1. No category

Lesson 7.2: Pricing European options using the
lognormal parameters.
ACTS 4302
Natalia A. Humphreys
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Acknowledgement
This work is based on the material in ASM MFE Study Manual for
Exam MFE/Exam 3F. Financial Economics (7th Edition), 2009, by
Abraham Weishaus.
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Pricing European options: Assumptions.
In this section we’ll make some progress towards pricing European
options. Suppose
1. S0 is the price of a stock
2. α is the continuously compounded expected rate of return on
a stock
3. σ is the volatility of the stock price
4. δ is the continuously compounded annual dividend return
5. K is the strike price
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Questions about the European options.
We’ll answer the following questions:
1. The probability that an option will pay off
2. Conditional payoff of the option, given that it pays off
3. Expected payoff of an option
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Probability that an option will pay off. Put.
A put will pay off if St < K :
St
K
Pr (St < K ) = Pr
<
=
S0
S0
K
St
< ln
=
= Pr ln
S0
S0

 K
ln
−
m
S0
K

= Pr Xt < ln
= Φ
S0
v
 
ln SK0 − (α − δ − 0.5σ 2 )t
=
√
= Φ
σ t


ln SK0 + (α − δ − 0.5σ 2 )t
 = Φ −dˆ2
√
= Φ −
σ t
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Probability that a put option will pay off. Final formula.
Pr (St < K ) = Φ −dˆ2 , where
ln SK0 + (α − δ − 0.5σ 2 )t
√
dˆ2 =
σ t
N(x) is the standard normal cumulative distribution function at x probability that a standard normal random variable X is less than
or equal to x:
Φ(x) = Pr (X < x) , X ∼ N(0, 1)
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Probability that an option will pay off. Call.
Similarly, for a call option, a call will pay off if St > K :
Pr (St > K ) = 1 − Pr (St < K ) = 1 − Φ −dˆ2 = Φ dˆ2
ln SK0 + (α − δ − 0.5σ 2 )t
√
dˆ2 =
σ t
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Example 7.2.1
A stock’s price follows a lognormal model. You are given the
following information about a stock:
(i) The initial price is 60.
(ii) The expected rate of return on the stock is 15%.
(iii) The stock pays dividends continuously at a rate proportional
to its price. The dividend yield is 5%.
(iv) The stocks volatility is 20%.
A European call option on the stock with strike price 70 expires in
3 months.
Calculate the probability that the option pays off.
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Example 7.2.1. Solution.
Solution.
Pr (St > K ) = Φ dˆ2
ln SK0 + (α − δ − 0.5σ 2 )t
√
dˆ2 =
=
σ t
ln 60
− 0.5 · 0.22 )0.25
70 + (0.15 − 0.05
√
=
= −1.3415
0.2 0.25
Pr (S0.25 > 70) = Φ (−1.3415) = 1 − Φ (1.3415) = 0.0901
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Conditional payoff of the option, given that it pays off.
Defn. The partial expectation of a continuous random variable X
having probability density function f (x), given that it is in the
interval [a, b] is the contribution to the expectation from values in
the interval [a, b]:
Z
PE [X |X ∈ [a, b]] =
b
xf (x) dx
a
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Conditional expectation
Recall the definition of a conditional probability:
P(B|A) =
P(A ∩ B)
P(A)
Then conditional expectation is:
E (X |Y ) =
PE (X |Y )
⇔ PE (X |Y ) = E (X |Y )P(Y )
P(Y )
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Conditional expectation. Lognormal r.v.
For a lognormal random variable X ∼ LND(m, v 2 ),
ln K − m − v 2
PE (X |X < K ) = E (X )Φ
v
Applying this for stocks,
PE[St |St < K ] = E
= S0 e
dˆ1 =
m+0.5v 2
ln
S0
K
Φ
St
S0
Φ
ln SK0 − m − v 2
ln SK0 − m − v 2
v
!
=
!
v
= S0 e (α−δ)t Φ(−dˆ1 )
+ (α − δ + 0.5σ 2 )t
√
σ t
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Expectation. Lognormal r.v.
Since
Pr (St < K ) = Φ(−dˆ2 ),
it follows that
E[St |St < K ] =
Note that
S0 e (α−δ)t Φ(−dˆ1 )
Φ(−dˆ2 )
√
dˆ2 = dˆ1 − σ t
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Expected payoff of a put option
One who owns a European put option has the following cash flows
at expiry:
1. Receipt of K if the stock price is below K ,
2. Payment of stock, if the stock price is below K :
E[−St |St < K ] = −
S0 e (α−δ)t Φ(−dˆ1 )
Φ(−dˆ2 )
3. 0, if the stock price is above K
The probability of the first two payments is
p = Pr (St < K ) = Φ(−dˆ2 )
Therefore, using the double expectation formula,
E[X ] = Eθ [EX [X |θ]], we obtain:
E[max(0, K − St )] = K Φ(−dˆ2 ) − S0 e (α−δ)t Φ(−dˆ1 )
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Expected payoff of a call option
One who owns a European call option has the following cash flows
at expiry:
1. Payment of K if the stock price is above K :
2. Receipt of stock, if the stock price is above K :
E[St |St > K ] =
S0 e (α−δ)t Φ(dˆ1 )
Φ(dˆ2 )
3. 0, if the stock price is below K
The probability of the first two payments is
p = Pr (St > K ) = Φ(dˆ2 )
Therefore, the expected call option payoff:
E[max(0, St − K )] = S0 e (α−δ)t Φ(dˆ1 ) − K Φ(dˆ2 )
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Example 7.2.2
A stock price follows a lognormal model. You are given the
following information about a non-dividend paying stock:
(i) The initial price is 50.
(ii) The expected rate of return on the stock is 15%.
(iii) The stocks volatility is 30%.
Determine the conditional expected value of the stock’s price after
3 months, given that it is higher than 75.
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Example 7.2.2. Solution.
Solution. By the above,
E[St |St > K ] =
S0 e (α−δ)t Φ(dˆ1 )
=∗
Φ(dˆ2 )
Let’s calculate dˆ1 and dˆ2 .
ln SK0 + (α − δ + 0.5σ 2 )t
√
dˆ1 =
=
σ t
+ 0.5 · 0.32 )0.25
ln 50
75 + (0.15√
= −2.3781
=
0.3 0.25
√
√
dˆ2 = dˆ1 − σ t = −2.3781 − 0.3 0.25 = −2.5281
Hence,
Φ(−2.3781)
Φ(−2.38)
≈ 50e 0.0375
=
Φ(−2.528)
Φ(−2.53)
0.0087
= 51.9106 ·
= 79.23
0.0057
∗ = 50e 0.15·0.25
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