The Greek Letters

1. The Greek Letters

2. Pricing Options • Both the Binomial Tree Approach and the Black Scholes approach produce the same option value as the number of steps in the Binomial tree becomes large. • For this section we will concentrate on the “Theoretical value” of the option – the black scholes solution.

3. Black Scholes • Value of Call Option = SN(d1)-Xe-rtN(d2) S = Current value of underlying asset X = Exercise price t = life until expiration of option r = riskless rate s2 = variance N(d ) = the cumulative normal distribution (the probability that a variable with a standard normal distribution will be less than d)

4. Black Scholes (Intuition) • Value of Call Option SN(d1) - Xe-rtN(d2) The expectedPV of costRisk Neutral Value of Sof investmentProbability of if S > XS > X

5. Black Scholes • Value of Call Option = SN(d1)-Xe-rtN(d2) Where:

6. Time Value of an Option • The time value of an option is the difference in the theoretical price of the option and the intrinsic value. • It represents the the possibility that the value of the option will increase over the time it is owned.

7. An Example1: • Assume that a financial institution has sold a European Call Option on a non dividend paying stock. • S= $49, X=$50, r = 0.05, s=0.20, t = 20 weeks =0.3846 years. Call option value = 2.372 • Assume that the institution has sold the option for $3 a share or .628 more than its theoretical value. • How can it hedge its risk? Hull Chapter 15 Fundementals of Futures and Options Markets

8. Naked vs. Covered position • The firm can do nothing and hold only the option (a naked position). It would then be forced to buy the shares if the owner of the option exercises it in 20 weeks. The profit diagram would look like the normal short call. • The firm can buy the stock today and have a covered call. This introduces a downside risk, if the value of the stock decreases the firm looses due to the decline in the value of the share.

9. Profit Diagram Covered Call Profit Long Spot Covered Call Short Call

10. Hedging with a Stop-Loss Strategy. • One possible solution is to develop a dynamic buying strategy for the share. For example the firm could buy shares whenever the stock price is greater than the exercise price, It could then sell the shares if the stock price drops below the exercise price. • It would then be hedged when the option will be exercised and unhedged when it will not be exercised.

11. Stop Loss Costs • The problem is that there are substantial transaction costs associated with the strategy. • Also there is uncertainty about the actual cost of the share. Therefore you are not buying and selling each time at the exercise price. • A better approach is to use the delta of the option

12. Delta of an option • The delta of the option shows how the theoretical price of the option will change with a small change in the underlying asset.

13. Time Value of the Option • Plotting the value of the option compared to the profit and or payoff provides a starting point to explaining delta. • Using the option above the following prices were obtained and graphed on the next slide. Stock Call Stock Call Stock Call 42 0.173 50 2.962 58 9.211 46 1.107 54 5.732 62 13.025

14. Time Value of Option

15. Call Option Value

16. Delta Graphically

17. Delta of an option • Intuitively a higher stock price should lead to a higher call price. The relationship between changes in the call price and the stock price is expressed by a single variable, delta. • The delta is the change in the call price for a very small change it the price of the underlying asset.

18. Calculating Delta • Delta can be found from the call price equation as: • Using delta hedging for a short position in a European call option would require keeping a long position of N(d1) shares at any given time. (and vice versa).

19. Delta explanation • Delta will be between 0 and 1. • A 1 cent change in the price of the underlying asset leads to a change of delta cents in the price of the option.

20. Delta and the stock price • For deep out-of-the money call options the delta will be close to zero. A small change in the stock price has little impact on the value of the option • For deep in the money options delta will be close to 1. A small change in the stock price will have an almost one to one change in the option price.

21. Delta vs. Share Price

22. Delta and Time to MaturityX=50 r=0.05 s=0.2

23. Delta X=35 r=0.05 s=.2

24. Example of Delta Hedging • Assume that we had sold the option in our example for 100,000 shares of stock. • Using the information from before: S= $49, X=$50, r = 0.05, s=0.20, t = 20 weeks =0.3846 years. Call option value = 2.3715 • Given 100,000 shares the value of the option is $237,150 • Assuming a share price of 49, the delta of the option is .5828594

25. The hedged position • The bank has a portfolio of delta shares for each share it has written an option on. • This implies it owns 100,000(.5828594) = 58,286 shares. • If the share price increases by $1 the value of the shares will increase by $58,286 • However the value of the option will decline.

26. Option value • The value of the option at a price of $50 is 2.926. • Therefore the value of the option will decrease by (2.926 – 2.3715)100,000 = $55,450

27. Total position Gain on Spot position = $58,286 Loss on Option = - $55,450 Net change in portfolio = $2,836 • They do not perfectly offset due to the size of the price change and rounding errors. The total value of the portfolio would change from $3,093,161.06 to $3,095,997.00

28. Dynamic Hedging • Since the value of delta changes at each stock price the amount of shares would need to be adjusted to keep the portfolio value hedged. • The larger the price change the less successful the hedge.

29. Delta of a portfolio • The delta of a portfolio of options is simply the weighted average of the individual deltas. Where the weight corresponds to the quantity of the option. • It is therefore possible to adjust the delta of a portfolio quickly by adjusting one or more of the option positions.

30. Delta of a put option • A long position in a put option should be hedged with a long position in the stock, (delta will be negative). • Delta for the put is given by N(d1) – 1 • Similar to call options, for deep in the money puts (Asset price is less than exercise price) the value of delta will be close to -1. For delta out of the money puts the delta will be close to zero.

31. Delta Hedging • The delta neutral portfolio removes much but not all of the risk associated with the position. • Looking at the value of the portfolio for a small range of prices changes provides a good indication of the ability of the hedge to remove the risk associated with a change in the stock price. • The change hedge is not perfect because the value of the option is not a linear function with relation to changes in the stock price. Consider the previous portfolio.

32. Value of Delta Neutral Portfolio(1 Short Call + Delta Shares) x 100,000 shares

33. Gamma • Gamma measures the curvature of the theoretical call option price line.

34. Gamma of an Option • The change in delta for a small change in the stock price is called the options gamma: • Call gamma

35. Gamma Graphically Gamma measures the amount of curvature In the call price relationship, The reason the portfolio Is not perfectly hedged is because delta provides only a linear estimate of the call price change. The hedge error is from the difference between the estimate from delta and the actual relationship

36. Gamma • If gamma is small it implies that delta changes slowly which implies the cost to adjust the portfolio will be small. • If gamma is large it implies that delta changes quickly and the cost to keep a portfolio delta neutral will be large.

37. Gamma Con’t • Gamma is the adjustment for the fact that the call option price dos not have a linear relationship with the spot price. • Delta provides a linear approximation of the change in the value of the call option that is less accurate the large the change in the stock price. • The impact of gamma is easy to see in our earlier example. • The impact of gamma will be the largest when the stock price is close to the exercise price

38. Gamma • The gamma of a non dividend paying stock option will always be positive (the larger the change in the stock price the larger the change in the value)

39. Gamma and Stock Price • The impact of gamma will be the largest when the stock price is close to the exercise price. • For deep in the money or deep out of the money call options gamma will be relatively small.

40. Gamma and Time to Maturity • Gamma will be highest for at the money options close to maturity. • Gamma will be low for both in the money and out of the money options that are close to maturity.

41. Gamma vs. Stock PriceX=50, r =0.05, s=.2, t=.5

42. Gamma vs Time to MaturityX=35, r=0.05, s=.2

43. Other Measures • The sensitivity of the value of the option to a change in the expiration of the option is measured by theta

44. Theta • Theta is generally negative for an option since as the time to maturity decreases the value of the option becomes less valuable. (Keeping everything else constant, as time passes the value of the option decreases).

45. ThetaX=35, r=0.05, t=.5, s=.2

46. Theta vs Time X=35, s=.2, r=0.05

47. Relationship between Delta, Theta and Gamma • From the derivation of the Black-Scholes Formula it can be shown that: (We will show this soon) • In a Delta Neutral portfolio delta =0 and the portfolio value remains relatively constant. This implies that if Theta is negative, Gamma needs to be of similar size and positive and vice versa. Therefore Theta is often considered as a proxy for Gamma.

48. Vega (or Kappa) • The rate of change of the option value with respect to the volatility of the underlying asset is given by the Vega (also sometimes called kappa) • The Black Scholes Model assumes that volatility is constant, so in theory this seems to be inconsistent with the model. • However variations of the Black Scholes do allow for stochastic volatility and their estimates of Vega are very close to those form the Black Scholes model so it serves as an approximation.

49. Vega • Vega will be highest for options that are at the money. As the option moves into or out of the money the impact of a volatility change is decreased.

50. Rho • The final measure is the change in the value of the option with respect to the change in the interest rate. As we have discussed the interest rate has the smallest impact on the value of the option. Therefore this is not used often in trading.