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Black-Scholes - Assignment Example

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First of all, reflect on the boundary condition for the call at expiration when T = 0. For one to accomplish this, reflect on the formula for the call option to be T → 0, that is as the time until maturity goes to 0. At…
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Black-Scholes
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Black-Scholes al Affiliation: Sensitivity Analysis In this question examine what happens to the price of the European call option when some of the parameters which enter the Black-Scholes formula vary.
Question One
1. Give the Black-Scholes formula for the price C0 of a European call option. Define carefully all the variables that enter into the formula

Where;
C0 is the theoretical call value
Is the estimated value of an option
S is the current stock price
Is the present value of options in the market such as New York State Exchange.
N is the cumulative standard normal probability distribution (Franke, Härdle and Hafner, 2015)
r is the risk free rate
Is the hypothetical rate of return of an option with no risk of fiscal loss.
T is time to maturity
Is the remaining lifespan of an option
X is the option strike price
The price at which call option can be exercised
AND
While
Therefore,
σ is the daily stock volatility
Is the level of deviation of a trading value series over time
Question Two
2. We are now interested in measuring the dollar change in the value of the option when the value of the stock changes. This amount is called “Delta” of the option:
Using the Black-Scholes formula, compute the Delta of a European call
From the Black-Scholes formula for a European call option today
t=0
Time to maturity t = T
Constant rate =(r ),
Constant volatility=(σ)
Strike price= K as:
EQUATION 1
&
C0 is the theoretical call value
S is the current stock price
N is the cumulative standard normal probability distribution DENOTED by ɸ herein after
Rf is the risk free rate
T-t is time to maturity
X is the option strike price
Determination of
To begin with, we apply the chain rule we get
The next step is to differentiate EQUATION 1 with respect to S to find
EQUATION 2
It now becomes apparent that
Therefore, we remain with the result
Question Three
3. Show that we always have
Beginning with a plain substitution for d2 and then computing from side to side in the algebra:
=
=
=
=
=
=
Thus from equation two above;

=

Thus, and
The boundaries condition proof
=0.5 when we assume r=0 and when t=T. This is because Φ(x)=0.5 if and only if K=0.
Consider the boundary conditions for the call option. First of all, reflect on the boundary condition for the call at expiration when T = 0. For one to accomplish this, reflect on the formula for the call option to be T → 0, that is as the time until maturity goes to 0. At maturity c (T) = max[S (T) − K, 0] and thus we need to show that T → 0 the formula becomes to c (0) = max[S(0) − K, 0]. If S(0) > K then ln(S(0) K ) > 0 so that as T → 0, d1 and d2 → +∞. Thus N (d1) and N(d2) → 1. Since e −rT → 1 as T → 0 it gets to c (0) → S(0) − K if S(0) > K.
References
Franke, J., Härdle, W. K., & Hafner, C. M. (2015). Black–Scholes Option Pricing Model. In Statistics of Financial Markets. Springer Berlin Heidelberg. Read More
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