Option Pricing: An Introduction. Ravi Mukkamala. Popular Models. Black-Scholes options pricing Binomial model (Cox, Ross, and Rubinstein) Implied volatility tree model (Deman and Kani) Implied binomial tree (Rubinstein). What are options?.
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(European option: Option to exercise at strike price at expiration only
American option: Option to exercise at strike price any time prior to the expiration
Asian option: Option to exercise at average price during the period prior to expiration)
Converting standard deviation from one time period to another
Rate of Return on an investment
N(x) = 1-0.5*(1+c1x+c2x2+c3x3+ c4x4)-4 if x>= 0
c1=0.196854; c2 = 0.115194; c3 = 0.000344; c4=0.019527
Prob(ST≤ K) = N([ln(St/K)+(T-t)(r-σ2/2))/σ√(T-t)])
current value of Ct = ΔtSt – e-r(T-t)Bt
Where (T-t) is the time to expiration and St is the current price of the underlying stock.
Bt is equal to K (strike price) times the probability of expiring in the money.
Recall that, Prob(ST≤ K) = N([ln(St/K)+(T-t)(r-σ2/2))/σ√(T-t)])
Δt = N([ln(St/K)+(T-t)(r+σ2/2))/σ√(T-t)])
Bt = K. N(([ln(St/K)+(T-t)(r-σ2/2))/σ√(T-t)])
Although the basic European call option valuation problem has a simple analytical solution (under the Black-Scholes assumptions), there are many variations of the problem that, currently, require the use of numerical methods. In particular, a rich variety of exotic options are traded. These options differ from the European call in that the payoff depends not only on the final time asset price, but also on its behavior over all or part of the time interval [0; T). For example, the payoff may depend on the maximum, minimum or average asset price and may knock-in or knock-out depending upon whether the asset price crosses a pre-determined barrier. Also, the option may have an early exercise facility, giving its holder the freedom to exercise before the expiry date. The design and analysis of numerical methods for valuing exotic options is still a very active research topic.
1) The stock pays no dividends during the option's life
2) European exercise terms are used
3) Markets are efficient
4) No commissions are charged
5) Interest rates remain constant and known
6) Returns are lognormally distributed
C0 = e-rΔt(pCu + (1-p)Cd) where Cu=ΔSu+ B and Cd=ΔSd+ B
pSu + (1-p) Sd = erΔtS0
p = (erΔtS0 - Sd)/(Su- Sd)= Risk-neutral probability
C(St, K, T-t, σ, r) = N(d1)St – N(d2)Ke-r(T-t)
When all other variables are known, determine σ using a numerical approximation technique (e.g. Newton-Raphson method)