Scholes to the Greeks c. Delta d. Theta e. Gamma f. Vega g. Rho 33
  34. 34. 1. Initial considerations (Pricing) Price of a call is a function of 1. Stock price Price of a stock is a function of 2. Interest rate 3. Volatility f t  f ( St , rt ,  t , K , ) 4. Strike price 5. Time The idea of pricing is finding f when the parameters change Example One monthPart 3. Hedging NON-Linear Risk S1=$95 K1=$30 (95-65) S0=$75 K=65 S1=$63 K1=$0 (63-65) Riskless Hedge Approach Option price? We find the optimal number of stocks to make both portfolios • Riskless portfolio: find the number of stocks “ϕ” equivalent $95ϕ-30Lecture 5 $95ϕ-30=63 ϕ – 0 S0=$75 Number of stocks $63ϕ-0
  35. 35. 1. Initial considerations (Pricing) Value of portfolio in three months $95ϕ-30 = 59.06 = 63 ϕ – 0 They are equivalent One leg Other leg Payoff will be the same for both scenarios $30  0   0.9375 Number of stocks $95  $63Part 3. Hedging NON-Linear Risk Payoff regardless the price Payoff of the of stock at t+1 0.9375 * 75 – 30 = 59.06 portfolio t+1 Then, the price of a call considering the present value of the call assuming r=6% and 1 month Transform the payoff to t 0.9375 * $75 - C = $59.06 * e –Rf*TLecture 5 0.9375 * $75 - C = $59.06 * e 0.06 x 0.833 C = $11.54
  36. 36. 1. Initial considerations (Pricing) Risk neutral approach • Each path has their own probability • We try to estimate these probabilities for a risk neutral individual and then use these risk neutral probabilities to price a call option. S0=95 pPart 3. Hedging NON-Linear Risk S0=75 1-p S0=63 •For a risk neutral investor, the current stock price is the expected payoff discounted at the risk-free rate of interest (Rf=6%) and T=0.083 (month)  R f *T 75  [95Pu  (1  Pu )63)] * e Generalization Today’s stock price is the S0  ( Pu Su  (1  Pd )Sd )e rt result of both legs •It is possible solve PuLecture 5 75e Rf *T  63 This is the risk neutral probability of Pu   0.387 the stock price increasing to $95 at 95  63 the end of the month
  37. 37. 1. Initial considerations (Pricing) A risk neutral individual would assess a 0.387 probability of receiving $30 and a 0.613 probability of receiving $ 0 Su=95 P=0.387 Cu=30 S0=75 C =65 1-P = 0.613 Sd=63 Cd=0Part 3. Hedging NON-Linear Risk The price of a call will have the same idea than in the previous slide C0  ( Pu Cu  (1  Pd )Cd )e rt C0  [0.384 * 30  (0.613) * 0]e0.6*0.83 The price of a call is the same! C = $11.54Lecture 5
  38. 38. 2. From Black-Scholes to the Greeks Using the Black – Scholes we know that the price of a call option depends on: •Price of the underlying asset (S) •Strike price (K) •Time to maturity, (T) •Interest rate, (r) andPart 3. Hedging NON-Linear Risk •Volatility, The first order approximation shows the effect of price when change some factors This show the effect of varying each of the parameters, S,T, r and σ by smallLecture 5 amounts δS, δT, δr and δσ, with K fixed. So each of the partial effect is given by a Greek letter
  39. 39. 2. From Black-Scholes to the Greeks Each of the partial effects is given a Greek letterPart 3. Hedging NON-Linear Risk Delta ∆ = δΠ/ δS Option price changes when the price of the underlying asset changes Theta Θ=- δΠ/ δT Option price changes as the time to maturity decreases. Rho ρ = δΠ/ δr Option price changes as the interest rate changes Vega ν = δΠ/ δσ Option price changes as the volatility changesLecture 5 Gamma Γ = δ2Π/ δS2 Measures the rate of change of the options as the price of the underlying changes (Acceleration) by a Greek letter
  40. 40. 3. Delta ∆ Delta (∆): how much will the price of an option move if the stock moves $1 • Delta varies from node to node • Defined as the first partial derivative with respect to price Part 3. Hedging NON-Linear Risk  S where is the option price and S is underlying asset price. • However, the relationship between option price and stock price is not linear. WHY?????Lecture 5 • Intuition the option costs much less than the stock!!
  41. 41. 3. Delta ∆ Relation Delta (∆), at/in/out the money Values of delta Delta   N(d1 ) Close to 1 when goes deep in the money Delta is close to 0.5 Close to 0 when goes deep out the money SPart 3. Hedging NON-Linear Risk Close to 0 when goes deep out the money Delta is close to -0.5   N(d1 )  1  Calls have positive delta (0 < C < 1) If the stock price goes up, the price for the callto -1 when goes deep in the money Close will go up.  Puts have a negative delta (-1< P < 0) If the stock goes up the price of the option will go down.  So...as expiration nears,Lecture 5 Delta for in-the-money calls will approach 1, reflecting a one-to- one reaction to price changes in the stock. Delta for out-of the-money calls will approach 0 and won’t react to price changes in the stock.
  42. 42. 3. Delta ∆ Relation Delta (∆), at/in/out the money Values of delta I have a call option • K= $50 60 days prior to expiration S=$50. (at-the-money option)  Δ should be 0.5 C=$2.Part 3. Hedging NON-Linear Risk • Case 1: St to $51, C goes up from to $2.50 ( S:C = 1:0.5 ) After 1 • Case 2: St+1, to $52? (Higher probability that the option will end up in-the-money at expiration) • What will happen to delta? … increases to 0.6 • C to $3.10 ($.60 move for a $1 movement in the stock) • Case 4: St to $49? • C to $1.50, reflecting the .50 delta After 4 • Case 5: St to $48, the option might go down to $1.10. • Delta would have gone down to .40 (lower probability the option willLecture 5 end up in-the-money at expiration).
  43. 43. 3. Delta ∆ Relation Delta (∆) time to maturity As expiration approaches, changes in the stock value will cause more dramatic changes in delta LogicalPart 3. Hedging NON-Linear Risk St = $50 K = $50 Two days from expiration Delta =.50 • Case 1: St+1= $51. Delta should be high (0.9) in just ONE day • Case 2: St+1= $49. Delta might change from .50 to .10 in ONE dayLecture 5 Delta reflects the probability that the option will finish in-the-money
  44. 44. 3. Delta ∆ In-the-money options will move more than out-of-the-money options (Remember the graph)  Short-term options will react more than longer-term options to the same price change in the stock.Part 3. Hedging NON-Linear Risk (From previous slide) Delta of a portfolio The delta of a portfolio of options is just the weighted sum of the individual deltasLecture 5 The weights wi equal the number of underlying option contracts
  45. 45. 3. Delta ∆ Delta (∆) The delta of an option depend on the kind of option • For a European call option on a non-dividend stock   N(d1 )Part 3. Hedging NON-Linear Risk • For a European put option on a non-dividend stock   N(d1 )  1 •For a European call option on a dividend-paying stock   eq N(d1 ) •For a European put option on a dividend-paying stock   e q  N(d1 )  1Lecture 5
  46. 46. 3. Delta hedging Delta neutral hedging is defined as keeping a portfolio’s value neutral to small changes in the underlying stock’s price. Stock price : $100 Call option : $10 Current delta : 0.4Part 3. Hedging NON-Linear Risk A financial institution sold 10 call option to its client, so that the client has right to buy 1,000 shares at time to maturity. To construct a delta hedge position • Financial institution should buy 0.4 x 1,000 = 400 shares of stock • If the stock price goes up to $1, the option price will go up by $0.4. In this situation, the financial institution has a $400 ($1 x 400 shares) gain in its stock position, and a $400 ($0.4 x 1,000 shares) loss in its option position.Lecture 5 • If the stock price goes down by $1, the option price will go down by $0.4. The total payoff of the financial institution is also zero. But...
  47. 47. 3. Delta hedging Delta changes over different stock price. If an investor wants to maintain his portfolio in delta neutral, he should adjust his hedged ratio periodically. The more frequently adjustment he does, the better delta-hedging he gets.Part 3. Hedging NON-Linear Risk Underlying stock price of $20, the investor will consider that his portfolio has no risk. As the underlying stock prices changes (up or down), the delta changes and he will have to use different delta hedging.Lecture 5 Delta measure can be combined with other risk measures to yield better risk measurement.
  48. 48. 4.Theta Θ Theta is the amount the price of calls and puts decrease for a one-day change in the time to expiration.  Rate of change of the option price respected  to the passage of time t If   T  t (time to maturity) this derivative is < 0    Part 3. Hedging NON-Linear Risk    (1) Note that t is different from τ t  t  This relation shows that: • The price of the option declines as maturity approaches Same idea •When time passes, the time value of the option decreases • Longer dated options are more valuable. BUT • The passage of time on an option is NOT uncertain,Lecture 5 It is not necessary to make a theta hedge portfolio against the effect of the passage of time.
  49. 49. Here we have a relation between time and option’s price. 4.Theta Θ How this relation changes when American Call is exercised early? Relation time and American call option American and European options: longer dated options give more opportunities for profit Early exercise of an American Option is NON OPTIMAL If an American Call is exercised before T, the payoff could bePart 3. Hedging NON-Linear Risk St – K Put – Call parity condition European call option notation C  P  (St  KerT ) Because P0 So... C  St  Ke rT Recall the restrictions on the value of a call option Lower bound C  max{ 0, S  Ke  rT } Also because r 0 C SK Now, the American Option worth more C A>C CA  C  S  KLecture 5 Ct  Ct  St  KerT  St  K Hence it will always be better to sell the A option rather than exercise it early What about PUT options???
  50. 50. 4.Theta Θ Early exercise of an American Option is NON OPTIMAL Intuitive reasonsPart 3. Hedging NON-Linear Risk 1. Delaying exercise delays the payment of the strike price. Option holder is able to earn interest on the strike price for a longer period of time. 2. More movements: Assume that you exercised your option today, what if tomorrow a big crazy thing will occur and the price of an underlying asset just shoots? Instead of exercising your American call option you should have sold it to someone elseLecture 5
  51. 51. 4.Theta Θ 90 DAYS: lose $.30 of its value in one month Time decay is 60-DAY option, lose $.40 of its value over stronger near the course of the following month. expiration 30-DAY option will lose the entire remaining $1 Time decay of an at-the-money call optionPart 3. Hedging NON-Linear Risk Time is more important ATM At the money V Out/In the money options At-the-money options will experience more significant dollarLecture 5 losses over time than in- or out-of- the-money options with the same underlying stock and expiration date.
  52. 52. 4.Theta Θ Theta is the amount the price of calls and puts decrease for a one-day change in the Summarizing time to expiration.Part 3. Hedging NON-Linear Risk For a European call option on a non-dividend stock, theta can be written as: St s   N(d1 )  rX  e r N(d 2 ) 2 Lecture 5 For a European put option on a non-dividend stock, theta can be shown as S    t s  N(d1 )  rX  e r N(d 2 ) 2 
  53. 53. 5. Gamma Γ Gamma is the rate that delta will change based on a $1 change in the stock price. Or The rate of change of delta respected to the rate of change of underlying asset price   2   2 S S Delta is the “SPEED” at which option prices change, gamma as the “ACCELARATION”Part 3. Hedging NON-Linear Risk  Gamma shows how often we should rebalance  If Γ is large then it will be necessary to change Δ by a large amount as S changes.  Options with the highest gamma are the most responsive to changes in the price of the underlying stock.Lecture 5
  54. 54. 5. Gamma Γ Delta is a dynamic number that changes as the stock price changes, doesn’t change at the same rate for every option based on a given stock. St = 50 K = 50 Delta = 0.5Part 3. Hedging NON-Linear Risk  The price of at-the-money options will change more significantly than the price of in- or out-of-the-money options.Lecture 5
  55. 55. Part 3. Hedging NON-Linear Risk 5. Gamma Γ The price of near-term at-the-money options will exhibit the most explosive response to price changes in the stock.  As your option moves in-the-money, delta will approach 1 more rapidly. If you’re an option buyer, high gamma is good as long as your forecast is correct.Lecture 5  If you’re an option seller and your forecast is incorrect, high gamma is the enemy. That’s because it can cause your position to work against you at a more accelerated rate
  56. 56. 5. Gamma Γ For a European call option on a non-dividend stock, theta can be written as: 1  N  d1  St s Part 3. Hedging NON-Linear Risk For a European put option on a non-dividend stock, theta can be shown as 1  N  d1  St s Lecture 5
  57. 57. 5. Gamma Γ Make a position gamma neutral  Suppose the gamma of a delta-neutral portfolio is Γ  Suppose the gamma of the option in this portfolio is ΓO,  The number of options added to the delta-neutral portfolio is w0.Part 3. Hedging NON-Linear Risk Then, the gamma of this new portfolio is Gamma of portfolio o o   To make a gamma-neutral portfolio, we should trade Gamma of portfolio o*   / o options Example Gamma of option Delta and gamma: 0.7 and 1.2. A delta-neutral portfolio has a gamma of -2,400.Lecture 5 To make a delta-neutral and gamma-neutral portfolio, we should add a long position of 2,400/1.2=2,000 shares and a short position of 2,000 x 0.7=1,400 shares in the original portfolio.
  58. 58. 5. Gamma Γ One more example  Suppose a portfolio is delta neutral with a gamma of -3000  Suppose the delta and gamma of the option is 0.62 and 1.50  Make a portfolio gamma neutral by buyingPart 3. Hedging NON-Linear Risk 3000  2000options o*   / o 1.5  This changes delta from 0 to 0.62 * 2000 = 1240  Sell 1240 shares of underlying to regain delta neutralityLecture 5
  59. 59. 5. Gamma Γ Relation gamma, delta and price of portfolio (Delta-gamma approximation) Given that the option value is not a linear function of underlying stock price Gamma makes the correction. 1 change in option value    change in stock price    (change in stock price)2 2Part 3. Hedging NON-Linear Risk St of XYZ = $657 This approximation comes from Call option = $120 the Taylor series expansion near Delta = 0.47 the initial stock price Gamma = 0.01. Price of the call option if XYZ stock price suddenly begins trading at $699 C(St+h) = C(St) + ∆ (Change St) + (1/2) (Change St)2 * Γ =Lecture 5 120 + 42 * 0.47 + (1/2) (422) * 0.01 = $148.56
  60. 60. LECTURE FIVEEnd Of The Lecture 60

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