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Fintor > talk > articles > sandbox | September 6 • 15:08 UTC • 5:08 PM SAST
The Black Scholes PDE
[edit]
1. The Black–Scholes model is a PDE which describes the evolution of the value of a option,
, through time,
, as related to changes in the underlying stock price,
with volatility
, and for a risk free rate
. The PDE is:
![{\displaystyle {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}=rV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c08ad5f350ae6037a16e03d37dde320a67768a8)
![{\displaystyle V=V(S,t)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dceb29777249f3a6b7ddc9230174b4a7f745fb6b)
2. For a call option, the solution to this PDE is the Black-Scholes formula; the option has strike price
, and time remaining until maturity
:
![{\displaystyle V(S,\tau )=S\mathbb {N} (d_{1})-Ke^{-r(\tau )}\mathbb {N} (d_{2})\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7442e4a7aea80afb1a814f763bb8d37fac2327fe)
- where:
![{\displaystyle d_{1}={\frac {\ln({\frac {S}{K}})+(r+{\frac {\sigma ^{2}}{2}})(\tau )}{\sigma {\sqrt {\tau }}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bf3edbd869983697639d7e5de2d0b308a12be3f)
![{\displaystyle d_{2}=d_{1}-\sigma {\sqrt {\tau }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03554086c75e3c9861d65ca0c49bbf7fbc2ca83d)
is the standard normal cumulative distribution function
(often written
).
3. To check whether this result is a solution, substitute the Black-Scholes formula into the Black-Scholes PDE.
- a. The partial derivatives are the Greeks:
, Theta, ![{\displaystyle =-{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\mathbb {N} (d_{2})\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d898eb42eeb037a64c9bc132c49e716bceb1277)
, Delta, ![{\displaystyle =\mathbb {N} (d_{1})\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9807a1eb22157a94bbe3f818cbd6bc4c56abe30)
, Gamma, ![{\displaystyle ={\frac {\mathbb {N} '(d_{1})}{S\sigma {\sqrt {\tau }}}}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/08008866f05ed7e7c3691377aac9da7c07a85d26)
- where:
is the standard normal Probability density function
(often written
).
- b. Substituting:
- LHS:
![{\displaystyle =-{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\mathbb {N} (d_{2})\ +{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\mathbb {N} '(d_{1})}{S\sigma {\sqrt {\tau }}}}+rS\mathbb {N} (d_{1})\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/55b0035f917b8060e71a8cfdcdc9549f8331089d)
![{\displaystyle =-{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\mathbb {N} (d_{2})\ +{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}+rS\mathbb {N} (d_{1})\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/725d978df05d400fd63e448e97cc0c4fad521a55)
.
.
- RHS:
![{\displaystyle =rV\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef07d7910ab3e1a0c04d76d55d2119302dd16fce)
- c. Conclusion: Since we have agreement, the Black-Scholes formula is a solution of the Black-Scholes PDE.
|
Calls |
Puts
|
value |
![{\displaystyle e^{-q\tau }S\mathbb {N} (d_{1})-e^{-r\tau }K\mathbb {N} (d_{2})\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca92d57c6768793210c79eac90d568800090dc61) |
|
delta |
![{\displaystyle e^{-q\tau }\mathbb {N} (d_{1})\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff2010c97152b380f40fc1ad495599ec69ab8358) |
|
vega |
|
theta |
![{\displaystyle -e^{-q\tau }{\frac {S\mathbb {N} '(d_{1})\sigma }{2{\sqrt {\tau }}}}-rKe^{-r\tau }\mathbb {N} (d_{2})+qSe^{-q\tau }\mathbb {N} (d_{1})\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb03c7981737c8fac8802b4540008fb7859c9cea) |
|
rho |
![{\displaystyle K\tau e^{-r\tau }\mathbb {N} (d_{2})\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14b51c0bd8b74d3300613f55c3243b233d2411b7) |
|
gamma |
|
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