Derivatives II

1
Financial Risk Management Option Pricing
2
Option Valuetime value intrinsic value
C(T2)
C(T1)
time
intrinsic
At maturity cMax (S-X,0) intrinsic value
Time 2 greater than Time 1.
3

• Notice, for an in-the-money option, the option
  value (price)intrinsic time (S-X) time
  value.
• Whereas, for the out-of-the-money option, option
  price intrinsic time 0 time value.

4
Underlying stock price movement
One Year Later
Today
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Binomial

• In the exterme, we can think of the underlying
  stock price as either increasing or just
  decreasing by some percentage.
• For example, let the stock price increase by 10
  or decrease by 10. The current stock price is
  20.

6
Cont.

• Call option. Strike price is assumed at 21.
  Maturity one year. Risk free rate (interest
  rate) is 3.
• Thus, the stock price one year from now can be
  either 20 x 1.1 22 or 20 x 0.9 18.
• The call value is Max (S-X,0). This means take
  the maximum value between S-X and 0.

7
Cont.

• The call value is (1/(1r))p x call(up) (1-p)
  x call(down
• p(1r) d/u-d 1.03 0.9/1.1
  0.90.65 (approx.)
• p is the probability of moving up. r is the
  riskfree rate (discount rate). u (1.1) is the
  up movement. d (0.9) is the down movement.

8
Cont.

• Call price (1/1.03)0.65 x 1 0.35 x 0
  (1/1.03)(0.65)

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One Year Later
1.1 x 20 22
CallMax(22-21,0)1
20 (today)
0.9 x 20 18
Call Max(18-21,0)0
10
One Year Later
pprobability of going up.
Call (up) Max(22-21,0)1
Call p x call (up) (1-p) call (down)
20 (today)
Discount by 1/(1r)
Call (down) Max(18-21,0)0
11
One Year Later
1.1 x 1.1 x 20
1.1 x 0.9 x 20 Or 0.9 x 1.1 x 20
20 (today)
0.9 x 0.9 x 20
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S
Trend
Deviations from trend generated by random
component
S0
Time
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Same as the binomial tree assume a process for
S and move forward. Then solve backwards to
obtain the option price.
discount
Mean of ST - X
Call
T
t
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Theory Meets Practice Nobel Economics Black-Sch
oles-Merton Model
S is spot price of underlying asset, X is
exercise price, s is the standard deviation of
underlying asset return, T-t is time to maturity,
C is call price, r is the riskfree rate, N(.)
cumulative normal prob. density function.
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Notice the equation simply subtracts the
underlying asset value from the strike/exercise
price (discounted). The only difference is the
weighting scheme attached to the value of the
underlying and the discounted strike
price. Think of SN(d1) exp(r(T-t)) as the
expected value of S given that the call is in
the money. N(d2) as the probability of being in
the money (probability of exercise).
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To obtain implied volatility, use the solver
function. Set the target sell to the call price,
the changing value cell to sigma, and set the
target equal to the call market price.
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Example (Nikkei Index)

• Numbers from the Nihon Keizai Shimbum, May 22
• Nikkei index 14176
• Exercise price 14000
• June (expiration, maturity) option premium is 435
• Nikkei historical volatility 22.3
• Riskfree rate 0.06 pa (TIBOR one month rate)
• Assume zero dividends.

19
Estimating volatility

• Recall, the volatility needs to be forecasted.
• For the binomial (for example)
  uexp(volatilitysqrt(time increment))
  dexp(-volatilitysqrt(time increment))
• Historical standard deviation
• Time Series models GARCH type models
• Implied volatility

20
Standard Deviation from Historical Returns
If we use daily data to obtain an estimate of
volatility, we can adjust this number by
multiplying by the square root of 250 (trading
days) to get an annual volatility.
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GARCH(1,1) Model
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Implied Volatilities

• Assume the market price for options is correct
  (informational efficiency).
• Assume all other market prices used are correct
  (underlying asset price and interest rate).
• Assume the model you use (Black-Scholes) is
  correct.

23
Cont.

• Obtain call price from model.
• cmodelc(S, X, r, T-t, volatility)
• Hold constant S, X, r, T-t.
• Keep changing the volatility until
  cmodelcmarket. cmarket is the market price of
  the call option (observed price).
• The volatility which equates cmodeland cmarket is
  our implied volatility.
• The volatility is implied from the observed
  market prices and the option pricing mode.
• This can be done using the solver in EXCEL.

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Theoretically, the volatility should be the
same. Empirically, we find (often) that it looks
like a smile.
volatility
empirical
theoretical
S/X