1. DERIVATIVES: Valuation Methods and Some Extra Stuff
Reference: John C. Hull, Options, Futures and Other Derivatives, Prentice Hall

2. K = Strike price, ST = Price of asset at maturity
Payoffs from Options (Terminal Value) European Options, Cost of option is NOT included
K = Strike price, ST = Price of asset at maturity
Payoff ST K
SHORT CALL
LONG CALL
SHORT PUT
LONG PUT 2

3. Review…
Valuation Approaches:
Binomial Trees
Black-Scholes
Monte Carlo Methods
Finite Differences

4. Binomial Trees

5. Basic Concept: RISK-NEUTRAL VALUATION
Assume that the expected return from all traded assets is the risk-free rate
Value the payoffs from the derivative by calculating their expected values, and then, use the risk-free rate to discount them

6. where s is the volatility and Dt is the length of the time step and,
In summary…
S0u p
where s is the volatility and Dt is the length of the time step and,
p = ( a - d) / ( u – d ) S0
S0d
1 - p 6 6

7. The Probability of an Up Move7 7

8. Example: Put Option, American Style
S0 = 50; K = 50; r =10%; s = 40%;
T = 5 months = ;
Dt = 1 month =
The parameters imply
u = ; d = ;
a = ; p = p = .4927 8 8

9. Value of stock
Value of option 9 9

10. [.5073 x x 5.45] exp{-.10 x .0833} = 2.66
There is no point in exercising the option at this node (S=K=50), thus early exercise is useless, better to wait, therefore, value is 2.66 10 10

11. (.5073 x x 14.64) exp{ -.10 x .083 } = 9.9
However, early exercise, generates 50 – = ; thus, the option should be exercised, the correct value for the option at the node is (10.31 > 9.9, implies that it is better to exercise NOW!) 11 11

12. (.5073 x x 14.64) exp{ -.10 x .083 } =
NOTE, it is not always better to exercise the option when the option is in the money… = 10.31… better NOT TO exercised at that node 12 12

13. Estimation of Delta at time Δt, that is, the sensitivity of the option price w/r to S [stock price]13 13

14. Black-Scholes

15. The Black-Scholes Formulas15 15

16. The N(x) Function
N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x 16 16

17. Monte Carlos

18. MONTE CARLO SIMULATION
We rely on the risk-neutral approach…
We generate a random path and then calculate the payoff of the option at time t=T; we discount the value using the risk-free rate
More formally…
(i) Generate a random path for S in a risk-neutral world
(ii) Determine the payoff at time t = T
(iii) Repeat the previous steps many times
(iiii) Calculate the mean of the payoffs (from the many paths generated)
(iiiii) “Bring the mean” to t= 0 using the risk free rate

19. ΔS = [Exp Mean S] S Δt + σ S ε sqrt (Δt)
Random sample from a distribution N(0, 1)
If S denotes the price of a stock that it does not pay a dividend between [0, T], the [Exp Mean S] = risk free rate. Otherwise, see before

20. In summary…
For European options, method is straightforward
For American options, we need to take care of an additional complication…

21. Consider a 3-year American put option (non-dividend-paying stock) than can be exercised at the end of year 1, 2, or 3. Risk-free rate= 6%; S0 = 1.00; K= 1.1 (We also know the volatility).
For simplicity, we will assume that we did a “mini Monte Carlo” (only 8 paths)

22. Cash flows had the option been exercised at t=3
But at the end of year 2, the put option was in the money for paths: 1, 3, 4, 6 and 7. Hence, we need to decide at this points if we exercise or continue…
For these paths we assume that
V = a + b S + c S2
S, stock value at t = 2; V is the value of continuing discounted to t=2 with the risk free rate

23. Cash flows had the option been exercised at t=3
V = a + b S + c S2
The values of S (t=2) are: 1.08; 1.07; .97; and .84
The values of V (t=2) are
0.00; 0.07exp(-.06 x 1); 0.18exp(-.06 x 1)
0.20exp(-.06 x 1); 0.09exp(-.06 x 1)
S, stock value at t = 2; V is the value of continuing discounted to t=2 with the risk free rate

24. We determined a, b and c. using a least-squares method
V = a + b S + c S2
We determined a, b and c. using a least-squares method
MIN sum(1 to 5) { Vi – a – b Si - c Si2 }2
The values of S (t=2) are: 1.08; 1.07; .97; and .84
The values of V (t=2) are
0.00; 0.07exp(-.06 x 1); 0.18exp(-.06 x 1)
0.20exp(-.06 x 1); 0.09exp(-.06 x 1)
S, stock value at t = 2; V is the value of continuing discounted to t=2 with the risk free rate

25. V = a + b S + c S2
We determined a, b and c. using a least-squares method
MIN sum(1 to 5) { Vi – a – b Si - c Si2 }2
WE GET IN THIS CASE
V = S S2
S, stock value at t = 2; V is the value of continuing discounted to t=2 with the risk free rate

26. V = a + b S + c S2
V = S S2
V gives the value, at point t=2, of continuing
That is, for paths (1, 3, 4, 6 and 7) we get
V = ( and )

27. That is, for paths (1, 3, 4, 6 and 7) we get
V = ( and )
But at t=2 the value of exercising is
.02
.03
.13
.33
.26 (( = .26))
We should only exercise at t=2 for paths 4, 5 and 6

28. Payoffs assuming that we have exercised at t=3 OR t=2 depending on what is best
We repeat the same procedure for time = 1 (when it was worth to exercise at t=1?) and we finally get the full payoff table

29. Full tableau of cash flows..
Thus, the value of the options is…
(1/8) [ 0.07 exp(-0.06 x 3) exp(-0.06 x 1)+ … exp(-0.06x1) ] =

30. Finite Differences

31. Attack the equation directly….

32. Consider the case of a Put Option, let us assume that the life of the option is T and Smax is a value sufficiently high (put has no value)
We want to find f (value of the option) solving the “real thing”

33. f = f ( t, S)
Time discretization: 0, Δt, 2 Δt, ……, T (N intervals, index i)
Stock-value discretization: 0, ΔS, 2 ΔS, ……, Smax (M intervals, index j)

34. Value of the option at time = T is fN,j = MAX (K – j Δ s, 0)
Stock Value
Index i, time; Index j, S
Smax
Value of the option at time = T
is fN,j = MAX (K – j Δ s, 0)
j =1, 2, …, M T
Time

35. Value of the option is K when S = 0 f i,0 = K for i=0,…, N
Stock Value
Index i, time; Index j, S
Smax
Value of the option is K when
S = 0
f i,0 = K for i=0,…, N T
Time

36. Value of the option is 0 when S = Smax f i, M = 0 for i=0,…, N
Stock Value
Index i, time; Index j, S
Smax
Value of the option is 0 when
S = Smax
f i, M = 0 for i=0,…, N T
Time

37. Stock Value
Index i, time; Index j, S
Smax
We know the value of f (options) along the red lines T
Time

38. Using a finite differences approach (explicit method)

39. Using a finite differences approach (explicit method)

40. Using a finite differences approach (explicit method)

41. Introducing the finite differences approximation into the B-S equation and re-arranging…
We get the following approx. to the B-S equation…

42. With…
And…

43. f i,j+1
f i,j
f i+1,j
f i,j-1

44. Stock Value
Value of f is known
We can set up a linear system of simultaneous eqs by using these points as i,j then we move to the left.. Etc etc
Smax T
Time
Value of f is unknown