| Математика
Key words: Options, Greeks, Taylor formula, Delta-Gamma approximation,Delta, Gamma, Vega, Theta, Convexity, Modified duration
Taylor series expansion

The Taylor series is mainly used to relate sensitivities to market value for derivative instruments.
An approximation is based on sensitivities (Greeks) such as delta, gamma, vega and theta.

In the real world prices are allowed to display a discontinuous behavior which results in abrupt movements and inadequacy of the Taylor series calculation.
Where do we apply the Taylor series?

A Brownian motion has an important continuity property which means a sample path should be a continuous function of time.
Theoretically, it can be said the Taylor series can be used when prices follow Brownian motions, i.e. no jumps.
Therefore, a reasonable level of accuracy can be achieved on very small continuous intervals.
Taylor expansion can’t be used in the context of simulation and stress testing.
From my opinion, accurate results can be obtained when approximating moves not more than 1% – 2% maximum.

A quadratic approximation to the price of the option can be constructed to easily revalue the market price:

Taylor series expansion

Taking into account Greeks letters we can rewrite the formula:

Taylor formula

A truncated version, also known as Delta-Gamma approximation is:

Delta-Gamma approximation

The gamma and other second order derivatives make the option price correction, i.e. option is not linear instrument.

The same logic can be applied to bond price; taking into account that bond price depends on time and changes in interest yields IR:

Bonds Delta-Gamma approximation

Divide the equation by price V:

Bonds Convexity Duration

Rewrite the formula:

change in bonds price

Here, convexity is a measure of how bond duration changes as interest rate change. As been written in the Article “Futures and forward convexity adjustment”, vanilla bonds, i.e. non-callable, have positive convexity thus the change of bond prices is none linear function as it follows from the formula above.

Exercise: Investor purchased a bond with a face value of $1000, coupon rate 5% paid semiannually, maturity is 2 years and yield to maturity is 5%.
What is the bond price if the yield increases by 1% (100 b.p.)?

Solution: It is easy to see, the PV is the same as the face value, $1 000.
To use the Taylor formula, we need to calculate modified duration and convexity.
We skip theta calculation because we need to estimate the market value change today, so ∆t is simply equals 0.

The solution is given in the attached Excel file, where we see that modified duration is 1.88 and convexity is 4.53.
Plugging these numbers into the formula gives us the result:

Market value of bond

The result is equal -$18.58, so the new price will be $1000-$18.58=$981.42. We also see that convexity adjustment is positive, $0.23; therefore it gives positive effect on the price change.

The closed formula solution for bond pricing gives us -$18.59. The difference is $0.01 which is a good estimation of market value change of the bond.

If the yield change is 2%, the difference is-$0.02, if yield changes by 3%, the difference will be $0.06 etc. As I already stated, changes should be as small as possible and good approximation achieved for moves up to 2%.

Exercise: Investor purchased an American vanilla call option on AXP US Equity at 3.3339% of the nominal. The underlying price is $89.695, time to expiry is 91 days, the strike is $89.695 ATM. Number of shares in the option is 1 and the implied volatility is 17.29%. What is the option price if the underlying increases by 1% ($90.59195)?

From Black-Scholes formula we have greeks calculation:
Delta(%): 50.89

Gamma(%): 4.6183

Vega(%): 0.18

Theta(1 day): -0.02

Sensitivities are calculated based on 1% change in the underlying price.

Solution: The dollar price of the bought option is 3.3339%*89.695=$2.99

The change of the option price in relative value is:

Option valuation

We don’t take into consideration second order derivatives, except gamma, as long as we have vanilla options.
The price will be 3.3339+0.531991=3.865%.

The new dollar price of the option is 3.865%*90.59195=$3.501

If we compare our approximation, $3.501 to the price derived from Black-Scholes formula we will see the difference is infinity small: $3.501-3.4995=$0.0015.

Now let’s do the same calculation but the new price is $91.695, i.e. 2.23 change%:

Option Taylor series approximation

The price will be 3.3339+1.2498=4.5836%.

The new dollar price of the option is 4.5836%*91.695=$4.20

Now we compare our result with closed formula solution. The difference is $4.20 – $4.21 = -$0.01 which is still a good approximation.

Theta and vega greeks are straightforward in the Taylor series, simply add or substract their values in the same way as for delta.