Give step-by-step solution with explanation and final answer:
A European put option provides the right to sell a stock at a pre
-
specified strike price
�
�
at maturity date
�
�
−
�
�
.
The Black
-
Scholes equation for pricing a put option is given as follows:
�
�
=
�
�
(
−
�
�
2
)
×
�
�
�
�
−
�
�
�
�
(
�
�
−
�
�
)
−
�
�
(
−
�
�
1
)
×
�
�
�
�
1
=
1
�
�
√
�
�
−
�
�
[
ln
(
�
�
�
�
)
+
(
�
�
�
�
+
�
�
2
2
)
(
�
�
−
�
�
)
]
�
�
2
=
�
�
1
−
�
�
√
�
�
−
�
d
)
You want to know the
“
implied volatility
”
for a put option with
2
6
weeks until expiration
(
assume
5
2
weeks in a year
)
and a current put option price of
�
�
=
$
1
.
5
0
.
The current stock price is
�
�
=
$
3
0
.
0
0
,
and the strike price of the put option is
�
�
=
$
2
7
.
0
0
.
The annual risk
-
free rate is
�
�
�
�
=
0
.
0
3
.
What is the value of
�
�
(
i
.
e
.
the implied volatility
)
for this put option?
Hint: using a spreadsheet, find the put option price using the Black
-
Scholes equation and a
“
guessed
”
value of
�
�
.
Keep adjusting your
�
�
value until you find that your answer for
�
�
equals $
1
.
5
0
.
When
�
�
=
$
1
.
5
0
,
the
�
�
value represents the stock price volatility implied by the current put option price.
Note: you obtained the implied volatilities for two NVDA call options from Yahoo! Finance in question
1
(
b
)
.
This current question illustrates how implied volatility is calculated.
Question:
Give step-by-step solution with explanation and final answer:
A European put option provides the right to sell a stock at a pre
-
specified strike price
�
�
at maturity date
�
�
−
�
�
.
The Black
-
Scholes equation for pricing a put option is given as follows:
�
�
=
�
�
(
−
�
�
2
)
×
�
�
�
�
−
�
�
�
�
(
�
�
−
�
�
)
−
�
�
(
−
�
�
1
)
×
�
�
�
�
1
=
1
�
�
√
�
�
−
�
�
[
ln
(
�
�
�
�
)
+
(
�
�
�
�
+
�
�
2
2
)
(
�
�
−
�
�
)
]
�
�
2
=
�
�
1
−
�
�
√
�
�
−
�
d
)
You want to know the
“
implied volatility
”
for a put option with
2
6
weeks until expiration
(
assume
5
2
weeks in a year
)
and a current put option price of
�
�
=
$
1
.
5
0
.
The current stock price is
�
�
=
$
3
0
.
0
0
,
and the strike price of the put option is
�
�
=
$
2
7
.
0
0
.
The annual risk
-
free rate is
�
�
�
�
=
0
.
0
3
.
What is the value of
�
�
(
i
.
e
.
the implied volatility
)
for this put option?
Hint: using a spreadsheet, find the put option price using the Black
-
Scholes equation and a
“
guessed
”
value of
�
�
.
Keep adjusting your
�
�
value until you find that your answer for
�
�
equals $
1
.
5
0
.
When
�
�
=
$
1
.
5
0
,
the
�
�
value represents the stock price volatility implied by the current put option price.
Note: you obtained the implied volatilities for two NVDA call options from Yahoo! Finance in question
1
(
b
)
.
This current question illustrates how implied volatility is calculated.
This is AI generated Response.
Asked by: Test User 100094
Created at: 2025-04-29 03:58:16
Final Answer:
RenderedRaw Text
Did you like this answer?
Let's work through the process step by step:
---
### **Step 1: Write Down the Black-Scholes Formula for a European Put Option**
\[
P = K e^{-rT} N(-d_2) - S N(-d_1)
\]
where:
- \( S \) = current stock price
- \( K \) = strike price
- \( r \) = annual risk-free rate (continuously compounded)
- \( T \) = time to maturity in years
- \( N(x) \) = cumulative distribution function of the standard normal distribution
- \( d_1 = \frac{ \ln(S/K) + (r + \frac{1}{2} \sigma^2 )T }{ \sigma \sqrt{T} } \)
- \( d_2 = d_1 - \sigma \sqrt{T} \)
- \( \sigma \) = volatility (this is the implied volatility we need to solve for)
---
### **Step 2: Plug in Known Values**
Given:
- \( S = \$30.00 \)
- \( K = \$27.00 \)
- \( r = 0.03 \)
- \( P = \$1.50 \)
- \( T = \frac{26}{52} = 0.5 \) years
---
### **Step 3: Set Up the Black-Scholes Equation**
\[
1.50 = 27 e^{-0.03 \times 0.5} N(-d_2) - 30 N(-d_1)
\]
where
\[
d_1 = \frac{ \ln(30/27) + (0.03 + 0.5 \sigma^2)(0.5) }{ \sigma \sqrt{0.5} }
\]
\[
d_2 = d_1 - \sigma \sqrt{0.5}
\]
---
### **Step 4: Solve for Implied Volatility (\( \sigma \))**
There is **no algebraic solution** for \( \sigma \); we must use a trial-and-error (numerical) method, such as the **goal seek** function in Excel, or iterate manually.
Let’s try a few typical values for \( \sigma \):
#### **First Guess: \( \sigma = 0.20 \) (20%)**
1. \( \sqrt{0.5} = 0.7071 \)
2. \( \ln(30/27) = \ln(1.1111) \approx 0.1054 \)
3. \( 0.5 \sigma^2 = 0.5 \times 0.04 = 0.02 \)
4. \( r + 0.5 \sigma^2 = 0.03 + 0.02 = 0.05 \)
5. \( (r + 0.5 \sigma^2) \times 0.5 = 0.025 \)
6. Numerator: \( 0.1054 + 0.025 = 0.1304 \)
7. Denominator: \( \sigma \sqrt{0.5} = 0.20 \times 0.7071 = 0.1414 \)
8. \( d_1 = 0.1304 / 0.1414 = 0.9224 \)
9. \( d_2 = 0.9224 - 0.1414 = 0.7810 \)
Now, \( N(-d_1) = N(-0.9224) \approx 0.1780 \), \( N(-d_2) = N(-0.7810) \approx 0.2175 \).
\( 27 e^{-0.015} = 27 \times 0.9851 = 26.597 \)
\[
P = 26.597 \times 0.2175 - 30 \times 0.1780
\]
\[
= 5.782 - 5.340 = 0.442
\]
This is **much less than \$1.50**, so the volatility guess is too low.
---
#### **Second Guess: \( \sigma = 0.40 \) (40%)**
- \( \sigma \sqrt{0.5} = 0.40 \times 0.7071 = 0.2828 \)
- \( 0.5 \sigma^2 = 0.5 \times 0.16 = 0.08 \)
- \( r + 0.5 \sigma^2 = 0.03 + 0.08 = 0.11 \)
- \( (r + 0.5 \sigma^2) \times 0.5 = 0.055 \)
- Numerator: \( 0.1054 + 0.055 = 0.1604 \)
- Denominator: \( 0.2828 \)
- \( d_1 = 0.1604 / 0.2828 = 0.5676 \)
- \( d_2 = 0.5676 - 0.2828 = 0.2848 \)
\( N(-d_1) = N(-0.5676) \approx 0.2853 \)
\( N(-d_2) = N(-0.2848) \approx 0.3887 \)
\[
P = 26.597 \times 0.3887 - 30 \times 0.2853 = 10.338 - 8.559 = 1.779
\]
This is **slightly higher than \$1.50**.
---
#### **Third Guess: \( \sigma = 0.35 \) (35%)**
- \( \sigma \sqrt{0.5} = 0.35 \times 0.7071 = 0.2475 \)
- \( 0.5 \sigma^2 = 0.5 \times 0.1225 = 0.06125 \)
- \( r + 0.5 \sigma^2 = 0.03 + 0.06125 = 0.09125 \)
- \( (r + 0.5 \sigma^2) \times 0.5 = 0.045625 \)
- Numerator: \( 0.1054 + 0.045625 = 0.151025 \)
- Denominator: \( 0.2475 \)
- \( d_1 = 0.151025 / 0.2475 = 0.6104 \)
- \( d_2 = 0.6104 - 0.2475 = 0.3629 \)
\( N(-0.6104) \approx 0.2707 \)
\( N(-0.3629) \approx 0.3584 \)
\[
P = 26.597 \times 0.3584 - 30 \times 0.2707 = 9.539 - 8.121 = 1.418
\]
Now, this is **slightly below \$1.50**.
---
#### **Interpolate for a More Accurate Value**
We can interpolate between \( \sigma = 0.35 \) and \( \sigma = 0.40 \):
\[
P_{0.35} = 1.418, \quad P_{0.40} = 1.779
\]
\[
\sigma = 0.35 + (0.40 - 0.35) \times \frac{1.50 - 1.418}{1.779 - 1.418}
\]
\[
= 0.35 + 0.05 \times \frac{0.082}{0.361} = 0.35 + 0.05 \times 0.227 = 0.35 + 0.0114 = 0.361
\]
So, implied volatility is **about 36.1%**.
---
### **Step 5: State the Final Answer**
**The implied volatility (\( \sigma \)) for the put option is approximately:**
\[
\boxed{36\%}
\]
---
## **Summary Table**
| \( \sigma \) | Calculated Put Price (\( P \)) |
|:---:|:---:|
| 0.20 | 0.44 |
| 0.35 | 1.42 |
| 0.40 | 1.78 |
Interpolated value for \( P = 1.50 \) gives \( \sigma \approx 0.36 \) or **36%**.
---
### **Explanation**
- You use the Black-Scholes formula for a put option.
- Plug in all known values except for volatility.
- Try different values for volatility until the calculated put price matches the observed price (\$1.50).
- The volatility that makes the Black-Scholes price equal the observed price is the **implied volatility**.
---
**Final Answer:**
\[
\boxed{\sigma \approx 36\%}
\]
This is the implied volatility for the given put option.
VIPSolutions focuses on clear, step‑by‑step explanations so you can learn quickly. Whether you need an AI‑generated walkthrough or a short hint to get unstuck, each solution is organized for fast reading and easy review later.
Search similar questions, compare approaches, and bookmark the best answers for revision. Our goal is simple: quick, reliable study help that feels natural—not noisy.
VIPSolutions