Group 17's Option Pricer

👍Contributions:

• Ran Xin

• Tian Yiping

• Li Lingxiao

How to Run

Environment requirements:

• Environment: Python 3.6

• Packages: tkinter : for UI, traceback: for capturing log, math: for some math calculations (root, square), scipy: for standard normal distribution density function, random: for generating Gaussian distribution random, numpy: for matrix operations

🔴To Run our option pricer, just put all the .py files in the same directory, and run gui.py.

GUI

Using a simple python tkinter GUI libray to build a simple GUI Option Pricer:

• Just like a Calculator the result answer shows in the above, highlight in yellow;

• The second line is Option Type, it's a combo box, list all avaliable option types: European, American, Asian, Basket; Also you can click the radio button to choose call or put option.

• The third line is alert information that informs you about which missing value you forget to input in order to make the calculation. We simply extract the error traceback message to achieve this.

• In the middle area is input value field, all the input parameters should be input here, left-hand side is most commonly input one, right-hand side is less common and also symmetric.

• In the bottom center (green) its a submit button, and the bottom left (red) is clear all input button.

How to calc?

Choose option type, here we select "Arithmetic basket option", call, input all the needed parameter, with variate:

if we forget input the correlation ρ , click submit will be informed,

after all correct, the answer will be calculated and display in above.

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Functionalities

parameter in code

spot price S, volatility sigma, risk-free rate r, time to maturity T, strike K, option premium V, Monte Carlo paths path, average steps n, repo rate q, correlation rho

European Option

european.py (S, sigma, r, T, K, type, q)

Using Lecture 3 Black-Scholes Formulas: with BS-PDE & Call-Put Parity

$$ C(S,t)=Se^{-qT}N(d_1)-Ke^{-rT}N(d_2) \qquad P(S,t)=Ke^{-rT}N(-d_2)-Se^{-qT}N(-d_1) $$

$$ in\ which:\qquad d_1={{ln(S/k)+(r-q)T}\over{\sigma \sqrt{T}}}+{1\over2}\sigma \sqrt{T} \qquad d_2={{ln(S/k)+(r-q)T}\over{\sigma \sqrt{T}}}-{1\over2}\sigma \sqrt{T} $$

Implied volatility

volatility.py (S, S_true, r, T, K, q, type)

Follow the formulas in "lecture 4" using Newton Method to solve the problem which converges quickly:

$$ \hat{\sigma}=\sqrt{2 \mid {{lnS_0/K+(r-q)T}\over T}\mid} \qquad vega={{Se^{-qT}\sqrt{T} \times e^{-0.5(d1^2)}}\over{\sqrt{2\pi}}} $$

American Option

american.py (S, sigma, r, T, K, n, type)

There is no closed formular, so we use bionominal tree in "lecture 6" to solve it. First step is to iterate the bionominal tree and calculate the stock price in each node. Second step is to backtrack the tree and calculate the price of call/put options.

$$ Se^{r\Delta t}=pS_u+(1-p)S_d \qquad p={{e^{r\Delta t}-d}\over{u-d}} \qquad u=e^{\sigma \sqrt{\Delta t}} \qquad d={1\over u} =e^{-\sigma \sqrt{\Delta t}} $$

Geometric Basket Option

baskt_geometric.py (S1, S2, sigma1, sigma2, rho, r, T, K, option_type)

Follow the formulas described in mathematical background in "assignment 3" geometric Brownian motion, we can figure out the call or put geometric basket option.

$$ B_g(t)=(\prod_{i=1}^n)^{1/n} \qquad \sigma_{B_g}={\sqrt{\sum_{i=1}^{n=2} \sum_{j=1}^{n=2} \sigma_i \sigma_j \rho_{i,j}}\over 2} \qquad \mu_{B_g}=r-{1\over 2}{\sum_{i=1}^{n=2} \sigma_i^2\over n}+{1\over 2}\sigma_{B_g}^2 \qquad n=2 $$

$$ C_{B_g}=e^{-rT}(B_g(0)e^{\mu T}N(\hat{d_1})-KN(\hat{d_2})) \qquad P_{B_g}=e^{-rT}(KN(-\hat{d_2})-B_g(0)e^{\mu T}N(-\hat{d_1})) $$

Arithmetic Basket Option

basket_arithmetic.py (path, S1, S2, sigma1, sigma2, rho, r, T, K, option_type, variate)

For arithmetic option price, there is no closed-form formulas so we need to use Monte Carlo to simulate the stock price. We use the method proved in assignment 2 (2.1) to generate two random varibles with correlation coefficient σ. Then, similar to Asian option, we can calculate the basket option price (95% confidence interval) with or without control variate.

$$ B_a(t)={1\over n}\sum_{i=1}^nSi(t) \qquad n=2 $$

Rest is similar to Asian-Arithmetic, 2 drift, 2 growthFactor. simulate multiple times output the 95% confidence interval.

Geometric Asian option

asian_geometric.py (S, sigma, r, T, K, n, option_type)

There is a closed formular in "lecture 5" we can simply implement it.

$$ \hat{\sigma}=\sigma \sqrt{{(n+1)(2n+1)}\over 6n^2} \qquad \hat{\mu}=(r-{1\over 2}\sigma^2){(n+1)\over 2n}+{1\over 2}\hat{\sigma}^2 $$

$$ C_{A_g}=e^{-rT}(S_0e^{\hat{\mu} T}N(\hat{d_1})-KN(\hat{d_2})) \qquad P_{A_g}=e^{-rT}(KN(-\hat{d_2})-S_0e^{\hat{\mu} T}N(-\hat{d_1})) $$

Arithmetic Asian option

asian_arithmetic.py (path, S, sigma, r, T, K, n, option_type, variate)

There is no closed-form formulas so we need to use Monte Carlo to simulate it.

$$ drift=e^{(r-0.5\sigma^2)\Delta t} \qquad growthFactor=drift \times e^{\sigma \sqrt{\Delta t}\times gaussRandom} $$

Then calculate the mean, std, cov of the payoff, simulate multiple times output the 95% confidence interval.

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Test cases and analysis

how each parameter affects the option price

r = 0.05, T = 3, and S(0) = 100. paths in Monte Carlo simulation is m = 100, 000. RandomSeed(1234)

Asian option:

σ	K	n	Type	Geometric	Arithmetic	Arithmetic(variate)
0.3	100	50	Put	8.482	7.681~7.818	7.799~7.807
0.3	100	100	Put	8.431	7.667~7.804	7.747~7.756
0.4	100	50	Put	12.558	11.125~11.304	11.277~11.292
0.3	100	50	Call	13.259	14.639~14.927	14.714~14.736
0.3	100	100	Call	13.138	14.482~14.768	14.598~14.620
0.4	100	50	Call	15.759	18.072~18.476	18.181~18.222

Geometric Asian	Call option	Put option
volatility (σ) ↑	↑↑	↑↑
average steps (n) ↑	↓	↓

Arithmetic Asian	Call option	Put option
volatility (σ) ↑	↑↑	↑↑
average steps (n) ↑	↓	↓

Basket options:

S1(0)	S2(0)	K	σ1	σ2	ρ	Type	Geometric	Arithmetic	Arithmetic(variate)
100	100	100	0.3	0.3	0.5	Put	11.491	10.469~10.658	10.563~10.587
100	100	100	0.3	0.3	0.9	Put	12.622	12.281~12.492	12.426~12.431
100	100	100	0.1	0.3	0.5	Put	6.586	5.479~5.593	5.508~5.525
100	100	80	0.3	0.3	0.5	Put	4.711	4.214~4.325	4.247~4.262
100	100	120	0.3	0.3	0.5	Put	21.289	19.712~19.980	19.867~19.900
100	100	100	0.5	0.5	0.5	Put	23.469	20.885~21.178	21.054~21.110
120	100	100	0.3	0.3	0.5	Put	8.915	7.975~8.144	8.043~8.068
120	120	100	0.3	0.3	0.5	Put	6.736	6.047~6.195	6.095~6.115
100	100	100	0.3	0.3	0.5	Put r=0.1	6.423	5.782~5.915	5.828~5.846
100	100	100	0.3	0.3	0.5	Put T=1	8.257	7.774~7.914	7.845~7.856
100	100	100	0.3	0.3	0.5	Call	22.102	24.334~24.818	24.461~24.523
100	100	100	0.3	0.3	0.9	Call	25.878	26.223~26.775	26.351~26.364
100	100	100	0.1	0.3	0.5	Call	17.924	19.296~19.641	19.414~19.451
100	100	80	0.3	0.3	0.5	Call	32.536	35.227~35.765	35.352~35.415
100	100	120	0.3	0.3	0.5	Call	14.685	16.434~16.854	16.559~16.617
100	100	100	0.5	0.5	0.5	Call	28.449	34.598~35.537	34.865~35.071
120	100	100	0.3	0.3	0.5	Call	28.753	31.815~32.377	31.946~32.024
120	120	100	0.3	0.3	0.5	Call	36.683	39.832~40.468	39.979~40.055
100	100	100	0.3	0.3	0.5	Call r=0.1	29.023	31.588~32.112	31.710~31.773
100	100	100	0.3	0.3	0.5	Call T=1	12.015	12.649~12.889	12.715~12.733

Geometric Basket	Call option	Put option
spot price (S1) ↑	↑	↓
spot price (S1) ↑ (S1)↑	↑↑	↓↓
strike (K) ↓	↑	↓
maturity (T) ↓	↓	↓
risk free rate (r) ↑	↑	↓
volatility (σ1) ↓	↓	↓
volatility (σ1) ↑ volatility (σ2) ↑	↑	↑
correlation coefficient (ρ) ↑	↑	↑

Arithmetic Basket	Call option	Put option
spot price (S) ↑	↑	↓
spot price (S1) ↑ (S1)↑	↑↑	↓↓
strike (K) ↓	↑	↓
maturity (T) ↓	↓	↓
risk free rate (r) ↑	↑	↓
volatility (σ) ↓	↓	↓
volatility (σ1) ↑ volatility (σ2) ↑	↑	↑
correlation coefficient (ρ) ↑	↑	↑

European:

S1(0)	K	σ1	r	T	Type	European
100	100	0.2	0.01	0.5	call	5.876024
100	120	0.2	0.01	0.5	call	0.774138
100	120	0.2	0.01	1	call	8.433318
100	100	0.3	0.01	0.5	call	8.677645
100	100	0.2	0.02	0.5	call	6.120654
100	100	0.2	0.01	0.5	put	5.377272
100	120	0.2	0.01	0.5	put	20.17563
100	120	0.2	0.01	1	put	7.438302
100	100	0.3	0.01	0.5	put	8.178893
100	100	0.2	0.02	0.5	put	5.125637

European	Call option	Put option
strike (K) ↑	↓↓	↑↑
maturity (T) ↑	↑	↓
risk free rate (r) ↑	↑	↓
volatility (σ) ↑	↑	↓

American:

S1(0)	K	σ1	r	T	Type	European
100	100	0.2	0.01	0.5	call	5.861975
100	120	0.2	0.01	0.5	call	0.776743
100	100	0.2	0.01	1	call	8.413504
100	100	0.3	0.01	0.5	call	8.656601
100	100	0.2	0.02	0.5	call	6.106614
100	100	0.2	0.01	0.5	put	5.363223
100	120	0.2	0.01	0.5	put	20.17824
100	100	0.2	0.01	1	put	7.418488
100	100	0.3	0.01	0.5	put	8.157849
100	100	0.2	0.02	0.5	put	5.111597

American	Call option	Put option
strike (K) ↑	↓↓	↑↑
maturity (T) ↑	↑	↓
risk free rate (r) ↑	↑	↓
volatility (σ) ↑	↑	↓