Download Adjusting the Black-Scholes Framework in the Presence of a Volatility Skew

Adjusting the Black-Scholes Framework
in the Presence of a Volatility Skew
Mentor: Christopher Prouty
Members: Ping An, Dawei Wang, Rui Yan
Shiyi Chen, Fanda Yang, Che Wang
Team Website:
http://sites.google.com/site/mfmmodelingprogramteam2/
UMN Master of Financial Mathematics Modeling Program Final Report
Table of Contents
1. Background ................................................................................................................. 1
2. Assumptions ................................................................................................................ 2
3. Our Model ................................................................................................................... 3
3.1 Workflow: ............................................................................................................... 3
3.2 Implied Volatility ..................................................................................................... 3
3.3 Cubic Spline Interpolation Method ......................................................................... 4
3.4 Extending Skew Curve ........................................................................................... 5
3.5. Market Implied Distribution.................................................................................... 7
3.6. Denoise MID ......................................................................................................... 8
4. Test (Monte Carlo) & Result ...................................................................................... 10
4.1 BS Test ................................................................................................................ 10
4.2 MID PDF RVs Test............................................................................................... 11
5. Improvement ............................................................................................................. 13
6. Conclusion ................................................................................................................ 15
UMN Master of Financial Mathematics Modeling Program Final Report
1. Background
Before the Black Monday in 1987, people think that Black-Scholes Model fits for the realized
market, or at least, not too bad. But when avalanche of out-of-the money options suddenly
became in-the-money, people began to realize that they needed to adjust the relationships
between volatility and other parameters.
Fig. 1.1 Volatility Skew before 1987
X-axis is the strike price (K), and Y-axis is the volatility (sigma).
People found the volatility changes with respect to strike price (K) and time to maturity (T). So
people need to trade lower-strike-option with a much higher volatility, and the volatility smile
looks like this:
Fig. 1.2 Volatility Skew after 1987
X-axis is the strike price (K), and Y-axis is the volatility (sigma).
But since we have a volatility smile, if the market moves, the seller of the option may lose or
make money on the option account even he/she has already delta-hedged. So the seller needs
to sell/buy extra underlying to remain delta-neutral. Our model is to calculate how much is the
extra delta that we need to take into consideration, and we called it “skew delta”.
1
UMN Master of Financial Mathematics Modeling Program Final Report
2. Assumptions
First of all, let’s see the Black-Scholes model’s assumptions:
•
It is possible to borrow and lend cash at a known constant risk-free interest rate.
•
The price follows a Geometric Brownian motion with constant drift and volatility.
•
There are no transaction costs.
•
The stock does not pay a dividend.
•
All securities are perfectly divisible (i.e. it is possible to buy any fraction of a share).
•
There are no restrictions on short selling.
•
There is no arbitrage opportunity.
Our model allows each assumption in the Black-Scholes but the second one. In our model, we
assume that:
•
The price follows a implied motion that we can know from our Market Implied Distribution
(MID) method.
•
The volatility smile doesn’t change its shape or increase/decrease, it just moves
paralleled to the left or right.
2
UMN Master of Financial Mathematics Modeling Program Final Report
3. Our Model
3.1 Workflow:
Our workflow looks like this:
Fig. 3.1 Workflow of our model
3.2 Implied Volatility
The basic idea of this process is trying to use different volatilities into Black-Scholes until a
predetermined option price is reached. The first method we tried to calculate implied volatility is
Newton’s method. This method is recognized by its rapid convergence. In order to obtain a
quick return in later Monte Carlo testing process that constitutes more than 1 million trials,
implementing Newton’s method seems appropriate here. However, due to the intrinsic
deficiencies, like local convergence, Newton’s method had a bad performance when dealing
with some extreme underlying prices. In some cases, it forced computer to crash, which did not
show any rapidity.
3
UMN Master of Financial Mathematics Modeling Program Final Report
Therefore, we turned to bisection method for its robustness. The actual method coded into our
program was modified to some extent. The first modification was placed to decide the starting
points – the lower bound and upper bound. Given an arbitrary distance between final lower and
upper bound, our program would set lower bound from 0.5 to some value that finally squeezes
the desired implied volatility between these two bounds. This process sacrifices time but gives
advantage of avoiding improper starting points set by human.
Another modification links to the thought of removing infeasible inputs. Given spot, strike,
interest rate and tenor constant, the option price function of volatility should have a theoretical
minimum output. But the market seems not always follow this rule. When such out-of-range spot
associated with strike was used to calculate the implied volatility, neither bisection nor Newton’s
method would give an ideal outcome. (Try this data: S = 30.66, K = 40.00, P = 9.40, r = 0.25%,
T = 0.25) Therefore, it is necessary to remove these extremes from dataset so as to avoid
endless loops in our program. The way we adapted is to listen an iteration counter and make
sure it never exceed an empirical number, say 100 in our program.
3.3 Cubic Spline Interpolation Method
This is a method to interpolate unknown values between given points using piecewise thirdorder polynomials. The essential idea is to fit a piecewise function of the form:
𝐹𝐹(𝑥𝑥) = �
𝑓𝑓1 (𝑥𝑥)
𝑓𝑓2 (𝑥𝑥)
𝑓𝑓𝑛𝑛−1 (𝑥𝑥)
For each 𝑓𝑓𝑖𝑖 is a third-order polynomial defined by:
𝑥𝑥1 ≤ 𝑥𝑥 < 𝑥𝑥2
𝑥𝑥2 ≤ 𝑥𝑥 < 𝑥𝑥3
…
𝑥𝑥𝑛𝑛−1 ≤ 𝑥𝑥 < 𝑥𝑥𝑛𝑛
𝑓𝑓𝑖𝑖 (𝑥𝑥) = 𝛼𝛼𝑖𝑖 𝑥𝑥 3 + 𝛽𝛽𝑖𝑖 𝑥𝑥 2 + 𝛾𝛾𝑖𝑖 𝑥𝑥 + 𝛿𝛿𝑖𝑖
(3.3.1)
(3.3.2)
𝑓𝑓𝑖𝑖 (𝑥𝑥𝑖𝑖 ) = 𝛼𝛼𝑖𝑖 𝑥𝑥𝑖𝑖 3 + 𝛽𝛽𝑖𝑖 𝑥𝑥𝑖𝑖 2 + 𝛾𝛾𝑖𝑖 𝑥𝑥𝑖𝑖 + 𝛿𝛿𝑖𝑖 = 𝑓𝑓𝑖𝑖−1 (𝑥𝑥𝑖𝑖 ) = ℎ𝑖𝑖
(3.3.3)
′
(𝑥𝑥)
𝑓𝑓𝑖𝑖′ (𝑥𝑥) = 3𝛼𝛼𝑖𝑖 (𝑥𝑥 − 𝑥𝑥𝑖𝑖 )2 + 𝛽𝛽𝑖𝑖 (𝑥𝑥 − 𝑥𝑥𝑖𝑖 ) + 𝛾𝛾𝑖𝑖 = 𝑓𝑓𝑖𝑖−1
(3.3.4)
Also, we want to make the function as soon as possible, therefore we make the first and second
derivatives agree at each adjacent segment.
′′
(𝑥𝑥)
𝑓𝑓𝑖𝑖′′ (𝑥𝑥) = 6𝑎𝑎𝑖𝑖 (𝑥𝑥 − 𝑥𝑥𝑖𝑖 ) + 2𝛽𝛽𝑖𝑖 = 𝑓𝑓𝑖𝑖−1
(3.3.5)
Cubic splines have good properties, first the spline functions will interpolate all data points;
secondly, interpolation function is continuous on each interval, so are the first and second
derivatives.
4
UMN Master of Financial Mathematics Modeling Program Final Report
Until now, we have 4𝑚𝑚 coefficients and 4𝑚𝑚 equations, therefore, from basic algebra, we could
get the coefficients, and the value between known points.
If we have 𝑚𝑚 + 1 observations(ℎ0 , ℎ1 , … … , ℎ𝑚𝑚 ), and 𝑚𝑚 + 1 inputs, we have
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜0
𝐴𝐴 = ⎜
⎜
⎜
⎜0
⎜
⎜
⎜
⎜
𝑥𝑥03 𝑥𝑥02 𝑥𝑥0 1 0 0 0 0
0 0 0 0 𝑥𝑥13 𝑥𝑥12 𝑥𝑥1 1
𝐵𝐵 = 𝐴𝐴−1 𝐻𝐻
…0 0 0 0 0 0 0
…0 0 0 0 0 0 0 0
⎞
⎟
3
2
0 0 0 0 0 0 0 0
… 𝑥𝑥𝑚𝑚 −2 𝑥𝑥𝑚𝑚 −2 𝑥𝑥𝑚𝑚−2 1 0 0 0 0 ⎟
⎟
3
2
0 0 0 0 0 0 0 0
… 𝑥𝑥𝑚𝑚
−1 𝑥𝑥𝑚𝑚−1 𝑥𝑥𝑚𝑚 −1 1 0 0 0 0 ⎟
…0 0 0 0 0 0 0 0 ⎟
𝑥𝑥13 𝑥𝑥12 𝑥𝑥1 1 −𝑥𝑥13 − 𝑥𝑥12 − 𝑥𝑥1
…
⎟
3
3
2
2
0 0 0 0 0 0 0 … 𝑥𝑥𝑚𝑚 −2 𝑥𝑥𝑚𝑚 −2 𝑥𝑥𝑚𝑚 −2 1 −𝑥𝑥𝑚𝑚−2 − 𝑥𝑥𝑚𝑚 −2 −𝑥𝑥𝑚𝑚 −2 1⎟
3𝑥𝑥12 2𝑥𝑥1 1 0 −3𝑥𝑥12 −2𝑥𝑥1 − 1 0
…0 0 0 0 0 0 0 0⎟
⎟
…
⎟
2
2
0 0 0 0 0 0 0
… 3𝑥𝑥𝑚𝑚
−2 2𝑥𝑥𝑚𝑚−2 1 0 −3𝑥𝑥𝑚𝑚−2 −2𝑥𝑥𝑚𝑚−2 − 1 0 ⎟
6𝑥𝑥1 2 0 0 −6𝑥𝑥1 − 2 0 0
…0 0 0 0 0 0 0 0 ⎟
⎟
…
… 6𝑥𝑥𝑚𝑚 −2 2 0 0 −6𝑥𝑥𝑚𝑚 −2 − 2 0 0 ⎟
0 0 0 0 0 0 0 0
⎟
6𝑥𝑥0 2 0 0 0 0 0 0
…0 0 0 0 0 0 0 0
… 0 0 0 0 6𝑥𝑥𝑚𝑚 −2 2 0 0 ⎠
⎝ 0 0 0 0 0 0 0 0
…
𝐵𝐵 = (𝛼𝛼0 , 𝛽𝛽0 , 𝛾𝛾0 , 𝛿𝛿0 , 𝛼𝛼1 , 𝛽𝛽1 , 𝛾𝛾1 , 𝛿𝛿1 , … , 𝛼𝛼𝑚𝑚 −1 , 𝛽𝛽𝑚𝑚 −1 , 𝛾𝛾𝑚𝑚 −1 , 𝛿𝛿𝑚𝑚 −1 )−1
𝐻𝐻 = (ℎ1 , ℎ2 , … , ℎ𝑚𝑚 , 0,0, … ,0)−1
(3.3.6)
(3.3.7a)
(3.3.7b)
(3.3.7c)
3.4 Extending Skew Curve
In order to simulate the implied volatility, we have to extend the skew curve that we have
obtained from Cubic Spline Interpolation. Now, we choose the Least Squares method to extend
the curve. The Least squares assumes that the best-fit curve of a given type is the curve that
has the minimal sum of the deviations squared (least square error) from a given set of data.
Suppose the curve is second curve to approximate the given set of data (𝑥𝑥1 , 𝑦𝑦1 ), (𝑥𝑥2 , 𝑦𝑦2 )… and
the equation is : 𝑦𝑦 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐𝑥𝑥 2 . The best fitting curve 𝑓𝑓(𝑥𝑥) = 𝑦𝑦 has the min least square error:
𝛱𝛱 = ∑𝑛𝑛𝑖𝑖=1[𝑦𝑦𝑖𝑖 − 𝑓𝑓(𝑥𝑥𝑖𝑖 )]2 = ∑𝑛𝑛𝑖𝑖=1[𝑦𝑦𝑖𝑖 − (𝑎𝑎 + 𝑏𝑏𝑥𝑥𝑖𝑖 + 𝑐𝑐𝑥𝑥𝑖𝑖2 )]2 = 𝑎𝑎 𝑚𝑚𝑚𝑚𝑚𝑚
The unknown coefficients a, b and c must yield zero first derivatives.
(3.4.1)
5
UMN Master of Financial Mathematics Modeling Program Final Report
𝑛𝑛
⎧ 𝜕𝜕𝜕𝜕 = 2 �[𝑦𝑦 − �𝑎𝑎 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐𝑥𝑥 2 �] = 0
𝑖𝑖
𝑖𝑖
𝑖𝑖
⎪ 𝜕𝜕𝜕𝜕
𝑖𝑖=1
⎪
𝑛𝑛
⎪
𝜕𝜕𝜕𝜕
= 2 � 𝑥𝑥𝑖𝑖 [𝑦𝑦𝑖𝑖 − �𝑎𝑎 + 𝑏𝑏𝑥𝑥𝑖𝑖 + 𝑐𝑐𝑥𝑥𝑖𝑖2 �] = 0
𝜕𝜕𝜕𝜕
⎨
𝑖𝑖=1
𝑛𝑛
⎪
⎪𝜕𝜕𝜕𝜕
⎪
= 2 � 𝑥𝑥𝑖𝑖2 [𝑦𝑦𝑖𝑖 − �𝑎𝑎 + 𝑏𝑏𝑥𝑥𝑖𝑖 + 𝑐𝑐𝑥𝑥𝑖𝑖2 �] = 0
𝜕𝜕𝜕𝜕
⎩
𝑖𝑖=1
(3.4.2)
Expanding the above equations, we have
𝑛𝑛
𝑛𝑛
𝑛𝑛
𝑛𝑛
⎧ � 𝑦𝑦 = 𝑎𝑎 � 1 + 𝑏𝑏 � 𝑥𝑥 + 𝑐𝑐 � 𝑥𝑥 2
𝑖𝑖
𝑖𝑖
𝑖𝑖
⎪
𝑖𝑖=1
𝑖𝑖=1
𝑖𝑖=1
⎪ 𝑖𝑖=1
𝑛𝑛
𝑛𝑛
𝑛𝑛
⎪ 𝑛𝑛
2
� 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 = 𝑎𝑎 � 𝑥𝑥𝑖𝑖 + 𝑏𝑏 � 𝑥𝑥𝑖𝑖 + 𝑐𝑐 � 𝑥𝑥𝑖𝑖3
⎨ 𝑖𝑖=1
𝑖𝑖=1
𝑖𝑖=1
𝑖𝑖=1
𝑛𝑛
𝑛𝑛
𝑛𝑛
⎪ 𝑛𝑛
⎪
3
2
2
⎪� 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 = 𝑎𝑎 � 𝑥𝑥𝑖𝑖 + 𝑏𝑏 � 𝑥𝑥𝑖𝑖 + 𝑐𝑐 � 𝑥𝑥𝑖𝑖4
⎩ 𝑖𝑖=1
𝑖𝑖=1
𝑖𝑖=1
𝑖𝑖=1
(3.4.3)
The unknown coefficients a, b and c can hence obtained by solving the above linear equations.
So,
𝑛𝑛
⎛
⎜ 𝑛𝑛
⎜
𝐴𝐴 = ⎜� 𝑥𝑥𝑖𝑖
⎜ 𝑖𝑖=1
⎜ 𝑛𝑛
� 𝑥𝑥𝑖𝑖2
⎝ 𝑖𝑖=1
𝑛𝑛
𝑛𝑛
� 𝑥𝑥𝑖𝑖
� 𝑥𝑥𝑖𝑖2
⎞
𝑖𝑖=1
𝑖𝑖=1
⎟
𝑛𝑛
𝑛𝑛
⎟
� 𝑥𝑥𝑖𝑖2 � 𝑥𝑥𝑖𝑖3 ⎟ , 𝑋𝑋
𝑖𝑖=1
𝑖𝑖=1
⎟
𝑛𝑛
𝑛𝑛
⎟
� 𝑥𝑥𝑖𝑖3 � 𝑥𝑥𝑖𝑖4
⎠
𝑖𝑖=1
𝑖𝑖=1
𝑛𝑛
� 𝑦𝑦𝑖𝑖
⎛
⎞
𝑖𝑖=1
⎜
⎟
𝑛𝑛
𝑎𝑎
⎜
⎟
= �𝑏𝑏 � , 𝑏𝑏 = ⎜ � 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 ⎟
𝑐𝑐
⎜ 𝑖𝑖=1
⎟
⎜ 𝑛𝑛
⎟
� 𝑥𝑥𝑖𝑖2 𝑦𝑦𝑖𝑖
⎝ 𝑖𝑖=1
⎠
𝑎𝑎
�𝑏𝑏 � = 𝐴𝐴−1 𝑏𝑏
𝑐𝑐
For example, if we have some data for strike price and implied volatilities (Table 3.4):
(3.4.4)
(3.4.5)
Table 3.4 Extending data example
K
Vol
5
1.22
7.5
1.2
10
0.9
12.5
0.82
15
0.74
17.5
0.6
20
0.58
22.5
0.5
We can use cubic splines and least squares to generate volatility skew curve for all strike prices
6
UMN Master of Financial Mathematics Modeling Program Final Report
Fig. 3.4 Extending Skew Curve
X-axis is the strike price (K), and Y-axis is the volatility (sigma).
3.5. Market Implied Distribution
First of all, our model use put option data rather than call option data. Though there is a put-call
parity that restricts the relationship between these two, but the real market doesn't follow it
perfectly.
Fig. 3.5 Volatility skew of put (left) and call (right)
X-axis is the strike price (K), and Y-axis is the volatility (sigma).
7
UMN Master of Financial Mathematics Modeling Program Final Report
As we can see from figure 3.5, using call data brings more uncertainty of concavity of the
volatility skew. When we use Market Implied Distribution method, we have to take the second
derivative of the skew, and the concavity causes a negative of probability which against the
definition of probability.
The price of a European put option on an asset with strike price K and maturity T is given by
∞
𝐾𝐾
𝐾𝐾
𝑃𝑃 = 𝑒𝑒 −𝑟𝑟𝑟𝑟 � (𝐾𝐾 − 𝑆𝑆)+𝑝𝑝(𝑆𝑆)𝑑𝑑𝑑𝑑 = 𝑒𝑒 −𝑟𝑟𝑟𝑟 � 𝐾𝐾𝐾𝐾(𝑆𝑆)𝑑𝑑𝑑𝑑 − 𝑒𝑒 −𝑟𝑟𝑟𝑟 � 𝑆𝑆𝑆𝑆(𝑆𝑆)𝑑𝑑𝑑𝑑
Which implies
0
0
𝐾𝐾
𝜕𝜕𝜕𝜕
= 𝑒𝑒 −𝑟𝑟𝑟𝑟 � 𝑝𝑝(𝑆𝑆)𝑑𝑑𝑑𝑑
𝜕𝜕𝜕𝜕
0
𝜕𝜕 2 𝑃𝑃
= 𝑒𝑒 −𝑟𝑟𝑟𝑟 𝑝𝑝(𝐾𝐾)
𝜕𝜕𝐾𝐾 2
𝑝𝑝(𝐾𝐾) = 𝑒𝑒 𝑟𝑟𝑟𝑟
𝜕𝜕 2 𝑃𝑃
𝜕𝜕𝐾𝐾 2
0
(3.5.1)
(3.5.2)
(3.5.3)
(3.5.4)
Where the second derivative of put prices with respect to strike 𝑃𝑃(𝐾𝐾) is the discounted market
implied density.
This risk-neutral probability distribution is estimated from implied volatility smiles. In this project,
we used finite differences to approximate the second derivatives.
𝜕𝜕 2 𝑃𝑃 𝑃𝑃(𝐾𝐾 + ∆𝐾𝐾) − 2𝑃𝑃(𝐾𝐾) + 𝑃𝑃(𝐾𝐾 − ∆𝐾𝐾)
≈
(∆𝐾𝐾)2
𝜕𝜕𝐾𝐾 2
(3.5.5)
3.6. Denoise MID
When we use market data to calculate the implied volatility, the skew curve is not absolute
convex at every points on the curve. Therefore, we might come up with negative market implied
density at certain strike price.
For this kind of “bad data”, we called them noise in our program. Then we made a method
called “Denoise”, this method threw away the “bad data” that have negative probabilities.
As figure 3.6A shows, after eliminating all “bad data” which cause a negative probability, we can
have a shape much more like a PDF. And we can see in figure 3.6B, after Denoise, our skew
curve looks much better than before (figure 3.5 left).
8
UMN Master of Financial Mathematics Modeling Program Final Report
Fig. 3.6A PDF before Denoise (left) and after Denoise (right)
X-axis is the underlying price (S), and Y-axis is the Probability (Pr).
Fig. 3.6B Volatility Skew after Denoise
X-axis is the strike price (K), and Y-axis is the volatility (sigma).
By now, we have the market implied distribution of the underlying price, we could find out the
expected underlying price at the maturity by calculating the integration below:
∞
𝐸𝐸𝐸𝐸 = � 𝑆𝑆 𝑝𝑝(𝑆𝑆) 𝑑𝑑𝑑𝑑
0
Then we just go back to our skew curve to find the implied volatility (𝐸𝐸𝐸𝐸) at 𝐸𝐸𝐸𝐸
(3.6.1)
9
UMN Master of Financial Mathematics Modeling Program Final Report
4. Test (Monte Carlo) & Result
4.1 BS Test
After we get the underlying price distribution through MID method, we want to use Monte Carlo
Method testing the validity of our model.
Firstly we generate the underlying price process from the B-S model’s assumption:
𝑑𝑑𝑑𝑑
= 𝜇𝜇𝜇𝜇𝜇𝜇 + 𝜎𝜎𝜎𝜎𝜎𝜎
𝑆𝑆
(4.1.1)
Because we assume the skew curve moved parallel, so we can got the skew curve of every day.
Use the skew curve to calculate implied volatility, then option price, Delta and Vega can be
calculated though the B-S model. Because we know the expectation of next period’s underlying
price, we can get the expectation of next period’s volatility. 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 × (𝐸𝐸𝐸𝐸 − 𝜎𝜎) is the loss of
portfolio from the change of volatility, and we have the expectation of underlying price
change (𝐸𝐸𝐸𝐸 − 𝑆𝑆), so we calculate the skew-delta as:
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑎𝑎𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 =
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 × (𝐸𝐸𝐸𝐸 − 𝜎𝜎)
𝐸𝐸𝐸𝐸 − 𝑆𝑆
(4.12)
to cover the loss. The new delta is equal to the old delta plus skew-delta. After we get all
information of the path, we calculate the P&L every day and get the statistics, mean and
standard deviation.
Undelying
Price (S)
Volatility
(σ)
new Delta (old
Delta + skewdelta)
Vega, Delta,
Option Price
Fig. 4.1 Test workflow
10
UMN Master of Financial Mathematics Modeling Program Final Report
We simulate underlying price process 1 million times, we get below result:
Table 4.1 Result from B-S Test
Mean of P&L Mean (BS Delta)
Mean of P&L Variance (BS Delta)
Mean of P&L Mean (New Delta)
Mean of P&L Variance (New Delta)
0.001029
0.00025853
0.0914
184710
The result of new delta is worse than the old delta, and the data of variance is so extreme. At
first we are confused by this bad result, and the reason why we have this bad simulation results
is that we based the simulation on the B-S model. The delta from B-S Model is one of the best
hedging strategy.
4.2 MID PDF RVs Test
After B-S Test, we use another method to generate the underlying price process. We get the
return distribution from MID method. Then we use formula
𝑆𝑆𝑡𝑡+1 = 𝑆𝑆𝑡𝑡 × 𝑒𝑒 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
to generate the underlying price process.
(4.2.1)
The first target is to get a random variable follow the given distribution. We calculate the CDF
through the PDF. Next step is to generate a random number in [0,1], then use the inverse CDF
function to get the corresponding random variable x.
Fig. 4.2 From PDF to CDF
X-axis is the strike price (K), and Y-axis is the Probability (Pr).
11
UMN Master of Financial Mathematics Modeling Program Final Report
We simulate 1oo thousand times, and then we get the result below:
Table 4.2 Result from MID PDF RVs Test
Mean of P&L Mean (BS Delta)
0.012278
Mean of P&L Std (BS Delta)
0.0000548
Mean of P&L Mean (New Delta)
0.011251
Mean of P&L Std (New Delta)
0.0024492
The new mean of P&L’s mean is better than the older one at the cost of the worse standard
deviation. The advantage is in a long run we will have a better hedging strategy result. But the
disadvantage is the much bigger profit or loss in the short run. We hope our result will be a
reference to traders who want to hedge options in a long run.
12
UMN Master of Financial Mathematics Modeling Program Final Report
5. Improvement
Until now, our model consider that the volatility skew doesn’t change its shape as time goes by
(which is the second assumption we made). But it is not true in the real world, so we need to
adjust our model to take the volatility surface into account. In addition to volatility skew, traders
use a volatility term structure when pricing options.
Using the Cubic Spline Interpolation method, we can interpolate between T, and then plot the 3D variation of implied volatilities with strike price and time to maturity. The implied volatility
surface simultaneously shows both volatility smile and term structure of volatility, Option traders
use an implied volatility plot to quickly determine the shape the shape of the implied volatility
surface.
Fig. 5.1 Volatility Surface
Option with different expiration dates and strikes whose underlying is SPY
For options of different maturities, we also see characteristic differences in implied volatility.
However, in this case, the dominant effect is related to the market’s implied impact of upcoming
events. For example, it is well-observed that realized volatility for stock prices rises significantly
on the day that a company reports its earnings.
In our previous MID result, we assume that the expected underlying movement (ES) tomorrow
is exactly the same as the expected movement on the maturity date. If we can calculate the
13
UMN Master of Financial Mathematics Modeling Program Final Report
options which expires every trading day, however, we can use them to get the expected
underlying movement on the next trading day. It will greatly improve the accuracy of our model.
14
UMN Master of Financial Mathematics Modeling Program Final Report
6. Conclusion
From this modeling program, we’ve learnt:
•
How to improve the B-S model if there is a volatility smile or even a volatility surface.
•
Why we need to use bisection method rather than Newton's method to get the volatility
smile from the market, and why sometimes both these two methods can’t work.
•
How to interpolate, extend or eliminate some points from a given data in order to
maintain its information in the greatest degree but still qualify our standard.
•
How to know the market movement of the future if we know today’s market information.
•
How to generate a bunch of random numbers that follows a given Probability Density
Function.
And we’ve also learnt:
•
How to work as a team to focus on a problem, discuss and solve it.
•
How to break programming task into pieces for everybody, define the standard and
make it up at last.
Though our result doesn’t improve the classical B-S model, we still learnt a lot from these ten
days, and hoping to have another ten days for further modeling. Our mentor, Chris Prouty, gave
us plenty of new ideas and suggestions which keep us standing closer to the industry. We really
appreciate his help and patient.
15