MFE409 Financial Risk Management Homewor 5 Solved.

Homework 5

Heyu Zhang

MFE409 Financial Risk Management

Valentin Haddad

#Question 1 The change in value of the option can be shown as a change in price of stock multiplied by its delta (using taylor’s series)

Mt+1− Mt = ∆1dS1 + ∆2dS2

= ∆1(µ1S1dt + σ1S1dWt1) + ∆2(µ2S2dt + σ2S2dWt2)

= (∆1µ1S1 + ∆2µ2S2)dt + (∆1σ1S1dWt1 + ∆2σ2S2dWt2)

So the dt term is the µc and the σc is the combined standard deviation of the two dWt terms. V aR = −µc + 2.326σc

q

                  = −(∆1µ1St,1 + ∆2µ2St,2) + 2.326(                               ∆21S2t,1σ21 + ∆22S2t,2σ12 + 2∆1∆2σ1σ2S1S2ρ)

#Question 2

For the delta gamma method, we need to introduce more terms from the taylor’s expansion. These terms will include the gamma of the stocks

Mt+1− Mt = ∆1dS1 + ∆2dS2 +  γ1dS12 +  γ2dS22 + γ1,2dS1dS2

= (∆1µ1S1 + ∆2µ2S2)dt + (∆1σ1S1dWt1 + ∆2σ2S2dWt2) +  γ1σ12S12dt +  γ2σ22S22dt + γ1,2S1S2σ1σ2ρdt

= (∆1µ1S1 + ∆2µ2S2 +  γ1σ12S12 +  γ2σ22S22 + γ1,2S1S2σ1σ2ρ)dt + (∆1σ1S1dWt1 + ∆2σ2S2dWt2) So the dt term is the µc and the σc is the combined standard deviation of the two dWt terms.

V aR = −µc + 2.326σc

= −(∆1µ1S1 + ∆2µ2S2 +  γ1σ12S12 +  γ2σ22S22 + γ1,2S1S2σ1σ2ρ

q

                                 +2.326(                      ∆21S2t,1σ21 + ∆22S2t,2σ12 + 2∆1∆2σ1σ2S1S2ρ))

#Question 3 The price of the option can be calculated by using the closed form solution provided in the question. All the values are annualized while using the closed form solution. The price of the option at t=0 is

1

## [1] 1.728485
#Question 4

The formula from question 1 and question 2 contain delta and gamma terms. this can be calculated by checking how the price of the option and the delta changes by increasing the stock price by 0.01. The cross gamma is calculated by increasing both the stock prices by 0.01 simultaneously.

The option price from both the methods are as follows:

Delta Approach

## [1] 0.472288
So the VaR using the delta approach is 47.23%.

Delta Gamma approach

## [1] 0.460004
The VaR using the delta gamma approach is 46%. As can be seen, the VaR due to the Delta gamma approach has reduced compared to the Delta approach. This is because convexity is always helpful and has increased the returns. The delta approach makes a big approximation which can be very wrong as the convexity increases.

#Question 5

The VaR from simulation can be calculated by finding the price of the option at t=0 and t=1 and then finding the return based on that. For calculating the price of the option at t=1, we need to simulate the price of S1 and S2 at t=1 using the provided stochastic equation. The value of VaR is

## [1] 0.4833693
This value of VaR is 48.33%, which is higher than that produced out of both the Gamma and Delta approaches. This is because simulation considers other types of risks as well (like vega, rho etc.). Due to this the VaR value is greater than others.

#Question 6 We would need to worry about the other greeks, for which we have assumed the values are the same. This involves the greeks of Rho (wrt interest rate), theta (wrt time decay) and Vega (wrt sigma). The value of these greeks are as below:

Vega1
Vega2
Theta
Rho
2.135479
1.614589
-8.555906
9.777822
In Option trading, one of the most important greek is the vega. Volatility is an important input into the black scholes equation and how option price changes with change in volatility is a key risk which needs to be considered.

#Interview Questions ##Question 1 Assume the ball’s radius is r. Then the distance between the center of√
the ball and the corner on the floor is             3r
##Question 2 The forward price will be the present price of the stock.

##Question 3

Split the coins to 2 piles. One pile contains 10 coins and the other contains 47 coins. Flip up every coin in the 10-coin pile then we will get what we want.
Reason: assume that there are x white-up coins in the 10-coin pile. Then there will be (10-x) white-up coins in the 47-coin pile. After flipping up the coins, there will be (10-x) white-up coins in the 10-coin pile which is the same as the 47-coin pile.
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