Adjusting the Black-Scholes Framework in the Presence of a ... · • The volatility smile...

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


1

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”.


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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.


3

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.


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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 third-order polynomials. The essential idea is to fit a piecewise function of the form:

𝐹𝐹(𝑥𝑥) = �

𝑓𝑓1(𝑥𝑥) 𝑥𝑥1 ≤ 𝑥𝑥 < 𝑥𝑥2𝑓𝑓2(𝑥𝑥) 𝑥𝑥2 ≤ 𝑥𝑥 < 𝑥𝑥3

…𝑓𝑓𝑛𝑛−1(𝑥𝑥) 𝑥𝑥𝑛𝑛−1 ≤ 𝑥𝑥 < 𝑥𝑥𝑛𝑛

� (3.3.1)

For each 𝑓𝑓𝑖𝑖 is a third-order polynomial defined by:

𝑓𝑓𝑖𝑖(𝑥𝑥) = 𝛼𝛼𝑖𝑖𝑥𝑥3 + 𝛽𝛽𝑖𝑖𝑥𝑥2 + 𝛾𝛾𝑖𝑖𝑥𝑥 + 𝛿𝛿𝑖𝑖 (3.3.2)

𝑓𝑓𝑖𝑖(𝑥𝑥𝑖𝑖) = 𝛼𝛼𝑖𝑖𝑥𝑥𝑖𝑖3 + 𝛽𝛽𝑖𝑖𝑥𝑥𝑖𝑖2 + 𝛾𝛾𝑖𝑖𝑥𝑥𝑖𝑖 + 𝛿𝛿𝑖𝑖 = 𝑓𝑓𝑖𝑖−1(𝑥𝑥𝑖𝑖) = ℎ𝑖𝑖 (3.3.3)

Also, we want to make the function as soon as possible, therefore we make the first and second derivatives agree at each adjacent segment.

𝑓𝑓𝑖𝑖′(𝑥𝑥) = 3𝛼𝛼𝑖𝑖(𝑥𝑥 − 𝑥𝑥𝑖𝑖)2 + 𝛽𝛽𝑖𝑖(𝑥𝑥 − 𝑥𝑥𝑖𝑖) + 𝛾𝛾𝑖𝑖 = 𝑓𝑓𝑖𝑖−1′ (𝑥𝑥) (3.3.4)

𝑓𝑓𝑖𝑖′′(𝑥𝑥) = 6𝑎𝑎𝑖𝑖(𝑥𝑥 − 𝑥𝑥𝑖𝑖) + 2𝛽𝛽𝑖𝑖 = 𝑓𝑓𝑖𝑖−1′′ (𝑥𝑥) (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.


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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

𝐵𝐵 = 𝐴𝐴−1𝐻𝐻 (3.3.6)

𝐴𝐴 =

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛

𝑥𝑥03 𝑥𝑥0

2 𝑥𝑥0 1 0 0 0 0 … 0 0 0 0 0 0 00 0 0 0 𝑥𝑥1

3 𝑥𝑥12 𝑥𝑥1 1 … 0 0 0 0 0 0 0 0

…0 0 0 0 0 0 0 0 … 𝑥𝑥𝑚𝑚−2

3 𝑥𝑥𝑚𝑚−22 𝑥𝑥𝑚𝑚−21 0 0 0 0

0 0 0 0 0 0 0 0 … 𝑥𝑥𝑚𝑚−13 𝑥𝑥𝑚𝑚−1

2 𝑥𝑥𝑚𝑚−11 0 0 0 0 𝑥𝑥1

3 𝑥𝑥12 𝑥𝑥1 1 −𝑥𝑥1

3 − 𝑥𝑥12 − 𝑥𝑥1 … 0 0 0 0 0 0 0 0

…0 0 0 0 0 0 0 0 … 𝑥𝑥𝑚𝑚−2

3 𝑥𝑥𝑚𝑚−22 𝑥𝑥𝑚𝑚−2 1 −𝑥𝑥𝑚𝑚−2

3 − 𝑥𝑥𝑚𝑚−22 −𝑥𝑥𝑚𝑚−2 1

3𝑥𝑥12 2𝑥𝑥1 1 0 −3𝑥𝑥1

2 −2𝑥𝑥1 − 1 0 … 0 0 0 0 0 0 0 0…

0 0 0 0 0 0 0 0 … 3𝑥𝑥𝑚𝑚−22 2𝑥𝑥𝑚𝑚−2 1 0 −3𝑥𝑥𝑚𝑚−2

2 −2𝑥𝑥𝑚𝑚−2 − 1 0 6𝑥𝑥1 2 0 0 −6𝑥𝑥1 − 2 0 0 … 0 0 0 0 0 0 0 0

…0 0 0 0 0 0 0 0 … 6𝑥𝑥𝑚𝑚−2 2 0 0 −6𝑥𝑥𝑚𝑚−2 − 2 0 0

6𝑥𝑥0 2 0 0 0 0 0 0 … 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 … 0 0 0 0 6𝑥𝑥𝑚𝑚−2 2 0 0 ⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞

(3.3.7a)

𝐵𝐵 = (𝛼𝛼0,𝛽𝛽0, 𝛾𝛾0,𝛿𝛿0,𝛼𝛼1,𝛽𝛽1,𝛾𝛾1,𝛿𝛿1, … ,𝛼𝛼𝑚𝑚−1,𝛽𝛽𝑚𝑚−1,𝛾𝛾𝑚𝑚−1,𝛿𝛿𝑚𝑚−1)−1 (3.3.7b)

𝐻𝐻 = (ℎ1,ℎ2, … ,ℎ𝑚𝑚 , 0,0, … ,0)−1 (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 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐𝑥𝑥2. The best fitting curve 𝑓𝑓(𝑥𝑥) = 𝑦𝑦 has the min least square error:

𝛱𝛱 = ∑ [𝑦𝑦𝑖𝑖 − 𝑓𝑓(𝑥𝑥𝑖𝑖)]2 =𝑛𝑛𝑖𝑖=1 ∑ [𝑦𝑦𝑖𝑖 − (𝑎𝑎 + 𝑏𝑏𝑥𝑥𝑖𝑖 + 𝑐𝑐𝑥𝑥𝑖𝑖2)]2 = 𝑎𝑎 𝑚𝑚𝑖𝑖𝑛𝑛𝑛𝑛

𝑖𝑖=1 (3.4.1)

The unknown coefficients a, b and c must yield zero first derivatives.


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⎩⎪⎪⎪⎨

⎪⎪⎪⎧ 𝜕𝜕𝛱𝛱

𝜕𝜕𝑎𝑎= 2�[𝑦𝑦𝑖𝑖 − �𝑎𝑎 + 𝑏𝑏𝑥𝑥𝑖𝑖 + 𝑐𝑐𝑥𝑥𝑖𝑖2�] = 0

𝑛𝑛

𝑖𝑖=1

𝜕𝜕𝛱𝛱𝜕𝜕𝑏𝑏

= 2�𝑥𝑥𝑖𝑖[𝑦𝑦𝑖𝑖 − �𝑎𝑎 + 𝑏𝑏𝑥𝑥𝑖𝑖 + 𝑐𝑐𝑥𝑥𝑖𝑖2�] = 0𝑛𝑛

𝑖𝑖=1

𝜕𝜕𝛱𝛱𝜕𝜕𝑐𝑐

= 2�𝑥𝑥𝑖𝑖2[𝑦𝑦𝑖𝑖 − �𝑎𝑎 + 𝑏𝑏𝑥𝑥𝑖𝑖 + 𝑐𝑐𝑥𝑥𝑖𝑖2�] = 0𝑛𝑛

𝑖𝑖=1

� (3.4.2)

Expanding the above equations, we have

⎩⎪⎪⎪⎨

⎪⎪⎪⎧ �𝑦𝑦𝑖𝑖 = 𝑎𝑎�1 + 𝑏𝑏�𝑥𝑥𝑖𝑖 + 𝑐𝑐�𝑥𝑥𝑖𝑖2

𝑛𝑛

𝑖𝑖=1

𝑛𝑛

𝑖𝑖=1

𝑛𝑛

𝑖𝑖=1

𝑛𝑛

𝑖𝑖=1

�𝑥𝑥𝑖𝑖𝑦𝑦𝑖𝑖 = 𝑎𝑎�𝑥𝑥𝑖𝑖 + 𝑏𝑏�𝑥𝑥𝑖𝑖2𝑛𝑛

𝑖𝑖=1

𝑛𝑛

𝑖𝑖=1

𝑛𝑛

𝑖𝑖=1

+ 𝑐𝑐�𝑥𝑥𝑖𝑖3𝑛𝑛

𝑖𝑖=1

�𝑥𝑥𝑖𝑖2𝑦𝑦𝑖𝑖 =𝑛𝑛

𝑖𝑖=1

𝑎𝑎�𝑥𝑥𝑖𝑖2𝑛𝑛

𝑖𝑖=1

+ 𝑏𝑏�𝑥𝑥𝑖𝑖3𝑛𝑛

𝑖𝑖=1

+ 𝑐𝑐�𝑥𝑥𝑖𝑖4𝑛𝑛

𝑖𝑖=1

� (3.4.3)

The unknown coefficients a, b and c can hence obtained by solving the above linear equations.

𝐴𝐴 =

⎝

⎜⎜⎜⎜⎜⎛𝑛𝑛 �𝑥𝑥𝑖𝑖

𝑛𝑛

𝑖𝑖=1

�𝑥𝑥𝑖𝑖2𝑛𝑛

𝑖𝑖=1

�𝑥𝑥𝑖𝑖

𝑛𝑛

𝑖𝑖=1


𝑖𝑖=1


𝑖𝑖=1


𝑖𝑖=1


𝑖𝑖=1


𝑖𝑖=1 ⎠

⎟⎟⎟⎟⎟⎞

,𝑋𝑋 = �𝑎𝑎𝑏𝑏𝑐𝑐� , 𝑏𝑏 =

⎝

⎜⎜⎜⎜⎜⎛

�𝑦𝑦𝑖𝑖

𝑛𝑛

𝑖𝑖=1

�𝑥𝑥𝑖𝑖𝑦𝑦𝑖𝑖

𝑛𝑛

𝑖𝑖=1

�𝑥𝑥𝑖𝑖2𝑦𝑦𝑖𝑖

𝑛𝑛

𝑖𝑖=1 ⎠

⎟⎟⎟⎟⎟⎞

(3.4.4)

So,

�𝑎𝑎𝑏𝑏𝑐𝑐� = 𝐴𝐴−1𝑏𝑏 (3.4.5)

For example, if we have some data for strike price and implied volatilities (Table 3.4):

Table 3.4 Extending data example K 5 7.5 10 12.5 15 17.5 20 22.5 Vol 1.22 1.2 0.9 0.82 0.74 0.6 0.58 0.5

We can use cubic splines and least squares to generate volatility skew curve for all strike prices


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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).


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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

𝑃𝑃 = 𝑒𝑒−𝑟𝑟𝑟𝑟 � (𝐾𝐾 − 𝑆𝑆)+𝑝𝑝(𝑆𝑆)𝑑𝑑𝑆𝑆∞

0= 𝑒𝑒−𝑟𝑟𝑟𝑟 � 𝐾𝐾𝑝𝑝(𝑆𝑆)𝑑𝑑𝑆𝑆

𝐾𝐾

0− 𝑒𝑒−𝑟𝑟𝑟𝑟 � 𝑆𝑆𝑝𝑝(𝑆𝑆)𝑑𝑑𝑆𝑆

𝐾𝐾

0 (3.5.1)

Which implies

𝜕𝜕𝑃𝑃𝜕𝜕𝐾𝐾

= 𝑒𝑒−𝑟𝑟𝑟𝑟 � 𝑝𝑝(𝑆𝑆)𝑑𝑑𝑆𝑆𝐾𝐾

0 (3.5.2)

𝜕𝜕2𝑃𝑃𝜕𝜕𝐾𝐾2 = 𝑒𝑒−𝑟𝑟𝑟𝑟𝑝𝑝(𝐾𝐾) (3.5.3)

𝑝𝑝(𝐾𝐾) = 𝑒𝑒𝑟𝑟𝑟𝑟𝜕𝜕2𝑃𝑃𝜕𝜕𝐾𝐾2 (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.

𝜕𝜕2𝑃𝑃𝜕𝜕𝐾𝐾2 ≈

𝑃𝑃(𝐾𝐾 + ∆𝐾𝐾) − 2𝑃𝑃(𝐾𝐾) + 𝑃𝑃(𝐾𝐾 − ∆𝐾𝐾)(∆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).


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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:

𝐸𝐸𝑆𝑆 = � 𝑆𝑆 𝑝𝑝(𝑆𝑆) 𝑑𝑑𝑆𝑆∞

0 (3.6.1)

Then we just go back to our skew curve to find the implied volatility (𝐸𝐸𝐸𝐸) at 𝐸𝐸𝑆𝑆


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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:

𝑑𝑑𝑆𝑆𝑆𝑆

= 𝜇𝜇𝑑𝑑𝜇𝜇 + 𝐸𝐸𝑑𝑑𝜎𝜎 (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. 𝑉𝑉𝑒𝑒𝑉𝑉𝑎𝑎 × (𝐸𝐸𝐸𝐸 − 𝐸𝐸) is the loss of portfolio from the change of volatility, and we have the expectation of underlying price change (𝐸𝐸𝑆𝑆 − 𝑆𝑆), so we calculate the skew-delta as:

𝐷𝐷𝑒𝑒𝐷𝐷𝜇𝜇𝑎𝑎𝑆𝑆𝑆𝑆𝑒𝑒𝜎𝜎 =𝑉𝑉𝑒𝑒𝑉𝑉𝑎𝑎 × (𝐸𝐸𝐸𝐸 − 𝐸𝐸)

𝐸𝐸𝑆𝑆 − 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.

Fig. 4.1 Test workflow

Undelying Price (S)

Volatility (σ)

Vega, Delta, Option Price

new Delta (old Delta + skew-

delta)


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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) 0.001029 Mean of P&L Variance (BS Delta) 0.00025853 Mean of P&L Mean (New Delta) 0.0914 Mean of P&L Variance (New Delta) 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

𝑆𝑆𝜇𝜇+1 = 𝑆𝑆𝜇𝜇 × 𝑒𝑒𝑟𝑟𝑎𝑎𝑛𝑛𝑑𝑑𝑟𝑟𝑚𝑚 𝑟𝑟𝑒𝑒𝜇𝜇𝑟𝑟𝑟𝑟𝑛𝑛 (4.2.1)

to generate the underlying price process.

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).


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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.


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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 3-D 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


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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.


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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.

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